/************************************************************************* Copyright (c) Sergey Bochkanov (ALGLIB project). >>> SOURCE LICENSE >>> This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation (www.fsf.org); either version 2 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. A copy of the GNU General Public License is available at http://www.fsf.org/licensing/licenses >>> END OF LICENSE >>> *************************************************************************/ #pragma warning disable 162 #pragma warning disable 219 using System; public partial class alglib { /************************************************************************* IDW interpolant. *************************************************************************/ public class idwinterpolant { // // Public declarations // public idwinterpolant() { _innerobj = new idwint.idwinterpolant(); } // // Although some of declarations below are public, you should not use them // They are intended for internal use only // private idwint.idwinterpolant _innerobj; public idwint.idwinterpolant innerobj { get { return _innerobj; } } public idwinterpolant(idwint.idwinterpolant obj) { _innerobj = obj; } } /************************************************************************* IDW interpolation INPUT PARAMETERS: Z - IDW interpolant built with one of model building subroutines. X - array[0..NX-1], interpolation point Result: IDW interpolant Z(X) -- ALGLIB -- Copyright 02.03.2010 by Bochkanov Sergey *************************************************************************/ public static double idwcalc(idwinterpolant z, double[] x) { double result = idwint.idwcalc(z.innerobj, x); return result; } /************************************************************************* IDW interpolant using modified Shepard method for uniform point distributions. INPUT PARAMETERS: XY - X and Y values, array[0..N-1,0..NX]. First NX columns contain X-values, last column contain Y-values. N - number of nodes, N>0. NX - space dimension, NX>=1. D - nodal function type, either: * 0 constant model. Just for demonstration only, worst model ever. * 1 linear model, least squares fitting. Simpe model for datasets too small for quadratic models * 2 quadratic model, least squares fitting. Best model available (if your dataset is large enough). * -1 "fast" linear model, use with caution!!! It is significantly faster than linear/quadratic and better than constant model. But it is less robust (especially in the presence of noise). NQ - number of points used to calculate nodal functions (ignored for constant models). NQ should be LARGER than: * max(1.5*(1+NX),2^NX+1) for linear model, * max(3/4*(NX+2)*(NX+1),2^NX+1) for quadratic model. Values less than this threshold will be silently increased. NW - number of points used to calculate weights and to interpolate. Required: >=2^NX+1, values less than this threshold will be silently increased. Recommended value: about 2*NQ OUTPUT PARAMETERS: Z - IDW interpolant. NOTES: * best results are obtained with quadratic models, worst - with constant models * when N is large, NQ and NW must be significantly smaller than N both to obtain optimal performance and to obtain optimal accuracy. In 2 or 3-dimensional tasks NQ=15 and NW=25 are good values to start with. * NQ and NW may be greater than N. In such cases they will be automatically decreased. * this subroutine is always succeeds (as long as correct parameters are passed). * see 'Multivariate Interpolation of Large Sets of Scattered Data' by Robert J. Renka for more information on this algorithm. * this subroutine assumes that point distribution is uniform at the small scales. If it isn't - for example, points are concentrated along "lines", but "lines" distribution is uniform at the larger scale - then you should use IDWBuildModifiedShepardR() -- ALGLIB PROJECT -- Copyright 02.03.2010 by Bochkanov Sergey *************************************************************************/ public static void idwbuildmodifiedshepard(double[,] xy, int n, int nx, int d, int nq, int nw, out idwinterpolant z) { z = new idwinterpolant(); idwint.idwbuildmodifiedshepard(xy, n, nx, d, nq, nw, z.innerobj); return; } /************************************************************************* IDW interpolant using modified Shepard method for non-uniform datasets. This type of model uses constant nodal functions and interpolates using all nodes which are closer than user-specified radius R. It may be used when points distribution is non-uniform at the small scale, but it is at the distances as large as R. INPUT PARAMETERS: XY - X and Y values, array[0..N-1,0..NX]. First NX columns contain X-values, last column contain Y-values. N - number of nodes, N>0. NX - space dimension, NX>=1. R - radius, R>0 OUTPUT PARAMETERS: Z - IDW interpolant. NOTES: * if there is less than IDWKMin points within R-ball, algorithm selects IDWKMin closest ones, so that continuity properties of interpolant are preserved even far from points. -- ALGLIB PROJECT -- Copyright 11.04.2010 by Bochkanov Sergey *************************************************************************/ public static void idwbuildmodifiedshepardr(double[,] xy, int n, int nx, double r, out idwinterpolant z) { z = new idwinterpolant(); idwint.idwbuildmodifiedshepardr(xy, n, nx, r, z.innerobj); return; } /************************************************************************* IDW model for noisy data. This subroutine may be used to handle noisy data, i.e. data with noise in OUTPUT values. It differs from IDWBuildModifiedShepard() in the following aspects: * nodal functions are not constrained to pass through nodes: Qi(xi)<>yi, i.e. we have fitting instead of interpolation. * weights which are used during least squares fitting stage are all equal to 1.0 (independently of distance) * "fast"-linear or constant nodal functions are not supported (either not robust enough or too rigid) This problem require far more complex tuning than interpolation problems. Below you can find some recommendations regarding this problem: * focus on tuning NQ; it controls noise reduction. As for NW, you can just make it equal to 2*NQ. * you can use cross-validation to determine optimal NQ. * optimal NQ is a result of complex tradeoff between noise level (more noise = larger NQ required) and underlying function complexity (given fixed N, larger NQ means smoothing of compex features in the data). For example, NQ=N will reduce noise to the minimum level possible, but you will end up with just constant/linear/quadratic (depending on D) least squares model for the whole dataset. INPUT PARAMETERS: XY - X and Y values, array[0..N-1,0..NX]. First NX columns contain X-values, last column contain Y-values. N - number of nodes, N>0. NX - space dimension, NX>=1. D - nodal function degree, either: * 1 linear model, least squares fitting. Simpe model for datasets too small for quadratic models (or for very noisy problems). * 2 quadratic model, least squares fitting. Best model available (if your dataset is large enough). NQ - number of points used to calculate nodal functions. NQ should be significantly larger than 1.5 times the number of coefficients in a nodal function to overcome effects of noise: * larger than 1.5*(1+NX) for linear model, * larger than 3/4*(NX+2)*(NX+1) for quadratic model. Values less than this threshold will be silently increased. NW - number of points used to calculate weights and to interpolate. Required: >=2^NX+1, values less than this threshold will be silently increased. Recommended value: about 2*NQ or larger OUTPUT PARAMETERS: Z - IDW interpolant. NOTES: * best results are obtained with quadratic models, linear models are not recommended to use unless you are pretty sure that it is what you want * this subroutine is always succeeds (as long as correct parameters are passed). * see 'Multivariate Interpolation of Large Sets of Scattered Data' by Robert J. Renka for more information on this algorithm. -- ALGLIB PROJECT -- Copyright 02.03.2010 by Bochkanov Sergey *************************************************************************/ public static void idwbuildnoisy(double[,] xy, int n, int nx, int d, int nq, int nw, out idwinterpolant z) { z = new idwinterpolant(); idwint.idwbuildnoisy(xy, n, nx, d, nq, nw, z.innerobj); return; } } public partial class alglib { /************************************************************************* Barycentric interpolant. *************************************************************************/ public class barycentricinterpolant { // // Public declarations // public barycentricinterpolant() { _innerobj = new ratint.barycentricinterpolant(); } // // Although some of declarations below are public, you should not use them // They are intended for internal use only // private ratint.barycentricinterpolant _innerobj; public ratint.barycentricinterpolant innerobj { get { return _innerobj; } } public barycentricinterpolant(ratint.barycentricinterpolant obj) { _innerobj = obj; } } /************************************************************************* Rational interpolation using barycentric formula F(t) = SUM(i=0,n-1,w[i]*f[i]/(t-x[i])) / SUM(i=0,n-1,w[i]/(t-x[i])) Input parameters: B - barycentric interpolant built with one of model building subroutines. T - interpolation point Result: barycentric interpolant F(t) -- ALGLIB -- Copyright 17.08.2009 by Bochkanov Sergey *************************************************************************/ public static double barycentriccalc(barycentricinterpolant b, double t) { double result = ratint.barycentriccalc(b.innerobj, t); return result; } /************************************************************************* Differentiation of barycentric interpolant: first derivative. Algorithm used in this subroutine is very robust and should not fail until provided with values too close to MaxRealNumber (usually MaxRealNumber/N or greater will overflow). INPUT PARAMETERS: B - barycentric interpolant built with one of model building subroutines. T - interpolation point OUTPUT PARAMETERS: F - barycentric interpolant at T DF - first derivative NOTE -- ALGLIB -- Copyright 17.08.2009 by Bochkanov Sergey *************************************************************************/ public static void barycentricdiff1(barycentricinterpolant b, double t, out double f, out double df) { f = 0; df = 0; ratint.barycentricdiff1(b.innerobj, t, ref f, ref df); return; } /************************************************************************* Differentiation of barycentric interpolant: first/second derivatives. INPUT PARAMETERS: B - barycentric interpolant built with one of model building subroutines. T - interpolation point OUTPUT PARAMETERS: F - barycentric interpolant at T DF - first derivative D2F - second derivative NOTE: this algorithm may fail due to overflow/underflor if used on data whose values are close to MaxRealNumber or MinRealNumber. Use more robust BarycentricDiff1() subroutine in such cases. -- ALGLIB -- Copyright 17.08.2009 by Bochkanov Sergey *************************************************************************/ public static void barycentricdiff2(barycentricinterpolant b, double t, out double f, out double df, out double d2f) { f = 0; df = 0; d2f = 0; ratint.barycentricdiff2(b.innerobj, t, ref f, ref df, ref d2f); return; } /************************************************************************* This subroutine performs linear transformation of the argument. INPUT PARAMETERS: B - rational interpolant in barycentric form CA, CB - transformation coefficients: x = CA*t + CB OUTPUT PARAMETERS: B - transformed interpolant with X replaced by T -- ALGLIB PROJECT -- Copyright 19.08.2009 by Bochkanov Sergey *************************************************************************/ public static void barycentriclintransx(barycentricinterpolant b, double ca, double cb) { ratint.barycentriclintransx(b.innerobj, ca, cb); return; } /************************************************************************* This subroutine performs linear transformation of the barycentric interpolant. INPUT PARAMETERS: B - rational interpolant in barycentric form CA, CB - transformation coefficients: B2(x) = CA*B(x) + CB OUTPUT PARAMETERS: B - transformed interpolant -- ALGLIB PROJECT -- Copyright 19.08.2009 by Bochkanov Sergey *************************************************************************/ public static void barycentriclintransy(barycentricinterpolant b, double ca, double cb) { ratint.barycentriclintransy(b.innerobj, ca, cb); return; } /************************************************************************* Extracts X/Y/W arrays from rational interpolant INPUT PARAMETERS: B - barycentric interpolant OUTPUT PARAMETERS: N - nodes count, N>0 X - interpolation nodes, array[0..N-1] F - function values, array[0..N-1] W - barycentric weights, array[0..N-1] -- ALGLIB -- Copyright 17.08.2009 by Bochkanov Sergey *************************************************************************/ public static void barycentricunpack(barycentricinterpolant b, out int n, out double[] x, out double[] y, out double[] w) { n = 0; x = new double[0]; y = new double[0]; w = new double[0]; ratint.barycentricunpack(b.innerobj, ref n, ref x, ref y, ref w); return; } /************************************************************************* Rational interpolant from X/Y/W arrays F(t) = SUM(i=0,n-1,w[i]*f[i]/(t-x[i])) / SUM(i=0,n-1,w[i]/(t-x[i])) INPUT PARAMETERS: X - interpolation nodes, array[0..N-1] F - function values, array[0..N-1] W - barycentric weights, array[0..N-1] N - nodes count, N>0 OUTPUT PARAMETERS: B - barycentric interpolant built from (X, Y, W) -- ALGLIB -- Copyright 17.08.2009 by Bochkanov Sergey *************************************************************************/ public static void barycentricbuildxyw(double[] x, double[] y, double[] w, int n, out barycentricinterpolant b) { b = new barycentricinterpolant(); ratint.barycentricbuildxyw(x, y, w, n, b.innerobj); return; } /************************************************************************* Rational interpolant without poles The subroutine constructs the rational interpolating function without real poles (see 'Barycentric rational interpolation with no poles and high rates of approximation', Michael S. Floater. and Kai Hormann, for more information on this subject). Input parameters: X - interpolation nodes, array[0..N-1]. Y - function values, array[0..N-1]. N - number of nodes, N>0. D - order of the interpolation scheme, 0 <= D <= N-1. D<0 will cause an error. D>=N it will be replaced with D=N-1. if you don't know what D to choose, use small value about 3-5. Output parameters: B - barycentric interpolant. Note: this algorithm always succeeds and calculates the weights with close to machine precision. -- ALGLIB PROJECT -- Copyright 17.06.2007 by Bochkanov Sergey *************************************************************************/ public static void barycentricbuildfloaterhormann(double[] x, double[] y, int n, int d, out barycentricinterpolant b) { b = new barycentricinterpolant(); ratint.barycentricbuildfloaterhormann(x, y, n, d, b.innerobj); return; } } public partial class alglib { /************************************************************************* Conversion from barycentric representation to Chebyshev basis. This function has O(N^2) complexity. INPUT PARAMETERS: P - polynomial in barycentric form A,B - base interval for Chebyshev polynomials (see below) A<>B OUTPUT PARAMETERS T - coefficients of Chebyshev representation; P(x) = sum { T[i]*Ti(2*(x-A)/(B-A)-1), i=0..N-1 }, where Ti - I-th Chebyshev polynomial. NOTES: barycentric interpolant passed as P may be either polynomial obtained from polynomial interpolation/ fitting or rational function which is NOT polynomial. We can't distinguish between these two cases, and this algorithm just tries to work assuming that P IS a polynomial. If not, algorithm will return results, but they won't have any meaning. -- ALGLIB -- Copyright 30.09.2010 by Bochkanov Sergey *************************************************************************/ public static void polynomialbar2cheb(barycentricinterpolant p, double a, double b, out double[] t) { t = new double[0]; polint.polynomialbar2cheb(p.innerobj, a, b, ref t); return; } /************************************************************************* Conversion from Chebyshev basis to barycentric representation. This function has O(N^2) complexity. INPUT PARAMETERS: T - coefficients of Chebyshev representation; P(x) = sum { T[i]*Ti(2*(x-A)/(B-A)-1), i=0..N }, where Ti - I-th Chebyshev polynomial. N - number of coefficients: * if given, only leading N elements of T are used * if not given, automatically determined from size of T A,B - base interval for Chebyshev polynomials (see above) A0. OUTPUT PARAMETERS A - coefficients, P(x) = sum { A[i]*((X-C)/S)^i, i=0..N-1 } N - number of coefficients (polynomial degree plus 1) NOTES: 1. this function accepts offset and scale, which can be set to improve numerical properties of polynomial. For example, if P was obtained as result of interpolation on [-1,+1], you can set C=0 and S=1 and represent P as sum of 1, x, x^2, x^3 and so on. In most cases you it is exactly what you need. However, if your interpolation model was built on [999,1001], you will see significant growth of numerical errors when using {1, x, x^2, x^3} as basis. Representing P as sum of 1, (x-1000), (x-1000)^2, (x-1000)^3 will be better option. Such representation can be obtained by using 1000.0 as offset C and 1.0 as scale S. 2. power basis is ill-conditioned and tricks described above can't solve this problem completely. This function will return coefficients in any case, but for N>8 they will become unreliable. However, N's less than 5 are pretty safe. 3. barycentric interpolant passed as P may be either polynomial obtained from polynomial interpolation/ fitting or rational function which is NOT polynomial. We can't distinguish between these two cases, and this algorithm just tries to work assuming that P IS a polynomial. If not, algorithm will return results, but they won't have any meaning. -- ALGLIB -- Copyright 30.09.2010 by Bochkanov Sergey *************************************************************************/ public static void polynomialbar2pow(barycentricinterpolant p, double c, double s, out double[] a) { a = new double[0]; polint.polynomialbar2pow(p.innerobj, c, s, ref a); return; } public static void polynomialbar2pow(barycentricinterpolant p, out double[] a) { double c; double s; a = new double[0]; c = 0; s = 1; polint.polynomialbar2pow(p.innerobj, c, s, ref a); return; } /************************************************************************* Conversion from power basis to barycentric representation. This function has O(N^2) complexity. INPUT PARAMETERS: A - coefficients, P(x) = sum { A[i]*((X-C)/S)^i, i=0..N-1 } N - number of coefficients (polynomial degree plus 1) * if given, only leading N elements of A are used * if not given, automatically determined from size of A C - offset (see below); 0.0 is used as default value. S - scale (see below); 1.0 is used as default value. S<>0. OUTPUT PARAMETERS P - polynomial in barycentric form NOTES: 1. this function accepts offset and scale, which can be set to improve numerical properties of polynomial. For example, if you interpolate on [-1,+1], you can set C=0 and S=1 and convert from sum of 1, x, x^2, x^3 and so on. In most cases you it is exactly what you need. However, if your interpolation model was built on [999,1001], you will see significant growth of numerical errors when using {1, x, x^2, x^3} as input basis. Converting from sum of 1, (x-1000), (x-1000)^2, (x-1000)^3 will be better option (you have to specify 1000.0 as offset C and 1.0 as scale S). 2. power basis is ill-conditioned and tricks described above can't solve this problem completely. This function will return barycentric model in any case, but for N>8 accuracy well degrade. However, N's less than 5 are pretty safe. -- ALGLIB -- Copyright 30.09.2010 by Bochkanov Sergey *************************************************************************/ public static void polynomialpow2bar(double[] a, int n, double c, double s, out barycentricinterpolant p) { p = new barycentricinterpolant(); polint.polynomialpow2bar(a, n, c, s, p.innerobj); return; } public static void polynomialpow2bar(double[] a, out barycentricinterpolant p) { int n; double c; double s; p = new barycentricinterpolant(); n = ap.len(a); c = 0; s = 1; polint.polynomialpow2bar(a, n, c, s, p.innerobj); return; } /************************************************************************* Lagrange intepolant: generation of the model on the general grid. This function has O(N^2) complexity. INPUT PARAMETERS: X - abscissas, array[0..N-1] Y - function values, array[0..N-1] N - number of points, N>=1 OUTPUT PARAMETERS P - barycentric model which represents Lagrange interpolant (see ratint unit info and BarycentricCalc() description for more information). -- ALGLIB -- Copyright 02.12.2009 by Bochkanov Sergey *************************************************************************/ public static void polynomialbuild(double[] x, double[] y, int n, out barycentricinterpolant p) { p = new barycentricinterpolant(); polint.polynomialbuild(x, y, n, p.innerobj); return; } public static void polynomialbuild(double[] x, double[] y, out barycentricinterpolant p) { int n; if( (ap.len(x)!=ap.len(y))) throw new alglibexception("Error while calling 'polynomialbuild': looks like one of arguments has wrong size"); p = new barycentricinterpolant(); n = ap.len(x); polint.polynomialbuild(x, y, n, p.innerobj); return; } /************************************************************************* Lagrange intepolant: generation of the model on equidistant grid. This function has O(N) complexity. INPUT PARAMETERS: A - left boundary of [A,B] B - right boundary of [A,B] Y - function values at the nodes, array[0..N-1] N - number of points, N>=1 for N=1 a constant model is constructed. OUTPUT PARAMETERS P - barycentric model which represents Lagrange interpolant (see ratint unit info and BarycentricCalc() description for more information). -- ALGLIB -- Copyright 03.12.2009 by Bochkanov Sergey *************************************************************************/ public static void polynomialbuildeqdist(double a, double b, double[] y, int n, out barycentricinterpolant p) { p = new barycentricinterpolant(); polint.polynomialbuildeqdist(a, b, y, n, p.innerobj); return; } public static void polynomialbuildeqdist(double a, double b, double[] y, out barycentricinterpolant p) { int n; p = new barycentricinterpolant(); n = ap.len(y); polint.polynomialbuildeqdist(a, b, y, n, p.innerobj); return; } /************************************************************************* Lagrange intepolant on Chebyshev grid (first kind). This function has O(N) complexity. INPUT PARAMETERS: A - left boundary of [A,B] B - right boundary of [A,B] Y - function values at the nodes, array[0..N-1], Y[I] = Y(0.5*(B+A) + 0.5*(B-A)*Cos(PI*(2*i+1)/(2*n))) N - number of points, N>=1 for N=1 a constant model is constructed. OUTPUT PARAMETERS P - barycentric model which represents Lagrange interpolant (see ratint unit info and BarycentricCalc() description for more information). -- ALGLIB -- Copyright 03.12.2009 by Bochkanov Sergey *************************************************************************/ public static void polynomialbuildcheb1(double a, double b, double[] y, int n, out barycentricinterpolant p) { p = new barycentricinterpolant(); polint.polynomialbuildcheb1(a, b, y, n, p.innerobj); return; } public static void polynomialbuildcheb1(double a, double b, double[] y, out barycentricinterpolant p) { int n; p = new barycentricinterpolant(); n = ap.len(y); polint.polynomialbuildcheb1(a, b, y, n, p.innerobj); return; } /************************************************************************* Lagrange intepolant on Chebyshev grid (second kind). This function has O(N) complexity. INPUT PARAMETERS: A - left boundary of [A,B] B - right boundary of [A,B] Y - function values at the nodes, array[0..N-1], Y[I] = Y(0.5*(B+A) + 0.5*(B-A)*Cos(PI*i/(n-1))) N - number of points, N>=1 for N=1 a constant model is constructed. OUTPUT PARAMETERS P - barycentric model which represents Lagrange interpolant (see ratint unit info and BarycentricCalc() description for more information). -- ALGLIB -- Copyright 03.12.2009 by Bochkanov Sergey *************************************************************************/ public static void polynomialbuildcheb2(double a, double b, double[] y, int n, out barycentricinterpolant p) { p = new barycentricinterpolant(); polint.polynomialbuildcheb2(a, b, y, n, p.innerobj); return; } public static void polynomialbuildcheb2(double a, double b, double[] y, out barycentricinterpolant p) { int n; p = new barycentricinterpolant(); n = ap.len(y); polint.polynomialbuildcheb2(a, b, y, n, p.innerobj); return; } /************************************************************************* Fast equidistant polynomial interpolation function with O(N) complexity INPUT PARAMETERS: A - left boundary of [A,B] B - right boundary of [A,B] F - function values, array[0..N-1] N - number of points on equidistant grid, N>=1 for N=1 a constant model is constructed. T - position where P(x) is calculated RESULT value of the Lagrange interpolant at T IMPORTANT this function provides fast interface which is not overflow-safe nor it is very precise. the best option is to use PolynomialBuildEqDist()/BarycentricCalc() subroutines unless you are pretty sure that your data will not result in overflow. -- ALGLIB -- Copyright 02.12.2009 by Bochkanov Sergey *************************************************************************/ public static double polynomialcalceqdist(double a, double b, double[] f, int n, double t) { double result = polint.polynomialcalceqdist(a, b, f, n, t); return result; } public static double polynomialcalceqdist(double a, double b, double[] f, double t) { int n; n = ap.len(f); double result = polint.polynomialcalceqdist(a, b, f, n, t); return result; } /************************************************************************* Fast polynomial interpolation function on Chebyshev points (first kind) with O(N) complexity. INPUT PARAMETERS: A - left boundary of [A,B] B - right boundary of [A,B] F - function values, array[0..N-1] N - number of points on Chebyshev grid (first kind), X[i] = 0.5*(B+A) + 0.5*(B-A)*Cos(PI*(2*i+1)/(2*n)) for N=1 a constant model is constructed. T - position where P(x) is calculated RESULT value of the Lagrange interpolant at T IMPORTANT this function provides fast interface which is not overflow-safe nor it is very precise. the best option is to use PolIntBuildCheb1()/BarycentricCalc() subroutines unless you are pretty sure that your data will not result in overflow. -- ALGLIB -- Copyright 02.12.2009 by Bochkanov Sergey *************************************************************************/ public static double polynomialcalccheb1(double a, double b, double[] f, int n, double t) { double result = polint.polynomialcalccheb1(a, b, f, n, t); return result; } public static double polynomialcalccheb1(double a, double b, double[] f, double t) { int n; n = ap.len(f); double result = polint.polynomialcalccheb1(a, b, f, n, t); return result; } /************************************************************************* Fast polynomial interpolation function on Chebyshev points (second kind) with O(N) complexity. INPUT PARAMETERS: A - left boundary of [A,B] B - right boundary of [A,B] F - function values, array[0..N-1] N - number of points on Chebyshev grid (second kind), X[i] = 0.5*(B+A) + 0.5*(B-A)*Cos(PI*i/(n-1)) for N=1 a constant model is constructed. T - position where P(x) is calculated RESULT value of the Lagrange interpolant at T IMPORTANT this function provides fast interface which is not overflow-safe nor it is very precise. the best option is to use PolIntBuildCheb2()/BarycentricCalc() subroutines unless you are pretty sure that your data will not result in overflow. -- ALGLIB -- Copyright 02.12.2009 by Bochkanov Sergey *************************************************************************/ public static double polynomialcalccheb2(double a, double b, double[] f, int n, double t) { double result = polint.polynomialcalccheb2(a, b, f, n, t); return result; } public static double polynomialcalccheb2(double a, double b, double[] f, double t) { int n; n = ap.len(f); double result = polint.polynomialcalccheb2(a, b, f, n, t); return result; } } public partial class alglib { /************************************************************************* 1-dimensional spline inteprolant *************************************************************************/ public class spline1dinterpolant { // // Public declarations // public spline1dinterpolant() { _innerobj = new spline1d.spline1dinterpolant(); } // // Although some of declarations below are public, you should not use them // They are intended for internal use only // private spline1d.spline1dinterpolant _innerobj; public spline1d.spline1dinterpolant innerobj { get { return _innerobj; } } public spline1dinterpolant(spline1d.spline1dinterpolant obj) { _innerobj = obj; } } /************************************************************************* This subroutine builds linear spline interpolant INPUT PARAMETERS: X - spline nodes, array[0..N-1] Y - function values, array[0..N-1] N - points count (optional): * N>=2 * if given, only first N points are used to build spline * if not given, automatically detected from X/Y sizes (len(X) must be equal to len(Y)) OUTPUT PARAMETERS: C - spline interpolant ORDER OF POINTS Subroutine automatically sorts points, so caller may pass unsorted array. -- ALGLIB PROJECT -- Copyright 24.06.2007 by Bochkanov Sergey *************************************************************************/ public static void spline1dbuildlinear(double[] x, double[] y, int n, out spline1dinterpolant c) { c = new spline1dinterpolant(); spline1d.spline1dbuildlinear(x, y, n, c.innerobj); return; } public static void spline1dbuildlinear(double[] x, double[] y, out spline1dinterpolant c) { int n; if( (ap.len(x)!=ap.len(y))) throw new alglibexception("Error while calling 'spline1dbuildlinear': looks like one of arguments has wrong size"); c = new spline1dinterpolant(); n = ap.len(x); spline1d.spline1dbuildlinear(x, y, n, c.innerobj); return; } /************************************************************************* This subroutine builds cubic spline interpolant. INPUT PARAMETERS: X - spline nodes, array[0..N-1]. Y - function values, array[0..N-1]. OPTIONAL PARAMETERS: N - points count: * N>=2 * if given, only first N points are used to build spline * if not given, automatically detected from X/Y sizes (len(X) must be equal to len(Y)) BoundLType - boundary condition type for the left boundary BoundL - left boundary condition (first or second derivative, depending on the BoundLType) BoundRType - boundary condition type for the right boundary BoundR - right boundary condition (first or second derivative, depending on the BoundRType) OUTPUT PARAMETERS: C - spline interpolant ORDER OF POINTS Subroutine automatically sorts points, so caller may pass unsorted array. SETTING BOUNDARY VALUES: The BoundLType/BoundRType parameters can have the following values: * -1, which corresonds to the periodic (cyclic) boundary conditions. In this case: * both BoundLType and BoundRType must be equal to -1. * BoundL/BoundR are ignored * Y[last] is ignored (it is assumed to be equal to Y[first]). * 0, which corresponds to the parabolically terminated spline (BoundL and/or BoundR are ignored). * 1, which corresponds to the first derivative boundary condition * 2, which corresponds to the second derivative boundary condition * by default, BoundType=0 is used PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS: Problems with periodic boundary conditions have Y[first_point]=Y[last_point]. However, this subroutine doesn't require you to specify equal values for the first and last points - it automatically forces them to be equal by copying Y[first_point] (corresponds to the leftmost, minimal X[]) to Y[last_point]. However it is recommended to pass consistent values of Y[], i.e. to make Y[first_point]=Y[last_point]. -- ALGLIB PROJECT -- Copyright 23.06.2007 by Bochkanov Sergey *************************************************************************/ public static void spline1dbuildcubic(double[] x, double[] y, int n, int boundltype, double boundl, int boundrtype, double boundr, out spline1dinterpolant c) { c = new spline1dinterpolant(); spline1d.spline1dbuildcubic(x, y, n, boundltype, boundl, boundrtype, boundr, c.innerobj); return; } public static void spline1dbuildcubic(double[] x, double[] y, out spline1dinterpolant c) { int n; int boundltype; double boundl; int boundrtype; double boundr; if( (ap.len(x)!=ap.len(y))) throw new alglibexception("Error while calling 'spline1dbuildcubic': looks like one of arguments has wrong size"); c = new spline1dinterpolant(); n = ap.len(x); boundltype = 0; boundl = 0; boundrtype = 0; boundr = 0; spline1d.spline1dbuildcubic(x, y, n, boundltype, boundl, boundrtype, boundr, c.innerobj); return; } /************************************************************************* This function solves following problem: given table y[] of function values at nodes x[], it calculates and returns table of function derivatives d[] (calculated at the same nodes x[]). This function yields same result as Spline1DBuildCubic() call followed by sequence of Spline1DDiff() calls, but it can be several times faster when called for ordered X[] and X2[]. INPUT PARAMETERS: X - spline nodes Y - function values OPTIONAL PARAMETERS: N - points count: * N>=2 * if given, only first N points are used * if not given, automatically detected from X/Y sizes (len(X) must be equal to len(Y)) BoundLType - boundary condition type for the left boundary BoundL - left boundary condition (first or second derivative, depending on the BoundLType) BoundRType - boundary condition type for the right boundary BoundR - right boundary condition (first or second derivative, depending on the BoundRType) OUTPUT PARAMETERS: D - derivative values at X[] ORDER OF POINTS Subroutine automatically sorts points, so caller may pass unsorted array. Derivative values are correctly reordered on return, so D[I] is always equal to S'(X[I]) independently of points order. SETTING BOUNDARY VALUES: The BoundLType/BoundRType parameters can have the following values: * -1, which corresonds to the periodic (cyclic) boundary conditions. In this case: * both BoundLType and BoundRType must be equal to -1. * BoundL/BoundR are ignored * Y[last] is ignored (it is assumed to be equal to Y[first]). * 0, which corresponds to the parabolically terminated spline (BoundL and/or BoundR are ignored). * 1, which corresponds to the first derivative boundary condition * 2, which corresponds to the second derivative boundary condition * by default, BoundType=0 is used PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS: Problems with periodic boundary conditions have Y[first_point]=Y[last_point]. However, this subroutine doesn't require you to specify equal values for the first and last points - it automatically forces them to be equal by copying Y[first_point] (corresponds to the leftmost, minimal X[]) to Y[last_point]. However it is recommended to pass consistent values of Y[], i.e. to make Y[first_point]=Y[last_point]. -- ALGLIB PROJECT -- Copyright 03.09.2010 by Bochkanov Sergey *************************************************************************/ public static void spline1dgriddiffcubic(double[] x, double[] y, int n, int boundltype, double boundl, int boundrtype, double boundr, out double[] d) { d = new double[0]; spline1d.spline1dgriddiffcubic(x, y, n, boundltype, boundl, boundrtype, boundr, ref d); return; } public static void spline1dgriddiffcubic(double[] x, double[] y, out double[] d) { int n; int boundltype; double boundl; int boundrtype; double boundr; if( (ap.len(x)!=ap.len(y))) throw new alglibexception("Error while calling 'spline1dgriddiffcubic': looks like one of arguments has wrong size"); d = new double[0]; n = ap.len(x); boundltype = 0; boundl = 0; boundrtype = 0; boundr = 0; spline1d.spline1dgriddiffcubic(x, y, n, boundltype, boundl, boundrtype, boundr, ref d); return; } /************************************************************************* This function solves following problem: given table y[] of function values at nodes x[], it calculates and returns tables of first and second function derivatives d1[] and d2[] (calculated at the same nodes x[]). This function yields same result as Spline1DBuildCubic() call followed by sequence of Spline1DDiff() calls, but it can be several times faster when called for ordered X[] and X2[]. INPUT PARAMETERS: X - spline nodes Y - function values OPTIONAL PARAMETERS: N - points count: * N>=2 * if given, only first N points are used * if not given, automatically detected from X/Y sizes (len(X) must be equal to len(Y)) BoundLType - boundary condition type for the left boundary BoundL - left boundary condition (first or second derivative, depending on the BoundLType) BoundRType - boundary condition type for the right boundary BoundR - right boundary condition (first or second derivative, depending on the BoundRType) OUTPUT PARAMETERS: D1 - S' values at X[] D2 - S'' values at X[] ORDER OF POINTS Subroutine automatically sorts points, so caller may pass unsorted array. Derivative values are correctly reordered on return, so D[I] is always equal to S'(X[I]) independently of points order. SETTING BOUNDARY VALUES: The BoundLType/BoundRType parameters can have the following values: * -1, which corresonds to the periodic (cyclic) boundary conditions. In this case: * both BoundLType and BoundRType must be equal to -1. * BoundL/BoundR are ignored * Y[last] is ignored (it is assumed to be equal to Y[first]). * 0, which corresponds to the parabolically terminated spline (BoundL and/or BoundR are ignored). * 1, which corresponds to the first derivative boundary condition * 2, which corresponds to the second derivative boundary condition * by default, BoundType=0 is used PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS: Problems with periodic boundary conditions have Y[first_point]=Y[last_point]. However, this subroutine doesn't require you to specify equal values for the first and last points - it automatically forces them to be equal by copying Y[first_point] (corresponds to the leftmost, minimal X[]) to Y[last_point]. However it is recommended to pass consistent values of Y[], i.e. to make Y[first_point]=Y[last_point]. -- ALGLIB PROJECT -- Copyright 03.09.2010 by Bochkanov Sergey *************************************************************************/ public static void spline1dgriddiff2cubic(double[] x, double[] y, int n, int boundltype, double boundl, int boundrtype, double boundr, out double[] d1, out double[] d2) { d1 = new double[0]; d2 = new double[0]; spline1d.spline1dgriddiff2cubic(x, y, n, boundltype, boundl, boundrtype, boundr, ref d1, ref d2); return; } public static void spline1dgriddiff2cubic(double[] x, double[] y, out double[] d1, out double[] d2) { int n; int boundltype; double boundl; int boundrtype; double boundr; if( (ap.len(x)!=ap.len(y))) throw new alglibexception("Error while calling 'spline1dgriddiff2cubic': looks like one of arguments has wrong size"); d1 = new double[0]; d2 = new double[0]; n = ap.len(x); boundltype = 0; boundl = 0; boundrtype = 0; boundr = 0; spline1d.spline1dgriddiff2cubic(x, y, n, boundltype, boundl, boundrtype, boundr, ref d1, ref d2); return; } /************************************************************************* This function solves following problem: given table y[] of function values at old nodes x[] and new nodes x2[], it calculates and returns table of function values y2[] (calculated at x2[]). This function yields same result as Spline1DBuildCubic() call followed by sequence of Spline1DDiff() calls, but it can be several times faster when called for ordered X[] and X2[]. INPUT PARAMETERS: X - old spline nodes Y - function values X2 - new spline nodes OPTIONAL PARAMETERS: N - points count: * N>=2 * if given, only first N points from X/Y are used * if not given, automatically detected from X/Y sizes (len(X) must be equal to len(Y)) BoundLType - boundary condition type for the left boundary BoundL - left boundary condition (first or second derivative, depending on the BoundLType) BoundRType - boundary condition type for the right boundary BoundR - right boundary condition (first or second derivative, depending on the BoundRType) N2 - new points count: * N2>=2 * if given, only first N2 points from X2 are used * if not given, automatically detected from X2 size OUTPUT PARAMETERS: F2 - function values at X2[] ORDER OF POINTS Subroutine automatically sorts points, so caller may pass unsorted array. Function values are correctly reordered on return, so F2[I] is always equal to S(X2[I]) independently of points order. SETTING BOUNDARY VALUES: The BoundLType/BoundRType parameters can have the following values: * -1, which corresonds to the periodic (cyclic) boundary conditions. In this case: * both BoundLType and BoundRType must be equal to -1. * BoundL/BoundR are ignored * Y[last] is ignored (it is assumed to be equal to Y[first]). * 0, which corresponds to the parabolically terminated spline (BoundL and/or BoundR are ignored). * 1, which corresponds to the first derivative boundary condition * 2, which corresponds to the second derivative boundary condition * by default, BoundType=0 is used PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS: Problems with periodic boundary conditions have Y[first_point]=Y[last_point]. However, this subroutine doesn't require you to specify equal values for the first and last points - it automatically forces them to be equal by copying Y[first_point] (corresponds to the leftmost, minimal X[]) to Y[last_point]. However it is recommended to pass consistent values of Y[], i.e. to make Y[first_point]=Y[last_point]. -- ALGLIB PROJECT -- Copyright 03.09.2010 by Bochkanov Sergey *************************************************************************/ public static void spline1dconvcubic(double[] x, double[] y, int n, int boundltype, double boundl, int boundrtype, double boundr, double[] x2, int n2, out double[] y2) { y2 = new double[0]; spline1d.spline1dconvcubic(x, y, n, boundltype, boundl, boundrtype, boundr, x2, n2, ref y2); return; } public static void spline1dconvcubic(double[] x, double[] y, double[] x2, out double[] y2) { int n; int boundltype; double boundl; int boundrtype; double boundr; int n2; if( (ap.len(x)!=ap.len(y))) throw new alglibexception("Error while calling 'spline1dconvcubic': looks like one of arguments has wrong size"); y2 = new double[0]; n = ap.len(x); boundltype = 0; boundl = 0; boundrtype = 0; boundr = 0; n2 = ap.len(x2); spline1d.spline1dconvcubic(x, y, n, boundltype, boundl, boundrtype, boundr, x2, n2, ref y2); return; } /************************************************************************* This function solves following problem: given table y[] of function values at old nodes x[] and new nodes x2[], it calculates and returns table of function values y2[] and derivatives d2[] (calculated at x2[]). This function yields same result as Spline1DBuildCubic() call followed by sequence of Spline1DDiff() calls, but it can be several times faster when called for ordered X[] and X2[]. INPUT PARAMETERS: X - old spline nodes Y - function values X2 - new spline nodes OPTIONAL PARAMETERS: N - points count: * N>=2 * if given, only first N points from X/Y are used * if not given, automatically detected from X/Y sizes (len(X) must be equal to len(Y)) BoundLType - boundary condition type for the left boundary BoundL - left boundary condition (first or second derivative, depending on the BoundLType) BoundRType - boundary condition type for the right boundary BoundR - right boundary condition (first or second derivative, depending on the BoundRType) N2 - new points count: * N2>=2 * if given, only first N2 points from X2 are used * if not given, automatically detected from X2 size OUTPUT PARAMETERS: F2 - function values at X2[] D2 - first derivatives at X2[] ORDER OF POINTS Subroutine automatically sorts points, so caller may pass unsorted array. Function values are correctly reordered on return, so F2[I] is always equal to S(X2[I]) independently of points order. SETTING BOUNDARY VALUES: The BoundLType/BoundRType parameters can have the following values: * -1, which corresonds to the periodic (cyclic) boundary conditions. In this case: * both BoundLType and BoundRType must be equal to -1. * BoundL/BoundR are ignored * Y[last] is ignored (it is assumed to be equal to Y[first]). * 0, which corresponds to the parabolically terminated spline (BoundL and/or BoundR are ignored). * 1, which corresponds to the first derivative boundary condition * 2, which corresponds to the second derivative boundary condition * by default, BoundType=0 is used PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS: Problems with periodic boundary conditions have Y[first_point]=Y[last_point]. However, this subroutine doesn't require you to specify equal values for the first and last points - it automatically forces them to be equal by copying Y[first_point] (corresponds to the leftmost, minimal X[]) to Y[last_point]. However it is recommended to pass consistent values of Y[], i.e. to make Y[first_point]=Y[last_point]. -- ALGLIB PROJECT -- Copyright 03.09.2010 by Bochkanov Sergey *************************************************************************/ public static void spline1dconvdiffcubic(double[] x, double[] y, int n, int boundltype, double boundl, int boundrtype, double boundr, double[] x2, int n2, out double[] y2, out double[] d2) { y2 = new double[0]; d2 = new double[0]; spline1d.spline1dconvdiffcubic(x, y, n, boundltype, boundl, boundrtype, boundr, x2, n2, ref y2, ref d2); return; } public static void spline1dconvdiffcubic(double[] x, double[] y, double[] x2, out double[] y2, out double[] d2) { int n; int boundltype; double boundl; int boundrtype; double boundr; int n2; if( (ap.len(x)!=ap.len(y))) throw new alglibexception("Error while calling 'spline1dconvdiffcubic': looks like one of arguments has wrong size"); y2 = new double[0]; d2 = new double[0]; n = ap.len(x); boundltype = 0; boundl = 0; boundrtype = 0; boundr = 0; n2 = ap.len(x2); spline1d.spline1dconvdiffcubic(x, y, n, boundltype, boundl, boundrtype, boundr, x2, n2, ref y2, ref d2); return; } /************************************************************************* This function solves following problem: given table y[] of function values at old nodes x[] and new nodes x2[], it calculates and returns table of function values y2[], first and second derivatives d2[] and dd2[] (calculated at x2[]). This function yields same result as Spline1DBuildCubic() call followed by sequence of Spline1DDiff() calls, but it can be several times faster when called for ordered X[] and X2[]. INPUT PARAMETERS: X - old spline nodes Y - function values X2 - new spline nodes OPTIONAL PARAMETERS: N - points count: * N>=2 * if given, only first N points from X/Y are used * if not given, automatically detected from X/Y sizes (len(X) must be equal to len(Y)) BoundLType - boundary condition type for the left boundary BoundL - left boundary condition (first or second derivative, depending on the BoundLType) BoundRType - boundary condition type for the right boundary BoundR - right boundary condition (first or second derivative, depending on the BoundRType) N2 - new points count: * N2>=2 * if given, only first N2 points from X2 are used * if not given, automatically detected from X2 size OUTPUT PARAMETERS: F2 - function values at X2[] D2 - first derivatives at X2[] DD2 - second derivatives at X2[] ORDER OF POINTS Subroutine automatically sorts points, so caller may pass unsorted array. Function values are correctly reordered on return, so F2[I] is always equal to S(X2[I]) independently of points order. SETTING BOUNDARY VALUES: The BoundLType/BoundRType parameters can have the following values: * -1, which corresonds to the periodic (cyclic) boundary conditions. In this case: * both BoundLType and BoundRType must be equal to -1. * BoundL/BoundR are ignored * Y[last] is ignored (it is assumed to be equal to Y[first]). * 0, which corresponds to the parabolically terminated spline (BoundL and/or BoundR are ignored). * 1, which corresponds to the first derivative boundary condition * 2, which corresponds to the second derivative boundary condition * by default, BoundType=0 is used PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS: Problems with periodic boundary conditions have Y[first_point]=Y[last_point]. However, this subroutine doesn't require you to specify equal values for the first and last points - it automatically forces them to be equal by copying Y[first_point] (corresponds to the leftmost, minimal X[]) to Y[last_point]. However it is recommended to pass consistent values of Y[], i.e. to make Y[first_point]=Y[last_point]. -- ALGLIB PROJECT -- Copyright 03.09.2010 by Bochkanov Sergey *************************************************************************/ public static void spline1dconvdiff2cubic(double[] x, double[] y, int n, int boundltype, double boundl, int boundrtype, double boundr, double[] x2, int n2, out double[] y2, out double[] d2, out double[] dd2) { y2 = new double[0]; d2 = new double[0]; dd2 = new double[0]; spline1d.spline1dconvdiff2cubic(x, y, n, boundltype, boundl, boundrtype, boundr, x2, n2, ref y2, ref d2, ref dd2); return; } public static void spline1dconvdiff2cubic(double[] x, double[] y, double[] x2, out double[] y2, out double[] d2, out double[] dd2) { int n; int boundltype; double boundl; int boundrtype; double boundr; int n2; if( (ap.len(x)!=ap.len(y))) throw new alglibexception("Error while calling 'spline1dconvdiff2cubic': looks like one of arguments has wrong size"); y2 = new double[0]; d2 = new double[0]; dd2 = new double[0]; n = ap.len(x); boundltype = 0; boundl = 0; boundrtype = 0; boundr = 0; n2 = ap.len(x2); spline1d.spline1dconvdiff2cubic(x, y, n, boundltype, boundl, boundrtype, boundr, x2, n2, ref y2, ref d2, ref dd2); return; } /************************************************************************* This subroutine builds Catmull-Rom spline interpolant. INPUT PARAMETERS: X - spline nodes, array[0..N-1]. Y - function values, array[0..N-1]. OPTIONAL PARAMETERS: N - points count: * N>=2 * if given, only first N points are used to build spline * if not given, automatically detected from X/Y sizes (len(X) must be equal to len(Y)) BoundType - boundary condition type: * -1 for periodic boundary condition * 0 for parabolically terminated spline (default) Tension - tension parameter: * tension=0 corresponds to classic Catmull-Rom spline (default) * 0=2 * if given, only first N points are used to build spline * if not given, automatically detected from X/Y sizes (len(X) must be equal to len(Y)) OUTPUT PARAMETERS: C - spline interpolant. ORDER OF POINTS Subroutine automatically sorts points, so caller may pass unsorted array. -- ALGLIB PROJECT -- Copyright 23.06.2007 by Bochkanov Sergey *************************************************************************/ public static void spline1dbuildhermite(double[] x, double[] y, double[] d, int n, out spline1dinterpolant c) { c = new spline1dinterpolant(); spline1d.spline1dbuildhermite(x, y, d, n, c.innerobj); return; } public static void spline1dbuildhermite(double[] x, double[] y, double[] d, out spline1dinterpolant c) { int n; if( (ap.len(x)!=ap.len(y)) || (ap.len(x)!=ap.len(d))) throw new alglibexception("Error while calling 'spline1dbuildhermite': looks like one of arguments has wrong size"); c = new spline1dinterpolant(); n = ap.len(x); spline1d.spline1dbuildhermite(x, y, d, n, c.innerobj); return; } /************************************************************************* This subroutine builds Akima spline interpolant INPUT PARAMETERS: X - spline nodes, array[0..N-1] Y - function values, array[0..N-1] N - points count (optional): * N>=5 * if given, only first N points are used to build spline * if not given, automatically detected from X/Y sizes (len(X) must be equal to len(Y)) OUTPUT PARAMETERS: C - spline interpolant ORDER OF POINTS Subroutine automatically sorts points, so caller may pass unsorted array. -- ALGLIB PROJECT -- Copyright 24.06.2007 by Bochkanov Sergey *************************************************************************/ public static void spline1dbuildakima(double[] x, double[] y, int n, out spline1dinterpolant c) { c = new spline1dinterpolant(); spline1d.spline1dbuildakima(x, y, n, c.innerobj); return; } public static void spline1dbuildakima(double[] x, double[] y, out spline1dinterpolant c) { int n; if( (ap.len(x)!=ap.len(y))) throw new alglibexception("Error while calling 'spline1dbuildakima': looks like one of arguments has wrong size"); c = new spline1dinterpolant(); n = ap.len(x); spline1d.spline1dbuildakima(x, y, n, c.innerobj); return; } /************************************************************************* This subroutine calculates the value of the spline at the given point X. INPUT PARAMETERS: C - spline interpolant X - point Result: S(x) -- ALGLIB PROJECT -- Copyright 23.06.2007 by Bochkanov Sergey *************************************************************************/ public static double spline1dcalc(spline1dinterpolant c, double x) { double result = spline1d.spline1dcalc(c.innerobj, x); return result; } /************************************************************************* This subroutine differentiates the spline. INPUT PARAMETERS: C - spline interpolant. X - point Result: S - S(x) DS - S'(x) D2S - S''(x) -- ALGLIB PROJECT -- Copyright 24.06.2007 by Bochkanov Sergey *************************************************************************/ public static void spline1ddiff(spline1dinterpolant c, double x, out double s, out double ds, out double d2s) { s = 0; ds = 0; d2s = 0; spline1d.spline1ddiff(c.innerobj, x, ref s, ref ds, ref d2s); return; } /************************************************************************* This subroutine unpacks the spline into the coefficients table. INPUT PARAMETERS: C - spline interpolant. X - point Result: Tbl - coefficients table, unpacked format, array[0..N-2, 0..5]. For I = 0...N-2: Tbl[I,0] = X[i] Tbl[I,1] = X[i+1] Tbl[I,2] = C0 Tbl[I,3] = C1 Tbl[I,4] = C2 Tbl[I,5] = C3 On [x[i], x[i+1]] spline is equals to: S(x) = C0 + C1*t + C2*t^2 + C3*t^3 t = x-x[i] -- ALGLIB PROJECT -- Copyright 29.06.2007 by Bochkanov Sergey *************************************************************************/ public static void spline1dunpack(spline1dinterpolant c, out int n, out double[,] tbl) { n = 0; tbl = new double[0,0]; spline1d.spline1dunpack(c.innerobj, ref n, ref tbl); return; } /************************************************************************* This subroutine performs linear transformation of the spline argument. INPUT PARAMETERS: C - spline interpolant. A, B- transformation coefficients: x = A*t + B Result: C - transformed spline -- ALGLIB PROJECT -- Copyright 30.06.2007 by Bochkanov Sergey *************************************************************************/ public static void spline1dlintransx(spline1dinterpolant c, double a, double b) { spline1d.spline1dlintransx(c.innerobj, a, b); return; } /************************************************************************* This subroutine performs linear transformation of the spline. INPUT PARAMETERS: C - spline interpolant. A, B- transformation coefficients: S2(x) = A*S(x) + B Result: C - transformed spline -- ALGLIB PROJECT -- Copyright 30.06.2007 by Bochkanov Sergey *************************************************************************/ public static void spline1dlintransy(spline1dinterpolant c, double a, double b) { spline1d.spline1dlintransy(c.innerobj, a, b); return; } /************************************************************************* This subroutine integrates the spline. INPUT PARAMETERS: C - spline interpolant. X - right bound of the integration interval [a, x], here 'a' denotes min(x[]) Result: integral(S(t)dt,a,x) -- ALGLIB PROJECT -- Copyright 23.06.2007 by Bochkanov Sergey *************************************************************************/ public static double spline1dintegrate(spline1dinterpolant c, double x) { double result = spline1d.spline1dintegrate(c.innerobj, x); return result; } } public partial class alglib { /************************************************************************* Polynomial fitting report: TaskRCond reciprocal of task's condition number RMSError RMS error AvgError average error AvgRelError average relative error (for non-zero Y[I]) MaxError maximum error *************************************************************************/ public class polynomialfitreport { // // Public declarations // public double taskrcond { get { return _innerobj.taskrcond; } set { _innerobj.taskrcond = value; } } public double rmserror { get { return _innerobj.rmserror; } set { _innerobj.rmserror = value; } } public double avgerror { get { return _innerobj.avgerror; } set { _innerobj.avgerror = value; } } public double avgrelerror { get { return _innerobj.avgrelerror; } set { _innerobj.avgrelerror = value; } } public double maxerror { get { return _innerobj.maxerror; } set { _innerobj.maxerror = value; } } public polynomialfitreport() { _innerobj = new lsfit.polynomialfitreport(); } // // Although some of declarations below are public, you should not use them // They are intended for internal use only // private lsfit.polynomialfitreport _innerobj; public lsfit.polynomialfitreport innerobj { get { return _innerobj; } } public polynomialfitreport(lsfit.polynomialfitreport obj) { _innerobj = obj; } } /************************************************************************* Barycentric fitting report: RMSError RMS error AvgError average error AvgRelError average relative error (for non-zero Y[I]) MaxError maximum error TaskRCond reciprocal of task's condition number *************************************************************************/ public class barycentricfitreport { // // Public declarations // public double taskrcond { get { return _innerobj.taskrcond; } set { _innerobj.taskrcond = value; } } public int dbest { get { return _innerobj.dbest; } set { _innerobj.dbest = value; } } public double rmserror { get { return _innerobj.rmserror; } set { _innerobj.rmserror = value; } } public double avgerror { get { return _innerobj.avgerror; } set { _innerobj.avgerror = value; } } public double avgrelerror { get { return _innerobj.avgrelerror; } set { _innerobj.avgrelerror = value; } } public double maxerror { get { return _innerobj.maxerror; } set { _innerobj.maxerror = value; } } public barycentricfitreport() { _innerobj = new lsfit.barycentricfitreport(); } // // Although some of declarations below are public, you should not use them // They are intended for internal use only // private lsfit.barycentricfitreport _innerobj; public lsfit.barycentricfitreport innerobj { get { return _innerobj; } } public barycentricfitreport(lsfit.barycentricfitreport obj) { _innerobj = obj; } } /************************************************************************* Spline fitting report: RMSError RMS error AvgError average error AvgRelError average relative error (for non-zero Y[I]) MaxError maximum error Fields below are filled by obsolete functions (Spline1DFitCubic, Spline1DFitHermite). Modern fitting functions do NOT fill these fields: TaskRCond reciprocal of task's condition number *************************************************************************/ public class spline1dfitreport { // // Public declarations // public double taskrcond { get { return _innerobj.taskrcond; } set { _innerobj.taskrcond = value; } } public double rmserror { get { return _innerobj.rmserror; } set { _innerobj.rmserror = value; } } public double avgerror { get { return _innerobj.avgerror; } set { _innerobj.avgerror = value; } } public double avgrelerror { get { return _innerobj.avgrelerror; } set { _innerobj.avgrelerror = value; } } public double maxerror { get { return _innerobj.maxerror; } set { _innerobj.maxerror = value; } } public spline1dfitreport() { _innerobj = new lsfit.spline1dfitreport(); } // // Although some of declarations below are public, you should not use them // They are intended for internal use only // private lsfit.spline1dfitreport _innerobj; public lsfit.spline1dfitreport innerobj { get { return _innerobj; } } public spline1dfitreport(lsfit.spline1dfitreport obj) { _innerobj = obj; } } /************************************************************************* Least squares fitting report: TaskRCond reciprocal of task's condition number IterationsCount number of internal iterations RMSError RMS error AvgError average error AvgRelError average relative error (for non-zero Y[I]) MaxError maximum error WRMSError weighted RMS error *************************************************************************/ public class lsfitreport { // // Public declarations // public double taskrcond { get { return _innerobj.taskrcond; } set { _innerobj.taskrcond = value; } } public int iterationscount { get { return _innerobj.iterationscount; } set { _innerobj.iterationscount = value; } } public double rmserror { get { return _innerobj.rmserror; } set { _innerobj.rmserror = value; } } public double avgerror { get { return _innerobj.avgerror; } set { _innerobj.avgerror = value; } } public double avgrelerror { get { return _innerobj.avgrelerror; } set { _innerobj.avgrelerror = value; } } public double maxerror { get { return _innerobj.maxerror; } set { _innerobj.maxerror = value; } } public double wrmserror { get { return _innerobj.wrmserror; } set { _innerobj.wrmserror = value; } } public lsfitreport() { _innerobj = new lsfit.lsfitreport(); } // // Although some of declarations below are public, you should not use them // They are intended for internal use only // private lsfit.lsfitreport _innerobj; public lsfit.lsfitreport innerobj { get { return _innerobj; } } public lsfitreport(lsfit.lsfitreport obj) { _innerobj = obj; } } /************************************************************************* Nonlinear fitter. You should use ALGLIB functions to work with fitter. Never try to access its fields directly! *************************************************************************/ public class lsfitstate { // // Public declarations // public bool needf { get { return _innerobj.needf; } set { _innerobj.needf = value; } } public bool needfg { get { return _innerobj.needfg; } set { _innerobj.needfg = value; } } public bool needfgh { get { return _innerobj.needfgh; } set { _innerobj.needfgh = value; } } public bool xupdated { get { return _innerobj.xupdated; } set { _innerobj.xupdated = value; } } public double[] c { get { return _innerobj.c; } } public double f { get { return _innerobj.f; } set { _innerobj.f = value; } } public double[] g { get { return _innerobj.g; } } public double[,] h { get { return _innerobj.h; } } public double[] x { get { return _innerobj.x; } } public lsfitstate() { _innerobj = new lsfit.lsfitstate(); } // // Although some of declarations below are public, you should not use them // They are intended for internal use only // private lsfit.lsfitstate _innerobj; public lsfit.lsfitstate innerobj { get { return _innerobj; } } public lsfitstate(lsfit.lsfitstate obj) { _innerobj = obj; } } /************************************************************************* Fitting by polynomials in barycentric form. This function provides simple unterface for unconstrained unweighted fitting. See PolynomialFitWC() if you need constrained fitting. Task is linear, so linear least squares solver is used. Complexity of this computational scheme is O(N*M^2), mostly dominated by least squares solver SEE ALSO: PolynomialFitWC() INPUT PARAMETERS: X - points, array[0..N-1]. Y - function values, array[0..N-1]. N - number of points, N>0 * if given, only leading N elements of X/Y are used * if not given, automatically determined from sizes of X/Y M - number of basis functions (= polynomial_degree + 1), M>=1 OUTPUT PARAMETERS: Info- same format as in LSFitLinearW() subroutine: * Info>0 task is solved * Info<=0 an error occured: -4 means inconvergence of internal SVD P - interpolant in barycentric form. Rep - report, same format as in LSFitLinearW() subroutine. Following fields are set: * RMSError rms error on the (X,Y). * AvgError average error on the (X,Y). * AvgRelError average relative error on the non-zero Y * MaxError maximum error NON-WEIGHTED ERRORS ARE CALCULATED NOTES: you can convert P from barycentric form to the power or Chebyshev basis with PolynomialBar2Pow() or PolynomialBar2Cheb() functions from POLINT subpackage. -- ALGLIB PROJECT -- Copyright 10.12.2009 by Bochkanov Sergey *************************************************************************/ public static void polynomialfit(double[] x, double[] y, int n, int m, out int info, out barycentricinterpolant p, out polynomialfitreport rep) { info = 0; p = new barycentricinterpolant(); rep = new polynomialfitreport(); lsfit.polynomialfit(x, y, n, m, ref info, p.innerobj, rep.innerobj); return; } public static void polynomialfit(double[] x, double[] y, int m, out int info, out barycentricinterpolant p, out polynomialfitreport rep) { int n; if( (ap.len(x)!=ap.len(y))) throw new alglibexception("Error while calling 'polynomialfit': looks like one of arguments has wrong size"); info = 0; p = new barycentricinterpolant(); rep = new polynomialfitreport(); n = ap.len(x); lsfit.polynomialfit(x, y, n, m, ref info, p.innerobj, rep.innerobj); return; } /************************************************************************* Weighted fitting by polynomials in barycentric form, with constraints on function values or first derivatives. Small regularizing term is used when solving constrained tasks (to improve stability). Task is linear, so linear least squares solver is used. Complexity of this computational scheme is O(N*M^2), mostly dominated by least squares solver SEE ALSO: PolynomialFit() INPUT PARAMETERS: X - points, array[0..N-1]. Y - function values, array[0..N-1]. W - weights, array[0..N-1] Each summand in square sum of approximation deviations from given values is multiplied by the square of corresponding weight. Fill it by 1's if you don't want to solve weighted task. N - number of points, N>0. * if given, only leading N elements of X/Y/W are used * if not given, automatically determined from sizes of X/Y/W XC - points where polynomial values/derivatives are constrained, array[0..K-1]. YC - values of constraints, array[0..K-1] DC - array[0..K-1], types of constraints: * DC[i]=0 means that P(XC[i])=YC[i] * DC[i]=1 means that P'(XC[i])=YC[i] SEE BELOW FOR IMPORTANT INFORMATION ON CONSTRAINTS K - number of constraints, 0<=K=1 OUTPUT PARAMETERS: Info- same format as in LSFitLinearW() subroutine: * Info>0 task is solved * Info<=0 an error occured: -4 means inconvergence of internal SVD -3 means inconsistent constraints P - interpolant in barycentric form. Rep - report, same format as in LSFitLinearW() subroutine. Following fields are set: * RMSError rms error on the (X,Y). * AvgError average error on the (X,Y). * AvgRelError average relative error on the non-zero Y * MaxError maximum error NON-WEIGHTED ERRORS ARE CALCULATED IMPORTANT: this subroitine doesn't calculate task's condition number for K<>0. NOTES: you can convert P from barycentric form to the power or Chebyshev basis with PolynomialBar2Pow() or PolynomialBar2Cheb() functions from POLINT subpackage. SETTING CONSTRAINTS - DANGERS AND OPPORTUNITIES: Setting constraints can lead to undesired results, like ill-conditioned behavior, or inconsistency being detected. From the other side, it allows us to improve quality of the fit. Here we summarize our experience with constrained regression splines: * even simple constraints can be inconsistent, see Wikipedia article on this subject: http://en.wikipedia.org/wiki/Birkhoff_interpolation * the greater is M (given fixed constraints), the more chances that constraints will be consistent * in the general case, consistency of constraints is NOT GUARANTEED. * in the one special cases, however, we can guarantee consistency. This case is: M>1 and constraints on the function values (NOT DERIVATIVES) Our final recommendation is to use constraints WHEN AND ONLY when you can't solve your task without them. Anything beyond special cases given above is not guaranteed and may result in inconsistency. -- ALGLIB PROJECT -- Copyright 10.12.2009 by Bochkanov Sergey *************************************************************************/ public static void polynomialfitwc(double[] x, double[] y, double[] w, int n, double[] xc, double[] yc, int[] dc, int k, int m, out int info, out barycentricinterpolant p, out polynomialfitreport rep) { info = 0; p = new barycentricinterpolant(); rep = new polynomialfitreport(); lsfit.polynomialfitwc(x, y, w, n, xc, yc, dc, k, m, ref info, p.innerobj, rep.innerobj); return; } public static void polynomialfitwc(double[] x, double[] y, double[] w, double[] xc, double[] yc, int[] dc, int m, out int info, out barycentricinterpolant p, out polynomialfitreport rep) { int n; int k; if( (ap.len(x)!=ap.len(y)) || (ap.len(x)!=ap.len(w))) throw new alglibexception("Error while calling 'polynomialfitwc': looks like one of arguments has wrong size"); if( (ap.len(xc)!=ap.len(yc)) || (ap.len(xc)!=ap.len(dc))) throw new alglibexception("Error while calling 'polynomialfitwc': looks like one of arguments has wrong size"); info = 0; p = new barycentricinterpolant(); rep = new polynomialfitreport(); n = ap.len(x); k = ap.len(xc); lsfit.polynomialfitwc(x, y, w, n, xc, yc, dc, k, m, ref info, p.innerobj, rep.innerobj); return; } /************************************************************************* Weghted rational least squares fitting using Floater-Hormann rational functions with optimal D chosen from [0,9], with constraints and individual weights. Equidistant grid with M node on [min(x),max(x)] is used to build basis functions. Different values of D are tried, optimal D (least WEIGHTED root mean square error) is chosen. Task is linear, so linear least squares solver is used. Complexity of this computational scheme is O(N*M^2) (mostly dominated by the least squares solver). SEE ALSO * BarycentricFitFloaterHormann(), "lightweight" fitting without invididual weights and constraints. INPUT PARAMETERS: X - points, array[0..N-1]. Y - function values, array[0..N-1]. W - weights, array[0..N-1] Each summand in square sum of approximation deviations from given values is multiplied by the square of corresponding weight. Fill it by 1's if you don't want to solve weighted task. N - number of points, N>0. XC - points where function values/derivatives are constrained, array[0..K-1]. YC - values of constraints, array[0..K-1] DC - array[0..K-1], types of constraints: * DC[i]=0 means that S(XC[i])=YC[i] * DC[i]=1 means that S'(XC[i])=YC[i] SEE BELOW FOR IMPORTANT INFORMATION ON CONSTRAINTS K - number of constraints, 0<=K=2. OUTPUT PARAMETERS: Info- same format as in LSFitLinearWC() subroutine. * Info>0 task is solved * Info<=0 an error occured: -4 means inconvergence of internal SVD -3 means inconsistent constraints -1 means another errors in parameters passed (N<=0, for example) B - barycentric interpolant. Rep - report, same format as in LSFitLinearWC() subroutine. Following fields are set: * DBest best value of the D parameter * RMSError rms error on the (X,Y). * AvgError average error on the (X,Y). * AvgRelError average relative error on the non-zero Y * MaxError maximum error NON-WEIGHTED ERRORS ARE CALCULATED IMPORTANT: this subroutine doesn't calculate task's condition number for K<>0. SETTING CONSTRAINTS - DANGERS AND OPPORTUNITIES: Setting constraints can lead to undesired results, like ill-conditioned behavior, or inconsistency being detected. From the other side, it allows us to improve quality of the fit. Here we summarize our experience with constrained barycentric interpolants: * excessive constraints can be inconsistent. Floater-Hormann basis functions aren't as flexible as splines (although they are very smooth). * the more evenly constraints are spread across [min(x),max(x)], the more chances that they will be consistent * the greater is M (given fixed constraints), the more chances that constraints will be consistent * in the general case, consistency of constraints IS NOT GUARANTEED. * in the several special cases, however, we CAN guarantee consistency. * one of this cases is constraints on the function VALUES at the interval boundaries. Note that consustency of the constraints on the function DERIVATIVES is NOT guaranteed (you can use in such cases cubic splines which are more flexible). * another special case is ONE constraint on the function value (OR, but not AND, derivative) anywhere in the interval Our final recommendation is to use constraints WHEN AND ONLY WHEN you can't solve your task without them. Anything beyond special cases given above is not guaranteed and may result in inconsistency. -- ALGLIB PROJECT -- Copyright 18.08.2009 by Bochkanov Sergey *************************************************************************/ public static void barycentricfitfloaterhormannwc(double[] x, double[] y, double[] w, int n, double[] xc, double[] yc, int[] dc, int k, int m, out int info, out barycentricinterpolant b, out barycentricfitreport rep) { info = 0; b = new barycentricinterpolant(); rep = new barycentricfitreport(); lsfit.barycentricfitfloaterhormannwc(x, y, w, n, xc, yc, dc, k, m, ref info, b.innerobj, rep.innerobj); return; } /************************************************************************* Rational least squares fitting using Floater-Hormann rational functions with optimal D chosen from [0,9]. Equidistant grid with M node on [min(x),max(x)] is used to build basis functions. Different values of D are tried, optimal D (least root mean square error) is chosen. Task is linear, so linear least squares solver is used. Complexity of this computational scheme is O(N*M^2) (mostly dominated by the least squares solver). INPUT PARAMETERS: X - points, array[0..N-1]. Y - function values, array[0..N-1]. N - number of points, N>0. M - number of basis functions ( = number_of_nodes), M>=2. OUTPUT PARAMETERS: Info- same format as in LSFitLinearWC() subroutine. * Info>0 task is solved * Info<=0 an error occured: -4 means inconvergence of internal SVD -3 means inconsistent constraints B - barycentric interpolant. Rep - report, same format as in LSFitLinearWC() subroutine. Following fields are set: * DBest best value of the D parameter * RMSError rms error on the (X,Y). * AvgError average error on the (X,Y). * AvgRelError average relative error on the non-zero Y * MaxError maximum error NON-WEIGHTED ERRORS ARE CALCULATED -- ALGLIB PROJECT -- Copyright 18.08.2009 by Bochkanov Sergey *************************************************************************/ public static void barycentricfitfloaterhormann(double[] x, double[] y, int n, int m, out int info, out barycentricinterpolant b, out barycentricfitreport rep) { info = 0; b = new barycentricinterpolant(); rep = new barycentricfitreport(); lsfit.barycentricfitfloaterhormann(x, y, n, m, ref info, b.innerobj, rep.innerobj); return; } /************************************************************************* Rational least squares fitting using Floater-Hormann rational functions with optimal D chosen from [0,9]. Equidistant grid with M node on [min(x),max(x)] is used to build basis functions. Different values of D are tried, optimal D (least root mean square error) is chosen. Task is linear, so linear least squares solver is used. Complexity of this computational scheme is O(N*M^2) (mostly dominated by the least squares solver). INPUT PARAMETERS: X - points, array[0..N-1]. Y - function values, array[0..N-1]. N - number of points, N>0. M - number of basis functions ( = number_of_nodes), M>=2. OUTPUT PARAMETERS: Info- same format as in LSFitLinearWC() subroutine. * Info>0 task is solved * Info<=0 an error occured: -4 means inconvergence of internal SVD -3 means inconsistent constraints B - barycentric interpolant. Rep - report, same format as in LSFitLinearWC() subroutine. Following fields are set: * DBest best value of the D parameter * RMSError rms error on the (X,Y). * AvgError average error on the (X,Y). * AvgRelError average relative error on the non-zero Y * MaxError maximum error NON-WEIGHTED ERRORS ARE CALCULATED -- ALGLIB PROJECT -- Copyright 18.08.2009 by Bochkanov Sergey *************************************************************************/ public static void spline1dfitpenalized(double[] x, double[] y, int n, int m, double rho, out int info, out spline1dinterpolant s, out spline1dfitreport rep) { info = 0; s = new spline1dinterpolant(); rep = new spline1dfitreport(); lsfit.spline1dfitpenalized(x, y, n, m, rho, ref info, s.innerobj, rep.innerobj); return; } public static void spline1dfitpenalized(double[] x, double[] y, int m, double rho, out int info, out spline1dinterpolant s, out spline1dfitreport rep) { int n; if( (ap.len(x)!=ap.len(y))) throw new alglibexception("Error while calling 'spline1dfitpenalized': looks like one of arguments has wrong size"); info = 0; s = new spline1dinterpolant(); rep = new spline1dfitreport(); n = ap.len(x); lsfit.spline1dfitpenalized(x, y, n, m, rho, ref info, s.innerobj, rep.innerobj); return; } /************************************************************************* Weighted fitting by penalized cubic spline. Equidistant grid with M nodes on [min(x,xc),max(x,xc)] is used to build basis functions. Basis functions are cubic splines with natural boundary conditions. Problem is regularized by adding non-linearity penalty to the usual least squares penalty function: S(x) = arg min { LS + P }, where LS = SUM { w[i]^2*(y[i] - S(x[i]))^2 } - least squares penalty P = C*10^rho*integral{ S''(x)^2*dx } - non-linearity penalty rho - tunable constant given by user C - automatically determined scale parameter, makes penalty invariant with respect to scaling of X, Y, W. INPUT PARAMETERS: X - points, array[0..N-1]. Y - function values, array[0..N-1]. W - weights, array[0..N-1] Each summand in square sum of approximation deviations from given values is multiplied by the square of corresponding weight. Fill it by 1's if you don't want to solve weighted problem. N - number of points (optional): * N>0 * if given, only first N elements of X/Y/W are processed * if not given, automatically determined from X/Y/W sizes M - number of basis functions ( = number_of_nodes), M>=4. Rho - regularization constant passed by user. It penalizes nonlinearity in the regression spline. It is logarithmically scaled, i.e. actual value of regularization constant is calculated as 10^Rho. It is automatically scaled so that: * Rho=2.0 corresponds to moderate amount of nonlinearity * generally, it should be somewhere in the [-8.0,+8.0] If you do not want to penalize nonlineary, pass small Rho. Values as low as -15 should work. OUTPUT PARAMETERS: Info- same format as in LSFitLinearWC() subroutine. * Info>0 task is solved * Info<=0 an error occured: -4 means inconvergence of internal SVD or Cholesky decomposition; problem may be too ill-conditioned (very rare) S - spline interpolant. Rep - Following fields are set: * RMSError rms error on the (X,Y). * AvgError average error on the (X,Y). * AvgRelError average relative error on the non-zero Y * MaxError maximum error NON-WEIGHTED ERRORS ARE CALCULATED IMPORTANT: this subroitine doesn't calculate task's condition number for K<>0. NOTE 1: additional nodes are added to the spline outside of the fitting interval to force linearity when xmax(x,xc). It is done for consistency - we penalize non-linearity at [min(x,xc),max(x,xc)], so it is natural to force linearity outside of this interval. NOTE 2: function automatically sorts points, so caller may pass unsorted array. -- ALGLIB PROJECT -- Copyright 19.10.2010 by Bochkanov Sergey *************************************************************************/ public static void spline1dfitpenalizedw(double[] x, double[] y, double[] w, int n, int m, double rho, out int info, out spline1dinterpolant s, out spline1dfitreport rep) { info = 0; s = new spline1dinterpolant(); rep = new spline1dfitreport(); lsfit.spline1dfitpenalizedw(x, y, w, n, m, rho, ref info, s.innerobj, rep.innerobj); return; } public static void spline1dfitpenalizedw(double[] x, double[] y, double[] w, int m, double rho, out int info, out spline1dinterpolant s, out spline1dfitreport rep) { int n; if( (ap.len(x)!=ap.len(y)) || (ap.len(x)!=ap.len(w))) throw new alglibexception("Error while calling 'spline1dfitpenalizedw': looks like one of arguments has wrong size"); info = 0; s = new spline1dinterpolant(); rep = new spline1dfitreport(); n = ap.len(x); lsfit.spline1dfitpenalizedw(x, y, w, n, m, rho, ref info, s.innerobj, rep.innerobj); return; } /************************************************************************* Weighted fitting by cubic spline, with constraints on function values or derivatives. Equidistant grid with M-2 nodes on [min(x,xc),max(x,xc)] is used to build basis functions. Basis functions are cubic splines with continuous second derivatives and non-fixed first derivatives at interval ends. Small regularizing term is used when solving constrained tasks (to improve stability). Task is linear, so linear least squares solver is used. Complexity of this computational scheme is O(N*M^2), mostly dominated by least squares solver SEE ALSO Spline1DFitHermiteWC() - fitting by Hermite splines (more flexible, less smooth) Spline1DFitCubic() - "lightweight" fitting by cubic splines, without invididual weights and constraints INPUT PARAMETERS: X - points, array[0..N-1]. Y - function values, array[0..N-1]. W - weights, array[0..N-1] Each summand in square sum of approximation deviations from given values is multiplied by the square of corresponding weight. Fill it by 1's if you don't want to solve weighted task. N - number of points (optional): * N>0 * if given, only first N elements of X/Y/W are processed * if not given, automatically determined from X/Y/W sizes XC - points where spline values/derivatives are constrained, array[0..K-1]. YC - values of constraints, array[0..K-1] DC - array[0..K-1], types of constraints: * DC[i]=0 means that S(XC[i])=YC[i] * DC[i]=1 means that S'(XC[i])=YC[i] SEE BELOW FOR IMPORTANT INFORMATION ON CONSTRAINTS K - number of constraints (optional): * 0<=K=4. OUTPUT PARAMETERS: Info- same format as in LSFitLinearWC() subroutine. * Info>0 task is solved * Info<=0 an error occured: -4 means inconvergence of internal SVD -3 means inconsistent constraints S - spline interpolant. Rep - report, same format as in LSFitLinearWC() subroutine. Following fields are set: * RMSError rms error on the (X,Y). * AvgError average error on the (X,Y). * AvgRelError average relative error on the non-zero Y * MaxError maximum error NON-WEIGHTED ERRORS ARE CALCULATED IMPORTANT: this subroitine doesn't calculate task's condition number for K<>0. ORDER OF POINTS Subroutine automatically sorts points, so caller may pass unsorted array. SETTING CONSTRAINTS - DANGERS AND OPPORTUNITIES: Setting constraints can lead to undesired results, like ill-conditioned behavior, or inconsistency being detected. From the other side, it allows us to improve quality of the fit. Here we summarize our experience with constrained regression splines: * excessive constraints can be inconsistent. Splines are piecewise cubic functions, and it is easy to create an example, where large number of constraints concentrated in small area will result in inconsistency. Just because spline is not flexible enough to satisfy all of them. And same constraints spread across the [min(x),max(x)] will be perfectly consistent. * the more evenly constraints are spread across [min(x),max(x)], the more chances that they will be consistent * the greater is M (given fixed constraints), the more chances that constraints will be consistent * in the general case, consistency of constraints IS NOT GUARANTEED. * in the several special cases, however, we CAN guarantee consistency. * one of this cases is constraints on the function values AND/OR its derivatives at the interval boundaries. * another special case is ONE constraint on the function value (OR, but not AND, derivative) anywhere in the interval Our final recommendation is to use constraints WHEN AND ONLY WHEN you can't solve your task without them. Anything beyond special cases given above is not guaranteed and may result in inconsistency. -- ALGLIB PROJECT -- Copyright 18.08.2009 by Bochkanov Sergey *************************************************************************/ public static void spline1dfitcubicwc(double[] x, double[] y, double[] w, int n, double[] xc, double[] yc, int[] dc, int k, int m, out int info, out spline1dinterpolant s, out spline1dfitreport rep) { info = 0; s = new spline1dinterpolant(); rep = new spline1dfitreport(); lsfit.spline1dfitcubicwc(x, y, w, n, xc, yc, dc, k, m, ref info, s.innerobj, rep.innerobj); return; } public static void spline1dfitcubicwc(double[] x, double[] y, double[] w, double[] xc, double[] yc, int[] dc, int m, out int info, out spline1dinterpolant s, out spline1dfitreport rep) { int n; int k; if( (ap.len(x)!=ap.len(y)) || (ap.len(x)!=ap.len(w))) throw new alglibexception("Error while calling 'spline1dfitcubicwc': looks like one of arguments has wrong size"); if( (ap.len(xc)!=ap.len(yc)) || (ap.len(xc)!=ap.len(dc))) throw new alglibexception("Error while calling 'spline1dfitcubicwc': looks like one of arguments has wrong size"); info = 0; s = new spline1dinterpolant(); rep = new spline1dfitreport(); n = ap.len(x); k = ap.len(xc); lsfit.spline1dfitcubicwc(x, y, w, n, xc, yc, dc, k, m, ref info, s.innerobj, rep.innerobj); return; } /************************************************************************* Weighted fitting by Hermite spline, with constraints on function values or first derivatives. Equidistant grid with M nodes on [min(x,xc),max(x,xc)] is used to build basis functions. Basis functions are Hermite splines. Small regularizing term is used when solving constrained tasks (to improve stability). Task is linear, so linear least squares solver is used. Complexity of this computational scheme is O(N*M^2), mostly dominated by least squares solver SEE ALSO Spline1DFitCubicWC() - fitting by Cubic splines (less flexible, more smooth) Spline1DFitHermite() - "lightweight" Hermite fitting, without invididual weights and constraints INPUT PARAMETERS: X - points, array[0..N-1]. Y - function values, array[0..N-1]. W - weights, array[0..N-1] Each summand in square sum of approximation deviations from given values is multiplied by the square of corresponding weight. Fill it by 1's if you don't want to solve weighted task. N - number of points (optional): * N>0 * if given, only first N elements of X/Y/W are processed * if not given, automatically determined from X/Y/W sizes XC - points where spline values/derivatives are constrained, array[0..K-1]. YC - values of constraints, array[0..K-1] DC - array[0..K-1], types of constraints: * DC[i]=0 means that S(XC[i])=YC[i] * DC[i]=1 means that S'(XC[i])=YC[i] SEE BELOW FOR IMPORTANT INFORMATION ON CONSTRAINTS K - number of constraints (optional): * 0<=K=4, M IS EVEN! OUTPUT PARAMETERS: Info- same format as in LSFitLinearW() subroutine: * Info>0 task is solved * Info<=0 an error occured: -4 means inconvergence of internal SVD -3 means inconsistent constraints -2 means odd M was passed (which is not supported) -1 means another errors in parameters passed (N<=0, for example) S - spline interpolant. Rep - report, same format as in LSFitLinearW() subroutine. Following fields are set: * RMSError rms error on the (X,Y). * AvgError average error on the (X,Y). * AvgRelError average relative error on the non-zero Y * MaxError maximum error NON-WEIGHTED ERRORS ARE CALCULATED IMPORTANT: this subroitine doesn't calculate task's condition number for K<>0. IMPORTANT: this subroitine supports only even M's ORDER OF POINTS Subroutine automatically sorts points, so caller may pass unsorted array. SETTING CONSTRAINTS - DANGERS AND OPPORTUNITIES: Setting constraints can lead to undesired results, like ill-conditioned behavior, or inconsistency being detected. From the other side, it allows us to improve quality of the fit. Here we summarize our experience with constrained regression splines: * excessive constraints can be inconsistent. Splines are piecewise cubic functions, and it is easy to create an example, where large number of constraints concentrated in small area will result in inconsistency. Just because spline is not flexible enough to satisfy all of them. And same constraints spread across the [min(x),max(x)] will be perfectly consistent. * the more evenly constraints are spread across [min(x),max(x)], the more chances that they will be consistent * the greater is M (given fixed constraints), the more chances that constraints will be consistent * in the general case, consistency of constraints is NOT GUARANTEED. * in the several special cases, however, we can guarantee consistency. * one of this cases is M>=4 and constraints on the function value (AND/OR its derivative) at the interval boundaries. * another special case is M>=4 and ONE constraint on the function value (OR, BUT NOT AND, derivative) anywhere in [min(x),max(x)] Our final recommendation is to use constraints WHEN AND ONLY when you can't solve your task without them. Anything beyond special cases given above is not guaranteed and may result in inconsistency. -- ALGLIB PROJECT -- Copyright 18.08.2009 by Bochkanov Sergey *************************************************************************/ public static void spline1dfithermitewc(double[] x, double[] y, double[] w, int n, double[] xc, double[] yc, int[] dc, int k, int m, out int info, out spline1dinterpolant s, out spline1dfitreport rep) { info = 0; s = new spline1dinterpolant(); rep = new spline1dfitreport(); lsfit.spline1dfithermitewc(x, y, w, n, xc, yc, dc, k, m, ref info, s.innerobj, rep.innerobj); return; } public static void spline1dfithermitewc(double[] x, double[] y, double[] w, double[] xc, double[] yc, int[] dc, int m, out int info, out spline1dinterpolant s, out spline1dfitreport rep) { int n; int k; if( (ap.len(x)!=ap.len(y)) || (ap.len(x)!=ap.len(w))) throw new alglibexception("Error while calling 'spline1dfithermitewc': looks like one of arguments has wrong size"); if( (ap.len(xc)!=ap.len(yc)) || (ap.len(xc)!=ap.len(dc))) throw new alglibexception("Error while calling 'spline1dfithermitewc': looks like one of arguments has wrong size"); info = 0; s = new spline1dinterpolant(); rep = new spline1dfitreport(); n = ap.len(x); k = ap.len(xc); lsfit.spline1dfithermitewc(x, y, w, n, xc, yc, dc, k, m, ref info, s.innerobj, rep.innerobj); return; } /************************************************************************* Least squares fitting by cubic spline. This subroutine is "lightweight" alternative for more complex and feature- rich Spline1DFitCubicWC(). See Spline1DFitCubicWC() for more information about subroutine parameters (we don't duplicate it here because of length) -- ALGLIB PROJECT -- Copyright 18.08.2009 by Bochkanov Sergey *************************************************************************/ public static void spline1dfitcubic(double[] x, double[] y, int n, int m, out int info, out spline1dinterpolant s, out spline1dfitreport rep) { info = 0; s = new spline1dinterpolant(); rep = new spline1dfitreport(); lsfit.spline1dfitcubic(x, y, n, m, ref info, s.innerobj, rep.innerobj); return; } public static void spline1dfitcubic(double[] x, double[] y, int m, out int info, out spline1dinterpolant s, out spline1dfitreport rep) { int n; if( (ap.len(x)!=ap.len(y))) throw new alglibexception("Error while calling 'spline1dfitcubic': looks like one of arguments has wrong size"); info = 0; s = new spline1dinterpolant(); rep = new spline1dfitreport(); n = ap.len(x); lsfit.spline1dfitcubic(x, y, n, m, ref info, s.innerobj, rep.innerobj); return; } /************************************************************************* Least squares fitting by Hermite spline. This subroutine is "lightweight" alternative for more complex and feature- rich Spline1DFitHermiteWC(). See Spline1DFitHermiteWC() description for more information about subroutine parameters (we don't duplicate it here because of length). -- ALGLIB PROJECT -- Copyright 18.08.2009 by Bochkanov Sergey *************************************************************************/ public static void spline1dfithermite(double[] x, double[] y, int n, int m, out int info, out spline1dinterpolant s, out spline1dfitreport rep) { info = 0; s = new spline1dinterpolant(); rep = new spline1dfitreport(); lsfit.spline1dfithermite(x, y, n, m, ref info, s.innerobj, rep.innerobj); return; } public static void spline1dfithermite(double[] x, double[] y, int m, out int info, out spline1dinterpolant s, out spline1dfitreport rep) { int n; if( (ap.len(x)!=ap.len(y))) throw new alglibexception("Error while calling 'spline1dfithermite': looks like one of arguments has wrong size"); info = 0; s = new spline1dinterpolant(); rep = new spline1dfitreport(); n = ap.len(x); lsfit.spline1dfithermite(x, y, n, m, ref info, s.innerobj, rep.innerobj); return; } /************************************************************************* Weighted linear least squares fitting. QR decomposition is used to reduce task to MxM, then triangular solver or SVD-based solver is used depending on condition number of the system. It allows to maximize speed and retain decent accuracy. INPUT PARAMETERS: Y - array[0..N-1] Function values in N points. W - array[0..N-1] Weights corresponding to function values. Each summand in square sum of approximation deviations from given values is multiplied by the square of corresponding weight. FMatrix - a table of basis functions values, array[0..N-1, 0..M-1]. FMatrix[I, J] - value of J-th basis function in I-th point. N - number of points used. N>=1. M - number of basis functions, M>=1. OUTPUT PARAMETERS: Info - error code: * -4 internal SVD decomposition subroutine failed (very rare and for degenerate systems only) * -1 incorrect N/M were specified * 1 task is solved C - decomposition coefficients, array[0..M-1] Rep - fitting report. Following fields are set: * Rep.TaskRCond reciprocal of condition number * RMSError rms error on the (X,Y). * AvgError average error on the (X,Y). * AvgRelError average relative error on the non-zero Y * MaxError maximum error NON-WEIGHTED ERRORS ARE CALCULATED -- ALGLIB -- Copyright 17.08.2009 by Bochkanov Sergey *************************************************************************/ public static void lsfitlinearw(double[] y, double[] w, double[,] fmatrix, int n, int m, out int info, out double[] c, out lsfitreport rep) { info = 0; c = new double[0]; rep = new lsfitreport(); lsfit.lsfitlinearw(y, w, fmatrix, n, m, ref info, ref c, rep.innerobj); return; } public static void lsfitlinearw(double[] y, double[] w, double[,] fmatrix, out int info, out double[] c, out lsfitreport rep) { int n; int m; if( (ap.len(y)!=ap.len(w)) || (ap.len(y)!=ap.rows(fmatrix))) throw new alglibexception("Error while calling 'lsfitlinearw': looks like one of arguments has wrong size"); info = 0; c = new double[0]; rep = new lsfitreport(); n = ap.len(y); m = ap.cols(fmatrix); lsfit.lsfitlinearw(y, w, fmatrix, n, m, ref info, ref c, rep.innerobj); return; } /************************************************************************* Weighted constained linear least squares fitting. This is variation of LSFitLinearW(), which searchs for min|A*x=b| given that K additional constaints C*x=bc are satisfied. It reduces original task to modified one: min|B*y-d| WITHOUT constraints, then LSFitLinearW() is called. INPUT PARAMETERS: Y - array[0..N-1] Function values in N points. W - array[0..N-1] Weights corresponding to function values. Each summand in square sum of approximation deviations from given values is multiplied by the square of corresponding weight. FMatrix - a table of basis functions values, array[0..N-1, 0..M-1]. FMatrix[I,J] - value of J-th basis function in I-th point. CMatrix - a table of constaints, array[0..K-1,0..M]. I-th row of CMatrix corresponds to I-th linear constraint: CMatrix[I,0]*C[0] + ... + CMatrix[I,M-1]*C[M-1] = CMatrix[I,M] N - number of points used. N>=1. M - number of basis functions, M>=1. K - number of constraints, 0 <= K < M K=0 corresponds to absence of constraints. OUTPUT PARAMETERS: Info - error code: * -4 internal SVD decomposition subroutine failed (very rare and for degenerate systems only) * -3 either too many constraints (M or more), degenerate constraints (some constraints are repetead twice) or inconsistent constraints were specified. * 1 task is solved C - decomposition coefficients, array[0..M-1] Rep - fitting report. Following fields are set: * RMSError rms error on the (X,Y). * AvgError average error on the (X,Y). * AvgRelError average relative error on the non-zero Y * MaxError maximum error NON-WEIGHTED ERRORS ARE CALCULATED IMPORTANT: this subroitine doesn't calculate task's condition number for K<>0. -- ALGLIB -- Copyright 07.09.2009 by Bochkanov Sergey *************************************************************************/ public static void lsfitlinearwc(double[] y, double[] w, double[,] fmatrix, double[,] cmatrix, int n, int m, int k, out int info, out double[] c, out lsfitreport rep) { info = 0; c = new double[0]; rep = new lsfitreport(); lsfit.lsfitlinearwc(y, w, fmatrix, cmatrix, n, m, k, ref info, ref c, rep.innerobj); return; } public static void lsfitlinearwc(double[] y, double[] w, double[,] fmatrix, double[,] cmatrix, out int info, out double[] c, out lsfitreport rep) { int n; int m; int k; if( (ap.len(y)!=ap.len(w)) || (ap.len(y)!=ap.rows(fmatrix))) throw new alglibexception("Error while calling 'lsfitlinearwc': looks like one of arguments has wrong size"); if( (ap.cols(fmatrix)!=ap.cols(cmatrix)-1)) throw new alglibexception("Error while calling 'lsfitlinearwc': looks like one of arguments has wrong size"); info = 0; c = new double[0]; rep = new lsfitreport(); n = ap.len(y); m = ap.cols(fmatrix); k = ap.rows(cmatrix); lsfit.lsfitlinearwc(y, w, fmatrix, cmatrix, n, m, k, ref info, ref c, rep.innerobj); return; } /************************************************************************* Linear least squares fitting. QR decomposition is used to reduce task to MxM, then triangular solver or SVD-based solver is used depending on condition number of the system. It allows to maximize speed and retain decent accuracy. INPUT PARAMETERS: Y - array[0..N-1] Function values in N points. FMatrix - a table of basis functions values, array[0..N-1, 0..M-1]. FMatrix[I, J] - value of J-th basis function in I-th point. N - number of points used. N>=1. M - number of basis functions, M>=1. OUTPUT PARAMETERS: Info - error code: * -4 internal SVD decomposition subroutine failed (very rare and for degenerate systems only) * 1 task is solved C - decomposition coefficients, array[0..M-1] Rep - fitting report. Following fields are set: * Rep.TaskRCond reciprocal of condition number * RMSError rms error on the (X,Y). * AvgError average error on the (X,Y). * AvgRelError average relative error on the non-zero Y * MaxError maximum error NON-WEIGHTED ERRORS ARE CALCULATED -- ALGLIB -- Copyright 17.08.2009 by Bochkanov Sergey *************************************************************************/ public static void lsfitlinear(double[] y, double[,] fmatrix, int n, int m, out int info, out double[] c, out lsfitreport rep) { info = 0; c = new double[0]; rep = new lsfitreport(); lsfit.lsfitlinear(y, fmatrix, n, m, ref info, ref c, rep.innerobj); return; } public static void lsfitlinear(double[] y, double[,] fmatrix, out int info, out double[] c, out lsfitreport rep) { int n; int m; if( (ap.len(y)!=ap.rows(fmatrix))) throw new alglibexception("Error while calling 'lsfitlinear': looks like one of arguments has wrong size"); info = 0; c = new double[0]; rep = new lsfitreport(); n = ap.len(y); m = ap.cols(fmatrix); lsfit.lsfitlinear(y, fmatrix, n, m, ref info, ref c, rep.innerobj); return; } /************************************************************************* Constained linear least squares fitting. This is variation of LSFitLinear(), which searchs for min|A*x=b| given that K additional constaints C*x=bc are satisfied. It reduces original task to modified one: min|B*y-d| WITHOUT constraints, then LSFitLinear() is called. INPUT PARAMETERS: Y - array[0..N-1] Function values in N points. FMatrix - a table of basis functions values, array[0..N-1, 0..M-1]. FMatrix[I,J] - value of J-th basis function in I-th point. CMatrix - a table of constaints, array[0..K-1,0..M]. I-th row of CMatrix corresponds to I-th linear constraint: CMatrix[I,0]*C[0] + ... + CMatrix[I,M-1]*C[M-1] = CMatrix[I,M] N - number of points used. N>=1. M - number of basis functions, M>=1. K - number of constraints, 0 <= K < M K=0 corresponds to absence of constraints. OUTPUT PARAMETERS: Info - error code: * -4 internal SVD decomposition subroutine failed (very rare and for degenerate systems only) * -3 either too many constraints (M or more), degenerate constraints (some constraints are repetead twice) or inconsistent constraints were specified. * 1 task is solved C - decomposition coefficients, array[0..M-1] Rep - fitting report. Following fields are set: * RMSError rms error on the (X,Y). * AvgError average error on the (X,Y). * AvgRelError average relative error on the non-zero Y * MaxError maximum error NON-WEIGHTED ERRORS ARE CALCULATED IMPORTANT: this subroitine doesn't calculate task's condition number for K<>0. -- ALGLIB -- Copyright 07.09.2009 by Bochkanov Sergey *************************************************************************/ public static void lsfitlinearc(double[] y, double[,] fmatrix, double[,] cmatrix, int n, int m, int k, out int info, out double[] c, out lsfitreport rep) { info = 0; c = new double[0]; rep = new lsfitreport(); lsfit.lsfitlinearc(y, fmatrix, cmatrix, n, m, k, ref info, ref c, rep.innerobj); return; } public static void lsfitlinearc(double[] y, double[,] fmatrix, double[,] cmatrix, out int info, out double[] c, out lsfitreport rep) { int n; int m; int k; if( (ap.len(y)!=ap.rows(fmatrix))) throw new alglibexception("Error while calling 'lsfitlinearc': looks like one of arguments has wrong size"); if( (ap.cols(fmatrix)!=ap.cols(cmatrix)-1)) throw new alglibexception("Error while calling 'lsfitlinearc': looks like one of arguments has wrong size"); info = 0; c = new double[0]; rep = new lsfitreport(); n = ap.len(y); m = ap.cols(fmatrix); k = ap.rows(cmatrix); lsfit.lsfitlinearc(y, fmatrix, cmatrix, n, m, k, ref info, ref c, rep.innerobj); return; } /************************************************************************* Weighted nonlinear least squares fitting using function values only. Combination of numerical differentiation and secant updates is used to obtain function Jacobian. Nonlinear task min(F(c)) is solved, where F(c) = (w[0]*(f(c,x[0])-y[0]))^2 + ... + (w[n-1]*(f(c,x[n-1])-y[n-1]))^2, * N is a number of points, * M is a dimension of a space points belong to, * K is a dimension of a space of parameters being fitted, * w is an N-dimensional vector of weight coefficients, * x is a set of N points, each of them is an M-dimensional vector, * c is a K-dimensional vector of parameters being fitted This subroutine uses only f(c,x[i]). INPUT PARAMETERS: X - array[0..N-1,0..M-1], points (one row = one point) Y - array[0..N-1], function values. W - weights, array[0..N-1] C - array[0..K-1], initial approximation to the solution, N - number of points, N>1 M - dimension of space K - number of parameters being fitted DiffStep- numerical differentiation step; should not be very small or large; large = loss of accuracy small = growth of round-off errors OUTPUT PARAMETERS: State - structure which stores algorithm state -- ALGLIB -- Copyright 18.10.2008 by Bochkanov Sergey *************************************************************************/ public static void lsfitcreatewf(double[,] x, double[] y, double[] w, double[] c, int n, int m, int k, double diffstep, out lsfitstate state) { state = new lsfitstate(); lsfit.lsfitcreatewf(x, y, w, c, n, m, k, diffstep, state.innerobj); return; } public static void lsfitcreatewf(double[,] x, double[] y, double[] w, double[] c, double diffstep, out lsfitstate state) { int n; int m; int k; if( (ap.rows(x)!=ap.len(y)) || (ap.rows(x)!=ap.len(w))) throw new alglibexception("Error while calling 'lsfitcreatewf': looks like one of arguments has wrong size"); state = new lsfitstate(); n = ap.rows(x); m = ap.cols(x); k = ap.len(c); lsfit.lsfitcreatewf(x, y, w, c, n, m, k, diffstep, state.innerobj); return; } /************************************************************************* Nonlinear least squares fitting using function values only. Combination of numerical differentiation and secant updates is used to obtain function Jacobian. Nonlinear task min(F(c)) is solved, where F(c) = (f(c,x[0])-y[0])^2 + ... + (f(c,x[n-1])-y[n-1])^2, * N is a number of points, * M is a dimension of a space points belong to, * K is a dimension of a space of parameters being fitted, * w is an N-dimensional vector of weight coefficients, * x is a set of N points, each of them is an M-dimensional vector, * c is a K-dimensional vector of parameters being fitted This subroutine uses only f(c,x[i]). INPUT PARAMETERS: X - array[0..N-1,0..M-1], points (one row = one point) Y - array[0..N-1], function values. C - array[0..K-1], initial approximation to the solution, N - number of points, N>1 M - dimension of space K - number of parameters being fitted DiffStep- numerical differentiation step; should not be very small or large; large = loss of accuracy small = growth of round-off errors OUTPUT PARAMETERS: State - structure which stores algorithm state -- ALGLIB -- Copyright 18.10.2008 by Bochkanov Sergey *************************************************************************/ public static void lsfitcreatef(double[,] x, double[] y, double[] c, int n, int m, int k, double diffstep, out lsfitstate state) { state = new lsfitstate(); lsfit.lsfitcreatef(x, y, c, n, m, k, diffstep, state.innerobj); return; } public static void lsfitcreatef(double[,] x, double[] y, double[] c, double diffstep, out lsfitstate state) { int n; int m; int k; if( (ap.rows(x)!=ap.len(y))) throw new alglibexception("Error while calling 'lsfitcreatef': looks like one of arguments has wrong size"); state = new lsfitstate(); n = ap.rows(x); m = ap.cols(x); k = ap.len(c); lsfit.lsfitcreatef(x, y, c, n, m, k, diffstep, state.innerobj); return; } /************************************************************************* Weighted nonlinear least squares fitting using gradient only. Nonlinear task min(F(c)) is solved, where F(c) = (w[0]*(f(c,x[0])-y[0]))^2 + ... + (w[n-1]*(f(c,x[n-1])-y[n-1]))^2, * N is a number of points, * M is a dimension of a space points belong to, * K is a dimension of a space of parameters being fitted, * w is an N-dimensional vector of weight coefficients, * x is a set of N points, each of them is an M-dimensional vector, * c is a K-dimensional vector of parameters being fitted This subroutine uses only f(c,x[i]) and its gradient. INPUT PARAMETERS: X - array[0..N-1,0..M-1], points (one row = one point) Y - array[0..N-1], function values. W - weights, array[0..N-1] C - array[0..K-1], initial approximation to the solution, N - number of points, N>1 M - dimension of space K - number of parameters being fitted CheapFG - boolean flag, which is: * True if both function and gradient calculation complexity are less than O(M^2). An improved algorithm can be used which corresponds to FGJ scheme from MINLM unit. * False otherwise. Standard Jacibian-bases Levenberg-Marquardt algo will be used (FJ scheme). OUTPUT PARAMETERS: State - structure which stores algorithm state See also: LSFitResults LSFitCreateFG (fitting without weights) LSFitCreateWFGH (fitting using Hessian) LSFitCreateFGH (fitting using Hessian, without weights) -- ALGLIB -- Copyright 17.08.2009 by Bochkanov Sergey *************************************************************************/ public static void lsfitcreatewfg(double[,] x, double[] y, double[] w, double[] c, int n, int m, int k, bool cheapfg, out lsfitstate state) { state = new lsfitstate(); lsfit.lsfitcreatewfg(x, y, w, c, n, m, k, cheapfg, state.innerobj); return; } public static void lsfitcreatewfg(double[,] x, double[] y, double[] w, double[] c, bool cheapfg, out lsfitstate state) { int n; int m; int k; if( (ap.rows(x)!=ap.len(y)) || (ap.rows(x)!=ap.len(w))) throw new alglibexception("Error while calling 'lsfitcreatewfg': looks like one of arguments has wrong size"); state = new lsfitstate(); n = ap.rows(x); m = ap.cols(x); k = ap.len(c); lsfit.lsfitcreatewfg(x, y, w, c, n, m, k, cheapfg, state.innerobj); return; } /************************************************************************* Nonlinear least squares fitting using gradient only, without individual weights. Nonlinear task min(F(c)) is solved, where F(c) = ((f(c,x[0])-y[0]))^2 + ... + ((f(c,x[n-1])-y[n-1]))^2, * N is a number of points, * M is a dimension of a space points belong to, * K is a dimension of a space of parameters being fitted, * x is a set of N points, each of them is an M-dimensional vector, * c is a K-dimensional vector of parameters being fitted This subroutine uses only f(c,x[i]) and its gradient. INPUT PARAMETERS: X - array[0..N-1,0..M-1], points (one row = one point) Y - array[0..N-1], function values. C - array[0..K-1], initial approximation to the solution, N - number of points, N>1 M - dimension of space K - number of parameters being fitted CheapFG - boolean flag, which is: * True if both function and gradient calculation complexity are less than O(M^2). An improved algorithm can be used which corresponds to FGJ scheme from MINLM unit. * False otherwise. Standard Jacibian-bases Levenberg-Marquardt algo will be used (FJ scheme). OUTPUT PARAMETERS: State - structure which stores algorithm state -- ALGLIB -- Copyright 17.08.2009 by Bochkanov Sergey *************************************************************************/ public static void lsfitcreatefg(double[,] x, double[] y, double[] c, int n, int m, int k, bool cheapfg, out lsfitstate state) { state = new lsfitstate(); lsfit.lsfitcreatefg(x, y, c, n, m, k, cheapfg, state.innerobj); return; } public static void lsfitcreatefg(double[,] x, double[] y, double[] c, bool cheapfg, out lsfitstate state) { int n; int m; int k; if( (ap.rows(x)!=ap.len(y))) throw new alglibexception("Error while calling 'lsfitcreatefg': looks like one of arguments has wrong size"); state = new lsfitstate(); n = ap.rows(x); m = ap.cols(x); k = ap.len(c); lsfit.lsfitcreatefg(x, y, c, n, m, k, cheapfg, state.innerobj); return; } /************************************************************************* Weighted nonlinear least squares fitting using gradient/Hessian. Nonlinear task min(F(c)) is solved, where F(c) = (w[0]*(f(c,x[0])-y[0]))^2 + ... + (w[n-1]*(f(c,x[n-1])-y[n-1]))^2, * N is a number of points, * M is a dimension of a space points belong to, * K is a dimension of a space of parameters being fitted, * w is an N-dimensional vector of weight coefficients, * x is a set of N points, each of them is an M-dimensional vector, * c is a K-dimensional vector of parameters being fitted This subroutine uses f(c,x[i]), its gradient and its Hessian. INPUT PARAMETERS: X - array[0..N-1,0..M-1], points (one row = one point) Y - array[0..N-1], function values. W - weights, array[0..N-1] C - array[0..K-1], initial approximation to the solution, N - number of points, N>1 M - dimension of space K - number of parameters being fitted OUTPUT PARAMETERS: State - structure which stores algorithm state -- ALGLIB -- Copyright 17.08.2009 by Bochkanov Sergey *************************************************************************/ public static void lsfitcreatewfgh(double[,] x, double[] y, double[] w, double[] c, int n, int m, int k, out lsfitstate state) { state = new lsfitstate(); lsfit.lsfitcreatewfgh(x, y, w, c, n, m, k, state.innerobj); return; } public static void lsfitcreatewfgh(double[,] x, double[] y, double[] w, double[] c, out lsfitstate state) { int n; int m; int k; if( (ap.rows(x)!=ap.len(y)) || (ap.rows(x)!=ap.len(w))) throw new alglibexception("Error while calling 'lsfitcreatewfgh': looks like one of arguments has wrong size"); state = new lsfitstate(); n = ap.rows(x); m = ap.cols(x); k = ap.len(c); lsfit.lsfitcreatewfgh(x, y, w, c, n, m, k, state.innerobj); return; } /************************************************************************* Nonlinear least squares fitting using gradient/Hessian, without individial weights. Nonlinear task min(F(c)) is solved, where F(c) = ((f(c,x[0])-y[0]))^2 + ... + ((f(c,x[n-1])-y[n-1]))^2, * N is a number of points, * M is a dimension of a space points belong to, * K is a dimension of a space of parameters being fitted, * x is a set of N points, each of them is an M-dimensional vector, * c is a K-dimensional vector of parameters being fitted This subroutine uses f(c,x[i]), its gradient and its Hessian. INPUT PARAMETERS: X - array[0..N-1,0..M-1], points (one row = one point) Y - array[0..N-1], function values. C - array[0..K-1], initial approximation to the solution, N - number of points, N>1 M - dimension of space K - number of parameters being fitted OUTPUT PARAMETERS: State - structure which stores algorithm state -- ALGLIB -- Copyright 17.08.2009 by Bochkanov Sergey *************************************************************************/ public static void lsfitcreatefgh(double[,] x, double[] y, double[] c, int n, int m, int k, out lsfitstate state) { state = new lsfitstate(); lsfit.lsfitcreatefgh(x, y, c, n, m, k, state.innerobj); return; } public static void lsfitcreatefgh(double[,] x, double[] y, double[] c, out lsfitstate state) { int n; int m; int k; if( (ap.rows(x)!=ap.len(y))) throw new alglibexception("Error while calling 'lsfitcreatefgh': looks like one of arguments has wrong size"); state = new lsfitstate(); n = ap.rows(x); m = ap.cols(x); k = ap.len(c); lsfit.lsfitcreatefgh(x, y, c, n, m, k, state.innerobj); return; } /************************************************************************* Stopping conditions for nonlinear least squares fitting. INPUT PARAMETERS: State - structure which stores algorithm state EpsF - stopping criterion. Algorithm stops if |F(k+1)-F(k)| <= EpsF*max{|F(k)|, |F(k+1)|, 1} EpsX - >=0 The subroutine finishes its work if on k+1-th iteration the condition |v|<=EpsX is fulfilled, where: * |.| means Euclidian norm * v - scaled step vector, v[i]=dx[i]/s[i] * dx - ste pvector, dx=X(k+1)-X(k) * s - scaling coefficients set by LSFitSetScale() MaxIts - maximum number of iterations. If MaxIts=0, the number of iterations is unlimited. Only Levenberg-Marquardt iterations are counted (L-BFGS/CG iterations are NOT counted because their cost is very low compared to that of LM). NOTE Passing EpsF=0, EpsX=0 and MaxIts=0 (simultaneously) will lead to automatic stopping criterion selection (according to the scheme used by MINLM unit). -- ALGLIB -- Copyright 17.08.2009 by Bochkanov Sergey *************************************************************************/ public static void lsfitsetcond(lsfitstate state, double epsf, double epsx, int maxits) { lsfit.lsfitsetcond(state.innerobj, epsf, epsx, maxits); return; } /************************************************************************* This function sets maximum step length INPUT PARAMETERS: State - structure which stores algorithm state StpMax - maximum step length, >=0. Set StpMax to 0.0, if you don't want to limit step length. Use this subroutine when you optimize target function which contains exp() or other fast growing functions, and optimization algorithm makes too large steps which leads to overflow. This function allows us to reject steps that are too large (and therefore expose us to the possible overflow) without actually calculating function value at the x+stp*d. NOTE: non-zero StpMax leads to moderate performance degradation because intermediate step of preconditioned L-BFGS optimization is incompatible with limits on step size. -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/ public static void lsfitsetstpmax(lsfitstate state, double stpmax) { lsfit.lsfitsetstpmax(state.innerobj, stpmax); return; } /************************************************************************* This function turns on/off reporting. INPUT PARAMETERS: State - structure which stores algorithm state NeedXRep- whether iteration reports are needed or not When reports are needed, State.C (current parameters) and State.F (current value of fitting function) are reported. -- ALGLIB -- Copyright 15.08.2010 by Bochkanov Sergey *************************************************************************/ public static void lsfitsetxrep(lsfitstate state, bool needxrep) { lsfit.lsfitsetxrep(state.innerobj, needxrep); return; } /************************************************************************* This function sets scaling coefficients for underlying optimizer. ALGLIB optimizers use scaling matrices to test stopping conditions (step size and gradient are scaled before comparison with tolerances). Scale of the I-th variable is a translation invariant measure of: a) "how large" the variable is b) how large the step should be to make significant changes in the function Generally, scale is NOT considered to be a form of preconditioner. But LM optimizer is unique in that it uses scaling matrix both in the stopping condition tests and as Marquardt damping factor. Proper scaling is very important for the algorithm performance. It is less important for the quality of results, but still has some influence (it is easier to converge when variables are properly scaled, so premature stopping is possible when very badly scalled variables are combined with relaxed stopping conditions). INPUT PARAMETERS: State - structure stores algorithm state S - array[N], non-zero scaling coefficients S[i] may be negative, sign doesn't matter. -- ALGLIB -- Copyright 14.01.2011 by Bochkanov Sergey *************************************************************************/ public static void lsfitsetscale(lsfitstate state, double[] s) { lsfit.lsfitsetscale(state.innerobj, s); return; } /************************************************************************* This function sets boundary constraints for underlying optimizer Boundary constraints are inactive by default (after initial creation). They are preserved until explicitly turned off with another SetBC() call. INPUT PARAMETERS: State - structure stores algorithm state BndL - lower bounds, array[K]. If some (all) variables are unbounded, you may specify very small number or -INF (latter is recommended because it will allow solver to use better algorithm). BndU - upper bounds, array[K]. If some (all) variables are unbounded, you may specify very large number or +INF (latter is recommended because it will allow solver to use better algorithm). NOTE 1: it is possible to specify BndL[i]=BndU[i]. In this case I-th variable will be "frozen" at X[i]=BndL[i]=BndU[i]. NOTE 2: unlike other constrained optimization algorithms, this solver has following useful properties: * bound constraints are always satisfied exactly * function is evaluated only INSIDE area specified by bound constraints -- ALGLIB -- Copyright 14.01.2011 by Bochkanov Sergey *************************************************************************/ public static void lsfitsetbc(lsfitstate state, double[] bndl, double[] bndu) { lsfit.lsfitsetbc(state.innerobj, bndl, bndu); return; } /************************************************************************* This function provides reverse communication interface Reverse communication interface is not documented or recommended to use. See below for functions which provide better documented API *************************************************************************/ public static bool lsfititeration(lsfitstate state) { bool result = lsfit.lsfititeration(state.innerobj); return result; } /************************************************************************* This family of functions is used to launcn iterations of nonlinear fitter These functions accept following parameters: func - callback which calculates function (or merit function) value func at given point x grad - callback which calculates function (or merit function) value func and gradient grad at given point x hess - callback which calculates function (or merit function) value func, gradient grad and Hessian hess at given point x rep - optional callback which is called after each iteration can be null obj - optional object which is passed to func/grad/hess/jac/rep can be null NOTES: 1. this algorithm is somewhat unusual because it works with parameterized function f(C,X), where X is a function argument (we have many points which are characterized by different argument values), and C is a parameter to fit. For example, if we want to do linear fit by f(c0,c1,x) = c0*x+c1, then x will be argument, and {c0,c1} will be parameters. It is important to understand that this algorithm finds minimum in the space of function PARAMETERS (not arguments), so it needs derivatives of f() with respect to C, not X. In the example above it will need f=c0*x+c1 and {df/dc0,df/dc1} = {x,1} instead of {df/dx} = {c0}. 2. Callback functions accept C as the first parameter, and X as the second 3. If state was created with LSFitCreateFG(), algorithm needs just function and its gradient, but if state was created with LSFitCreateFGH(), algorithm will need function, gradient and Hessian. According to the said above, there ase several versions of this function, which accept different sets of callbacks. This flexibility opens way to subtle errors - you may create state with LSFitCreateFGH() (optimization using Hessian), but call function which does not accept Hessian. So when algorithm will request Hessian, there will be no callback to call. In this case exception will be thrown. Be careful to avoid such errors because there is no way to find them at compile time - you can see them at runtime only. -- ALGLIB -- Copyright 17.08.2009 by Bochkanov Sergey *************************************************************************/ public static void lsfitfit(lsfitstate state, ndimensional_pfunc func, ndimensional_rep rep, object obj) { if( func==null ) throw new alglibexception("ALGLIB: error in 'lsfitfit()' (func is null)"); while( alglib.lsfititeration(state) ) { if( state.needf ) { func(state.c, state.x, ref state.innerobj.f, obj); continue; } if( state.innerobj.xupdated ) { if( rep!=null ) rep(state.innerobj.c, state.innerobj.f, obj); continue; } throw new alglibexception("ALGLIB: error in 'lsfitfit' (some derivatives were not provided?)"); } } public static void lsfitfit(lsfitstate state, ndimensional_pfunc func, ndimensional_pgrad grad, ndimensional_rep rep, object obj) { if( func==null ) throw new alglibexception("ALGLIB: error in 'lsfitfit()' (func is null)"); if( grad==null ) throw new alglibexception("ALGLIB: error in 'lsfitfit()' (grad is null)"); while( alglib.lsfititeration(state) ) { if( state.needf ) { func(state.c, state.x, ref state.innerobj.f, obj); continue; } if( state.needfg ) { grad(state.c, state.x, ref state.innerobj.f, state.innerobj.g, obj); continue; } if( state.innerobj.xupdated ) { if( rep!=null ) rep(state.innerobj.c, state.innerobj.f, obj); continue; } throw new alglibexception("ALGLIB: error in 'lsfitfit' (some derivatives were not provided?)"); } } public static void lsfitfit(lsfitstate state, ndimensional_pfunc func, ndimensional_pgrad grad, ndimensional_phess hess, ndimensional_rep rep, object obj) { if( func==null ) throw new alglibexception("ALGLIB: error in 'lsfitfit()' (func is null)"); if( grad==null ) throw new alglibexception("ALGLIB: error in 'lsfitfit()' (grad is null)"); if( hess==null ) throw new alglibexception("ALGLIB: error in 'lsfitfit()' (hess is null)"); while( alglib.lsfititeration(state) ) { if( state.needf ) { func(state.c, state.x, ref state.innerobj.f, obj); continue; } if( state.needfg ) { grad(state.c, state.x, ref state.innerobj.f, state.innerobj.g, obj); continue; } if( state.needfgh ) { hess(state.c, state.x, ref state.innerobj.f, state.innerobj.g, state.innerobj.h, obj); continue; } if( state.innerobj.xupdated ) { if( rep!=null ) rep(state.innerobj.c, state.innerobj.f, obj); continue; } throw new alglibexception("ALGLIB: error in 'lsfitfit' (some derivatives were not provided?)"); } } /************************************************************************* Nonlinear least squares fitting results. Called after return from LSFitFit(). INPUT PARAMETERS: State - algorithm state OUTPUT PARAMETERS: Info - completetion code: * 1 relative function improvement is no more than EpsF. * 2 relative step is no more than EpsX. * 4 gradient norm is no more than EpsG * 5 MaxIts steps was taken * 7 stopping conditions are too stringent, further improvement is impossible C - array[0..K-1], solution Rep - optimization report. Following fields are set: * Rep.TerminationType completetion code: * RMSError rms error on the (X,Y). * AvgError average error on the (X,Y). * AvgRelError average relative error on the non-zero Y * MaxError maximum error NON-WEIGHTED ERRORS ARE CALCULATED * WRMSError weighted rms error on the (X,Y). -- ALGLIB -- Copyright 17.08.2009 by Bochkanov Sergey *************************************************************************/ public static void lsfitresults(lsfitstate state, out int info, out double[] c, out lsfitreport rep) { info = 0; c = new double[0]; rep = new lsfitreport(); lsfit.lsfitresults(state.innerobj, ref info, ref c, rep.innerobj); return; } } public partial class alglib { /************************************************************************* Parametric spline inteprolant: 2-dimensional curve. You should not try to access its members directly - use PSpline2XXXXXXXX() functions instead. *************************************************************************/ public class pspline2interpolant { // // Public declarations // public pspline2interpolant() { _innerobj = new pspline.pspline2interpolant(); } // // Although some of declarations below are public, you should not use them // They are intended for internal use only // private pspline.pspline2interpolant _innerobj; public pspline.pspline2interpolant innerobj { get { return _innerobj; } } public pspline2interpolant(pspline.pspline2interpolant obj) { _innerobj = obj; } } /************************************************************************* Parametric spline inteprolant: 3-dimensional curve. You should not try to access its members directly - use PSpline3XXXXXXXX() functions instead. *************************************************************************/ public class pspline3interpolant { // // Public declarations // public pspline3interpolant() { _innerobj = new pspline.pspline3interpolant(); } // // Although some of declarations below are public, you should not use them // They are intended for internal use only // private pspline.pspline3interpolant _innerobj; public pspline.pspline3interpolant innerobj { get { return _innerobj; } } public pspline3interpolant(pspline.pspline3interpolant obj) { _innerobj = obj; } } /************************************************************************* This function builds non-periodic 2-dimensional parametric spline which starts at (X[0],Y[0]) and ends at (X[N-1],Y[N-1]). INPUT PARAMETERS: XY - points, array[0..N-1,0..1]. XY[I,0:1] corresponds to the Ith point. Order of points is important! N - points count, N>=5 for Akima splines, N>=2 for other types of splines. ST - spline type: * 0 Akima spline * 1 parabolically terminated Catmull-Rom spline (Tension=0) * 2 parabolically terminated cubic spline PT - parameterization type: * 0 uniform * 1 chord length * 2 centripetal OUTPUT PARAMETERS: P - parametric spline interpolant NOTES: * this function assumes that there all consequent points are distinct. I.e. (x0,y0)<>(x1,y1), (x1,y1)<>(x2,y2), (x2,y2)<>(x3,y3) and so on. However, non-consequent points may coincide, i.e. we can have (x0,y0)= =(x2,y2). -- ALGLIB PROJECT -- Copyright 28.05.2010 by Bochkanov Sergey *************************************************************************/ public static void pspline2build(double[,] xy, int n, int st, int pt, out pspline2interpolant p) { p = new pspline2interpolant(); pspline.pspline2build(xy, n, st, pt, p.innerobj); return; } /************************************************************************* This function builds non-periodic 3-dimensional parametric spline which starts at (X[0],Y[0],Z[0]) and ends at (X[N-1],Y[N-1],Z[N-1]). Same as PSpline2Build() function, but for 3D, so we won't duplicate its description here. -- ALGLIB PROJECT -- Copyright 28.05.2010 by Bochkanov Sergey *************************************************************************/ public static void pspline3build(double[,] xy, int n, int st, int pt, out pspline3interpolant p) { p = new pspline3interpolant(); pspline.pspline3build(xy, n, st, pt, p.innerobj); return; } /************************************************************************* This function builds periodic 2-dimensional parametric spline which starts at (X[0],Y[0]), goes through all points to (X[N-1],Y[N-1]) and then back to (X[0],Y[0]). INPUT PARAMETERS: XY - points, array[0..N-1,0..1]. XY[I,0:1] corresponds to the Ith point. XY[N-1,0:1] must be different from XY[0,0:1]. Order of points is important! N - points count, N>=3 for other types of splines. ST - spline type: * 1 Catmull-Rom spline (Tension=0) with cyclic boundary conditions * 2 cubic spline with cyclic boundary conditions PT - parameterization type: * 0 uniform * 1 chord length * 2 centripetal OUTPUT PARAMETERS: P - parametric spline interpolant NOTES: * this function assumes that there all consequent points are distinct. I.e. (x0,y0)<>(x1,y1), (x1,y1)<>(x2,y2), (x2,y2)<>(x3,y3) and so on. However, non-consequent points may coincide, i.e. we can have (x0,y0)= =(x2,y2). * last point of sequence is NOT equal to the first point. You shouldn't make curve "explicitly periodic" by making them equal. -- ALGLIB PROJECT -- Copyright 28.05.2010 by Bochkanov Sergey *************************************************************************/ public static void pspline2buildperiodic(double[,] xy, int n, int st, int pt, out pspline2interpolant p) { p = new pspline2interpolant(); pspline.pspline2buildperiodic(xy, n, st, pt, p.innerobj); return; } /************************************************************************* This function builds periodic 3-dimensional parametric spline which starts at (X[0],Y[0],Z[0]), goes through all points to (X[N-1],Y[N-1],Z[N-1]) and then back to (X[0],Y[0],Z[0]). Same as PSpline2Build() function, but for 3D, so we won't duplicate its description here. -- ALGLIB PROJECT -- Copyright 28.05.2010 by Bochkanov Sergey *************************************************************************/ public static void pspline3buildperiodic(double[,] xy, int n, int st, int pt, out pspline3interpolant p) { p = new pspline3interpolant(); pspline.pspline3buildperiodic(xy, n, st, pt, p.innerobj); return; } /************************************************************************* This function returns vector of parameter values correspoding to points. I.e. for P created from (X[0],Y[0])...(X[N-1],Y[N-1]) and U=TValues(P) we have (X[0],Y[0]) = PSpline2Calc(P,U[0]), (X[1],Y[1]) = PSpline2Calc(P,U[1]), (X[2],Y[2]) = PSpline2Calc(P,U[2]), ... INPUT PARAMETERS: P - parametric spline interpolant OUTPUT PARAMETERS: N - array size T - array[0..N-1] NOTES: * for non-periodic splines U[0]=0, U[0]1) correspond to parts of the curve before the first (after the last) point * for periodic splines T<0 (or T>1) are projected into [0,1] by making T=T-floor(T). OUTPUT PARAMETERS: X - X-position Y - Y-position -- ALGLIB PROJECT -- Copyright 28.05.2010 by Bochkanov Sergey *************************************************************************/ public static void pspline2calc(pspline2interpolant p, double t, out double x, out double y) { x = 0; y = 0; pspline.pspline2calc(p.innerobj, t, ref x, ref y); return; } /************************************************************************* This function calculates the value of the parametric spline for a given value of parameter T. INPUT PARAMETERS: P - parametric spline interpolant T - point: * T in [0,1] corresponds to interval spanned by points * for non-periodic splines T<0 (or T>1) correspond to parts of the curve before the first (after the last) point * for periodic splines T<0 (or T>1) are projected into [0,1] by making T=T-floor(T). OUTPUT PARAMETERS: X - X-position Y - Y-position Z - Z-position -- ALGLIB PROJECT -- Copyright 28.05.2010 by Bochkanov Sergey *************************************************************************/ public static void pspline3calc(pspline3interpolant p, double t, out double x, out double y, out double z) { x = 0; y = 0; z = 0; pspline.pspline3calc(p.innerobj, t, ref x, ref y, ref z); return; } /************************************************************************* This function calculates tangent vector for a given value of parameter T INPUT PARAMETERS: P - parametric spline interpolant T - point: * T in [0,1] corresponds to interval spanned by points * for non-periodic splines T<0 (or T>1) correspond to parts of the curve before the first (after the last) point * for periodic splines T<0 (or T>1) are projected into [0,1] by making T=T-floor(T). OUTPUT PARAMETERS: X - X-component of tangent vector (normalized) Y - Y-component of tangent vector (normalized) NOTE: X^2+Y^2 is either 1 (for non-zero tangent vector) or 0. -- ALGLIB PROJECT -- Copyright 28.05.2010 by Bochkanov Sergey *************************************************************************/ public static void pspline2tangent(pspline2interpolant p, double t, out double x, out double y) { x = 0; y = 0; pspline.pspline2tangent(p.innerobj, t, ref x, ref y); return; } /************************************************************************* This function calculates tangent vector for a given value of parameter T INPUT PARAMETERS: P - parametric spline interpolant T - point: * T in [0,1] corresponds to interval spanned by points * for non-periodic splines T<0 (or T>1) correspond to parts of the curve before the first (after the last) point * for periodic splines T<0 (or T>1) are projected into [0,1] by making T=T-floor(T). OUTPUT PARAMETERS: X - X-component of tangent vector (normalized) Y - Y-component of tangent vector (normalized) Z - Z-component of tangent vector (normalized) NOTE: X^2+Y^2+Z^2 is either 1 (for non-zero tangent vector) or 0. -- ALGLIB PROJECT -- Copyright 28.05.2010 by Bochkanov Sergey *************************************************************************/ public static void pspline3tangent(pspline3interpolant p, double t, out double x, out double y, out double z) { x = 0; y = 0; z = 0; pspline.pspline3tangent(p.innerobj, t, ref x, ref y, ref z); return; } /************************************************************************* This function calculates derivative, i.e. it returns (dX/dT,dY/dT). INPUT PARAMETERS: P - parametric spline interpolant T - point: * T in [0,1] corresponds to interval spanned by points * for non-periodic splines T<0 (or T>1) correspond to parts of the curve before the first (after the last) point * for periodic splines T<0 (or T>1) are projected into [0,1] by making T=T-floor(T). OUTPUT PARAMETERS: X - X-value DX - X-derivative Y - Y-value DY - Y-derivative -- ALGLIB PROJECT -- Copyright 28.05.2010 by Bochkanov Sergey *************************************************************************/ public static void pspline2diff(pspline2interpolant p, double t, out double x, out double dx, out double y, out double dy) { x = 0; dx = 0; y = 0; dy = 0; pspline.pspline2diff(p.innerobj, t, ref x, ref dx, ref y, ref dy); return; } /************************************************************************* This function calculates derivative, i.e. it returns (dX/dT,dY/dT,dZ/dT). INPUT PARAMETERS: P - parametric spline interpolant T - point: * T in [0,1] corresponds to interval spanned by points * for non-periodic splines T<0 (or T>1) correspond to parts of the curve before the first (after the last) point * for periodic splines T<0 (or T>1) are projected into [0,1] by making T=T-floor(T). OUTPUT PARAMETERS: X - X-value DX - X-derivative Y - Y-value DY - Y-derivative Z - Z-value DZ - Z-derivative -- ALGLIB PROJECT -- Copyright 28.05.2010 by Bochkanov Sergey *************************************************************************/ public static void pspline3diff(pspline3interpolant p, double t, out double x, out double dx, out double y, out double dy, out double z, out double dz) { x = 0; dx = 0; y = 0; dy = 0; z = 0; dz = 0; pspline.pspline3diff(p.innerobj, t, ref x, ref dx, ref y, ref dy, ref z, ref dz); return; } /************************************************************************* This function calculates first and second derivative with respect to T. INPUT PARAMETERS: P - parametric spline interpolant T - point: * T in [0,1] corresponds to interval spanned by points * for non-periodic splines T<0 (or T>1) correspond to parts of the curve before the first (after the last) point * for periodic splines T<0 (or T>1) are projected into [0,1] by making T=T-floor(T). OUTPUT PARAMETERS: X - X-value DX - derivative D2X - second derivative Y - Y-value DY - derivative D2Y - second derivative -- ALGLIB PROJECT -- Copyright 28.05.2010 by Bochkanov Sergey *************************************************************************/ public static void pspline2diff2(pspline2interpolant p, double t, out double x, out double dx, out double d2x, out double y, out double dy, out double d2y) { x = 0; dx = 0; d2x = 0; y = 0; dy = 0; d2y = 0; pspline.pspline2diff2(p.innerobj, t, ref x, ref dx, ref d2x, ref y, ref dy, ref d2y); return; } /************************************************************************* This function calculates first and second derivative with respect to T. INPUT PARAMETERS: P - parametric spline interpolant T - point: * T in [0,1] corresponds to interval spanned by points * for non-periodic splines T<0 (or T>1) correspond to parts of the curve before the first (after the last) point * for periodic splines T<0 (or T>1) are projected into [0,1] by making T=T-floor(T). OUTPUT PARAMETERS: X - X-value DX - derivative D2X - second derivative Y - Y-value DY - derivative D2Y - second derivative Z - Z-value DZ - derivative D2Z - second derivative -- ALGLIB PROJECT -- Copyright 28.05.2010 by Bochkanov Sergey *************************************************************************/ public static void pspline3diff2(pspline3interpolant p, double t, out double x, out double dx, out double d2x, out double y, out double dy, out double d2y, out double z, out double dz, out double d2z) { x = 0; dx = 0; d2x = 0; y = 0; dy = 0; d2y = 0; z = 0; dz = 0; d2z = 0; pspline.pspline3diff2(p.innerobj, t, ref x, ref dx, ref d2x, ref y, ref dy, ref d2y, ref z, ref dz, ref d2z); return; } /************************************************************************* This function calculates arc length, i.e. length of curve between t=a and t=b. INPUT PARAMETERS: P - parametric spline interpolant A,B - parameter values corresponding to arc ends: * B>A will result in positive length returned * BA will result in positive length returned * B=2, N>=2 Output parameters: C - spline interpolant -- ALGLIB PROJECT -- Copyright 05.07.2007 by Bochkanov Sergey *************************************************************************/ public static void spline2dbuildbilinear(double[] x, double[] y, double[,] f, int m, int n, out spline2dinterpolant c) { c = new spline2dinterpolant(); spline2d.spline2dbuildbilinear(x, y, f, m, n, c.innerobj); return; } /************************************************************************* This subroutine builds bicubic spline coefficients table. Input parameters: X - spline abscissas, array[0..N-1] Y - spline ordinates, array[0..M-1] F - function values, array[0..M-1,0..N-1] M,N - grid size, M>=2, N>=2 Output parameters: C - spline interpolant -- ALGLIB PROJECT -- Copyright 05.07.2007 by Bochkanov Sergey *************************************************************************/ public static void spline2dbuildbicubic(double[] x, double[] y, double[,] f, int m, int n, out spline2dinterpolant c) { c = new spline2dinterpolant(); spline2d.spline2dbuildbicubic(x, y, f, m, n, c.innerobj); return; } /************************************************************************* This subroutine calculates the value of the bilinear or bicubic spline at the given point X. Input parameters: C - coefficients table. Built by BuildBilinearSpline or BuildBicubicSpline. X, Y- point Result: S(x,y) -- ALGLIB PROJECT -- Copyright 05.07.2007 by Bochkanov Sergey *************************************************************************/ public static double spline2dcalc(spline2dinterpolant c, double x, double y) { double result = spline2d.spline2dcalc(c.innerobj, x, y); return result; } /************************************************************************* This subroutine calculates the value of the bilinear or bicubic spline at the given point X and its derivatives. Input parameters: C - spline interpolant. X, Y- point Output parameters: F - S(x,y) FX - dS(x,y)/dX FY - dS(x,y)/dY FXY - d2S(x,y)/dXdY -- ALGLIB PROJECT -- Copyright 05.07.2007 by Bochkanov Sergey *************************************************************************/ public static void spline2ddiff(spline2dinterpolant c, double x, double y, out double f, out double fx, out double fy, out double fxy) { f = 0; fx = 0; fy = 0; fxy = 0; spline2d.spline2ddiff(c.innerobj, x, y, ref f, ref fx, ref fy, ref fxy); return; } /************************************************************************* This subroutine unpacks two-dimensional spline into the coefficients table Input parameters: C - spline interpolant. Result: M, N- grid size (x-axis and y-axis) Tbl - coefficients table, unpacked format, [0..(N-1)*(M-1)-1, 0..19]. For I = 0...M-2, J=0..N-2: K = I*(N-1)+J Tbl[K,0] = X[j] Tbl[K,1] = X[j+1] Tbl[K,2] = Y[i] Tbl[K,3] = Y[i+1] Tbl[K,4] = C00 Tbl[K,5] = C01 Tbl[K,6] = C02 Tbl[K,7] = C03 Tbl[K,8] = C10 Tbl[K,9] = C11 ... Tbl[K,19] = C33 On each grid square spline is equals to: S(x) = SUM(c[i,j]*(x^i)*(y^j), i=0..3, j=0..3) t = x-x[j] u = y-y[i] -- ALGLIB PROJECT -- Copyright 29.06.2007 by Bochkanov Sergey *************************************************************************/ public static void spline2dunpack(spline2dinterpolant c, out int m, out int n, out double[,] tbl) { m = 0; n = 0; tbl = new double[0,0]; spline2d.spline2dunpack(c.innerobj, ref m, ref n, ref tbl); return; } /************************************************************************* This subroutine performs linear transformation of the spline argument. Input parameters: C - spline interpolant AX, BX - transformation coefficients: x = A*t + B AY, BY - transformation coefficients: y = A*u + B Result: C - transformed spline -- ALGLIB PROJECT -- Copyright 30.06.2007 by Bochkanov Sergey *************************************************************************/ public static void spline2dlintransxy(spline2dinterpolant c, double ax, double bx, double ay, double by) { spline2d.spline2dlintransxy(c.innerobj, ax, bx, ay, by); return; } /************************************************************************* This subroutine performs linear transformation of the spline. Input parameters: C - spline interpolant. A, B- transformation coefficients: S2(x,y) = A*S(x,y) + B Output parameters: C - transformed spline -- ALGLIB PROJECT -- Copyright 30.06.2007 by Bochkanov Sergey *************************************************************************/ public static void spline2dlintransf(spline2dinterpolant c, double a, double b) { spline2d.spline2dlintransf(c.innerobj, a, b); return; } /************************************************************************* Bicubic spline resampling Input parameters: A - function values at the old grid, array[0..OldHeight-1, 0..OldWidth-1] OldHeight - old grid height, OldHeight>1 OldWidth - old grid width, OldWidth>1 NewHeight - new grid height, NewHeight>1 NewWidth - new grid width, NewWidth>1 Output parameters: B - function values at the new grid, array[0..NewHeight-1, 0..NewWidth-1] -- ALGLIB routine -- 15 May, 2007 Copyright by Bochkanov Sergey *************************************************************************/ public static void spline2dresamplebicubic(double[,] a, int oldheight, int oldwidth, out double[,] b, int newheight, int newwidth) { b = new double[0,0]; spline2d.spline2dresamplebicubic(a, oldheight, oldwidth, ref b, newheight, newwidth); return; } /************************************************************************* Bilinear spline resampling Input parameters: A - function values at the old grid, array[0..OldHeight-1, 0..OldWidth-1] OldHeight - old grid height, OldHeight>1 OldWidth - old grid width, OldWidth>1 NewHeight - new grid height, NewHeight>1 NewWidth - new grid width, NewWidth>1 Output parameters: B - function values at the new grid, array[0..NewHeight-1, 0..NewWidth-1] -- ALGLIB routine -- 09.07.2007 Copyright by Bochkanov Sergey *************************************************************************/ public static void spline2dresamplebilinear(double[,] a, int oldheight, int oldwidth, out double[,] b, int newheight, int newwidth) { b = new double[0,0]; spline2d.spline2dresamplebilinear(a, oldheight, oldwidth, ref b, newheight, newwidth); return; } } public partial class alglib { public class idwint { /************************************************************************* IDW interpolant. *************************************************************************/ public class idwinterpolant { public int n; public int nx; public int d; public double r; public int nw; public nearestneighbor.kdtree tree; public int modeltype; public double[,] q; public double[] xbuf; public int[] tbuf; public double[] rbuf; public double[,] xybuf; public int debugsolverfailures; public double debugworstrcond; public double debugbestrcond; public idwinterpolant() { tree = new nearestneighbor.kdtree(); q = new double[0,0]; xbuf = new double[0]; tbuf = new int[0]; rbuf = new double[0]; xybuf = new double[0,0]; } }; public const double idwqfactor = 1.5; public const int idwkmin = 5; /************************************************************************* IDW interpolation INPUT PARAMETERS: Z - IDW interpolant built with one of model building subroutines. X - array[0..NX-1], interpolation point Result: IDW interpolant Z(X) -- ALGLIB -- Copyright 02.03.2010 by Bochkanov Sergey *************************************************************************/ public static double idwcalc(idwinterpolant z, double[] x) { double result = 0; int nx = 0; int i = 0; int k = 0; double r = 0; double s = 0; double w = 0; double v1 = 0; double v2 = 0; double d0 = 0; double di = 0; // // these initializers are not really necessary, // but without them compiler complains about uninitialized locals // k = 0; // // Query // if( z.modeltype==0 ) { // // NQ/NW-based model // nx = z.nx; k = nearestneighbor.kdtreequeryknn(z.tree, x, z.nw, true); nearestneighbor.kdtreequeryresultsdistances(z.tree, ref z.rbuf); nearestneighbor.kdtreequeryresultstags(z.tree, ref z.tbuf); } if( z.modeltype==1 ) { // // R-based model // nx = z.nx; k = nearestneighbor.kdtreequeryrnn(z.tree, x, z.r, true); nearestneighbor.kdtreequeryresultsdistances(z.tree, ref z.rbuf); nearestneighbor.kdtreequeryresultstags(z.tree, ref z.tbuf); if( kd0 // // NOTE 2: self-match is USED for this query // // NOTE 3: last point almost always gain zero weight, but it MUST // be used for fitting because sometimes it will gain NON-ZERO // weight - for example, when all distances are equal. // r = z.rbuf[k-1]; d0 = z.rbuf[0]; result = 0; s = 0; for(i=0; i<=k-1; i++) { di = z.rbuf[i]; if( (double)(di)==(double)(d0) ) { // // distance is equal to shortest, set it 1.0 // without explicitly calculating (which would give // us same result, but 'll expose us to the risk of // division by zero). // w = 1; } else { // // use normalized formula // v1 = (r-di)/(r-d0); v2 = d0/di; w = math.sqr(v1*v2); } result = result+w*idwcalcq(z, x, z.tbuf[i]); s = s+w; } result = result/s; return result; } /************************************************************************* IDW interpolant using modified Shepard method for uniform point distributions. INPUT PARAMETERS: XY - X and Y values, array[0..N-1,0..NX]. First NX columns contain X-values, last column contain Y-values. N - number of nodes, N>0. NX - space dimension, NX>=1. D - nodal function type, either: * 0 constant model. Just for demonstration only, worst model ever. * 1 linear model, least squares fitting. Simpe model for datasets too small for quadratic models * 2 quadratic model, least squares fitting. Best model available (if your dataset is large enough). * -1 "fast" linear model, use with caution!!! It is significantly faster than linear/quadratic and better than constant model. But it is less robust (especially in the presence of noise). NQ - number of points used to calculate nodal functions (ignored for constant models). NQ should be LARGER than: * max(1.5*(1+NX),2^NX+1) for linear model, * max(3/4*(NX+2)*(NX+1),2^NX+1) for quadratic model. Values less than this threshold will be silently increased. NW - number of points used to calculate weights and to interpolate. Required: >=2^NX+1, values less than this threshold will be silently increased. Recommended value: about 2*NQ OUTPUT PARAMETERS: Z - IDW interpolant. NOTES: * best results are obtained with quadratic models, worst - with constant models * when N is large, NQ and NW must be significantly smaller than N both to obtain optimal performance and to obtain optimal accuracy. In 2 or 3-dimensional tasks NQ=15 and NW=25 are good values to start with. * NQ and NW may be greater than N. In such cases they will be automatically decreased. * this subroutine is always succeeds (as long as correct parameters are passed). * see 'Multivariate Interpolation of Large Sets of Scattered Data' by Robert J. Renka for more information on this algorithm. * this subroutine assumes that point distribution is uniform at the small scales. If it isn't - for example, points are concentrated along "lines", but "lines" distribution is uniform at the larger scale - then you should use IDWBuildModifiedShepardR() -- ALGLIB PROJECT -- Copyright 02.03.2010 by Bochkanov Sergey *************************************************************************/ public static void idwbuildmodifiedshepard(double[,] xy, int n, int nx, int d, int nq, int nw, idwinterpolant z) { int i = 0; int j = 0; int k = 0; int j2 = 0; int j3 = 0; double v = 0; double r = 0; double s = 0; double d0 = 0; double di = 0; double v1 = 0; double v2 = 0; int nc = 0; int offs = 0; double[] x = new double[0]; double[] qrbuf = new double[0]; double[,] qxybuf = new double[0,0]; double[] y = new double[0]; double[,] fmatrix = new double[0,0]; double[] w = new double[0]; double[] qsol = new double[0]; double[] temp = new double[0]; int[] tags = new int[0]; int info = 0; double taskrcond = 0; int i_ = 0; // // these initializers are not really necessary, // but without them compiler complains about uninitialized locals // nc = 0; // // assertions // ap.assert(n>0, "IDWBuildModifiedShepard: N<=0!"); ap.assert(nx>=1, "IDWBuildModifiedShepard: NX<1!"); ap.assert(d>=-1 & d<=2, "IDWBuildModifiedShepard: D<>-1 and D<>0 and D<>1 and D<>2!"); // // Correct parameters if needed // if( d==1 ) { nq = Math.Max(nq, (int)Math.Ceiling(idwqfactor*(1+nx))+1); nq = Math.Max(nq, (int)Math.Round(Math.Pow(2, nx))+1); } if( d==2 ) { nq = Math.Max(nq, (int)Math.Ceiling(idwqfactor*(nx+2)*(nx+1)/2)+1); nq = Math.Max(nq, (int)Math.Round(Math.Pow(2, nx))+1); } nw = Math.Max(nw, (int)Math.Round(Math.Pow(2, nx))+1); nq = Math.Min(nq, n); nw = Math.Min(nw, n); // // primary initialization of Z // idwinit1(n, nx, d, nq, nw, z); z.modeltype = 0; // // Create KD-tree // tags = new int[n]; for(i=0; i<=n-1; i++) { tags[i] = i; } nearestneighbor.kdtreebuildtagged(xy, tags, n, nx, 1, 2, z.tree); // // build nodal functions // temp = new double[nq+1]; x = new double[nx]; qrbuf = new double[nq]; qxybuf = new double[nq, nx+1]; if( d==-1 ) { w = new double[nq]; } if( d==1 ) { y = new double[nq]; w = new double[nq]; qsol = new double[nx]; // // NX for linear members, // 1 for temporary storage // fmatrix = new double[nq, nx+1]; } if( d==2 ) { y = new double[nq]; w = new double[nq]; qsol = new double[nx+(int)Math.Round(nx*(nx+1)*0.5)]; // // NX for linear members, // Round(NX*(NX+1)*0.5) for quadratic model, // 1 for temporary storage // fmatrix = new double[nq, nx+(int)Math.Round(nx*(nx+1)*0.5)+1]; } for(i=0; i<=n-1; i++) { // // Initialize center and function value. // If D=0 it is all what we need // for(i_=0; i_<=nx;i_++) { z.q[i,i_] = xy[i,i_]; } if( d==0 ) { continue; } // // calculate weights for linear/quadratic members calculation. // // NOTE 1: weights are calculated using NORMALIZED modified // Shepard's formula. Original formula is w(i) = sqr((R-di)/(R*di)), // where di is i-th distance, R is max(di). Modified formula have // following form: // w_mod(i) = 1, if di=d0 // w_mod(i) = w(i)/w(0), if di<>d0 // // NOTE 2: self-match is NOT used for this query // // NOTE 3: last point almost always gain zero weight, but it MUST // be used for fitting because sometimes it will gain NON-ZERO // weight - for example, when all distances are equal. // for(i_=0; i_<=nx-1;i_++) { x[i_] = xy[i,i_]; } k = nearestneighbor.kdtreequeryknn(z.tree, x, nq, false); nearestneighbor.kdtreequeryresultsxy(z.tree, ref qxybuf); nearestneighbor.kdtreequeryresultsdistances(z.tree, ref qrbuf); r = qrbuf[k-1]; d0 = qrbuf[0]; for(j=0; j<=k-1; j++) { di = qrbuf[j]; if( (double)(di)==(double)(d0) ) { // // distance is equal to shortest, set it 1.0 // without explicitly calculating (which would give // us same result, but 'll expose us to the risk of // division by zero). // w[j] = 1; } else { // // use normalized formula // v1 = (r-di)/(r-d0); v2 = d0/di; w[j] = math.sqr(v1*v2); } } // // calculate linear/quadratic members // if( d==-1 ) { // // "Fast" linear nodal function calculated using // inverse distance weighting // for(j=0; j<=nx-1; j++) { x[j] = 0; } s = 0; for(j=0; j<=k-1; j++) { // // calculate J-th inverse distance weighted gradient: // grad_k = (y_j-y_k)*(x_j-x_k)/sqr(norm(x_j-x_k)) // grad = sum(wk*grad_k)/sum(w_k) // v = 0; for(j2=0; j2<=nx-1; j2++) { v = v+math.sqr(qxybuf[j,j2]-xy[i,j2]); } // // Although x_j<>x_k, sqr(norm(x_j-x_k)) may be zero due to // underflow. If it is, we assume than J-th gradient is zero // (i.e. don't add anything) // if( (double)(v)!=(double)(0) ) { for(j2=0; j2<=nx-1; j2++) { x[j2] = x[j2]+w[j]*(qxybuf[j,nx]-xy[i,nx])*(qxybuf[j,j2]-xy[i,j2])/v; } } s = s+w[j]; } for(j=0; j<=nx-1; j++) { z.q[i,nx+1+j] = x[j]/s; } } else { // // Least squares models: build // if( d==1 ) { // // Linear nodal function calculated using // least squares fitting to its neighbors // for(j=0; j<=k-1; j++) { for(j2=0; j2<=nx-1; j2++) { fmatrix[j,j2] = qxybuf[j,j2]-xy[i,j2]; } y[j] = qxybuf[j,nx]-xy[i,nx]; } nc = nx; } if( d==2 ) { // // Quadratic nodal function calculated using // least squares fitting to its neighbors // for(j=0; j<=k-1; j++) { offs = 0; for(j2=0; j2<=nx-1; j2++) { fmatrix[j,offs] = qxybuf[j,j2]-xy[i,j2]; offs = offs+1; } for(j2=0; j2<=nx-1; j2++) { for(j3=j2; j3<=nx-1; j3++) { fmatrix[j,offs] = (qxybuf[j,j2]-xy[i,j2])*(qxybuf[j,j3]-xy[i,j3]); offs = offs+1; } } y[j] = qxybuf[j,nx]-xy[i,nx]; } nc = nx+(int)Math.Round(nx*(nx+1)*0.5); } idwinternalsolver(ref y, ref w, ref fmatrix, ref temp, k, nc, ref info, ref qsol, ref taskrcond); // // Least squares models: copy results // if( info>0 ) { // // LLS task is solved, copy results // z.debugworstrcond = Math.Min(z.debugworstrcond, taskrcond); z.debugbestrcond = Math.Max(z.debugbestrcond, taskrcond); for(j=0; j<=nc-1; j++) { z.q[i,nx+1+j] = qsol[j]; } } else { // // Solver failure, very strange, but we will use // zero values to handle it. // z.debugsolverfailures = z.debugsolverfailures+1; for(j=0; j<=nc-1; j++) { z.q[i,nx+1+j] = 0; } } } } } /************************************************************************* IDW interpolant using modified Shepard method for non-uniform datasets. This type of model uses constant nodal functions and interpolates using all nodes which are closer than user-specified radius R. It may be used when points distribution is non-uniform at the small scale, but it is at the distances as large as R. INPUT PARAMETERS: XY - X and Y values, array[0..N-1,0..NX]. First NX columns contain X-values, last column contain Y-values. N - number of nodes, N>0. NX - space dimension, NX>=1. R - radius, R>0 OUTPUT PARAMETERS: Z - IDW interpolant. NOTES: * if there is less than IDWKMin points within R-ball, algorithm selects IDWKMin closest ones, so that continuity properties of interpolant are preserved even far from points. -- ALGLIB PROJECT -- Copyright 11.04.2010 by Bochkanov Sergey *************************************************************************/ public static void idwbuildmodifiedshepardr(double[,] xy, int n, int nx, double r, idwinterpolant z) { int i = 0; int[] tags = new int[0]; int i_ = 0; // // assertions // ap.assert(n>0, "IDWBuildModifiedShepardR: N<=0!"); ap.assert(nx>=1, "IDWBuildModifiedShepardR: NX<1!"); ap.assert((double)(r)>(double)(0), "IDWBuildModifiedShepardR: R<=0!"); // // primary initialization of Z // idwinit1(n, nx, 0, 0, n, z); z.modeltype = 1; z.r = r; // // Create KD-tree // tags = new int[n]; for(i=0; i<=n-1; i++) { tags[i] = i; } nearestneighbor.kdtreebuildtagged(xy, tags, n, nx, 1, 2, z.tree); // // build nodal functions // for(i=0; i<=n-1; i++) { for(i_=0; i_<=nx;i_++) { z.q[i,i_] = xy[i,i_]; } } } /************************************************************************* IDW model for noisy data. This subroutine may be used to handle noisy data, i.e. data with noise in OUTPUT values. It differs from IDWBuildModifiedShepard() in the following aspects: * nodal functions are not constrained to pass through nodes: Qi(xi)<>yi, i.e. we have fitting instead of interpolation. * weights which are used during least squares fitting stage are all equal to 1.0 (independently of distance) * "fast"-linear or constant nodal functions are not supported (either not robust enough or too rigid) This problem require far more complex tuning than interpolation problems. Below you can find some recommendations regarding this problem: * focus on tuning NQ; it controls noise reduction. As for NW, you can just make it equal to 2*NQ. * you can use cross-validation to determine optimal NQ. * optimal NQ is a result of complex tradeoff between noise level (more noise = larger NQ required) and underlying function complexity (given fixed N, larger NQ means smoothing of compex features in the data). For example, NQ=N will reduce noise to the minimum level possible, but you will end up with just constant/linear/quadratic (depending on D) least squares model for the whole dataset. INPUT PARAMETERS: XY - X and Y values, array[0..N-1,0..NX]. First NX columns contain X-values, last column contain Y-values. N - number of nodes, N>0. NX - space dimension, NX>=1. D - nodal function degree, either: * 1 linear model, least squares fitting. Simpe model for datasets too small for quadratic models (or for very noisy problems). * 2 quadratic model, least squares fitting. Best model available (if your dataset is large enough). NQ - number of points used to calculate nodal functions. NQ should be significantly larger than 1.5 times the number of coefficients in a nodal function to overcome effects of noise: * larger than 1.5*(1+NX) for linear model, * larger than 3/4*(NX+2)*(NX+1) for quadratic model. Values less than this threshold will be silently increased. NW - number of points used to calculate weights and to interpolate. Required: >=2^NX+1, values less than this threshold will be silently increased. Recommended value: about 2*NQ or larger OUTPUT PARAMETERS: Z - IDW interpolant. NOTES: * best results are obtained with quadratic models, linear models are not recommended to use unless you are pretty sure that it is what you want * this subroutine is always succeeds (as long as correct parameters are passed). * see 'Multivariate Interpolation of Large Sets of Scattered Data' by Robert J. Renka for more information on this algorithm. -- ALGLIB PROJECT -- Copyright 02.03.2010 by Bochkanov Sergey *************************************************************************/ public static void idwbuildnoisy(double[,] xy, int n, int nx, int d, int nq, int nw, idwinterpolant z) { int i = 0; int j = 0; int k = 0; int j2 = 0; int j3 = 0; double v = 0; int nc = 0; int offs = 0; double taskrcond = 0; double[] x = new double[0]; double[] qrbuf = new double[0]; double[,] qxybuf = new double[0,0]; double[] y = new double[0]; double[] w = new double[0]; double[,] fmatrix = new double[0,0]; double[] qsol = new double[0]; int[] tags = new int[0]; double[] temp = new double[0]; int info = 0; int i_ = 0; // // these initializers are not really necessary, // but without them compiler complains about uninitialized locals // nc = 0; // // assertions // ap.assert(n>0, "IDWBuildNoisy: N<=0!"); ap.assert(nx>=1, "IDWBuildNoisy: NX<1!"); ap.assert(d>=1 & d<=2, "IDWBuildNoisy: D<>1 and D<>2!"); // // Correct parameters if needed // if( d==1 ) { nq = Math.Max(nq, (int)Math.Ceiling(idwqfactor*(1+nx))+1); } if( d==2 ) { nq = Math.Max(nq, (int)Math.Ceiling(idwqfactor*(nx+2)*(nx+1)/2)+1); } nw = Math.Max(nw, (int)Math.Round(Math.Pow(2, nx))+1); nq = Math.Min(nq, n); nw = Math.Min(nw, n); // // primary initialization of Z // idwinit1(n, nx, d, nq, nw, z); z.modeltype = 0; // // Create KD-tree // tags = new int[n]; for(i=0; i<=n-1; i++) { tags[i] = i; } nearestneighbor.kdtreebuildtagged(xy, tags, n, nx, 1, 2, z.tree); // // build nodal functions // (special algorithm for noisy data is used) // temp = new double[nq+1]; x = new double[nx]; qrbuf = new double[nq]; qxybuf = new double[nq, nx+1]; if( d==1 ) { y = new double[nq]; w = new double[nq]; qsol = new double[1+nx]; // // 1 for constant member, // NX for linear members, // 1 for temporary storage // fmatrix = new double[nq, 1+nx+1]; } if( d==2 ) { y = new double[nq]; w = new double[nq]; qsol = new double[1+nx+(int)Math.Round(nx*(nx+1)*0.5)]; // // 1 for constant member, // NX for linear members, // Round(NX*(NX+1)*0.5) for quadratic model, // 1 for temporary storage // fmatrix = new double[nq, 1+nx+(int)Math.Round(nx*(nx+1)*0.5)+1]; } for(i=0; i<=n-1; i++) { // // Initialize center. // for(i_=0; i_<=nx-1;i_++) { z.q[i,i_] = xy[i,i_]; } // // Calculate linear/quadratic members // using least squares fit // NOTE 1: all weight are equal to 1.0 // NOTE 2: self-match is USED for this query // for(i_=0; i_<=nx-1;i_++) { x[i_] = xy[i,i_]; } k = nearestneighbor.kdtreequeryknn(z.tree, x, nq, true); nearestneighbor.kdtreequeryresultsxy(z.tree, ref qxybuf); nearestneighbor.kdtreequeryresultsdistances(z.tree, ref qrbuf); if( d==1 ) { // // Linear nodal function calculated using // least squares fitting to its neighbors // for(j=0; j<=k-1; j++) { fmatrix[j,0] = 1.0; for(j2=0; j2<=nx-1; j2++) { fmatrix[j,1+j2] = qxybuf[j,j2]-xy[i,j2]; } y[j] = qxybuf[j,nx]; w[j] = 1; } nc = 1+nx; } if( d==2 ) { // // Quadratic nodal function calculated using // least squares fitting to its neighbors // for(j=0; j<=k-1; j++) { fmatrix[j,0] = 1; offs = 1; for(j2=0; j2<=nx-1; j2++) { fmatrix[j,offs] = qxybuf[j,j2]-xy[i,j2]; offs = offs+1; } for(j2=0; j2<=nx-1; j2++) { for(j3=j2; j3<=nx-1; j3++) { fmatrix[j,offs] = (qxybuf[j,j2]-xy[i,j2])*(qxybuf[j,j3]-xy[i,j3]); offs = offs+1; } } y[j] = qxybuf[j,nx]; w[j] = 1; } nc = 1+nx+(int)Math.Round(nx*(nx+1)*0.5); } idwinternalsolver(ref y, ref w, ref fmatrix, ref temp, k, nc, ref info, ref qsol, ref taskrcond); // // Least squares models: copy results // if( info>0 ) { // // LLS task is solved, copy results // z.debugworstrcond = Math.Min(z.debugworstrcond, taskrcond); z.debugbestrcond = Math.Max(z.debugbestrcond, taskrcond); for(j=0; j<=nc-1; j++) { z.q[i,nx+j] = qsol[j]; } } else { // // Solver failure, very strange, but we will use // zero values to handle it. // z.debugsolverfailures = z.debugsolverfailures+1; v = 0; for(j=0; j<=k-1; j++) { v = v+qxybuf[j,nx]; } z.q[i,nx] = v/k; for(j=0; j<=nc-2; j++) { z.q[i,nx+1+j] = 0; } } } } /************************************************************************* Internal subroutine: K-th nodal function calculation -- ALGLIB -- Copyright 02.03.2010 by Bochkanov Sergey *************************************************************************/ private static double idwcalcq(idwinterpolant z, double[] x, int k) { double result = 0; int nx = 0; int i = 0; int j = 0; int offs = 0; nx = z.nx; // // constant member // result = z.q[k,nx]; // // linear members // if( z.d>=1 ) { for(i=0; i<=nx-1; i++) { result = result+z.q[k,nx+1+i]*(x[i]-z.q[k,i]); } } // // quadratic members // if( z.d>=2 ) { offs = nx+1+nx; for(i=0; i<=nx-1; i++) { for(j=i; j<=nx-1; j++) { result = result+z.q[k,offs]*(x[i]-z.q[k,i])*(x[j]-z.q[k,j]); offs = offs+1; } } } return result; } /************************************************************************* Initialization of internal structures. It assumes correctness of all parameters. -- ALGLIB -- Copyright 02.03.2010 by Bochkanov Sergey *************************************************************************/ private static void idwinit1(int n, int nx, int d, int nq, int nw, idwinterpolant z) { z.debugsolverfailures = 0; z.debugworstrcond = 1.0; z.debugbestrcond = 0; z.n = n; z.nx = nx; z.d = 0; if( d==1 ) { z.d = 1; } if( d==2 ) { z.d = 2; } if( d==-1 ) { z.d = 1; } z.nw = nw; if( d==-1 ) { z.q = new double[n, nx+1+nx]; } if( d==0 ) { z.q = new double[n, nx+1]; } if( d==1 ) { z.q = new double[n, nx+1+nx]; } if( d==2 ) { z.q = new double[n, nx+1+nx+(int)Math.Round(nx*(nx+1)*0.5)]; } z.tbuf = new int[nw]; z.rbuf = new double[nw]; z.xybuf = new double[nw, nx+1]; z.xbuf = new double[nx]; } /************************************************************************* Linear least squares solver for small tasks. Works faster than standard ALGLIB solver in non-degenerate cases (due to absense of internal allocations and optimized row/colums). In degenerate cases it calls standard solver, which results in small performance penalty associated with preliminary steps. INPUT PARAMETERS: Y array[0..N-1] W array[0..N-1] FMatrix array[0..N-1,0..M], have additional column for temporary values Temp array[0..N] *************************************************************************/ private static void idwinternalsolver(ref double[] y, ref double[] w, ref double[,] fmatrix, ref double[] temp, int n, int m, ref int info, ref double[] x, ref double taskrcond) { int i = 0; int j = 0; double v = 0; double tau = 0; double[] b = new double[0]; densesolver.densesolverlsreport srep = new densesolver.densesolverlsreport(); int i_ = 0; int i1_ = 0; info = 0; // // set up info // info = 1; // // prepare matrix // for(i=0; i<=n-1; i++) { fmatrix[i,m] = y[i]; v = w[i]; for(i_=0; i_<=m;i_++) { fmatrix[i,i_] = v*fmatrix[i,i_]; } } // // use either fast algorithm or general algorithm // if( m<=n ) { // // QR decomposition // We assume that M<=N (we would have called LSFit() otherwise) // for(i=0; i<=m-1; i++) { if( i(double)(10000*n*math.machineepsilon) ) { // // solve triangular system R*x = FMatrix[0:M-1,M] // using fast algorithm, then exit // x[m-1] = fmatrix[m-1,m]/fmatrix[m-1,m-1]; for(i=m-2; i>=0; i--) { v = 0.0; for(i_=i+1; i_<=m-1;i_++) { v += fmatrix[i,i_]*x[i_]; } x[i] = (fmatrix[i,m]-v)/fmatrix[i,i]; } } else { // // use more general algorithm // b = new double[m]; for(i=0; i<=m-1; i++) { for(j=0; j<=i-1; j++) { fmatrix[i,j] = 0.0; } b[i] = fmatrix[i,m]; } densesolver.rmatrixsolvels(fmatrix, m, m, b, 10000*math.machineepsilon, ref info, srep, ref x); } } else { // // use more general algorithm // b = new double[n]; for(i=0; i<=n-1; i++) { b[i] = fmatrix[i,m]; } densesolver.rmatrixsolvels(fmatrix, n, m, b, 10000*math.machineepsilon, ref info, srep, ref x); taskrcond = srep.r2; } } } public class ratint { /************************************************************************* Barycentric interpolant. *************************************************************************/ public class barycentricinterpolant { public int n; public double sy; public double[] x; public double[] y; public double[] w; public barycentricinterpolant() { x = new double[0]; y = new double[0]; w = new double[0]; } }; /************************************************************************* Rational interpolation using barycentric formula F(t) = SUM(i=0,n-1,w[i]*f[i]/(t-x[i])) / SUM(i=0,n-1,w[i]/(t-x[i])) Input parameters: B - barycentric interpolant built with one of model building subroutines. T - interpolation point Result: barycentric interpolant F(t) -- ALGLIB -- Copyright 17.08.2009 by Bochkanov Sergey *************************************************************************/ public static double barycentriccalc(barycentricinterpolant b, double t) { double result = 0; double s1 = 0; double s2 = 0; double s = 0; double v = 0; int i = 0; ap.assert(!Double.IsInfinity(t), "BarycentricCalc: infinite T!"); // // special case: NaN // if( Double.IsNaN(t) ) { result = Double.NaN; return result; } // // special case: N=1 // if( b.n==1 ) { result = b.sy*b.y[0]; return result; } // // Here we assume that task is normalized, i.e.: // 1. abs(Y[i])<=1 // 2. abs(W[i])<=1 // 3. X[] is ordered // s = Math.Abs(t-b.x[0]); for(i=0; i<=b.n-1; i++) { v = b.x[i]; if( (double)(v)==(double)(t) ) { result = b.sy*b.y[i]; return result; } v = Math.Abs(t-v); if( (double)(v)<(double)(s) ) { s = v; } } s1 = 0; s2 = 0; for(i=0; i<=b.n-1; i++) { v = s/(t-b.x[i]); v = v*b.w[i]; s1 = s1+v*b.y[i]; s2 = s2+v; } result = b.sy*s1/s2; return result; } /************************************************************************* Differentiation of barycentric interpolant: first derivative. Algorithm used in this subroutine is very robust and should not fail until provided with values too close to MaxRealNumber (usually MaxRealNumber/N or greater will overflow). INPUT PARAMETERS: B - barycentric interpolant built with one of model building subroutines. T - interpolation point OUTPUT PARAMETERS: F - barycentric interpolant at T DF - first derivative NOTE -- ALGLIB -- Copyright 17.08.2009 by Bochkanov Sergey *************************************************************************/ public static void barycentricdiff1(barycentricinterpolant b, double t, ref double f, ref double df) { double v = 0; double vv = 0; int i = 0; int k = 0; double n0 = 0; double n1 = 0; double d0 = 0; double d1 = 0; double s0 = 0; double s1 = 0; double xk = 0; double xi = 0; double xmin = 0; double xmax = 0; double xscale1 = 0; double xoffs1 = 0; double xscale2 = 0; double xoffs2 = 0; double xprev = 0; f = 0; df = 0; ap.assert(!Double.IsInfinity(t), "BarycentricDiff1: infinite T!"); // // special case: NaN // if( Double.IsNaN(t) ) { f = Double.NaN; df = Double.NaN; return; } // // special case: N=1 // if( b.n==1 ) { f = b.sy*b.y[0]; df = 0; return; } if( (double)(b.sy)==(double)(0) ) { f = 0; df = 0; return; } ap.assert((double)(b.sy)>(double)(0), "BarycentricDiff1: internal error"); // // We assume than N>1 and B.SY>0. Find: // 1. pivot point (X[i] closest to T) // 2. width of interval containing X[i] // v = Math.Abs(b.x[0]-t); k = 0; xmin = b.x[0]; xmax = b.x[0]; for(i=1; i<=b.n-1; i++) { vv = b.x[i]; if( (double)(Math.Abs(vv-t))<(double)(v) ) { v = Math.Abs(vv-t); k = i; } xmin = Math.Min(xmin, vv); xmax = Math.Max(xmax, vv); } // // pivot point found, calculate dNumerator and dDenominator // xscale1 = 1/(xmax-xmin); xoffs1 = -(xmin/(xmax-xmin))+1; xscale2 = 2; xoffs2 = -3; t = t*xscale1+xoffs1; t = t*xscale2+xoffs2; xk = b.x[k]; xk = xk*xscale1+xoffs1; xk = xk*xscale2+xoffs2; v = t-xk; n0 = 0; n1 = 0; d0 = 0; d1 = 0; xprev = -2; for(i=0; i<=b.n-1; i++) { xi = b.x[i]; xi = xi*xscale1+xoffs1; xi = xi*xscale2+xoffs2; ap.assert((double)(xi)>(double)(xprev), "BarycentricDiff1: points are too close!"); xprev = xi; if( i!=k ) { vv = math.sqr(t-xi); s0 = (t-xk)/(t-xi); s1 = (xk-xi)/vv; } else { s0 = 1; s1 = 0; } vv = b.w[i]*b.y[i]; n0 = n0+s0*vv; n1 = n1+s1*vv; vv = b.w[i]; d0 = d0+s0*vv; d1 = d1+s1*vv; } f = b.sy*n0/d0; df = (n1*d0-n0*d1)/math.sqr(d0); if( (double)(df)!=(double)(0) ) { df = Math.Sign(df)*Math.Exp(Math.Log(Math.Abs(df))+Math.Log(b.sy)+Math.Log(xscale1)+Math.Log(xscale2)); } } /************************************************************************* Differentiation of barycentric interpolant: first/second derivatives. INPUT PARAMETERS: B - barycentric interpolant built with one of model building subroutines. T - interpolation point OUTPUT PARAMETERS: F - barycentric interpolant at T DF - first derivative D2F - second derivative NOTE: this algorithm may fail due to overflow/underflor if used on data whose values are close to MaxRealNumber or MinRealNumber. Use more robust BarycentricDiff1() subroutine in such cases. -- ALGLIB -- Copyright 17.08.2009 by Bochkanov Sergey *************************************************************************/ public static void barycentricdiff2(barycentricinterpolant b, double t, ref double f, ref double df, ref double d2f) { double v = 0; double vv = 0; int i = 0; int k = 0; double n0 = 0; double n1 = 0; double n2 = 0; double d0 = 0; double d1 = 0; double d2 = 0; double s0 = 0; double s1 = 0; double s2 = 0; double xk = 0; double xi = 0; f = 0; df = 0; d2f = 0; ap.assert(!Double.IsInfinity(t), "BarycentricDiff1: infinite T!"); // // special case: NaN // if( Double.IsNaN(t) ) { f = Double.NaN; df = Double.NaN; d2f = Double.NaN; return; } // // special case: N=1 // if( b.n==1 ) { f = b.sy*b.y[0]; df = 0; d2f = 0; return; } if( (double)(b.sy)==(double)(0) ) { f = 0; df = 0; d2f = 0; return; } // // We assume than N>1 and B.SY>0. Find: // 1. pivot point (X[i] closest to T) // 2. width of interval containing X[i] // ap.assert((double)(b.sy)>(double)(0), "BarycentricDiff: internal error"); f = 0; df = 0; d2f = 0; v = Math.Abs(b.x[0]-t); k = 0; for(i=1; i<=b.n-1; i++) { vv = b.x[i]; if( (double)(Math.Abs(vv-t))<(double)(v) ) { v = Math.Abs(vv-t); k = i; } } // // pivot point found, calculate dNumerator and dDenominator // xk = b.x[k]; v = t-xk; n0 = 0; n1 = 0; n2 = 0; d0 = 0; d1 = 0; d2 = 0; for(i=0; i<=b.n-1; i++) { if( i!=k ) { xi = b.x[i]; vv = math.sqr(t-xi); s0 = (t-xk)/(t-xi); s1 = (xk-xi)/vv; s2 = -(2*(xk-xi)/(vv*(t-xi))); } else { s0 = 1; s1 = 0; s2 = 0; } vv = b.w[i]*b.y[i]; n0 = n0+s0*vv; n1 = n1+s1*vv; n2 = n2+s2*vv; vv = b.w[i]; d0 = d0+s0*vv; d1 = d1+s1*vv; d2 = d2+s2*vv; } f = b.sy*n0/d0; df = b.sy*(n1*d0-n0*d1)/math.sqr(d0); d2f = b.sy*((n2*d0-n0*d2)*math.sqr(d0)-(n1*d0-n0*d1)*2*d0*d1)/math.sqr(math.sqr(d0)); } /************************************************************************* This subroutine performs linear transformation of the argument. INPUT PARAMETERS: B - rational interpolant in barycentric form CA, CB - transformation coefficients: x = CA*t + CB OUTPUT PARAMETERS: B - transformed interpolant with X replaced by T -- ALGLIB PROJECT -- Copyright 19.08.2009 by Bochkanov Sergey *************************************************************************/ public static void barycentriclintransx(barycentricinterpolant b, double ca, double cb) { int i = 0; int j = 0; double v = 0; // // special case, replace by constant F(CB) // if( (double)(ca)==(double)(0) ) { b.sy = barycentriccalc(b, cb); v = 1; for(i=0; i<=b.n-1; i++) { b.y[i] = 1; b.w[i] = v; v = -v; } return; } // // general case: CA<>0 // for(i=0; i<=b.n-1; i++) { b.x[i] = (b.x[i]-cb)/ca; } if( (double)(ca)<(double)(0) ) { for(i=0; i<=b.n-1; i++) { if( i(double)(0) ) { v = 1/b.sy; for(i_=0; i_<=b.n-1;i_++) { b.y[i_] = v*b.y[i_]; } } } /************************************************************************* Extracts X/Y/W arrays from rational interpolant INPUT PARAMETERS: B - barycentric interpolant OUTPUT PARAMETERS: N - nodes count, N>0 X - interpolation nodes, array[0..N-1] F - function values, array[0..N-1] W - barycentric weights, array[0..N-1] -- ALGLIB -- Copyright 17.08.2009 by Bochkanov Sergey *************************************************************************/ public static void barycentricunpack(barycentricinterpolant b, ref int n, ref double[] x, ref double[] y, ref double[] w) { double v = 0; int i_ = 0; n = 0; x = new double[0]; y = new double[0]; w = new double[0]; n = b.n; x = new double[n]; y = new double[n]; w = new double[n]; v = b.sy; for(i_=0; i_<=n-1;i_++) { x[i_] = b.x[i_]; } for(i_=0; i_<=n-1;i_++) { y[i_] = v*b.y[i_]; } for(i_=0; i_<=n-1;i_++) { w[i_] = b.w[i_]; } } /************************************************************************* Rational interpolant from X/Y/W arrays F(t) = SUM(i=0,n-1,w[i]*f[i]/(t-x[i])) / SUM(i=0,n-1,w[i]/(t-x[i])) INPUT PARAMETERS: X - interpolation nodes, array[0..N-1] F - function values, array[0..N-1] W - barycentric weights, array[0..N-1] N - nodes count, N>0 OUTPUT PARAMETERS: B - barycentric interpolant built from (X, Y, W) -- ALGLIB -- Copyright 17.08.2009 by Bochkanov Sergey *************************************************************************/ public static void barycentricbuildxyw(double[] x, double[] y, double[] w, int n, barycentricinterpolant b) { int i_ = 0; ap.assert(n>0, "BarycentricBuildXYW: incorrect N!"); // // fill X/Y/W // b.x = new double[n]; b.y = new double[n]; b.w = new double[n]; for(i_=0; i_<=n-1;i_++) { b.x[i_] = x[i_]; } for(i_=0; i_<=n-1;i_++) { b.y[i_] = y[i_]; } for(i_=0; i_<=n-1;i_++) { b.w[i_] = w[i_]; } b.n = n; // // Normalize // barycentricnormalize(b); } /************************************************************************* Rational interpolant without poles The subroutine constructs the rational interpolating function without real poles (see 'Barycentric rational interpolation with no poles and high rates of approximation', Michael S. Floater. and Kai Hormann, for more information on this subject). Input parameters: X - interpolation nodes, array[0..N-1]. Y - function values, array[0..N-1]. N - number of nodes, N>0. D - order of the interpolation scheme, 0 <= D <= N-1. D<0 will cause an error. D>=N it will be replaced with D=N-1. if you don't know what D to choose, use small value about 3-5. Output parameters: B - barycentric interpolant. Note: this algorithm always succeeds and calculates the weights with close to machine precision. -- ALGLIB PROJECT -- Copyright 17.06.2007 by Bochkanov Sergey *************************************************************************/ public static void barycentricbuildfloaterhormann(double[] x, double[] y, int n, int d, barycentricinterpolant b) { double s0 = 0; double s = 0; double v = 0; int i = 0; int j = 0; int k = 0; int[] perm = new int[0]; double[] wtemp = new double[0]; double[] sortrbuf = new double[0]; double[] sortrbuf2 = new double[0]; int i_ = 0; ap.assert(n>0, "BarycentricFloaterHormann: N<=0!"); ap.assert(d>=0, "BarycentricFloaterHormann: incorrect D!"); // // Prepare // if( d>n-1 ) { d = n-1; } b.n = n; // // special case: N=1 // if( n==1 ) { b.x = new double[n]; b.y = new double[n]; b.w = new double[n]; b.x[0] = x[0]; b.y[0] = y[0]; b.w[0] = 1; barycentricnormalize(b); return; } // // Fill X/Y // b.x = new double[n]; b.y = new double[n]; for(i_=0; i_<=n-1;i_++) { b.x[i_] = x[i_]; } for(i_=0; i_<=n-1;i_++) { b.y[i_] = y[i_]; } tsort.tagsortfastr(ref b.x, ref b.y, ref sortrbuf, ref sortrbuf2, n); // // Calculate Wk // b.w = new double[n]; s0 = 1; for(k=1; k<=d; k++) { s0 = -s0; } for(k=0; k<=n-1; k++) { // // Wk // s = 0; for(i=Math.Max(k-d, 0); i<=Math.Min(k, n-1-d); i++) { v = 1; for(j=i; j<=i+d; j++) { if( j!=k ) { v = v/Math.Abs(b.x[k]-b.x[j]); } } s = s+v; } b.w[k] = s0*s; // // Next S0 // s0 = -s0; } // // Normalize // barycentricnormalize(b); } /************************************************************************* Copying of the barycentric interpolant (for internal use only) INPUT PARAMETERS: B - barycentric interpolant OUTPUT PARAMETERS: B2 - copy(B1) -- ALGLIB -- Copyright 17.08.2009 by Bochkanov Sergey *************************************************************************/ public static void barycentriccopy(barycentricinterpolant b, barycentricinterpolant b2) { int i_ = 0; b2.n = b.n; b2.sy = b.sy; b2.x = new double[b2.n]; b2.y = new double[b2.n]; b2.w = new double[b2.n]; for(i_=0; i_<=b2.n-1;i_++) { b2.x[i_] = b.x[i_]; } for(i_=0; i_<=b2.n-1;i_++) { b2.y[i_] = b.y[i_]; } for(i_=0; i_<=b2.n-1;i_++) { b2.w[i_] = b.w[i_]; } } /************************************************************************* Normalization of barycentric interpolant: * B.N, B.X, B.Y and B.W are initialized * B.SY is NOT initialized * Y[] is normalized, scaling coefficient is stored in B.SY * W[] is normalized, no scaling coefficient is stored * X[] is sorted Internal subroutine. *************************************************************************/ private static void barycentricnormalize(barycentricinterpolant b) { int[] p1 = new int[0]; int[] p2 = new int[0]; int i = 0; int j = 0; int j2 = 0; double v = 0; int i_ = 0; // // Normalize task: |Y|<=1, |W|<=1, sort X[] // b.sy = 0; for(i=0; i<=b.n-1; i++) { b.sy = Math.Max(b.sy, Math.Abs(b.y[i])); } if( (double)(b.sy)>(double)(0) & (double)(Math.Abs(b.sy-1))>(double)(10*math.machineepsilon) ) { v = 1/b.sy; for(i_=0; i_<=b.n-1;i_++) { b.y[i_] = v*b.y[i_]; } } v = 0; for(i=0; i<=b.n-1; i++) { v = Math.Max(v, Math.Abs(b.w[i])); } if( (double)(v)>(double)(0) & (double)(Math.Abs(v-1))>(double)(10*math.machineepsilon) ) { v = 1/v; for(i_=0; i_<=b.n-1;i_++) { b.w[i_] = v*b.w[i_]; } } for(i=0; i<=b.n-2; i++) { if( (double)(b.x[i+1])<(double)(b.x[i]) ) { tsort.tagsort(ref b.x, b.n, ref p1, ref p2); for(j=0; j<=b.n-1; j++) { j2 = p2[j]; v = b.y[j]; b.y[j] = b.y[j2]; b.y[j2] = v; v = b.w[j]; b.w[j] = b.w[j2]; b.w[j2] = v; } break; } } } } public class polint { /************************************************************************* Conversion from barycentric representation to Chebyshev basis. This function has O(N^2) complexity. INPUT PARAMETERS: P - polynomial in barycentric form A,B - base interval for Chebyshev polynomials (see below) A<>B OUTPUT PARAMETERS T - coefficients of Chebyshev representation; P(x) = sum { T[i]*Ti(2*(x-A)/(B-A)-1), i=0..N-1 }, where Ti - I-th Chebyshev polynomial. NOTES: barycentric interpolant passed as P may be either polynomial obtained from polynomial interpolation/ fitting or rational function which is NOT polynomial. We can't distinguish between these two cases, and this algorithm just tries to work assuming that P IS a polynomial. If not, algorithm will return results, but they won't have any meaning. -- ALGLIB -- Copyright 30.09.2010 by Bochkanov Sergey *************************************************************************/ public static void polynomialbar2cheb(ratint.barycentricinterpolant p, double a, double b, ref double[] t) { int i = 0; int k = 0; double[] vp = new double[0]; double[] vx = new double[0]; double[] tk = new double[0]; double[] tk1 = new double[0]; double v = 0; int i_ = 0; t = new double[0]; ap.assert(math.isfinite(a), "PolynomialBar2Cheb: A is not finite!"); ap.assert(math.isfinite(b), "PolynomialBar2Cheb: B is not finite!"); ap.assert((double)(a)!=(double)(b), "PolynomialBar2Cheb: A=B!"); ap.assert(p.n>0, "PolynomialBar2Cheb: P is not correctly initialized barycentric interpolant!"); // // Calculate function values on a Chebyshev grid // vp = new double[p.n]; vx = new double[p.n]; for(i=0; i<=p.n-1; i++) { vx[i] = Math.Cos(Math.PI*(i+0.5)/p.n); vp[i] = ratint.barycentriccalc(p, 0.5*(vx[i]+1)*(b-a)+a); } // // T[0] // t = new double[p.n]; v = 0; for(i=0; i<=p.n-1; i++) { v = v+vp[i]; } t[0] = v/p.n; // // other T's. // // NOTES: // 1. TK stores T{k} on VX, TK1 stores T{k-1} on VX // 2. we can do same calculations with fast DCT, but it // * adds dependencies // * still leaves us with O(N^2) algorithm because // preparation of function values is O(N^2) process // if( p.n>1 ) { tk = new double[p.n]; tk1 = new double[p.n]; for(i=0; i<=p.n-1; i++) { tk[i] = vx[i]; tk1[i] = 1; } for(k=1; k<=p.n-1; k++) { // // calculate discrete product of function vector and TK // v = 0.0; for(i_=0; i_<=p.n-1;i_++) { v += tk[i_]*vp[i_]; } t[k] = v/(0.5*p.n); // // Update TK and TK1 // for(i=0; i<=p.n-1; i++) { v = 2*vx[i]*tk[i]-tk1[i]; tk1[i] = tk[i]; tk[i] = v; } } } } /************************************************************************* Conversion from Chebyshev basis to barycentric representation. This function has O(N^2) complexity. INPUT PARAMETERS: T - coefficients of Chebyshev representation; P(x) = sum { T[i]*Ti(2*(x-A)/(B-A)-1), i=0..N }, where Ti - I-th Chebyshev polynomial. N - number of coefficients: * if given, only leading N elements of T are used * if not given, automatically determined from size of T A,B - base interval for Chebyshev polynomials (see above) A=1, "PolynomialBar2Cheb: N<1"); ap.assert(ap.len(t)>=n, "PolynomialBar2Cheb: Length(T)0. OUTPUT PARAMETERS A - coefficients, P(x) = sum { A[i]*((X-C)/S)^i, i=0..N-1 } N - number of coefficients (polynomial degree plus 1) NOTES: 1. this function accepts offset and scale, which can be set to improve numerical properties of polynomial. For example, if P was obtained as result of interpolation on [-1,+1], you can set C=0 and S=1 and represent P as sum of 1, x, x^2, x^3 and so on. In most cases you it is exactly what you need. However, if your interpolation model was built on [999,1001], you will see significant growth of numerical errors when using {1, x, x^2, x^3} as basis. Representing P as sum of 1, (x-1000), (x-1000)^2, (x-1000)^3 will be better option. Such representation can be obtained by using 1000.0 as offset C and 1.0 as scale S. 2. power basis is ill-conditioned and tricks described above can't solve this problem completely. This function will return coefficients in any case, but for N>8 they will become unreliable. However, N's less than 5 are pretty safe. 3. barycentric interpolant passed as P may be either polynomial obtained from polynomial interpolation/ fitting or rational function which is NOT polynomial. We can't distinguish between these two cases, and this algorithm just tries to work assuming that P IS a polynomial. If not, algorithm will return results, but they won't have any meaning. -- ALGLIB -- Copyright 30.09.2010 by Bochkanov Sergey *************************************************************************/ public static void polynomialbar2pow(ratint.barycentricinterpolant p, double c, double s, ref double[] a) { int i = 0; int k = 0; double e = 0; double d = 0; double[] vp = new double[0]; double[] vx = new double[0]; double[] tk = new double[0]; double[] tk1 = new double[0]; double[] t = new double[0]; double v = 0; int i_ = 0; a = new double[0]; ap.assert(math.isfinite(c), "PolynomialBar2Pow: C is not finite!"); ap.assert(math.isfinite(s), "PolynomialBar2Pow: S is not finite!"); ap.assert((double)(s)!=(double)(0), "PolynomialBar2Pow: S=0!"); ap.assert(p.n>0, "PolynomialBar2Pow: P is not correctly initialized barycentric interpolant!"); // // Calculate function values on a Chebyshev grid // vp = new double[p.n]; vx = new double[p.n]; for(i=0; i<=p.n-1; i++) { vx[i] = Math.Cos(Math.PI*(i+0.5)/p.n); vp[i] = ratint.barycentriccalc(p, s*vx[i]+c); } // // T[0] // t = new double[p.n]; v = 0; for(i=0; i<=p.n-1; i++) { v = v+vp[i]; } t[0] = v/p.n; // // other T's. // // NOTES: // 1. TK stores T{k} on VX, TK1 stores T{k-1} on VX // 2. we can do same calculations with fast DCT, but it // * adds dependencies // * still leaves us with O(N^2) algorithm because // preparation of function values is O(N^2) process // if( p.n>1 ) { tk = new double[p.n]; tk1 = new double[p.n]; for(i=0; i<=p.n-1; i++) { tk[i] = vx[i]; tk1[i] = 1; } for(k=1; k<=p.n-1; k++) { // // calculate discrete product of function vector and TK // v = 0.0; for(i_=0; i_<=p.n-1;i_++) { v += tk[i_]*vp[i_]; } t[k] = v/(0.5*p.n); // // Update TK and TK1 // for(i=0; i<=p.n-1; i++) { v = 2*vx[i]*tk[i]-tk1[i]; tk1[i] = tk[i]; tk[i] = v; } } } // // Convert from Chebyshev basis to power basis // a = new double[p.n]; for(i=0; i<=p.n-1; i++) { a[i] = 0; } d = 0; for(i=0; i<=p.n-1; i++) { for(k=i; k<=p.n-1; k++) { e = a[k]; a[k] = 0; if( i<=1 & k==i ) { a[k] = 1; } else { if( i!=0 ) { a[k] = 2*d; } if( k>i+1 ) { a[k] = a[k]-a[k-2]; } } d = e; } d = a[i]; e = 0; k = i; while( k<=p.n-1 ) { e = e+a[k]*t[k]; k = k+2; } a[i] = e; } } /************************************************************************* Conversion from power basis to barycentric representation. This function has O(N^2) complexity. INPUT PARAMETERS: A - coefficients, P(x) = sum { A[i]*((X-C)/S)^i, i=0..N-1 } N - number of coefficients (polynomial degree plus 1) * if given, only leading N elements of A are used * if not given, automatically determined from size of A C - offset (see below); 0.0 is used as default value. S - scale (see below); 1.0 is used as default value. S<>0. OUTPUT PARAMETERS P - polynomial in barycentric form NOTES: 1. this function accepts offset and scale, which can be set to improve numerical properties of polynomial. For example, if you interpolate on [-1,+1], you can set C=0 and S=1 and convert from sum of 1, x, x^2, x^3 and so on. In most cases you it is exactly what you need. However, if your interpolation model was built on [999,1001], you will see significant growth of numerical errors when using {1, x, x^2, x^3} as input basis. Converting from sum of 1, (x-1000), (x-1000)^2, (x-1000)^3 will be better option (you have to specify 1000.0 as offset C and 1.0 as scale S). 2. power basis is ill-conditioned and tricks described above can't solve this problem completely. This function will return barycentric model in any case, but for N>8 accuracy well degrade. However, N's less than 5 are pretty safe. -- ALGLIB -- Copyright 30.09.2010 by Bochkanov Sergey *************************************************************************/ public static void polynomialpow2bar(double[] a, int n, double c, double s, ratint.barycentricinterpolant p) { int i = 0; int k = 0; double[] y = new double[0]; double vx = 0; double vy = 0; double px = 0; ap.assert(math.isfinite(c), "PolynomialPow2Bar: C is not finite!"); ap.assert(math.isfinite(s), "PolynomialPow2Bar: S is not finite!"); ap.assert((double)(s)!=(double)(0), "PolynomialPow2Bar: S is zero!"); ap.assert(n>=1, "PolynomialPow2Bar: N<1"); ap.assert(ap.len(a)>=n, "PolynomialPow2Bar: Length(A)=1 OUTPUT PARAMETERS P - barycentric model which represents Lagrange interpolant (see ratint unit info and BarycentricCalc() description for more information). -- ALGLIB -- Copyright 02.12.2009 by Bochkanov Sergey *************************************************************************/ public static void polynomialbuild(double[] x, double[] y, int n, ratint.barycentricinterpolant p) { int j = 0; int k = 0; double[] w = new double[0]; double b = 0; double a = 0; double v = 0; double mx = 0; double[] sortrbuf = new double[0]; double[] sortrbuf2 = new double[0]; int i_ = 0; x = (double[])x.Clone(); y = (double[])y.Clone(); ap.assert(n>0, "PolynomialBuild: N<=0!"); ap.assert(ap.len(x)>=n, "PolynomialBuild: Length(X)=n, "PolynomialBuild: Length(Y)=1 for N=1 a constant model is constructed. OUTPUT PARAMETERS P - barycentric model which represents Lagrange interpolant (see ratint unit info and BarycentricCalc() description for more information). -- ALGLIB -- Copyright 03.12.2009 by Bochkanov Sergey *************************************************************************/ public static void polynomialbuildeqdist(double a, double b, double[] y, int n, ratint.barycentricinterpolant p) { int i = 0; double[] w = new double[0]; double[] x = new double[0]; double v = 0; ap.assert(n>0, "PolynomialBuildEqDist: N<=0!"); ap.assert(ap.len(y)>=n, "PolynomialBuildEqDist: Length(Y)=1 for N=1 a constant model is constructed. OUTPUT PARAMETERS P - barycentric model which represents Lagrange interpolant (see ratint unit info and BarycentricCalc() description for more information). -- ALGLIB -- Copyright 03.12.2009 by Bochkanov Sergey *************************************************************************/ public static void polynomialbuildcheb1(double a, double b, double[] y, int n, ratint.barycentricinterpolant p) { int i = 0; double[] w = new double[0]; double[] x = new double[0]; double v = 0; double t = 0; ap.assert(n>0, "PolynomialBuildCheb1: N<=0!"); ap.assert(ap.len(y)>=n, "PolynomialBuildCheb1: Length(Y)=1 for N=1 a constant model is constructed. OUTPUT PARAMETERS P - barycentric model which represents Lagrange interpolant (see ratint unit info and BarycentricCalc() description for more information). -- ALGLIB -- Copyright 03.12.2009 by Bochkanov Sergey *************************************************************************/ public static void polynomialbuildcheb2(double a, double b, double[] y, int n, ratint.barycentricinterpolant p) { int i = 0; double[] w = new double[0]; double[] x = new double[0]; double v = 0; ap.assert(n>0, "PolynomialBuildCheb2: N<=0!"); ap.assert(ap.len(y)>=n, "PolynomialBuildCheb2: Length(Y)=1 for N=1 a constant model is constructed. T - position where P(x) is calculated RESULT value of the Lagrange interpolant at T IMPORTANT this function provides fast interface which is not overflow-safe nor it is very precise. the best option is to use PolynomialBuildEqDist()/BarycentricCalc() subroutines unless you are pretty sure that your data will not result in overflow. -- ALGLIB -- Copyright 02.12.2009 by Bochkanov Sergey *************************************************************************/ public static double polynomialcalceqdist(double a, double b, double[] f, int n, double t) { double result = 0; double s1 = 0; double s2 = 0; double v = 0; double threshold = 0; double s = 0; double h = 0; int i = 0; int j = 0; double w = 0; double x = 0; ap.assert(n>0, "PolynomialCalcEqDist: N<=0!"); ap.assert(ap.len(f)>=n, "PolynomialCalcEqDist: Length(F)(double)(threshold) ) { // // use fast formula // j = -1; s = 1.0; } // // Calculate using safe or fast barycentric formula // s1 = 0; s2 = 0; w = 1.0; h = (b-a)/(n-1); for(i=0; i<=n-1; i++) { if( i!=j ) { v = s*w/(t-(a+i*h)); s1 = s1+v*f[i]; s2 = s2+v; } else { v = w; s1 = s1+v*f[i]; s2 = s2+v; } w = -(w*(n-1-i)); w = w/(i+1); } result = s1/s2; return result; } /************************************************************************* Fast polynomial interpolation function on Chebyshev points (first kind) with O(N) complexity. INPUT PARAMETERS: A - left boundary of [A,B] B - right boundary of [A,B] F - function values, array[0..N-1] N - number of points on Chebyshev grid (first kind), X[i] = 0.5*(B+A) + 0.5*(B-A)*Cos(PI*(2*i+1)/(2*n)) for N=1 a constant model is constructed. T - position where P(x) is calculated RESULT value of the Lagrange interpolant at T IMPORTANT this function provides fast interface which is not overflow-safe nor it is very precise. the best option is to use PolIntBuildCheb1()/BarycentricCalc() subroutines unless you are pretty sure that your data will not result in overflow. -- ALGLIB -- Copyright 02.12.2009 by Bochkanov Sergey *************************************************************************/ public static double polynomialcalccheb1(double a, double b, double[] f, int n, double t) { double result = 0; double s1 = 0; double s2 = 0; double v = 0; double threshold = 0; double s = 0; int i = 0; int j = 0; double a0 = 0; double delta = 0; double alpha = 0; double beta = 0; double ca = 0; double sa = 0; double tempc = 0; double temps = 0; double x = 0; double w = 0; double p1 = 0; ap.assert(n>0, "PolynomialCalcCheb1: N<=0!"); ap.assert(ap.len(f)>=n, "PolynomialCalcCheb1: Length(F)(double)(threshold) ) { // // use fast formula // j = -1; s = 1.0; } // // Calculate using safe or fast barycentric formula // s1 = 0; s2 = 0; ca = Math.Cos(a0); sa = Math.Sin(a0); p1 = 1.0; for(i=0; i<=n-1; i++) { // // Calculate X[i], W[i] // x = ca; w = p1*sa; // // Proceed // if( i!=j ) { v = s*w/(t-x); s1 = s1+v*f[i]; s2 = s2+v; } else { v = w; s1 = s1+v*f[i]; s2 = s2+v; } // // Next CA, SA, P1 // temps = sa-(alpha*sa-beta*ca); tempc = ca-(alpha*ca+beta*sa); sa = temps; ca = tempc; p1 = -p1; } result = s1/s2; return result; } /************************************************************************* Fast polynomial interpolation function on Chebyshev points (second kind) with O(N) complexity. INPUT PARAMETERS: A - left boundary of [A,B] B - right boundary of [A,B] F - function values, array[0..N-1] N - number of points on Chebyshev grid (second kind), X[i] = 0.5*(B+A) + 0.5*(B-A)*Cos(PI*i/(n-1)) for N=1 a constant model is constructed. T - position where P(x) is calculated RESULT value of the Lagrange interpolant at T IMPORTANT this function provides fast interface which is not overflow-safe nor it is very precise. the best option is to use PolIntBuildCheb2()/BarycentricCalc() subroutines unless you are pretty sure that your data will not result in overflow. -- ALGLIB -- Copyright 02.12.2009 by Bochkanov Sergey *************************************************************************/ public static double polynomialcalccheb2(double a, double b, double[] f, int n, double t) { double result = 0; double s1 = 0; double s2 = 0; double v = 0; double threshold = 0; double s = 0; int i = 0; int j = 0; double a0 = 0; double delta = 0; double alpha = 0; double beta = 0; double ca = 0; double sa = 0; double tempc = 0; double temps = 0; double x = 0; double w = 0; double p1 = 0; ap.assert(n>0, "PolynomialCalcCheb2: N<=0!"); ap.assert(ap.len(f)>=n, "PolynomialCalcCheb2: Length(F)(double)(threshold) ) { // // use fast formula // j = -1; s = 1.0; } // // Calculate using safe or fast barycentric formula // s1 = 0; s2 = 0; ca = Math.Cos(a0); sa = Math.Sin(a0); p1 = 1.0; for(i=0; i<=n-1; i++) { // // Calculate X[i], W[i] // x = ca; if( i==0 | i==n-1 ) { w = 0.5*p1; } else { w = 1.0*p1; } // // Proceed // if( i!=j ) { v = s*w/(t-x); s1 = s1+v*f[i]; s2 = s2+v; } else { v = w; s1 = s1+v*f[i]; s2 = s2+v; } // // Next CA, SA, P1 // temps = sa-(alpha*sa-beta*ca); tempc = ca-(alpha*ca+beta*sa); sa = temps; ca = tempc; p1 = -p1; } result = s1/s2; return result; } } public class spline1d { /************************************************************************* 1-dimensional spline inteprolant *************************************************************************/ public class spline1dinterpolant { public bool periodic; public int n; public int k; public double[] x; public double[] c; public spline1dinterpolant() { x = new double[0]; c = new double[0]; } }; /************************************************************************* This subroutine builds linear spline interpolant INPUT PARAMETERS: X - spline nodes, array[0..N-1] Y - function values, array[0..N-1] N - points count (optional): * N>=2 * if given, only first N points are used to build spline * if not given, automatically detected from X/Y sizes (len(X) must be equal to len(Y)) OUTPUT PARAMETERS: C - spline interpolant ORDER OF POINTS Subroutine automatically sorts points, so caller may pass unsorted array. -- ALGLIB PROJECT -- Copyright 24.06.2007 by Bochkanov Sergey *************************************************************************/ public static void spline1dbuildlinear(double[] x, double[] y, int n, spline1dinterpolant c) { int i = 0; x = (double[])x.Clone(); y = (double[])y.Clone(); ap.assert(n>1, "Spline1DBuildLinear: N<2!"); ap.assert(ap.len(x)>=n, "Spline1DBuildLinear: Length(X)=n, "Spline1DBuildLinear: Length(Y)=2 * if given, only first N points are used to build spline * if not given, automatically detected from X/Y sizes (len(X) must be equal to len(Y)) BoundLType - boundary condition type for the left boundary BoundL - left boundary condition (first or second derivative, depending on the BoundLType) BoundRType - boundary condition type for the right boundary BoundR - right boundary condition (first or second derivative, depending on the BoundRType) OUTPUT PARAMETERS: C - spline interpolant ORDER OF POINTS Subroutine automatically sorts points, so caller may pass unsorted array. SETTING BOUNDARY VALUES: The BoundLType/BoundRType parameters can have the following values: * -1, which corresonds to the periodic (cyclic) boundary conditions. In this case: * both BoundLType and BoundRType must be equal to -1. * BoundL/BoundR are ignored * Y[last] is ignored (it is assumed to be equal to Y[first]). * 0, which corresponds to the parabolically terminated spline (BoundL and/or BoundR are ignored). * 1, which corresponds to the first derivative boundary condition * 2, which corresponds to the second derivative boundary condition * by default, BoundType=0 is used PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS: Problems with periodic boundary conditions have Y[first_point]=Y[last_point]. However, this subroutine doesn't require you to specify equal values for the first and last points - it automatically forces them to be equal by copying Y[first_point] (corresponds to the leftmost, minimal X[]) to Y[last_point]. However it is recommended to pass consistent values of Y[], i.e. to make Y[first_point]=Y[last_point]. -- ALGLIB PROJECT -- Copyright 23.06.2007 by Bochkanov Sergey *************************************************************************/ public static void spline1dbuildcubic(double[] x, double[] y, int n, int boundltype, double boundl, int boundrtype, double boundr, spline1dinterpolant c) { double[] a1 = new double[0]; double[] a2 = new double[0]; double[] a3 = new double[0]; double[] b = new double[0]; double[] dt = new double[0]; double[] d = new double[0]; int[] p = new int[0]; int ylen = 0; x = (double[])x.Clone(); y = (double[])y.Clone(); // // check correctness of boundary conditions // ap.assert(((boundltype==-1 | boundltype==0) | boundltype==1) | boundltype==2, "Spline1DBuildCubic: incorrect BoundLType!"); ap.assert(((boundrtype==-1 | boundrtype==0) | boundrtype==1) | boundrtype==2, "Spline1DBuildCubic: incorrect BoundRType!"); ap.assert((boundrtype==-1 & boundltype==-1) | (boundrtype!=-1 & boundltype!=-1), "Spline1DBuildCubic: incorrect BoundLType/BoundRType!"); if( boundltype==1 | boundltype==2 ) { ap.assert(math.isfinite(boundl), "Spline1DBuildCubic: BoundL is infinite or NAN!"); } if( boundrtype==1 | boundrtype==2 ) { ap.assert(math.isfinite(boundr), "Spline1DBuildCubic: BoundR is infinite or NAN!"); } // // check lengths of arguments // ap.assert(n>=2, "Spline1DBuildCubic: N<2!"); ap.assert(ap.len(x)>=n, "Spline1DBuildCubic: Length(X)=n, "Spline1DBuildCubic: Length(Y)=2 * if given, only first N points are used * if not given, automatically detected from X/Y sizes (len(X) must be equal to len(Y)) BoundLType - boundary condition type for the left boundary BoundL - left boundary condition (first or second derivative, depending on the BoundLType) BoundRType - boundary condition type for the right boundary BoundR - right boundary condition (first or second derivative, depending on the BoundRType) OUTPUT PARAMETERS: D - derivative values at X[] ORDER OF POINTS Subroutine automatically sorts points, so caller may pass unsorted array. Derivative values are correctly reordered on return, so D[I] is always equal to S'(X[I]) independently of points order. SETTING BOUNDARY VALUES: The BoundLType/BoundRType parameters can have the following values: * -1, which corresonds to the periodic (cyclic) boundary conditions. In this case: * both BoundLType and BoundRType must be equal to -1. * BoundL/BoundR are ignored * Y[last] is ignored (it is assumed to be equal to Y[first]). * 0, which corresponds to the parabolically terminated spline (BoundL and/or BoundR are ignored). * 1, which corresponds to the first derivative boundary condition * 2, which corresponds to the second derivative boundary condition * by default, BoundType=0 is used PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS: Problems with periodic boundary conditions have Y[first_point]=Y[last_point]. However, this subroutine doesn't require you to specify equal values for the first and last points - it automatically forces them to be equal by copying Y[first_point] (corresponds to the leftmost, minimal X[]) to Y[last_point]. However it is recommended to pass consistent values of Y[], i.e. to make Y[first_point]=Y[last_point]. -- ALGLIB PROJECT -- Copyright 03.09.2010 by Bochkanov Sergey *************************************************************************/ public static void spline1dgriddiffcubic(double[] x, double[] y, int n, int boundltype, double boundl, int boundrtype, double boundr, ref double[] d) { double[] a1 = new double[0]; double[] a2 = new double[0]; double[] a3 = new double[0]; double[] b = new double[0]; double[] dt = new double[0]; int[] p = new int[0]; int i = 0; int ylen = 0; int i_ = 0; x = (double[])x.Clone(); y = (double[])y.Clone(); d = new double[0]; // // check correctness of boundary conditions // ap.assert(((boundltype==-1 | boundltype==0) | boundltype==1) | boundltype==2, "Spline1DGridDiffCubic: incorrect BoundLType!"); ap.assert(((boundrtype==-1 | boundrtype==0) | boundrtype==1) | boundrtype==2, "Spline1DGridDiffCubic: incorrect BoundRType!"); ap.assert((boundrtype==-1 & boundltype==-1) | (boundrtype!=-1 & boundltype!=-1), "Spline1DGridDiffCubic: incorrect BoundLType/BoundRType!"); if( boundltype==1 | boundltype==2 ) { ap.assert(math.isfinite(boundl), "Spline1DGridDiffCubic: BoundL is infinite or NAN!"); } if( boundrtype==1 | boundrtype==2 ) { ap.assert(math.isfinite(boundr), "Spline1DGridDiffCubic: BoundR is infinite or NAN!"); } // // check lengths of arguments // ap.assert(n>=2, "Spline1DGridDiffCubic: N<2!"); ap.assert(ap.len(x)>=n, "Spline1DGridDiffCubic: Length(X)=n, "Spline1DGridDiffCubic: Length(Y)=2 * if given, only first N points are used * if not given, automatically detected from X/Y sizes (len(X) must be equal to len(Y)) BoundLType - boundary condition type for the left boundary BoundL - left boundary condition (first or second derivative, depending on the BoundLType) BoundRType - boundary condition type for the right boundary BoundR - right boundary condition (first or second derivative, depending on the BoundRType) OUTPUT PARAMETERS: D1 - S' values at X[] D2 - S'' values at X[] ORDER OF POINTS Subroutine automatically sorts points, so caller may pass unsorted array. Derivative values are correctly reordered on return, so D[I] is always equal to S'(X[I]) independently of points order. SETTING BOUNDARY VALUES: The BoundLType/BoundRType parameters can have the following values: * -1, which corresonds to the periodic (cyclic) boundary conditions. In this case: * both BoundLType and BoundRType must be equal to -1. * BoundL/BoundR are ignored * Y[last] is ignored (it is assumed to be equal to Y[first]). * 0, which corresponds to the parabolically terminated spline (BoundL and/or BoundR are ignored). * 1, which corresponds to the first derivative boundary condition * 2, which corresponds to the second derivative boundary condition * by default, BoundType=0 is used PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS: Problems with periodic boundary conditions have Y[first_point]=Y[last_point]. However, this subroutine doesn't require you to specify equal values for the first and last points - it automatically forces them to be equal by copying Y[first_point] (corresponds to the leftmost, minimal X[]) to Y[last_point]. However it is recommended to pass consistent values of Y[], i.e. to make Y[first_point]=Y[last_point]. -- ALGLIB PROJECT -- Copyright 03.09.2010 by Bochkanov Sergey *************************************************************************/ public static void spline1dgriddiff2cubic(double[] x, double[] y, int n, int boundltype, double boundl, int boundrtype, double boundr, ref double[] d1, ref double[] d2) { double[] a1 = new double[0]; double[] a2 = new double[0]; double[] a3 = new double[0]; double[] b = new double[0]; double[] dt = new double[0]; int[] p = new int[0]; int i = 0; int ylen = 0; double delta = 0; double delta2 = 0; double delta3 = 0; double s0 = 0; double s1 = 0; double s2 = 0; double s3 = 0; int i_ = 0; x = (double[])x.Clone(); y = (double[])y.Clone(); d1 = new double[0]; d2 = new double[0]; // // check correctness of boundary conditions // ap.assert(((boundltype==-1 | boundltype==0) | boundltype==1) | boundltype==2, "Spline1DGridDiff2Cubic: incorrect BoundLType!"); ap.assert(((boundrtype==-1 | boundrtype==0) | boundrtype==1) | boundrtype==2, "Spline1DGridDiff2Cubic: incorrect BoundRType!"); ap.assert((boundrtype==-1 & boundltype==-1) | (boundrtype!=-1 & boundltype!=-1), "Spline1DGridDiff2Cubic: incorrect BoundLType/BoundRType!"); if( boundltype==1 | boundltype==2 ) { ap.assert(math.isfinite(boundl), "Spline1DGridDiff2Cubic: BoundL is infinite or NAN!"); } if( boundrtype==1 | boundrtype==2 ) { ap.assert(math.isfinite(boundr), "Spline1DGridDiff2Cubic: BoundR is infinite or NAN!"); } // // check lengths of arguments // ap.assert(n>=2, "Spline1DGridDiff2Cubic: N<2!"); ap.assert(ap.len(x)>=n, "Spline1DGridDiff2Cubic: Length(X)=n, "Spline1DGridDiff2Cubic: Length(Y)=2 * if given, only first N points from X/Y are used * if not given, automatically detected from X/Y sizes (len(X) must be equal to len(Y)) BoundLType - boundary condition type for the left boundary BoundL - left boundary condition (first or second derivative, depending on the BoundLType) BoundRType - boundary condition type for the right boundary BoundR - right boundary condition (first or second derivative, depending on the BoundRType) N2 - new points count: * N2>=2 * if given, only first N2 points from X2 are used * if not given, automatically detected from X2 size OUTPUT PARAMETERS: F2 - function values at X2[] ORDER OF POINTS Subroutine automatically sorts points, so caller may pass unsorted array. Function values are correctly reordered on return, so F2[I] is always equal to S(X2[I]) independently of points order. SETTING BOUNDARY VALUES: The BoundLType/BoundRType parameters can have the following values: * -1, which corresonds to the periodic (cyclic) boundary conditions. In this case: * both BoundLType and BoundRType must be equal to -1. * BoundL/BoundR are ignored * Y[last] is ignored (it is assumed to be equal to Y[first]). * 0, which corresponds to the parabolically terminated spline (BoundL and/or BoundR are ignored). * 1, which corresponds to the first derivative boundary condition * 2, which corresponds to the second derivative boundary condition * by default, BoundType=0 is used PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS: Problems with periodic boundary conditions have Y[first_point]=Y[last_point]. However, this subroutine doesn't require you to specify equal values for the first and last points - it automatically forces them to be equal by copying Y[first_point] (corresponds to the leftmost, minimal X[]) to Y[last_point]. However it is recommended to pass consistent values of Y[], i.e. to make Y[first_point]=Y[last_point]. -- ALGLIB PROJECT -- Copyright 03.09.2010 by Bochkanov Sergey *************************************************************************/ public static void spline1dconvcubic(double[] x, double[] y, int n, int boundltype, double boundl, int boundrtype, double boundr, double[] x2, int n2, ref double[] y2) { double[] a1 = new double[0]; double[] a2 = new double[0]; double[] a3 = new double[0]; double[] b = new double[0]; double[] d = new double[0]; double[] dt = new double[0]; double[] d1 = new double[0]; double[] d2 = new double[0]; int[] p = new int[0]; int[] p2 = new int[0]; int i = 0; int ylen = 0; double t = 0; double t2 = 0; int i_ = 0; x = (double[])x.Clone(); y = (double[])y.Clone(); x2 = (double[])x2.Clone(); y2 = new double[0]; // // check correctness of boundary conditions // ap.assert(((boundltype==-1 | boundltype==0) | boundltype==1) | boundltype==2, "Spline1DConvCubic: incorrect BoundLType!"); ap.assert(((boundrtype==-1 | boundrtype==0) | boundrtype==1) | boundrtype==2, "Spline1DConvCubic: incorrect BoundRType!"); ap.assert((boundrtype==-1 & boundltype==-1) | (boundrtype!=-1 & boundltype!=-1), "Spline1DConvCubic: incorrect BoundLType/BoundRType!"); if( boundltype==1 | boundltype==2 ) { ap.assert(math.isfinite(boundl), "Spline1DConvCubic: BoundL is infinite or NAN!"); } if( boundrtype==1 | boundrtype==2 ) { ap.assert(math.isfinite(boundr), "Spline1DConvCubic: BoundR is infinite or NAN!"); } // // check lengths of arguments // ap.assert(n>=2, "Spline1DConvCubic: N<2!"); ap.assert(ap.len(x)>=n, "Spline1DConvCubic: Length(X)=n, "Spline1DConvCubic: Length(Y)=2, "Spline1DConvCubic: N2<2!"); ap.assert(ap.len(x2)>=n2, "Spline1DConvCubic: Length(X2)=n2, "Spline1DConvCubic: internal error!"); for(i=0; i<=n2-1; i++) { dt[p2[i]] = y2[i]; } for(i_=0; i_<=n2-1;i_++) { y2[i_] = dt[i_]; } } /************************************************************************* This function solves following problem: given table y[] of function values at old nodes x[] and new nodes x2[], it calculates and returns table of function values y2[] and derivatives d2[] (calculated at x2[]). This function yields same result as Spline1DBuildCubic() call followed by sequence of Spline1DDiff() calls, but it can be several times faster when called for ordered X[] and X2[]. INPUT PARAMETERS: X - old spline nodes Y - function values X2 - new spline nodes OPTIONAL PARAMETERS: N - points count: * N>=2 * if given, only first N points from X/Y are used * if not given, automatically detected from X/Y sizes (len(X) must be equal to len(Y)) BoundLType - boundary condition type for the left boundary BoundL - left boundary condition (first or second derivative, depending on the BoundLType) BoundRType - boundary condition type for the right boundary BoundR - right boundary condition (first or second derivative, depending on the BoundRType) N2 - new points count: * N2>=2 * if given, only first N2 points from X2 are used * if not given, automatically detected from X2 size OUTPUT PARAMETERS: F2 - function values at X2[] D2 - first derivatives at X2[] ORDER OF POINTS Subroutine automatically sorts points, so caller may pass unsorted array. Function values are correctly reordered on return, so F2[I] is always equal to S(X2[I]) independently of points order. SETTING BOUNDARY VALUES: The BoundLType/BoundRType parameters can have the following values: * -1, which corresonds to the periodic (cyclic) boundary conditions. In this case: * both BoundLType and BoundRType must be equal to -1. * BoundL/BoundR are ignored * Y[last] is ignored (it is assumed to be equal to Y[first]). * 0, which corresponds to the parabolically terminated spline (BoundL and/or BoundR are ignored). * 1, which corresponds to the first derivative boundary condition * 2, which corresponds to the second derivative boundary condition * by default, BoundType=0 is used PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS: Problems with periodic boundary conditions have Y[first_point]=Y[last_point]. However, this subroutine doesn't require you to specify equal values for the first and last points - it automatically forces them to be equal by copying Y[first_point] (corresponds to the leftmost, minimal X[]) to Y[last_point]. However it is recommended to pass consistent values of Y[], i.e. to make Y[first_point]=Y[last_point]. -- ALGLIB PROJECT -- Copyright 03.09.2010 by Bochkanov Sergey *************************************************************************/ public static void spline1dconvdiffcubic(double[] x, double[] y, int n, int boundltype, double boundl, int boundrtype, double boundr, double[] x2, int n2, ref double[] y2, ref double[] d2) { double[] a1 = new double[0]; double[] a2 = new double[0]; double[] a3 = new double[0]; double[] b = new double[0]; double[] d = new double[0]; double[] dt = new double[0]; double[] rt1 = new double[0]; int[] p = new int[0]; int[] p2 = new int[0]; int i = 0; int ylen = 0; double t = 0; double t2 = 0; int i_ = 0; x = (double[])x.Clone(); y = (double[])y.Clone(); x2 = (double[])x2.Clone(); y2 = new double[0]; d2 = new double[0]; // // check correctness of boundary conditions // ap.assert(((boundltype==-1 | boundltype==0) | boundltype==1) | boundltype==2, "Spline1DConvDiffCubic: incorrect BoundLType!"); ap.assert(((boundrtype==-1 | boundrtype==0) | boundrtype==1) | boundrtype==2, "Spline1DConvDiffCubic: incorrect BoundRType!"); ap.assert((boundrtype==-1 & boundltype==-1) | (boundrtype!=-1 & boundltype!=-1), "Spline1DConvDiffCubic: incorrect BoundLType/BoundRType!"); if( boundltype==1 | boundltype==2 ) { ap.assert(math.isfinite(boundl), "Spline1DConvDiffCubic: BoundL is infinite or NAN!"); } if( boundrtype==1 | boundrtype==2 ) { ap.assert(math.isfinite(boundr), "Spline1DConvDiffCubic: BoundR is infinite or NAN!"); } // // check lengths of arguments // ap.assert(n>=2, "Spline1DConvDiffCubic: N<2!"); ap.assert(ap.len(x)>=n, "Spline1DConvDiffCubic: Length(X)=n, "Spline1DConvDiffCubic: Length(Y)=2, "Spline1DConvDiffCubic: N2<2!"); ap.assert(ap.len(x2)>=n2, "Spline1DConvDiffCubic: Length(X2)=n2, "Spline1DConvDiffCubic: internal error!"); for(i=0; i<=n2-1; i++) { dt[p2[i]] = y2[i]; } for(i_=0; i_<=n2-1;i_++) { y2[i_] = dt[i_]; } for(i=0; i<=n2-1; i++) { dt[p2[i]] = d2[i]; } for(i_=0; i_<=n2-1;i_++) { d2[i_] = dt[i_]; } } /************************************************************************* This function solves following problem: given table y[] of function values at old nodes x[] and new nodes x2[], it calculates and returns table of function values y2[], first and second derivatives d2[] and dd2[] (calculated at x2[]). This function yields same result as Spline1DBuildCubic() call followed by sequence of Spline1DDiff() calls, but it can be several times faster when called for ordered X[] and X2[]. INPUT PARAMETERS: X - old spline nodes Y - function values X2 - new spline nodes OPTIONAL PARAMETERS: N - points count: * N>=2 * if given, only first N points from X/Y are used * if not given, automatically detected from X/Y sizes (len(X) must be equal to len(Y)) BoundLType - boundary condition type for the left boundary BoundL - left boundary condition (first or second derivative, depending on the BoundLType) BoundRType - boundary condition type for the right boundary BoundR - right boundary condition (first or second derivative, depending on the BoundRType) N2 - new points count: * N2>=2 * if given, only first N2 points from X2 are used * if not given, automatically detected from X2 size OUTPUT PARAMETERS: F2 - function values at X2[] D2 - first derivatives at X2[] DD2 - second derivatives at X2[] ORDER OF POINTS Subroutine automatically sorts points, so caller may pass unsorted array. Function values are correctly reordered on return, so F2[I] is always equal to S(X2[I]) independently of points order. SETTING BOUNDARY VALUES: The BoundLType/BoundRType parameters can have the following values: * -1, which corresonds to the periodic (cyclic) boundary conditions. In this case: * both BoundLType and BoundRType must be equal to -1. * BoundL/BoundR are ignored * Y[last] is ignored (it is assumed to be equal to Y[first]). * 0, which corresponds to the parabolically terminated spline (BoundL and/or BoundR are ignored). * 1, which corresponds to the first derivative boundary condition * 2, which corresponds to the second derivative boundary condition * by default, BoundType=0 is used PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS: Problems with periodic boundary conditions have Y[first_point]=Y[last_point]. However, this subroutine doesn't require you to specify equal values for the first and last points - it automatically forces them to be equal by copying Y[first_point] (corresponds to the leftmost, minimal X[]) to Y[last_point]. However it is recommended to pass consistent values of Y[], i.e. to make Y[first_point]=Y[last_point]. -- ALGLIB PROJECT -- Copyright 03.09.2010 by Bochkanov Sergey *************************************************************************/ public static void spline1dconvdiff2cubic(double[] x, double[] y, int n, int boundltype, double boundl, int boundrtype, double boundr, double[] x2, int n2, ref double[] y2, ref double[] d2, ref double[] dd2) { double[] a1 = new double[0]; double[] a2 = new double[0]; double[] a3 = new double[0]; double[] b = new double[0]; double[] d = new double[0]; double[] dt = new double[0]; int[] p = new int[0]; int[] p2 = new int[0]; int i = 0; int ylen = 0; double t = 0; double t2 = 0; int i_ = 0; x = (double[])x.Clone(); y = (double[])y.Clone(); x2 = (double[])x2.Clone(); y2 = new double[0]; d2 = new double[0]; dd2 = new double[0]; // // check correctness of boundary conditions // ap.assert(((boundltype==-1 | boundltype==0) | boundltype==1) | boundltype==2, "Spline1DConvDiff2Cubic: incorrect BoundLType!"); ap.assert(((boundrtype==-1 | boundrtype==0) | boundrtype==1) | boundrtype==2, "Spline1DConvDiff2Cubic: incorrect BoundRType!"); ap.assert((boundrtype==-1 & boundltype==-1) | (boundrtype!=-1 & boundltype!=-1), "Spline1DConvDiff2Cubic: incorrect BoundLType/BoundRType!"); if( boundltype==1 | boundltype==2 ) { ap.assert(math.isfinite(boundl), "Spline1DConvDiff2Cubic: BoundL is infinite or NAN!"); } if( boundrtype==1 | boundrtype==2 ) { ap.assert(math.isfinite(boundr), "Spline1DConvDiff2Cubic: BoundR is infinite or NAN!"); } // // check lengths of arguments // ap.assert(n>=2, "Spline1DConvDiff2Cubic: N<2!"); ap.assert(ap.len(x)>=n, "Spline1DConvDiff2Cubic: Length(X)=n, "Spline1DConvDiff2Cubic: Length(Y)=2, "Spline1DConvDiff2Cubic: N2<2!"); ap.assert(ap.len(x2)>=n2, "Spline1DConvDiff2Cubic: Length(X2)=n2, "Spline1DConvDiff2Cubic: internal error!"); for(i=0; i<=n2-1; i++) { dt[p2[i]] = y2[i]; } for(i_=0; i_<=n2-1;i_++) { y2[i_] = dt[i_]; } for(i=0; i<=n2-1; i++) { dt[p2[i]] = d2[i]; } for(i_=0; i_<=n2-1;i_++) { d2[i_] = dt[i_]; } for(i=0; i<=n2-1; i++) { dt[p2[i]] = dd2[i]; } for(i_=0; i_<=n2-1;i_++) { dd2[i_] = dt[i_]; } } /************************************************************************* This subroutine builds Catmull-Rom spline interpolant. INPUT PARAMETERS: X - spline nodes, array[0..N-1]. Y - function values, array[0..N-1]. OPTIONAL PARAMETERS: N - points count: * N>=2 * if given, only first N points are used to build spline * if not given, automatically detected from X/Y sizes (len(X) must be equal to len(Y)) BoundType - boundary condition type: * -1 for periodic boundary condition * 0 for parabolically terminated spline (default) Tension - tension parameter: * tension=0 corresponds to classic Catmull-Rom spline (default) * 0=2, "Spline1DBuildCatmullRom: N<2!"); ap.assert(boundtype==-1 | boundtype==0, "Spline1DBuildCatmullRom: incorrect BoundType!"); ap.assert((double)(tension)>=(double)(0), "Spline1DBuildCatmullRom: Tension<0!"); ap.assert((double)(tension)<=(double)(1), "Spline1DBuildCatmullRom: Tension>1!"); ap.assert(ap.len(x)>=n, "Spline1DBuildCatmullRom: Length(X)=n, "Spline1DBuildCatmullRom: Length(Y)=2 * if given, only first N points are used to build spline * if not given, automatically detected from X/Y sizes (len(X) must be equal to len(Y)) OUTPUT PARAMETERS: C - spline interpolant. ORDER OF POINTS Subroutine automatically sorts points, so caller may pass unsorted array. -- ALGLIB PROJECT -- Copyright 23.06.2007 by Bochkanov Sergey *************************************************************************/ public static void spline1dbuildhermite(double[] x, double[] y, double[] d, int n, spline1dinterpolant c) { int i = 0; double delta = 0; double delta2 = 0; double delta3 = 0; x = (double[])x.Clone(); y = (double[])y.Clone(); d = (double[])d.Clone(); ap.assert(n>=2, "Spline1DBuildHermite: N<2!"); ap.assert(ap.len(x)>=n, "Spline1DBuildHermite: Length(X)=n, "Spline1DBuildHermite: Length(Y)=n, "Spline1DBuildHermite: Length(D)=5 * if given, only first N points are used to build spline * if not given, automatically detected from X/Y sizes (len(X) must be equal to len(Y)) OUTPUT PARAMETERS: C - spline interpolant ORDER OF POINTS Subroutine automatically sorts points, so caller may pass unsorted array. -- ALGLIB PROJECT -- Copyright 24.06.2007 by Bochkanov Sergey *************************************************************************/ public static void spline1dbuildakima(double[] x, double[] y, int n, spline1dinterpolant c) { int i = 0; double[] d = new double[0]; double[] w = new double[0]; double[] diff = new double[0]; x = (double[])x.Clone(); y = (double[])y.Clone(); ap.assert(n>=5, "Spline1DBuildAkima: N<5!"); ap.assert(ap.len(x)>=n, "Spline1DBuildAkima: Length(X)=n, "Spline1DBuildAkima: Length(Y)=x ) { r = m; } else { l = m; } } // // Interpolation // x = x-c.x[l]; m = 4*l; result = c.c[m]+x*(c.c[m+1]+x*(c.c[m+2]+x*c.c[m+3])); return result; } /************************************************************************* This subroutine differentiates the spline. INPUT PARAMETERS: C - spline interpolant. X - point Result: S - S(x) DS - S'(x) D2S - S''(x) -- ALGLIB PROJECT -- Copyright 24.06.2007 by Bochkanov Sergey *************************************************************************/ public static void spline1ddiff(spline1dinterpolant c, double x, ref double s, ref double ds, ref double d2s) { int l = 0; int r = 0; int m = 0; double t = 0; s = 0; ds = 0; d2s = 0; ap.assert(c.k==3, "Spline1DDiff: internal error"); ap.assert(!Double.IsInfinity(x), "Spline1DDiff: infinite X!"); // // special case: NaN // if( Double.IsNaN(x) ) { s = Double.NaN; ds = Double.NaN; d2s = Double.NaN; return; } // // correct if periodic // if( c.periodic ) { apserv.apperiodicmap(ref x, c.x[0], c.x[c.n-1], ref t); } // // Binary search // l = 0; r = c.n-2+1; while( l!=r-1 ) { m = (l+r)/2; if( c.x[m]>=x ) { r = m; } else { l = m; } } // // Differentiation // x = x-c.x[l]; m = 4*l; s = c.c[m]+x*(c.c[m+1]+x*(c.c[m+2]+x*c.c[m+3])); ds = c.c[m+1]+2*x*c.c[m+2]+3*math.sqr(x)*c.c[m+3]; d2s = 2*c.c[m+2]+6*x*c.c[m+3]; } /************************************************************************* This subroutine makes the copy of the spline. INPUT PARAMETERS: C - spline interpolant. Result: CC - spline copy -- ALGLIB PROJECT -- Copyright 29.06.2007 by Bochkanov Sergey *************************************************************************/ public static void spline1dcopy(spline1dinterpolant c, spline1dinterpolant cc) { int i_ = 0; cc.periodic = c.periodic; cc.n = c.n; cc.k = c.k; cc.x = new double[cc.n]; for(i_=0; i_<=cc.n-1;i_++) { cc.x[i_] = c.x[i_]; } cc.c = new double[(cc.k+1)*(cc.n-1)]; for(i_=0; i_<=(cc.k+1)*(cc.n-1)-1;i_++) { cc.c[i_] = c.c[i_]; } } /************************************************************************* This subroutine unpacks the spline into the coefficients table. INPUT PARAMETERS: C - spline interpolant. X - point Result: Tbl - coefficients table, unpacked format, array[0..N-2, 0..5]. For I = 0...N-2: Tbl[I,0] = X[i] Tbl[I,1] = X[i+1] Tbl[I,2] = C0 Tbl[I,3] = C1 Tbl[I,4] = C2 Tbl[I,5] = C3 On [x[i], x[i+1]] spline is equals to: S(x) = C0 + C1*t + C2*t^2 + C3*t^3 t = x-x[i] -- ALGLIB PROJECT -- Copyright 29.06.2007 by Bochkanov Sergey *************************************************************************/ public static void spline1dunpack(spline1dinterpolant c, ref int n, ref double[,] tbl) { int i = 0; int j = 0; n = 0; tbl = new double[0,0]; tbl = new double[c.n-2+1, 2+c.k+1]; n = c.n; // // Fill // for(i=0; i<=n-2; i++) { tbl[i,0] = c.x[i]; tbl[i,1] = c.x[i+1]; for(j=0; j<=c.k; j++) { tbl[i,2+j] = c.c[(c.k+1)*i+j]; } } } /************************************************************************* This subroutine performs linear transformation of the spline argument. INPUT PARAMETERS: C - spline interpolant. A, B- transformation coefficients: x = A*t + B Result: C - transformed spline -- ALGLIB PROJECT -- Copyright 30.06.2007 by Bochkanov Sergey *************************************************************************/ public static void spline1dlintransx(spline1dinterpolant c, double a, double b) { int i = 0; int j = 0; int n = 0; double v = 0; double dv = 0; double d2v = 0; double[] x = new double[0]; double[] y = new double[0]; double[] d = new double[0]; n = c.n; // // Special case: A=0 // if( (double)(a)==(double)(0) ) { v = spline1dcalc(c, b); for(i=0; i<=n-2; i++) { c.c[(c.k+1)*i] = v; for(j=1; j<=c.k; j++) { c.c[(c.k+1)*i+j] = 0; } } return; } // // General case: A<>0. // Unpack, X, Y, dY/dX. // Scale and pack again. // ap.assert(c.k==3, "Spline1DLinTransX: internal error"); x = new double[n-1+1]; y = new double[n-1+1]; d = new double[n-1+1]; for(i=0; i<=n-1; i++) { x[i] = c.x[i]; spline1ddiff(c, x[i], ref v, ref dv, ref d2v); x[i] = (x[i]-b)/a; y[i] = v; d[i] = a*dv; } spline1dbuildhermite(x, y, d, n, c); } /************************************************************************* This subroutine performs linear transformation of the spline. INPUT PARAMETERS: C - spline interpolant. A, B- transformation coefficients: S2(x) = A*S(x) + B Result: C - transformed spline -- ALGLIB PROJECT -- Copyright 30.06.2007 by Bochkanov Sergey *************************************************************************/ public static void spline1dlintransy(spline1dinterpolant c, double a, double b) { int i = 0; int j = 0; int n = 0; n = c.n; for(i=0; i<=n-2; i++) { c.c[(c.k+1)*i] = a*c.c[(c.k+1)*i]+b; for(j=1; j<=c.k; j++) { c.c[(c.k+1)*i+j] = a*c.c[(c.k+1)*i+j]; } } } /************************************************************************* This subroutine integrates the spline. INPUT PARAMETERS: C - spline interpolant. X - right bound of the integration interval [a, x], here 'a' denotes min(x[]) Result: integral(S(t)dt,a,x) -- ALGLIB PROJECT -- Copyright 23.06.2007 by Bochkanov Sergey *************************************************************************/ public static double spline1dintegrate(spline1dinterpolant c, double x) { double result = 0; int n = 0; int i = 0; int j = 0; int l = 0; int r = 0; int m = 0; double w = 0; double v = 0; double t = 0; double intab = 0; double additionalterm = 0; n = c.n; // // Periodic splines require special treatment. We make // following transformation: // // integral(S(t)dt,A,X) = integral(S(t)dt,A,Z)+AdditionalTerm // // here X may lie outside of [A,B], Z lies strictly in [A,B], // AdditionalTerm is equals to integral(S(t)dt,A,B) times some // integer number (may be zero). // if( c.periodic & ((double)(x)<(double)(c.x[0]) | (double)(x)>(double)(c.x[c.n-1])) ) { // // compute integral(S(x)dx,A,B) // intab = 0; for(i=0; i<=c.n-2; i++) { w = c.x[i+1]-c.x[i]; m = (c.k+1)*i; intab = intab+c.c[m]*w; v = w; for(j=1; j<=c.k; j++) { v = v*w; intab = intab+c.c[m+j]*v/(j+1); } } // // map X into [A,B] // apserv.apperiodicmap(ref x, c.x[0], c.x[c.n-1], ref t); additionalterm = t*intab; } else { additionalterm = 0; } // // Binary search in the [ x[0], ..., x[n-2] ] (x[n-1] is not included) // l = 0; r = n-2+1; while( l!=r-1 ) { m = (l+r)/2; if( (double)(c.x[m])>=(double)(x) ) { r = m; } else { l = m; } } // // Integration // result = 0; for(i=0; i<=l-1; i++) { w = c.x[i+1]-c.x[i]; m = (c.k+1)*i; result = result+c.c[m]*w; v = w; for(j=1; j<=c.k; j++) { v = v*w; result = result+c.c[m+j]*v/(j+1); } } w = x-c.x[l]; m = (c.k+1)*l; v = w; result = result+c.c[m]*w; for(j=1; j<=c.k; j++) { v = v*w; result = result+c.c[m+j]*v/(j+1); } result = result+additionalterm; return result; } /************************************************************************* Internal version of Spline1DConvDiff Converts from Hermite spline given by grid XOld to new grid X2 INPUT PARAMETERS: XOld - old grid YOld - values at old grid DOld - first derivative at old grid N - grid size X2 - new grid N2 - new grid size Y - possibly preallocated output array (reallocate if too small) NeedY - do we need Y? D1 - possibly preallocated output array (reallocate if too small) NeedD1 - do we need D1? D2 - possibly preallocated output array (reallocate if too small) NeedD2 - do we need D1? OUTPUT ARRAYS: Y - values, if needed D1 - first derivative, if needed D2 - second derivative, if needed -- ALGLIB PROJECT -- Copyright 03.09.2010 by Bochkanov Sergey *************************************************************************/ public static void spline1dconvdiffinternal(double[] xold, double[] yold, double[] dold, int n, double[] x2, int n2, ref double[] y, bool needy, ref double[] d1, bool needd1, ref double[] d2, bool needd2) { int intervalindex = 0; int pointindex = 0; bool havetoadvance = new bool(); double c0 = 0; double c1 = 0; double c2 = 0; double c3 = 0; double a = 0; double b = 0; double w = 0; double w2 = 0; double w3 = 0; double fa = 0; double fb = 0; double da = 0; double db = 0; double t = 0; // // Prepare space // if( needy & ap.len(y)=n2 ) { break; } t = x2[pointindex]; // // do we need to advance interval? // havetoadvance = false; if( intervalindex==-1 ) { havetoadvance = true; } else { if( intervalindex=(double)(b); } } if( havetoadvance ) { intervalindex = intervalindex+1; a = xold[intervalindex]; b = xold[intervalindex+1]; w = b-a; w2 = w*w; w3 = w*w2; fa = yold[intervalindex]; fb = yold[intervalindex+1]; da = dold[intervalindex]; db = dold[intervalindex+1]; c0 = fa; c1 = da; c2 = (3*(fb-fa)-2*da*w-db*w)/w2; c3 = (2*(fa-fb)+da*w+db*w)/w3; continue; } // // Calculate spline and its derivatives using power basis // t = t-a; if( needy ) { y[pointindex] = c0+t*(c1+t*(c2+t*c3)); } if( needd1 ) { d1[pointindex] = c1+2*t*c2+3*t*t*c3; } if( needd2 ) { d2[pointindex] = 2*c2+6*t*c3; } pointindex = pointindex+1; } } /************************************************************************* Internal subroutine. Heap sort. *************************************************************************/ public static void heapsortdpoints(ref double[] x, ref double[] y, ref double[] d, int n) { double[] rbuf = new double[0]; int[] ibuf = new int[0]; double[] rbuf2 = new double[0]; int[] ibuf2 = new int[0]; int i = 0; int i_ = 0; ibuf = new int[n]; rbuf = new double[n]; for(i=0; i<=n-1; i++) { ibuf[i] = i; } tsort.tagsortfasti(ref x, ref ibuf, ref rbuf2, ref ibuf2, n); for(i=0; i<=n-1; i++) { rbuf[i] = y[ibuf[i]]; } for(i_=0; i_<=n-1;i_++) { y[i_] = rbuf[i_]; } for(i=0; i<=n-1; i++) { rbuf[i] = d[ibuf[i]]; } for(i_=0; i_<=n-1;i_++) { d[i_] = rbuf[i_]; } } /************************************************************************* Internal version of Spline1DGridDiffCubic. Accepts pre-ordered X/Y, temporary arrays (which may be preallocated, if you want to save time, or not) and output array (which may be preallocated too). Y is passed as var-parameter because we may need to force last element to be equal to the first one (if periodic boundary conditions are specified). -- ALGLIB PROJECT -- Copyright 03.09.2010 by Bochkanov Sergey *************************************************************************/ private static void spline1dgriddiffcubicinternal(double[] x, ref double[] y, int n, int boundltype, double boundl, int boundrtype, double boundr, ref double[] d, ref double[] a1, ref double[] a2, ref double[] a3, ref double[] b, ref double[] dt) { int i = 0; int i_ = 0; // // allocate arrays // if( ap.len(d)=0; k--) { x[k] = (d[k]-c[k]*x[k+1])/b[k]; } } /************************************************************************* Internal subroutine. Cyclic tridiagonal solver. Solves ( B[0] C[0] A[0] ) ( A[1] B[1] C[1] ) ( A[2] B[2] C[2] ) ( .......... ) * X = D ( .......... ) ( A[N-2] B[N-2] C[N-2] ) ( C[N-1] A[N-1] B[N-1] ) *************************************************************************/ private static void solvecyclictridiagonal(double[] a, double[] b, double[] c, double[] d, int n, ref double[] x) { int k = 0; double alpha = 0; double beta = 0; double gamma = 0; double[] y = new double[0]; double[] z = new double[0]; double[] u = new double[0]; b = (double[])b.Clone(); if( ap.len(x)0 * if given, only leading N elements of X/Y are used * if not given, automatically determined from sizes of X/Y M - number of basis functions (= polynomial_degree + 1), M>=1 OUTPUT PARAMETERS: Info- same format as in LSFitLinearW() subroutine: * Info>0 task is solved * Info<=0 an error occured: -4 means inconvergence of internal SVD P - interpolant in barycentric form. Rep - report, same format as in LSFitLinearW() subroutine. Following fields are set: * RMSError rms error on the (X,Y). * AvgError average error on the (X,Y). * AvgRelError average relative error on the non-zero Y * MaxError maximum error NON-WEIGHTED ERRORS ARE CALCULATED NOTES: you can convert P from barycentric form to the power or Chebyshev basis with PolynomialBar2Pow() or PolynomialBar2Cheb() functions from POLINT subpackage. -- ALGLIB PROJECT -- Copyright 10.12.2009 by Bochkanov Sergey *************************************************************************/ public static void polynomialfit(double[] x, double[] y, int n, int m, ref int info, ratint.barycentricinterpolant p, polynomialfitreport rep) { int i = 0; double[] w = new double[0]; double[] xc = new double[0]; double[] yc = new double[0]; int[] dc = new int[0]; info = 0; ap.assert(n>0, "PolynomialFit: N<=0!"); ap.assert(m>0, "PolynomialFit: M<=0!"); ap.assert(ap.len(x)>=n, "PolynomialFit: Length(X)=n, "PolynomialFit: Length(Y)0. * if given, only leading N elements of X/Y/W are used * if not given, automatically determined from sizes of X/Y/W XC - points where polynomial values/derivatives are constrained, array[0..K-1]. YC - values of constraints, array[0..K-1] DC - array[0..K-1], types of constraints: * DC[i]=0 means that P(XC[i])=YC[i] * DC[i]=1 means that P'(XC[i])=YC[i] SEE BELOW FOR IMPORTANT INFORMATION ON CONSTRAINTS K - number of constraints, 0<=K=1 OUTPUT PARAMETERS: Info- same format as in LSFitLinearW() subroutine: * Info>0 task is solved * Info<=0 an error occured: -4 means inconvergence of internal SVD -3 means inconsistent constraints P - interpolant in barycentric form. Rep - report, same format as in LSFitLinearW() subroutine. Following fields are set: * RMSError rms error on the (X,Y). * AvgError average error on the (X,Y). * AvgRelError average relative error on the non-zero Y * MaxError maximum error NON-WEIGHTED ERRORS ARE CALCULATED IMPORTANT: this subroitine doesn't calculate task's condition number for K<>0. NOTES: you can convert P from barycentric form to the power or Chebyshev basis with PolynomialBar2Pow() or PolynomialBar2Cheb() functions from POLINT subpackage. SETTING CONSTRAINTS - DANGERS AND OPPORTUNITIES: Setting constraints can lead to undesired results, like ill-conditioned behavior, or inconsistency being detected. From the other side, it allows us to improve quality of the fit. Here we summarize our experience with constrained regression splines: * even simple constraints can be inconsistent, see Wikipedia article on this subject: http://en.wikipedia.org/wiki/Birkhoff_interpolation * the greater is M (given fixed constraints), the more chances that constraints will be consistent * in the general case, consistency of constraints is NOT GUARANTEED. * in the one special cases, however, we can guarantee consistency. This case is: M>1 and constraints on the function values (NOT DERIVATIVES) Our final recommendation is to use constraints WHEN AND ONLY when you can't solve your task without them. Anything beyond special cases given above is not guaranteed and may result in inconsistency. -- ALGLIB PROJECT -- Copyright 10.12.2009 by Bochkanov Sergey *************************************************************************/ public static void polynomialfitwc(double[] x, double[] y, double[] w, int n, double[] xc, double[] yc, int[] dc, int k, int m, ref int info, ratint.barycentricinterpolant p, polynomialfitreport rep) { double xa = 0; double xb = 0; double sa = 0; double sb = 0; double[] xoriginal = new double[0]; double[] yoriginal = new double[0]; double[] y2 = new double[0]; double[] w2 = new double[0]; double[] tmp = new double[0]; double[] tmp2 = new double[0]; double[] bx = new double[0]; double[] by = new double[0]; double[] bw = new double[0]; int i = 0; int j = 0; double u = 0; double v = 0; double s = 0; int relcnt = 0; lsfitreport lrep = new lsfitreport(); x = (double[])x.Clone(); y = (double[])y.Clone(); w = (double[])w.Clone(); xc = (double[])xc.Clone(); yc = (double[])yc.Clone(); info = 0; ap.assert(n>0, "PolynomialFitWC: N<=0!"); ap.assert(m>0, "PolynomialFitWC: M<=0!"); ap.assert(k>=0, "PolynomialFitWC: K<0!"); ap.assert(k=M!"); ap.assert(ap.len(x)>=n, "PolynomialFitWC: Length(X)=n, "PolynomialFitWC: Length(Y)=n, "PolynomialFitWC: Length(W)=k, "PolynomialFitWC: Length(XC)=k, "PolynomialFitWC: Length(YC)=k, "PolynomialFitWC: Length(DC)0. XC - points where function values/derivatives are constrained, array[0..K-1]. YC - values of constraints, array[0..K-1] DC - array[0..K-1], types of constraints: * DC[i]=0 means that S(XC[i])=YC[i] * DC[i]=1 means that S'(XC[i])=YC[i] SEE BELOW FOR IMPORTANT INFORMATION ON CONSTRAINTS K - number of constraints, 0<=K=2. OUTPUT PARAMETERS: Info- same format as in LSFitLinearWC() subroutine. * Info>0 task is solved * Info<=0 an error occured: -4 means inconvergence of internal SVD -3 means inconsistent constraints -1 means another errors in parameters passed (N<=0, for example) B - barycentric interpolant. Rep - report, same format as in LSFitLinearWC() subroutine. Following fields are set: * DBest best value of the D parameter * RMSError rms error on the (X,Y). * AvgError average error on the (X,Y). * AvgRelError average relative error on the non-zero Y * MaxError maximum error NON-WEIGHTED ERRORS ARE CALCULATED IMPORTANT: this subroutine doesn't calculate task's condition number for K<>0. SETTING CONSTRAINTS - DANGERS AND OPPORTUNITIES: Setting constraints can lead to undesired results, like ill-conditioned behavior, or inconsistency being detected. From the other side, it allows us to improve quality of the fit. Here we summarize our experience with constrained barycentric interpolants: * excessive constraints can be inconsistent. Floater-Hormann basis functions aren't as flexible as splines (although they are very smooth). * the more evenly constraints are spread across [min(x),max(x)], the more chances that they will be consistent * the greater is M (given fixed constraints), the more chances that constraints will be consistent * in the general case, consistency of constraints IS NOT GUARANTEED. * in the several special cases, however, we CAN guarantee consistency. * one of this cases is constraints on the function VALUES at the interval boundaries. Note that consustency of the constraints on the function DERIVATIVES is NOT guaranteed (you can use in such cases cubic splines which are more flexible). * another special case is ONE constraint on the function value (OR, but not AND, derivative) anywhere in the interval Our final recommendation is to use constraints WHEN AND ONLY WHEN you can't solve your task without them. Anything beyond special cases given above is not guaranteed and may result in inconsistency. -- ALGLIB PROJECT -- Copyright 18.08.2009 by Bochkanov Sergey *************************************************************************/ public static void barycentricfitfloaterhormannwc(double[] x, double[] y, double[] w, int n, double[] xc, double[] yc, int[] dc, int k, int m, ref int info, ratint.barycentricinterpolant b, barycentricfitreport rep) { int d = 0; int i = 0; double wrmscur = 0; double wrmsbest = 0; ratint.barycentricinterpolant locb = new ratint.barycentricinterpolant(); barycentricfitreport locrep = new barycentricfitreport(); int locinfo = 0; info = 0; ap.assert(n>0, "BarycentricFitFloaterHormannWC: N<=0!"); ap.assert(m>0, "BarycentricFitFloaterHormannWC: M<=0!"); ap.assert(k>=0, "BarycentricFitFloaterHormannWC: K<0!"); ap.assert(k=M!"); ap.assert(ap.len(x)>=n, "BarycentricFitFloaterHormannWC: Length(X)=n, "BarycentricFitFloaterHormannWC: Length(Y)=n, "BarycentricFitFloaterHormannWC: Length(W)=k, "BarycentricFitFloaterHormannWC: Length(XC)=k, "BarycentricFitFloaterHormannWC: Length(YC)=k, "BarycentricFitFloaterHormannWC: Length(DC)0, "BarycentricFitFloaterHormannWC: unexpected result from BarycentricFitWCFixedD!"); if( locinfo>0 ) { // // Calculate weghted RMS // wrmscur = 0; for(i=0; i<=n-1; i++) { wrmscur = wrmscur+math.sqr(w[i]*(y[i]-ratint.barycentriccalc(locb, x[i]))); } wrmscur = Math.Sqrt(wrmscur/n); if( (double)(wrmscur)<(double)(wrmsbest) | rep.dbest<0 ) { ratint.barycentriccopy(locb, b); rep.dbest = d; info = 1; rep.rmserror = locrep.rmserror; rep.avgerror = locrep.avgerror; rep.avgrelerror = locrep.avgrelerror; rep.maxerror = locrep.maxerror; rep.taskrcond = locrep.taskrcond; wrmsbest = wrmscur; } } else { if( locinfo!=-3 & info<0 ) { info = locinfo; } } } } /************************************************************************* Rational least squares fitting using Floater-Hormann rational functions with optimal D chosen from [0,9]. Equidistant grid with M node on [min(x),max(x)] is used to build basis functions. Different values of D are tried, optimal D (least root mean square error) is chosen. Task is linear, so linear least squares solver is used. Complexity of this computational scheme is O(N*M^2) (mostly dominated by the least squares solver). INPUT PARAMETERS: X - points, array[0..N-1]. Y - function values, array[0..N-1]. N - number of points, N>0. M - number of basis functions ( = number_of_nodes), M>=2. OUTPUT PARAMETERS: Info- same format as in LSFitLinearWC() subroutine. * Info>0 task is solved * Info<=0 an error occured: -4 means inconvergence of internal SVD -3 means inconsistent constraints B - barycentric interpolant. Rep - report, same format as in LSFitLinearWC() subroutine. Following fields are set: * DBest best value of the D parameter * RMSError rms error on the (X,Y). * AvgError average error on the (X,Y). * AvgRelError average relative error on the non-zero Y * MaxError maximum error NON-WEIGHTED ERRORS ARE CALCULATED -- ALGLIB PROJECT -- Copyright 18.08.2009 by Bochkanov Sergey *************************************************************************/ public static void barycentricfitfloaterhormann(double[] x, double[] y, int n, int m, ref int info, ratint.barycentricinterpolant b, barycentricfitreport rep) { double[] w = new double[0]; double[] xc = new double[0]; double[] yc = new double[0]; int[] dc = new int[0]; int i = 0; info = 0; ap.assert(n>0, "BarycentricFitFloaterHormann: N<=0!"); ap.assert(m>0, "BarycentricFitFloaterHormann: M<=0!"); ap.assert(ap.len(x)>=n, "BarycentricFitFloaterHormann: Length(X)=n, "BarycentricFitFloaterHormann: Length(Y)0. M - number of basis functions ( = number_of_nodes), M>=2. OUTPUT PARAMETERS: Info- same format as in LSFitLinearWC() subroutine. * Info>0 task is solved * Info<=0 an error occured: -4 means inconvergence of internal SVD -3 means inconsistent constraints B - barycentric interpolant. Rep - report, same format as in LSFitLinearWC() subroutine. Following fields are set: * DBest best value of the D parameter * RMSError rms error on the (X,Y). * AvgError average error on the (X,Y). * AvgRelError average relative error on the non-zero Y * MaxError maximum error NON-WEIGHTED ERRORS ARE CALCULATED -- ALGLIB PROJECT -- Copyright 18.08.2009 by Bochkanov Sergey *************************************************************************/ public static void spline1dfitpenalized(double[] x, double[] y, int n, int m, double rho, ref int info, spline1d.spline1dinterpolant s, spline1dfitreport rep) { double[] w = new double[0]; int i = 0; x = (double[])x.Clone(); y = (double[])y.Clone(); info = 0; ap.assert(n>=1, "Spline1DFitPenalized: N<1!"); ap.assert(m>=4, "Spline1DFitPenalized: M<4!"); ap.assert(ap.len(x)>=n, "Spline1DFitPenalized: Length(X)=n, "Spline1DFitPenalized: Length(Y)0 * if given, only first N elements of X/Y/W are processed * if not given, automatically determined from X/Y/W sizes M - number of basis functions ( = number_of_nodes), M>=4. Rho - regularization constant passed by user. It penalizes nonlinearity in the regression spline. It is logarithmically scaled, i.e. actual value of regularization constant is calculated as 10^Rho. It is automatically scaled so that: * Rho=2.0 corresponds to moderate amount of nonlinearity * generally, it should be somewhere in the [-8.0,+8.0] If you do not want to penalize nonlineary, pass small Rho. Values as low as -15 should work. OUTPUT PARAMETERS: Info- same format as in LSFitLinearWC() subroutine. * Info>0 task is solved * Info<=0 an error occured: -4 means inconvergence of internal SVD or Cholesky decomposition; problem may be too ill-conditioned (very rare) S - spline interpolant. Rep - Following fields are set: * RMSError rms error on the (X,Y). * AvgError average error on the (X,Y). * AvgRelError average relative error on the non-zero Y * MaxError maximum error NON-WEIGHTED ERRORS ARE CALCULATED IMPORTANT: this subroitine doesn't calculate task's condition number for K<>0. NOTE 1: additional nodes are added to the spline outside of the fitting interval to force linearity when xmax(x,xc). It is done for consistency - we penalize non-linearity at [min(x,xc),max(x,xc)], so it is natural to force linearity outside of this interval. NOTE 2: function automatically sorts points, so caller may pass unsorted array. -- ALGLIB PROJECT -- Copyright 19.10.2010 by Bochkanov Sergey *************************************************************************/ public static void spline1dfitpenalizedw(double[] x, double[] y, double[] w, int n, int m, double rho, ref int info, spline1d.spline1dinterpolant s, spline1dfitreport rep) { int i = 0; int j = 0; int b = 0; double v = 0; double relcnt = 0; double xa = 0; double xb = 0; double sa = 0; double sb = 0; double[] xoriginal = new double[0]; double[] yoriginal = new double[0]; double pdecay = 0; double tdecay = 0; double[,] fmatrix = new double[0,0]; double[] fcolumn = new double[0]; double[] y2 = new double[0]; double[] w2 = new double[0]; double[] xc = new double[0]; double[] yc = new double[0]; int[] dc = new int[0]; double fdmax = 0; double admax = 0; double[,] amatrix = new double[0,0]; double[,] d2matrix = new double[0,0]; double fa = 0; double ga = 0; double fb = 0; double gb = 0; double lambdav = 0; double[] bx = new double[0]; double[] by = new double[0]; double[] bd1 = new double[0]; double[] bd2 = new double[0]; double[] tx = new double[0]; double[] ty = new double[0]; double[] td = new double[0]; spline1d.spline1dinterpolant bs = new spline1d.spline1dinterpolant(); double[,] nmatrix = new double[0,0]; double[] rightpart = new double[0]; fbls.fblslincgstate cgstate = new fbls.fblslincgstate(); double[] c = new double[0]; double[] tmp0 = new double[0]; int i_ = 0; int i1_ = 0; x = (double[])x.Clone(); y = (double[])y.Clone(); w = (double[])w.Clone(); info = 0; ap.assert(n>=1, "Spline1DFitPenalizedW: N<1!"); ap.assert(m>=4, "Spline1DFitPenalizedW: M<4!"); ap.assert(ap.len(x)>=n, "Spline1DFitPenalizedW: Length(X)=n, "Spline1DFitPenalizedW: Length(Y)=n, "Spline1DFitPenalizedW: Length(W)(double)(v) ) { rho = v; } lambdav = Math.Pow(10, rho); // // Sort X, Y, W // spline1d.heapsortdpoints(ref x, ref y, ref w, n); // // Scale X, Y, XC, YC // lsfitscalexy(ref x, ref y, ref w, n, ref xc, ref yc, dc, 0, ref xa, ref xb, ref sa, ref sb, ref xoriginal, ref yoriginal); // // Allocate space // fmatrix = new double[n, m]; amatrix = new double[m, m]; d2matrix = new double[m, m]; bx = new double[m]; by = new double[m]; fcolumn = new double[n]; nmatrix = new double[m, m]; rightpart = new double[m]; tmp0 = new double[Math.Max(m, n)]; c = new double[m]; // // Fill: // * FMatrix by values of basis functions // * TmpAMatrix by second derivatives of I-th function at J-th point // * CMatrix by constraints // fdmax = 0; for(b=0; b<=m-1; b++) { // // Prepare I-th basis function // for(j=0; j<=m-1; j++) { bx[j] = (double)(2*j)/(double)(m-1)-1; by[j] = 0; } by[b] = 1; spline1d.spline1dgriddiff2cubic(bx, by, m, 2, 0.0, 2, 0.0, ref bd1, ref bd2); spline1d.spline1dbuildcubic(bx, by, m, 2, 0.0, 2, 0.0, bs); // // Calculate B-th column of FMatrix // Update FDMax (maximum column norm) // spline1d.spline1dconvcubic(bx, by, m, 2, 0.0, 2, 0.0, x, n, ref fcolumn); for(i_=0; i_<=n-1;i_++) { fmatrix[i_,b] = fcolumn[i_]; } v = 0; for(i=0; i<=n-1; i++) { v = v+math.sqr(w[i]*fcolumn[i]); } fdmax = Math.Max(fdmax, v); // // Fill temporary with second derivatives of basis function // for(i_=0; i_<=m-1;i_++) { d2matrix[b,i_] = bd2[i_]; } } // // * calculate penalty matrix A // * calculate max of diagonal elements of A // * calculate PDecay - coefficient before penalty matrix // for(i=0; i<=m-1; i++) { for(j=i; j<=m-1; j++) { // // calculate integral(B_i''*B_j'') where B_i and B_j are // i-th and j-th basis splines. // B_i and B_j are piecewise linear functions. // v = 0; for(b=0; b<=m-2; b++) { fa = d2matrix[i,b]; fb = d2matrix[i,b+1]; ga = d2matrix[j,b]; gb = d2matrix[j,b+1]; v = v+(bx[b+1]-bx[b])*(fa*ga+(fa*(gb-ga)+ga*(fb-fa))/2+(fb-fa)*(gb-ga)/3); } amatrix[i,j] = v; amatrix[j,i] = v; } } admax = 0; for(i=0; i<=m-1; i++) { admax = Math.Max(admax, Math.Abs(amatrix[i,i])); } pdecay = lambdav*fdmax/admax; // // Calculate TDecay for Tikhonov regularization // tdecay = fdmax*(1+pdecay)*10*math.machineepsilon; // // Prepare system // // NOTE: FMatrix is spoiled during this process // for(i=0; i<=n-1; i++) { v = w[i]; for(i_=0; i_<=m-1;i_++) { fmatrix[i,i_] = v*fmatrix[i,i_]; } } ablas.rmatrixgemm(m, m, n, 1.0, fmatrix, 0, 0, 1, fmatrix, 0, 0, 0, 0.0, ref nmatrix, 0, 0); for(i=0; i<=m-1; i++) { for(j=0; j<=m-1; j++) { nmatrix[i,j] = nmatrix[i,j]+pdecay*amatrix[i,j]; } } for(i=0; i<=m-1; i++) { nmatrix[i,i] = nmatrix[i,i]+tdecay; } for(i=0; i<=m-1; i++) { rightpart[i] = 0; } for(i=0; i<=n-1; i++) { v = y[i]*w[i]; for(i_=0; i_<=m-1;i_++) { rightpart[i_] = rightpart[i_] + v*fmatrix[i,i_]; } } // // Solve system // if( !trfac.spdmatrixcholesky(ref nmatrix, m, true) ) { info = -4; return; } fbls.fblscholeskysolve(nmatrix, 1.0, m, true, ref rightpart, ref tmp0); for(i_=0; i_<=m-1;i_++) { c[i_] = rightpart[i_]; } // // add nodes to force linearity outside of the fitting interval // spline1d.spline1dgriddiffcubic(bx, c, m, 2, 0.0, 2, 0.0, ref bd1); tx = new double[m+2]; ty = new double[m+2]; td = new double[m+2]; i1_ = (0) - (1); for(i_=1; i_<=m;i_++) { tx[i_] = bx[i_+i1_]; } i1_ = (0) - (1); for(i_=1; i_<=m;i_++) { ty[i_] = rightpart[i_+i1_]; } i1_ = (0) - (1); for(i_=1; i_<=m;i_++) { td[i_] = bd1[i_+i1_]; } tx[0] = tx[1]-(tx[2]-tx[1]); ty[0] = ty[1]-td[1]*(tx[2]-tx[1]); td[0] = td[1]; tx[m+1] = tx[m]+(tx[m]-tx[m-1]); ty[m+1] = ty[m]+td[m]*(tx[m]-tx[m-1]); td[m+1] = td[m]; spline1d.spline1dbuildhermite(tx, ty, td, m+2, s); spline1d.spline1dlintransx(s, 2/(xb-xa), -((xa+xb)/(xb-xa))); spline1d.spline1dlintransy(s, sb-sa, sa); info = 1; // // Fill report // rep.rmserror = 0; rep.avgerror = 0; rep.avgrelerror = 0; rep.maxerror = 0; relcnt = 0; spline1d.spline1dconvcubic(bx, rightpart, m, 2, 0.0, 2, 0.0, x, n, ref fcolumn); for(i=0; i<=n-1; i++) { v = (sb-sa)*fcolumn[i]+sa; rep.rmserror = rep.rmserror+math.sqr(v-yoriginal[i]); rep.avgerror = rep.avgerror+Math.Abs(v-yoriginal[i]); if( (double)(yoriginal[i])!=(double)(0) ) { rep.avgrelerror = rep.avgrelerror+Math.Abs(v-yoriginal[i])/Math.Abs(yoriginal[i]); relcnt = relcnt+1; } rep.maxerror = Math.Max(rep.maxerror, Math.Abs(v-yoriginal[i])); } rep.rmserror = Math.Sqrt(rep.rmserror/n); rep.avgerror = rep.avgerror/n; if( (double)(relcnt)!=(double)(0) ) { rep.avgrelerror = rep.avgrelerror/relcnt; } } /************************************************************************* Weighted fitting by cubic spline, with constraints on function values or derivatives. Equidistant grid with M-2 nodes on [min(x,xc),max(x,xc)] is used to build basis functions. Basis functions are cubic splines with continuous second derivatives and non-fixed first derivatives at interval ends. Small regularizing term is used when solving constrained tasks (to improve stability). Task is linear, so linear least squares solver is used. Complexity of this computational scheme is O(N*M^2), mostly dominated by least squares solver SEE ALSO Spline1DFitHermiteWC() - fitting by Hermite splines (more flexible, less smooth) Spline1DFitCubic() - "lightweight" fitting by cubic splines, without invididual weights and constraints INPUT PARAMETERS: X - points, array[0..N-1]. Y - function values, array[0..N-1]. W - weights, array[0..N-1] Each summand in square sum of approximation deviations from given values is multiplied by the square of corresponding weight. Fill it by 1's if you don't want to solve weighted task. N - number of points (optional): * N>0 * if given, only first N elements of X/Y/W are processed * if not given, automatically determined from X/Y/W sizes XC - points where spline values/derivatives are constrained, array[0..K-1]. YC - values of constraints, array[0..K-1] DC - array[0..K-1], types of constraints: * DC[i]=0 means that S(XC[i])=YC[i] * DC[i]=1 means that S'(XC[i])=YC[i] SEE BELOW FOR IMPORTANT INFORMATION ON CONSTRAINTS K - number of constraints (optional): * 0<=K=4. OUTPUT PARAMETERS: Info- same format as in LSFitLinearWC() subroutine. * Info>0 task is solved * Info<=0 an error occured: -4 means inconvergence of internal SVD -3 means inconsistent constraints S - spline interpolant. Rep - report, same format as in LSFitLinearWC() subroutine. Following fields are set: * RMSError rms error on the (X,Y). * AvgError average error on the (X,Y). * AvgRelError average relative error on the non-zero Y * MaxError maximum error NON-WEIGHTED ERRORS ARE CALCULATED IMPORTANT: this subroitine doesn't calculate task's condition number for K<>0. ORDER OF POINTS Subroutine automatically sorts points, so caller may pass unsorted array. SETTING CONSTRAINTS - DANGERS AND OPPORTUNITIES: Setting constraints can lead to undesired results, like ill-conditioned behavior, or inconsistency being detected. From the other side, it allows us to improve quality of the fit. Here we summarize our experience with constrained regression splines: * excessive constraints can be inconsistent. Splines are piecewise cubic functions, and it is easy to create an example, where large number of constraints concentrated in small area will result in inconsistency. Just because spline is not flexible enough to satisfy all of them. And same constraints spread across the [min(x),max(x)] will be perfectly consistent. * the more evenly constraints are spread across [min(x),max(x)], the more chances that they will be consistent * the greater is M (given fixed constraints), the more chances that constraints will be consistent * in the general case, consistency of constraints IS NOT GUARANTEED. * in the several special cases, however, we CAN guarantee consistency. * one of this cases is constraints on the function values AND/OR its derivatives at the interval boundaries. * another special case is ONE constraint on the function value (OR, but not AND, derivative) anywhere in the interval Our final recommendation is to use constraints WHEN AND ONLY WHEN you can't solve your task without them. Anything beyond special cases given above is not guaranteed and may result in inconsistency. -- ALGLIB PROJECT -- Copyright 18.08.2009 by Bochkanov Sergey *************************************************************************/ public static void spline1dfitcubicwc(double[] x, double[] y, double[] w, int n, double[] xc, double[] yc, int[] dc, int k, int m, ref int info, spline1d.spline1dinterpolant s, spline1dfitreport rep) { int i = 0; info = 0; ap.assert(n>=1, "Spline1DFitCubicWC: N<1!"); ap.assert(m>=4, "Spline1DFitCubicWC: M<4!"); ap.assert(k>=0, "Spline1DFitCubicWC: K<0!"); ap.assert(k=M!"); ap.assert(ap.len(x)>=n, "Spline1DFitCubicWC: Length(X)=n, "Spline1DFitCubicWC: Length(Y)=n, "Spline1DFitCubicWC: Length(W)=k, "Spline1DFitCubicWC: Length(XC)=k, "Spline1DFitCubicWC: Length(YC)=k, "Spline1DFitCubicWC: Length(DC)0 * if given, only first N elements of X/Y/W are processed * if not given, automatically determined from X/Y/W sizes XC - points where spline values/derivatives are constrained, array[0..K-1]. YC - values of constraints, array[0..K-1] DC - array[0..K-1], types of constraints: * DC[i]=0 means that S(XC[i])=YC[i] * DC[i]=1 means that S'(XC[i])=YC[i] SEE BELOW FOR IMPORTANT INFORMATION ON CONSTRAINTS K - number of constraints (optional): * 0<=K=4, M IS EVEN! OUTPUT PARAMETERS: Info- same format as in LSFitLinearW() subroutine: * Info>0 task is solved * Info<=0 an error occured: -4 means inconvergence of internal SVD -3 means inconsistent constraints -2 means odd M was passed (which is not supported) -1 means another errors in parameters passed (N<=0, for example) S - spline interpolant. Rep - report, same format as in LSFitLinearW() subroutine. Following fields are set: * RMSError rms error on the (X,Y). * AvgError average error on the (X,Y). * AvgRelError average relative error on the non-zero Y * MaxError maximum error NON-WEIGHTED ERRORS ARE CALCULATED IMPORTANT: this subroitine doesn't calculate task's condition number for K<>0. IMPORTANT: this subroitine supports only even M's ORDER OF POINTS Subroutine automatically sorts points, so caller may pass unsorted array. SETTING CONSTRAINTS - DANGERS AND OPPORTUNITIES: Setting constraints can lead to undesired results, like ill-conditioned behavior, or inconsistency being detected. From the other side, it allows us to improve quality of the fit. Here we summarize our experience with constrained regression splines: * excessive constraints can be inconsistent. Splines are piecewise cubic functions, and it is easy to create an example, where large number of constraints concentrated in small area will result in inconsistency. Just because spline is not flexible enough to satisfy all of them. And same constraints spread across the [min(x),max(x)] will be perfectly consistent. * the more evenly constraints are spread across [min(x),max(x)], the more chances that they will be consistent * the greater is M (given fixed constraints), the more chances that constraints will be consistent * in the general case, consistency of constraints is NOT GUARANTEED. * in the several special cases, however, we can guarantee consistency. * one of this cases is M>=4 and constraints on the function value (AND/OR its derivative) at the interval boundaries. * another special case is M>=4 and ONE constraint on the function value (OR, BUT NOT AND, derivative) anywhere in [min(x),max(x)] Our final recommendation is to use constraints WHEN AND ONLY when you can't solve your task without them. Anything beyond special cases given above is not guaranteed and may result in inconsistency. -- ALGLIB PROJECT -- Copyright 18.08.2009 by Bochkanov Sergey *************************************************************************/ public static void spline1dfithermitewc(double[] x, double[] y, double[] w, int n, double[] xc, double[] yc, int[] dc, int k, int m, ref int info, spline1d.spline1dinterpolant s, spline1dfitreport rep) { int i = 0; info = 0; ap.assert(n>=1, "Spline1DFitHermiteWC: N<1!"); ap.assert(m>=4, "Spline1DFitHermiteWC: M<4!"); ap.assert(m%2==0, "Spline1DFitHermiteWC: M is odd!"); ap.assert(k>=0, "Spline1DFitHermiteWC: K<0!"); ap.assert(k=M!"); ap.assert(ap.len(x)>=n, "Spline1DFitHermiteWC: Length(X)=n, "Spline1DFitHermiteWC: Length(Y)=n, "Spline1DFitHermiteWC: Length(W)=k, "Spline1DFitHermiteWC: Length(XC)=k, "Spline1DFitHermiteWC: Length(YC)=k, "Spline1DFitHermiteWC: Length(DC)=1, "Spline1DFitCubic: N<1!"); ap.assert(m>=4, "Spline1DFitCubic: M<4!"); ap.assert(ap.len(x)>=n, "Spline1DFitCubic: Length(X)=n, "Spline1DFitCubic: Length(Y)=1, "Spline1DFitHermite: N<1!"); ap.assert(m>=4, "Spline1DFitHermite: M<4!"); ap.assert(m%2==0, "Spline1DFitHermite: M is odd!"); ap.assert(ap.len(x)>=n, "Spline1DFitHermite: Length(X)=n, "Spline1DFitHermite: Length(Y)=1. M - number of basis functions, M>=1. OUTPUT PARAMETERS: Info - error code: * -4 internal SVD decomposition subroutine failed (very rare and for degenerate systems only) * -1 incorrect N/M were specified * 1 task is solved C - decomposition coefficients, array[0..M-1] Rep - fitting report. Following fields are set: * Rep.TaskRCond reciprocal of condition number * RMSError rms error on the (X,Y). * AvgError average error on the (X,Y). * AvgRelError average relative error on the non-zero Y * MaxError maximum error NON-WEIGHTED ERRORS ARE CALCULATED -- ALGLIB -- Copyright 17.08.2009 by Bochkanov Sergey *************************************************************************/ public static void lsfitlinearw(double[] y, double[] w, double[,] fmatrix, int n, int m, ref int info, ref double[] c, lsfitreport rep) { info = 0; c = new double[0]; ap.assert(n>=1, "LSFitLinearW: N<1!"); ap.assert(m>=1, "LSFitLinearW: M<1!"); ap.assert(ap.len(y)>=n, "LSFitLinearW: length(Y)=n, "LSFitLinearW: length(W)=n, "LSFitLinearW: rows(FMatrix)=m, "LSFitLinearW: cols(FMatrix)=1. M - number of basis functions, M>=1. K - number of constraints, 0 <= K < M K=0 corresponds to absence of constraints. OUTPUT PARAMETERS: Info - error code: * -4 internal SVD decomposition subroutine failed (very rare and for degenerate systems only) * -3 either too many constraints (M or more), degenerate constraints (some constraints are repetead twice) or inconsistent constraints were specified. * 1 task is solved C - decomposition coefficients, array[0..M-1] Rep - fitting report. Following fields are set: * RMSError rms error on the (X,Y). * AvgError average error on the (X,Y). * AvgRelError average relative error on the non-zero Y * MaxError maximum error NON-WEIGHTED ERRORS ARE CALCULATED IMPORTANT: this subroitine doesn't calculate task's condition number for K<>0. -- ALGLIB -- Copyright 07.09.2009 by Bochkanov Sergey *************************************************************************/ public static void lsfitlinearwc(double[] y, double[] w, double[,] fmatrix, double[,] cmatrix, int n, int m, int k, ref int info, ref double[] c, lsfitreport rep) { int i = 0; int j = 0; double[] tau = new double[0]; double[,] q = new double[0,0]; double[,] f2 = new double[0,0]; double[] tmp = new double[0]; double[] c0 = new double[0]; double v = 0; int i_ = 0; y = (double[])y.Clone(); cmatrix = (double[,])cmatrix.Clone(); info = 0; c = new double[0]; ap.assert(n>=1, "LSFitLinearWC: N<1!"); ap.assert(m>=1, "LSFitLinearWC: M<1!"); ap.assert(k>=0, "LSFitLinearWC: K<0!"); ap.assert(ap.len(y)>=n, "LSFitLinearWC: length(Y)=n, "LSFitLinearWC: length(W)=n, "LSFitLinearWC: rows(FMatrix)=m, "LSFitLinearWC: cols(FMatrix)=k, "LSFitLinearWC: rows(CMatrix)=m+1 | k==0, "LSFitLinearWC: cols(CMatrix)=m ) { info = -3; return; } // // Solve // if( k==0 ) { // // no constraints // lsfitlinearinternal(y, w, fmatrix, n, m, ref info, ref c, rep); } else { // // First, find general form solution of constraints system: // * factorize C = L*Q // * unpack Q // * fill upper part of C with zeros (for RCond) // // We got C=C0+Q2'*y where Q2 is lower M-K rows of Q. // ortfac.rmatrixlq(ref cmatrix, k, m, ref tau); ortfac.rmatrixlqunpackq(cmatrix, k, m, tau, m, ref q); for(i=0; i<=k-1; i++) { for(j=i+1; j<=m-1; j++) { cmatrix[i,j] = 0.0; } } if( (double)(rcond.rmatrixlurcondinf(cmatrix, k))<(double)(1000*math.machineepsilon) ) { info = -3; return; } tmp = new double[k]; for(i=0; i<=k-1; i++) { if( i>0 ) { v = 0.0; for(i_=0; i_<=i-1;i_++) { v += cmatrix[i,i_]*tmp[i_]; } } else { v = 0; } tmp[i] = (cmatrix[i,m]-v)/cmatrix[i,i]; } c0 = new double[m]; for(i=0; i<=m-1; i++) { c0[i] = 0; } for(i=0; i<=k-1; i++) { v = tmp[i]; for(i_=0; i_<=m-1;i_++) { c0[i_] = c0[i_] + v*q[i,i_]; } } // // Second, prepare modified matrix F2 = F*Q2' and solve modified task // tmp = new double[Math.Max(n, m)+1]; f2 = new double[n, m-k]; blas.matrixvectormultiply(fmatrix, 0, n-1, 0, m-1, false, c0, 0, m-1, -1.0, ref y, 0, n-1, 1.0); blas.matrixmatrixmultiply(fmatrix, 0, n-1, 0, m-1, false, q, k, m-1, 0, m-1, true, 1.0, ref f2, 0, n-1, 0, m-k-1, 0.0, ref tmp); lsfitlinearinternal(y, w, f2, n, m-k, ref info, ref tmp, rep); rep.taskrcond = -1; if( info<=0 ) { return; } // // then, convert back to original answer: C = C0 + Q2'*Y0 // c = new double[m]; for(i_=0; i_<=m-1;i_++) { c[i_] = c0[i_]; } blas.matrixvectormultiply(q, k, m-1, 0, m-1, true, tmp, 0, m-k-1, 1.0, ref c, 0, m-1, 1.0); } } /************************************************************************* Linear least squares fitting. QR decomposition is used to reduce task to MxM, then triangular solver or SVD-based solver is used depending on condition number of the system. It allows to maximize speed and retain decent accuracy. INPUT PARAMETERS: Y - array[0..N-1] Function values in N points. FMatrix - a table of basis functions values, array[0..N-1, 0..M-1]. FMatrix[I, J] - value of J-th basis function in I-th point. N - number of points used. N>=1. M - number of basis functions, M>=1. OUTPUT PARAMETERS: Info - error code: * -4 internal SVD decomposition subroutine failed (very rare and for degenerate systems only) * 1 task is solved C - decomposition coefficients, array[0..M-1] Rep - fitting report. Following fields are set: * Rep.TaskRCond reciprocal of condition number * RMSError rms error on the (X,Y). * AvgError average error on the (X,Y). * AvgRelError average relative error on the non-zero Y * MaxError maximum error NON-WEIGHTED ERRORS ARE CALCULATED -- ALGLIB -- Copyright 17.08.2009 by Bochkanov Sergey *************************************************************************/ public static void lsfitlinear(double[] y, double[,] fmatrix, int n, int m, ref int info, ref double[] c, lsfitreport rep) { double[] w = new double[0]; int i = 0; info = 0; c = new double[0]; ap.assert(n>=1, "LSFitLinear: N<1!"); ap.assert(m>=1, "LSFitLinear: M<1!"); ap.assert(ap.len(y)>=n, "LSFitLinear: length(Y)=n, "LSFitLinear: rows(FMatrix)=m, "LSFitLinear: cols(FMatrix)=1. M - number of basis functions, M>=1. K - number of constraints, 0 <= K < M K=0 corresponds to absence of constraints. OUTPUT PARAMETERS: Info - error code: * -4 internal SVD decomposition subroutine failed (very rare and for degenerate systems only) * -3 either too many constraints (M or more), degenerate constraints (some constraints are repetead twice) or inconsistent constraints were specified. * 1 task is solved C - decomposition coefficients, array[0..M-1] Rep - fitting report. Following fields are set: * RMSError rms error on the (X,Y). * AvgError average error on the (X,Y). * AvgRelError average relative error on the non-zero Y * MaxError maximum error NON-WEIGHTED ERRORS ARE CALCULATED IMPORTANT: this subroitine doesn't calculate task's condition number for K<>0. -- ALGLIB -- Copyright 07.09.2009 by Bochkanov Sergey *************************************************************************/ public static void lsfitlinearc(double[] y, double[,] fmatrix, double[,] cmatrix, int n, int m, int k, ref int info, ref double[] c, lsfitreport rep) { double[] w = new double[0]; int i = 0; y = (double[])y.Clone(); info = 0; c = new double[0]; ap.assert(n>=1, "LSFitLinearC: N<1!"); ap.assert(m>=1, "LSFitLinearC: M<1!"); ap.assert(k>=0, "LSFitLinearC: K<0!"); ap.assert(ap.len(y)>=n, "LSFitLinearC: length(Y)=n, "LSFitLinearC: rows(FMatrix)=m, "LSFitLinearC: cols(FMatrix)=k, "LSFitLinearC: rows(CMatrix)=m+1 | k==0, "LSFitLinearC: cols(CMatrix)1 M - dimension of space K - number of parameters being fitted DiffStep- numerical differentiation step; should not be very small or large; large = loss of accuracy small = growth of round-off errors OUTPUT PARAMETERS: State - structure which stores algorithm state -- ALGLIB -- Copyright 18.10.2008 by Bochkanov Sergey *************************************************************************/ public static void lsfitcreatewf(double[,] x, double[] y, double[] w, double[] c, int n, int m, int k, double diffstep, lsfitstate state) { int i = 0; int i_ = 0; ap.assert(n>=1, "LSFitCreateWF: N<1!"); ap.assert(m>=1, "LSFitCreateWF: M<1!"); ap.assert(k>=1, "LSFitCreateWF: K<1!"); ap.assert(ap.len(c)>=k, "LSFitCreateWF: length(C)=n, "LSFitCreateWF: length(Y)=n, "LSFitCreateWF: length(W)=n, "LSFitCreateWF: rows(X)=m, "LSFitCreateWF: cols(X)(double)(0), "LSFitCreateWF: DiffStep<=0!"); state.npoints = n; state.nweights = n; state.wkind = 1; state.m = m; state.k = k; lsfitsetcond(state, 0.0, 0.0, 0); lsfitsetstpmax(state, 0.0); lsfitsetxrep(state, false); state.taskx = new double[n, m]; state.tasky = new double[n]; state.w = new double[n]; state.c = new double[k]; state.x = new double[m]; for(i_=0; i_<=k-1;i_++) { state.c[i_] = c[i_]; } for(i_=0; i_<=n-1;i_++) { state.w[i_] = w[i_]; } for(i=0; i<=n-1; i++) { for(i_=0; i_<=m-1;i_++) { state.taskx[i,i_] = x[i,i_]; } state.tasky[i] = y[i]; } state.s = new double[k]; state.bndl = new double[k]; state.bndu = new double[k]; for(i=0; i<=k-1; i++) { state.s[i] = 1.0; state.bndl[i] = Double.NegativeInfinity; state.bndu[i] = Double.PositiveInfinity; } state.optalgo = 0; state.prevnpt = -1; state.prevalgo = -1; minlm.minlmcreatev(k, n, state.c, diffstep, state.optstate); lsfitclearrequestfields(state); state.rstate.ia = new int[4+1]; state.rstate.ra = new double[2+1]; state.rstate.stage = -1; } /************************************************************************* Nonlinear least squares fitting using function values only. Combination of numerical differentiation and secant updates is used to obtain function Jacobian. Nonlinear task min(F(c)) is solved, where F(c) = (f(c,x[0])-y[0])^2 + ... + (f(c,x[n-1])-y[n-1])^2, * N is a number of points, * M is a dimension of a space points belong to, * K is a dimension of a space of parameters being fitted, * w is an N-dimensional vector of weight coefficients, * x is a set of N points, each of them is an M-dimensional vector, * c is a K-dimensional vector of parameters being fitted This subroutine uses only f(c,x[i]). INPUT PARAMETERS: X - array[0..N-1,0..M-1], points (one row = one point) Y - array[0..N-1], function values. C - array[0..K-1], initial approximation to the solution, N - number of points, N>1 M - dimension of space K - number of parameters being fitted DiffStep- numerical differentiation step; should not be very small or large; large = loss of accuracy small = growth of round-off errors OUTPUT PARAMETERS: State - structure which stores algorithm state -- ALGLIB -- Copyright 18.10.2008 by Bochkanov Sergey *************************************************************************/ public static void lsfitcreatef(double[,] x, double[] y, double[] c, int n, int m, int k, double diffstep, lsfitstate state) { int i = 0; int i_ = 0; ap.assert(n>=1, "LSFitCreateF: N<1!"); ap.assert(m>=1, "LSFitCreateF: M<1!"); ap.assert(k>=1, "LSFitCreateF: K<1!"); ap.assert(ap.len(c)>=k, "LSFitCreateF: length(C)=n, "LSFitCreateF: length(Y)=n, "LSFitCreateF: rows(X)=m, "LSFitCreateF: cols(X)=n, "LSFitCreateF: rows(X)=m, "LSFitCreateF: cols(X)(double)(0), "LSFitCreateF: DiffStep<=0!"); state.npoints = n; state.wkind = 0; state.m = m; state.k = k; lsfitsetcond(state, 0.0, 0.0, 0); lsfitsetstpmax(state, 0.0); lsfitsetxrep(state, false); state.taskx = new double[n, m]; state.tasky = new double[n]; state.c = new double[k]; state.x = new double[m]; for(i_=0; i_<=k-1;i_++) { state.c[i_] = c[i_]; } for(i=0; i<=n-1; i++) { for(i_=0; i_<=m-1;i_++) { state.taskx[i,i_] = x[i,i_]; } state.tasky[i] = y[i]; } state.s = new double[k]; state.bndl = new double[k]; state.bndu = new double[k]; for(i=0; i<=k-1; i++) { state.s[i] = 1.0; state.bndl[i] = Double.NegativeInfinity; state.bndu[i] = Double.PositiveInfinity; } state.optalgo = 0; state.prevnpt = -1; state.prevalgo = -1; minlm.minlmcreatev(k, n, state.c, diffstep, state.optstate); lsfitclearrequestfields(state); state.rstate.ia = new int[4+1]; state.rstate.ra = new double[2+1]; state.rstate.stage = -1; } /************************************************************************* Weighted nonlinear least squares fitting using gradient only. Nonlinear task min(F(c)) is solved, where F(c) = (w[0]*(f(c,x[0])-y[0]))^2 + ... + (w[n-1]*(f(c,x[n-1])-y[n-1]))^2, * N is a number of points, * M is a dimension of a space points belong to, * K is a dimension of a space of parameters being fitted, * w is an N-dimensional vector of weight coefficients, * x is a set of N points, each of them is an M-dimensional vector, * c is a K-dimensional vector of parameters being fitted This subroutine uses only f(c,x[i]) and its gradient. INPUT PARAMETERS: X - array[0..N-1,0..M-1], points (one row = one point) Y - array[0..N-1], function values. W - weights, array[0..N-1] C - array[0..K-1], initial approximation to the solution, N - number of points, N>1 M - dimension of space K - number of parameters being fitted CheapFG - boolean flag, which is: * True if both function and gradient calculation complexity are less than O(M^2). An improved algorithm can be used which corresponds to FGJ scheme from MINLM unit. * False otherwise. Standard Jacibian-bases Levenberg-Marquardt algo will be used (FJ scheme). OUTPUT PARAMETERS: State - structure which stores algorithm state See also: LSFitResults LSFitCreateFG (fitting without weights) LSFitCreateWFGH (fitting using Hessian) LSFitCreateFGH (fitting using Hessian, without weights) -- ALGLIB -- Copyright 17.08.2009 by Bochkanov Sergey *************************************************************************/ public static void lsfitcreatewfg(double[,] x, double[] y, double[] w, double[] c, int n, int m, int k, bool cheapfg, lsfitstate state) { int i = 0; int i_ = 0; ap.assert(n>=1, "LSFitCreateWFG: N<1!"); ap.assert(m>=1, "LSFitCreateWFG: M<1!"); ap.assert(k>=1, "LSFitCreateWFG: K<1!"); ap.assert(ap.len(c)>=k, "LSFitCreateWFG: length(C)=n, "LSFitCreateWFG: length(Y)=n, "LSFitCreateWFG: length(W)=n, "LSFitCreateWFG: rows(X)=m, "LSFitCreateWFG: cols(X)1 M - dimension of space K - number of parameters being fitted CheapFG - boolean flag, which is: * True if both function and gradient calculation complexity are less than O(M^2). An improved algorithm can be used which corresponds to FGJ scheme from MINLM unit. * False otherwise. Standard Jacibian-bases Levenberg-Marquardt algo will be used (FJ scheme). OUTPUT PARAMETERS: State - structure which stores algorithm state -- ALGLIB -- Copyright 17.08.2009 by Bochkanov Sergey *************************************************************************/ public static void lsfitcreatefg(double[,] x, double[] y, double[] c, int n, int m, int k, bool cheapfg, lsfitstate state) { int i = 0; int i_ = 0; ap.assert(n>=1, "LSFitCreateFG: N<1!"); ap.assert(m>=1, "LSFitCreateFG: M<1!"); ap.assert(k>=1, "LSFitCreateFG: K<1!"); ap.assert(ap.len(c)>=k, "LSFitCreateFG: length(C)=n, "LSFitCreateFG: length(Y)=n, "LSFitCreateFG: rows(X)=m, "LSFitCreateFG: cols(X)=n, "LSFitCreateFG: rows(X)=m, "LSFitCreateFG: cols(X)1 M - dimension of space K - number of parameters being fitted OUTPUT PARAMETERS: State - structure which stores algorithm state -- ALGLIB -- Copyright 17.08.2009 by Bochkanov Sergey *************************************************************************/ public static void lsfitcreatewfgh(double[,] x, double[] y, double[] w, double[] c, int n, int m, int k, lsfitstate state) { int i = 0; int i_ = 0; ap.assert(n>=1, "LSFitCreateWFGH: N<1!"); ap.assert(m>=1, "LSFitCreateWFGH: M<1!"); ap.assert(k>=1, "LSFitCreateWFGH: K<1!"); ap.assert(ap.len(c)>=k, "LSFitCreateWFGH: length(C)=n, "LSFitCreateWFGH: length(Y)=n, "LSFitCreateWFGH: length(W)=n, "LSFitCreateWFGH: rows(X)=m, "LSFitCreateWFGH: cols(X)1 M - dimension of space K - number of parameters being fitted OUTPUT PARAMETERS: State - structure which stores algorithm state -- ALGLIB -- Copyright 17.08.2009 by Bochkanov Sergey *************************************************************************/ public static void lsfitcreatefgh(double[,] x, double[] y, double[] c, int n, int m, int k, lsfitstate state) { int i = 0; int i_ = 0; ap.assert(n>=1, "LSFitCreateFGH: N<1!"); ap.assert(m>=1, "LSFitCreateFGH: M<1!"); ap.assert(k>=1, "LSFitCreateFGH: K<1!"); ap.assert(ap.len(c)>=k, "LSFitCreateFGH: length(C)=n, "LSFitCreateFGH: length(Y)=n, "LSFitCreateFGH: rows(X)=m, "LSFitCreateFGH: cols(X)=0 The subroutine finishes its work if on k+1-th iteration the condition |v|<=EpsX is fulfilled, where: * |.| means Euclidian norm * v - scaled step vector, v[i]=dx[i]/s[i] * dx - ste pvector, dx=X(k+1)-X(k) * s - scaling coefficients set by LSFitSetScale() MaxIts - maximum number of iterations. If MaxIts=0, the number of iterations is unlimited. Only Levenberg-Marquardt iterations are counted (L-BFGS/CG iterations are NOT counted because their cost is very low compared to that of LM). NOTE Passing EpsF=0, EpsX=0 and MaxIts=0 (simultaneously) will lead to automatic stopping criterion selection (according to the scheme used by MINLM unit). -- ALGLIB -- Copyright 17.08.2009 by Bochkanov Sergey *************************************************************************/ public static void lsfitsetcond(lsfitstate state, double epsf, double epsx, int maxits) { ap.assert(math.isfinite(epsf), "LSFitSetCond: EpsF is not finite!"); ap.assert((double)(epsf)>=(double)(0), "LSFitSetCond: negative EpsF!"); ap.assert(math.isfinite(epsx), "LSFitSetCond: EpsX is not finite!"); ap.assert((double)(epsx)>=(double)(0), "LSFitSetCond: negative EpsX!"); ap.assert(maxits>=0, "LSFitSetCond: negative MaxIts!"); state.epsf = epsf; state.epsx = epsx; state.maxits = maxits; } /************************************************************************* This function sets maximum step length INPUT PARAMETERS: State - structure which stores algorithm state StpMax - maximum step length, >=0. Set StpMax to 0.0, if you don't want to limit step length. Use this subroutine when you optimize target function which contains exp() or other fast growing functions, and optimization algorithm makes too large steps which leads to overflow. This function allows us to reject steps that are too large (and therefore expose us to the possible overflow) without actually calculating function value at the x+stp*d. NOTE: non-zero StpMax leads to moderate performance degradation because intermediate step of preconditioned L-BFGS optimization is incompatible with limits on step size. -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/ public static void lsfitsetstpmax(lsfitstate state, double stpmax) { ap.assert((double)(stpmax)>=(double)(0), "LSFitSetStpMax: StpMax<0!"); state.stpmax = stpmax; } /************************************************************************* This function turns on/off reporting. INPUT PARAMETERS: State - structure which stores algorithm state NeedXRep- whether iteration reports are needed or not When reports are needed, State.C (current parameters) and State.F (current value of fitting function) are reported. -- ALGLIB -- Copyright 15.08.2010 by Bochkanov Sergey *************************************************************************/ public static void lsfitsetxrep(lsfitstate state, bool needxrep) { state.xrep = needxrep; } /************************************************************************* This function sets scaling coefficients for underlying optimizer. ALGLIB optimizers use scaling matrices to test stopping conditions (step size and gradient are scaled before comparison with tolerances). Scale of the I-th variable is a translation invariant measure of: a) "how large" the variable is b) how large the step should be to make significant changes in the function Generally, scale is NOT considered to be a form of preconditioner. But LM optimizer is unique in that it uses scaling matrix both in the stopping condition tests and as Marquardt damping factor. Proper scaling is very important for the algorithm performance. It is less important for the quality of results, but still has some influence (it is easier to converge when variables are properly scaled, so premature stopping is possible when very badly scalled variables are combined with relaxed stopping conditions). INPUT PARAMETERS: State - structure stores algorithm state S - array[N], non-zero scaling coefficients S[i] may be negative, sign doesn't matter. -- ALGLIB -- Copyright 14.01.2011 by Bochkanov Sergey *************************************************************************/ public static void lsfitsetscale(lsfitstate state, double[] s) { int i = 0; ap.assert(ap.len(s)>=state.k, "LSFitSetScale: Length(S)=k, "LSFitSetBC: Length(BndL)=k, "LSFitSetBC: Length(BndU)BndU[i]"); } state.bndl[i] = bndl[i]; state.bndu[i] = bndu[i]; } } /************************************************************************* NOTES: 1. this algorithm is somewhat unusual because it works with parameterized function f(C,X), where X is a function argument (we have many points which are characterized by different argument values), and C is a parameter to fit. For example, if we want to do linear fit by f(c0,c1,x) = c0*x+c1, then x will be argument, and {c0,c1} will be parameters. It is important to understand that this algorithm finds minimum in the space of function PARAMETERS (not arguments), so it needs derivatives of f() with respect to C, not X. In the example above it will need f=c0*x+c1 and {df/dc0,df/dc1} = {x,1} instead of {df/dx} = {c0}. 2. Callback functions accept C as the first parameter, and X as the second 3. If state was created with LSFitCreateFG(), algorithm needs just function and its gradient, but if state was created with LSFitCreateFGH(), algorithm will need function, gradient and Hessian. According to the said above, there ase several versions of this function, which accept different sets of callbacks. This flexibility opens way to subtle errors - you may create state with LSFitCreateFGH() (optimization using Hessian), but call function which does not accept Hessian. So when algorithm will request Hessian, there will be no callback to call. In this case exception will be thrown. Be careful to avoid such errors because there is no way to find them at compile time - you can see them at runtime only. -- ALGLIB -- Copyright 17.08.2009 by Bochkanov Sergey *************************************************************************/ public static bool lsfititeration(lsfitstate state) { bool result = new bool(); int n = 0; int m = 0; int k = 0; int i = 0; int j = 0; double v = 0; double vv = 0; double relcnt = 0; int i_ = 0; // // Reverse communication preparations // I know it looks ugly, but it works the same way // anywhere from C++ to Python. // // This code initializes locals by: // * random values determined during code // generation - on first subroutine call // * values from previous call - on subsequent calls // if( state.rstate.stage>=0 ) { n = state.rstate.ia[0]; m = state.rstate.ia[1]; k = state.rstate.ia[2]; i = state.rstate.ia[3]; j = state.rstate.ia[4]; v = state.rstate.ra[0]; vv = state.rstate.ra[1]; relcnt = state.rstate.ra[2]; } else { n = -983; m = -989; k = -834; i = 900; j = -287; v = 364; vv = 214; relcnt = -338; } if( state.rstate.stage==0 ) { goto lbl_0; } if( state.rstate.stage==1 ) { goto lbl_1; } if( state.rstate.stage==2 ) { goto lbl_2; } if( state.rstate.stage==3 ) { goto lbl_3; } if( state.rstate.stage==4 ) { goto lbl_4; } if( state.rstate.stage==5 ) { goto lbl_5; } if( state.rstate.stage==6 ) { goto lbl_6; } // // Routine body // // // init // if( state.wkind==1 ) { ap.assert(state.npoints==state.nweights, "LSFitFit: number of points is not equal to the number of weights"); } n = state.npoints; m = state.m; k = state.k; minlm.minlmsetcond(state.optstate, 0.0, state.epsf, state.epsx, state.maxits); minlm.minlmsetstpmax(state.optstate, state.stpmax); minlm.minlmsetxrep(state.optstate, state.xrep); minlm.minlmsetscale(state.optstate, state.s); minlm.minlmsetbc(state.optstate, state.bndl, state.bndu); // // Optimize // lbl_7: if( !minlm.minlmiteration(state.optstate) ) { goto lbl_8; } if( !state.optstate.needfi ) { goto lbl_9; } // // calculate f[] = wi*(f(xi,c)-yi) // i = 0; lbl_11: if( i>n-1 ) { goto lbl_13; } for(i_=0; i_<=k-1;i_++) { state.c[i_] = state.optstate.x[i_]; } for(i_=0; i_<=m-1;i_++) { state.x[i_] = state.taskx[i,i_]; } state.pointindex = i; lsfitclearrequestfields(state); state.needf = true; state.rstate.stage = 0; goto lbl_rcomm; lbl_0: state.needf = false; if( state.wkind==1 ) { vv = state.w[i]; } else { vv = 1.0; } state.optstate.fi[i] = vv*(state.f-state.tasky[i]); i = i+1; goto lbl_11; lbl_13: goto lbl_7; lbl_9: if( !state.optstate.needf ) { goto lbl_14; } // // calculate F = sum (wi*(f(xi,c)-yi))^2 // state.optstate.f = 0; i = 0; lbl_16: if( i>n-1 ) { goto lbl_18; } for(i_=0; i_<=k-1;i_++) { state.c[i_] = state.optstate.x[i_]; } for(i_=0; i_<=m-1;i_++) { state.x[i_] = state.taskx[i,i_]; } state.pointindex = i; lsfitclearrequestfields(state); state.needf = true; state.rstate.stage = 1; goto lbl_rcomm; lbl_1: state.needf = false; if( state.wkind==1 ) { vv = state.w[i]; } else { vv = 1.0; } state.optstate.f = state.optstate.f+math.sqr(vv*(state.f-state.tasky[i])); i = i+1; goto lbl_16; lbl_18: goto lbl_7; lbl_14: if( !state.optstate.needfg ) { goto lbl_19; } // // calculate F/gradF // state.optstate.f = 0; for(i=0; i<=k-1; i++) { state.optstate.g[i] = 0; } i = 0; lbl_21: if( i>n-1 ) { goto lbl_23; } for(i_=0; i_<=k-1;i_++) { state.c[i_] = state.optstate.x[i_]; } for(i_=0; i_<=m-1;i_++) { state.x[i_] = state.taskx[i,i_]; } state.pointindex = i; lsfitclearrequestfields(state); state.needfg = true; state.rstate.stage = 2; goto lbl_rcomm; lbl_2: state.needfg = false; if( state.wkind==1 ) { vv = state.w[i]; } else { vv = 1.0; } state.optstate.f = state.optstate.f+math.sqr(vv*(state.f-state.tasky[i])); v = math.sqr(vv)*2*(state.f-state.tasky[i]); for(i_=0; i_<=k-1;i_++) { state.optstate.g[i_] = state.optstate.g[i_] + v*state.g[i_]; } i = i+1; goto lbl_21; lbl_23: goto lbl_7; lbl_19: if( !state.optstate.needfij ) { goto lbl_24; } // // calculate Fi/jac(Fi) // i = 0; lbl_26: if( i>n-1 ) { goto lbl_28; } for(i_=0; i_<=k-1;i_++) { state.c[i_] = state.optstate.x[i_]; } for(i_=0; i_<=m-1;i_++) { state.x[i_] = state.taskx[i,i_]; } state.pointindex = i; lsfitclearrequestfields(state); state.needfg = true; state.rstate.stage = 3; goto lbl_rcomm; lbl_3: state.needfg = false; if( state.wkind==1 ) { vv = state.w[i]; } else { vv = 1.0; } state.optstate.fi[i] = vv*(state.f-state.tasky[i]); for(i_=0; i_<=k-1;i_++) { state.optstate.j[i,i_] = vv*state.g[i_]; } i = i+1; goto lbl_26; lbl_28: goto lbl_7; lbl_24: if( !state.optstate.needfgh ) { goto lbl_29; } // // calculate F/grad(F)/hess(F) // state.optstate.f = 0; for(i=0; i<=k-1; i++) { state.optstate.g[i] = 0; } for(i=0; i<=k-1; i++) { for(j=0; j<=k-1; j++) { state.optstate.h[i,j] = 0; } } i = 0; lbl_31: if( i>n-1 ) { goto lbl_33; } for(i_=0; i_<=k-1;i_++) { state.c[i_] = state.optstate.x[i_]; } for(i_=0; i_<=m-1;i_++) { state.x[i_] = state.taskx[i,i_]; } state.pointindex = i; lsfitclearrequestfields(state); state.needfgh = true; state.rstate.stage = 4; goto lbl_rcomm; lbl_4: state.needfgh = false; if( state.wkind==1 ) { vv = state.w[i]; } else { vv = 1.0; } state.optstate.f = state.optstate.f+math.sqr(vv*(state.f-state.tasky[i])); v = math.sqr(vv)*2*(state.f-state.tasky[i]); for(i_=0; i_<=k-1;i_++) { state.optstate.g[i_] = state.optstate.g[i_] + v*state.g[i_]; } for(j=0; j<=k-1; j++) { v = 2*math.sqr(vv)*state.g[j]; for(i_=0; i_<=k-1;i_++) { state.optstate.h[j,i_] = state.optstate.h[j,i_] + v*state.g[i_]; } v = 2*math.sqr(vv)*(state.f-state.tasky[i]); for(i_=0; i_<=k-1;i_++) { state.optstate.h[j,i_] = state.optstate.h[j,i_] + v*state.h[j,i_]; } } i = i+1; goto lbl_31; lbl_33: goto lbl_7; lbl_29: if( !state.optstate.xupdated ) { goto lbl_34; } // // Report new iteration // for(i_=0; i_<=k-1;i_++) { state.c[i_] = state.optstate.x[i_]; } state.f = state.optstate.f; lsfitclearrequestfields(state); state.xupdated = true; state.rstate.stage = 5; goto lbl_rcomm; lbl_5: state.xupdated = false; goto lbl_7; lbl_34: goto lbl_7; lbl_8: minlm.minlmresults(state.optstate, ref state.c, state.optrep); state.repterminationtype = state.optrep.terminationtype; state.repiterationscount = state.optrep.iterationscount; // // calculate errors // if( state.repterminationtype<=0 ) { goto lbl_36; } state.reprmserror = 0; state.repwrmserror = 0; state.repavgerror = 0; state.repavgrelerror = 0; state.repmaxerror = 0; relcnt = 0; i = 0; lbl_38: if( i>n-1 ) { goto lbl_40; } for(i_=0; i_<=k-1;i_++) { state.c[i_] = state.c[i_]; } for(i_=0; i_<=m-1;i_++) { state.x[i_] = state.taskx[i,i_]; } state.pointindex = i; lsfitclearrequestfields(state); state.needf = true; state.rstate.stage = 6; goto lbl_rcomm; lbl_6: state.needf = false; v = state.f; if( state.wkind==1 ) { vv = state.w[i]; } else { vv = 1.0; } state.reprmserror = state.reprmserror+math.sqr(v-state.tasky[i]); state.repwrmserror = state.repwrmserror+math.sqr(vv*(v-state.tasky[i])); state.repavgerror = state.repavgerror+Math.Abs(v-state.tasky[i]); if( (double)(state.tasky[i])!=(double)(0) ) { state.repavgrelerror = state.repavgrelerror+Math.Abs(v-state.tasky[i])/Math.Abs(state.tasky[i]); relcnt = relcnt+1; } state.repmaxerror = Math.Max(state.repmaxerror, Math.Abs(v-state.tasky[i])); i = i+1; goto lbl_38; lbl_40: state.reprmserror = Math.Sqrt(state.reprmserror/n); state.repwrmserror = Math.Sqrt(state.repwrmserror/n); state.repavgerror = state.repavgerror/n; if( (double)(relcnt)!=(double)(0) ) { state.repavgrelerror = state.repavgrelerror/relcnt; } lbl_36: result = false; return result; // // Saving state // lbl_rcomm: result = true; state.rstate.ia[0] = n; state.rstate.ia[1] = m; state.rstate.ia[2] = k; state.rstate.ia[3] = i; state.rstate.ia[4] = j; state.rstate.ra[0] = v; state.rstate.ra[1] = vv; state.rstate.ra[2] = relcnt; return result; } /************************************************************************* Nonlinear least squares fitting results. Called after return from LSFitFit(). INPUT PARAMETERS: State - algorithm state OUTPUT PARAMETERS: Info - completetion code: * 1 relative function improvement is no more than EpsF. * 2 relative step is no more than EpsX. * 4 gradient norm is no more than EpsG * 5 MaxIts steps was taken * 7 stopping conditions are too stringent, further improvement is impossible C - array[0..K-1], solution Rep - optimization report. Following fields are set: * Rep.TerminationType completetion code: * RMSError rms error on the (X,Y). * AvgError average error on the (X,Y). * AvgRelError average relative error on the non-zero Y * MaxError maximum error NON-WEIGHTED ERRORS ARE CALCULATED * WRMSError weighted rms error on the (X,Y). -- ALGLIB -- Copyright 17.08.2009 by Bochkanov Sergey *************************************************************************/ public static void lsfitresults(lsfitstate state, ref int info, ref double[] c, lsfitreport rep) { int i_ = 0; info = 0; c = new double[0]; info = state.repterminationtype; if( info>0 ) { c = new double[state.k]; for(i_=0; i_<=state.k-1;i_++) { c[i_] = state.c[i_]; } rep.rmserror = state.reprmserror; rep.wrmserror = state.repwrmserror; rep.avgerror = state.repavgerror; rep.avgrelerror = state.repavgrelerror; rep.maxerror = state.repmaxerror; rep.iterationscount = state.repiterationscount; } } /************************************************************************* Internal subroutine: automatic scaling for LLS tasks. NEVER CALL IT DIRECTLY! Maps abscissas to [-1,1], standartizes ordinates and correspondingly scales constraints. It also scales weights so that max(W[i])=1 Transformations performed: * X, XC [XA,XB] => [-1,+1] transformation makes min(X)=-1, max(X)=+1 * Y [SA,SB] => [0,1] transformation makes mean(Y)=0, stddev(Y)=1 * YC transformed accordingly to SA, SB, DC[I] -- ALGLIB PROJECT -- Copyright 08.09.2009 by Bochkanov Sergey *************************************************************************/ public static void lsfitscalexy(ref double[] x, ref double[] y, ref double[] w, int n, ref double[] xc, ref double[] yc, int[] dc, int k, ref double xa, ref double xb, ref double sa, ref double sb, ref double[] xoriginal, ref double[] yoriginal) { double xmin = 0; double xmax = 0; int i = 0; double mx = 0; int i_ = 0; xa = 0; xb = 0; sa = 0; sb = 0; xoriginal = new double[0]; yoriginal = new double[0]; ap.assert(n>=1, "LSFitScaleXY: incorrect N"); ap.assert(k>=0, "LSFitScaleXY: incorrect K"); // // Calculate xmin/xmax. // Force xmin<>xmax. // xmin = x[0]; xmax = x[0]; for(i=1; i<=n-1; i++) { xmin = Math.Min(xmin, x[i]); xmax = Math.Max(xmax, x[i]); } for(i=0; i<=k-1; i++) { xmin = Math.Min(xmin, xc[i]); xmax = Math.Max(xmax, xc[i]); } if( (double)(xmin)==(double)(xmax) ) { if( (double)(xmin)==(double)(0) ) { xmin = -1; xmax = 1; } else { if( (double)(xmin)>(double)(0) ) { xmin = 0.5*xmin; } else { xmax = 0.5*xmax; } } } // // Transform abscissas: map [XA,XB] to [0,1] // // Store old X[] in XOriginal[] (it will be used // to calculate relative error). // xoriginal = new double[n]; for(i_=0; i_<=n-1;i_++) { xoriginal[i_] = x[i_]; } xa = xmin; xb = xmax; for(i=0; i<=n-1; i++) { x[i] = 2*(x[i]-0.5*(xa+xb))/(xb-xa); } for(i=0; i<=k-1; i++) { ap.assert(dc[i]>=0, "LSFitScaleXY: internal error!"); xc[i] = 2*(xc[i]-0.5*(xa+xb))/(xb-xa); yc[i] = yc[i]*Math.Pow(0.5*(xb-xa), dc[i]); } // // Transform function values: map [SA,SB] to [0,1] // SA = mean(Y), // SB = SA+stddev(Y). // // Store old Y[] in YOriginal[] (it will be used // to calculate relative error). // yoriginal = new double[n]; for(i_=0; i_<=n-1;i_++) { yoriginal[i_] = y[i_]; } sa = 0; for(i=0; i<=n-1; i++) { sa = sa+y[i]; } sa = sa/n; sb = 0; for(i=0; i<=n-1; i++) { sb = sb+math.sqr(y[i]-sa); } sb = Math.Sqrt(sb/n)+sa; if( (double)(sb)==(double)(sa) ) { sb = 2*sa; } if( (double)(sb)==(double)(sa) ) { sb = sa+1; } for(i=0; i<=n-1; i++) { y[i] = (y[i]-sa)/(sb-sa); } for(i=0; i<=k-1; i++) { if( dc[i]==0 ) { yc[i] = (yc[i]-sa)/(sb-sa); } else { yc[i] = yc[i]/(sb-sa); } } // // Scale weights // mx = 0; for(i=0; i<=n-1; i++) { mx = Math.Max(mx, Math.Abs(w[i])); } if( (double)(mx)!=(double)(0) ) { for(i=0; i<=n-1; i++) { w[i] = w[i]/mx; } } } /************************************************************************* Internal spline fitting subroutine -- ALGLIB PROJECT -- Copyright 08.09.2009 by Bochkanov Sergey *************************************************************************/ private static void spline1dfitinternal(int st, double[] x, double[] y, double[] w, int n, double[] xc, double[] yc, int[] dc, int k, int m, ref int info, spline1d.spline1dinterpolant s, spline1dfitreport rep) { double[,] fmatrix = new double[0,0]; double[,] cmatrix = new double[0,0]; double[] y2 = new double[0]; double[] w2 = new double[0]; double[] sx = new double[0]; double[] sy = new double[0]; double[] sd = new double[0]; double[] tmp = new double[0]; double[] xoriginal = new double[0]; double[] yoriginal = new double[0]; lsfitreport lrep = new lsfitreport(); double v0 = 0; double v1 = 0; double v2 = 0; double mx = 0; spline1d.spline1dinterpolant s2 = new spline1d.spline1dinterpolant(); int i = 0; int j = 0; int relcnt = 0; double xa = 0; double xb = 0; double sa = 0; double sb = 0; double bl = 0; double br = 0; double decay = 0; int i_ = 0; x = (double[])x.Clone(); y = (double[])y.Clone(); w = (double[])w.Clone(); xc = (double[])xc.Clone(); yc = (double[])yc.Clone(); info = 0; ap.assert(st==0 | st==1, "Spline1DFit: internal error!"); if( st==0 & m<4 ) { info = -1; return; } if( st==1 & m<4 ) { info = -1; return; } if( (n<1 | k<0) | k>=m ) { info = -1; return; } for(i=0; i<=k-1; i++) { info = 0; if( dc[i]<0 ) { info = -1; } if( dc[i]>1 ) { info = -1; } if( info<0 ) { return; } } if( st==1 & m%2!=0 ) { // // Hermite fitter must have even number of basis functions // info = -2; return; } // // weight decay for correct handling of task which becomes // degenerate after constraints are applied // decay = 10000*math.machineepsilon; // // Scale X, Y, XC, YC // lsfitscalexy(ref x, ref y, ref w, n, ref xc, ref yc, dc, k, ref xa, ref xb, ref sa, ref sb, ref xoriginal, ref yoriginal); // // allocate space, initialize: // * SX - grid for basis functions // * SY - values of basis functions at grid points // * FMatrix- values of basis functions at X[] // * CMatrix- values (derivatives) of basis functions at XC[] // y2 = new double[n+m]; w2 = new double[n+m]; fmatrix = new double[n+m, m]; if( k>0 ) { cmatrix = new double[k, m+1]; } if( st==0 ) { // // allocate space for cubic spline // sx = new double[m-2]; sy = new double[m-2]; for(j=0; j<=m-2-1; j++) { sx[j] = (double)(2*j)/(double)(m-2-1)-1; } } if( st==1 ) { // // allocate space for Hermite spline // sx = new double[m/2]; sy = new double[m/2]; sd = new double[m/2]; for(j=0; j<=m/2-1; j++) { sx[j] = (double)(2*j)/(double)(m/2-1)-1; } } // // Prepare design and constraints matrices: // * fill constraints matrix // * fill first N rows of design matrix with values // * fill next M rows of design matrix with regularizing term // * append M zeros to Y // * append M elements, mean(abs(W)) each, to W // for(j=0; j<=m-1; j++) { // // prepare Jth basis function // if( st==0 ) { // // cubic spline basis // for(i=0; i<=m-2-1; i++) { sy[i] = 0; } bl = 0; br = 0; if( j=0 & dc[i]<=2, "Spline1DFit: internal error!"); spline1d.spline1ddiff(s2, xc[i], ref v0, ref v1, ref v2); if( dc[i]==0 ) { cmatrix[i,j] = v0; } if( dc[i]==1 ) { cmatrix[i,j] = v1; } if( dc[i]==2 ) { cmatrix[i,j] = v2; } } } for(i=0; i<=k-1; i++) { cmatrix[i,m] = yc[i]; } for(i=0; i<=m-1; i++) { for(j=0; j<=m-1; j++) { if( i==j ) { fmatrix[n+i,j] = decay; } else { fmatrix[n+i,j] = 0; } } } y2 = new double[n+m]; w2 = new double[n+m]; for(i_=0; i_<=n-1;i_++) { y2[i_] = y[i_]; } for(i_=0; i_<=n-1;i_++) { w2[i_] = w[i_]; } mx = 0; for(i=0; i<=n-1; i++) { mx = mx+Math.Abs(w[i]); } mx = mx/n; for(i=0; i<=m-1; i++) { y2[n+i] = 0; w2[n+i] = mx; } // // Solve constrained task // if( k>0 ) { // // solve using regularization // lsfitlinearwc(y2, w2, fmatrix, cmatrix, n+m, m, k, ref info, ref tmp, lrep); } else { // // no constraints, no regularization needed // lsfitlinearwc(y, w, fmatrix, cmatrix, n, m, k, ref info, ref tmp, lrep); } if( info<0 ) { return; } // // Generate spline and scale it // if( st==0 ) { // // cubic spline basis // for(i_=0; i_<=m-2-1;i_++) { sy[i_] = tmp[i_]; } spline1d.spline1dbuildcubic(sx, sy, m-2, 1, tmp[m-2], 1, tmp[m-1], s); } if( st==1 ) { // // Hermite basis // for(i=0; i<=m/2-1; i++) { sy[i] = tmp[2*i]; sd[i] = tmp[2*i+1]; } spline1d.spline1dbuildhermite(sx, sy, sd, m/2, s); } spline1d.spline1dlintransx(s, 2/(xb-xa), -((xa+xb)/(xb-xa))); spline1d.spline1dlintransy(s, sb-sa, sa); // // Scale absolute errors obtained from LSFitLinearW. // Relative error should be calculated separately // (because of shifting/scaling of the task) // rep.taskrcond = lrep.taskrcond; rep.rmserror = lrep.rmserror*(sb-sa); rep.avgerror = lrep.avgerror*(sb-sa); rep.maxerror = lrep.maxerror*(sb-sa); rep.avgrelerror = 0; relcnt = 0; for(i=0; i<=n-1; i++) { if( (double)(yoriginal[i])!=(double)(0) ) { rep.avgrelerror = rep.avgrelerror+Math.Abs(spline1d.spline1dcalc(s, xoriginal[i])-yoriginal[i])/Math.Abs(yoriginal[i]); relcnt = relcnt+1; } } if( relcnt!=0 ) { rep.avgrelerror = rep.avgrelerror/relcnt; } } /************************************************************************* Internal fitting subroutine *************************************************************************/ private static void lsfitlinearinternal(double[] y, double[] w, double[,] fmatrix, int n, int m, ref int info, ref double[] c, lsfitreport rep) { double threshold = 0; double[,] ft = new double[0,0]; double[,] q = new double[0,0]; double[,] l = new double[0,0]; double[,] r = new double[0,0]; double[] b = new double[0]; double[] wmod = new double[0]; double[] tau = new double[0]; int i = 0; int j = 0; double v = 0; double[] sv = new double[0]; double[,] u = new double[0,0]; double[,] vt = new double[0,0]; double[] tmp = new double[0]; double[] utb = new double[0]; double[] sutb = new double[0]; int relcnt = 0; int i_ = 0; info = 0; c = new double[0]; if( n<1 | m<1 ) { info = -1; return; } info = 1; threshold = Math.Sqrt(math.machineepsilon); // // Degenerate case, needs special handling // if( n=M. Generate design matrix and reduce to N=M using // QR decomposition. // ft = new double[n, m]; b = new double[n]; for(j=0; j<=n-1; j++) { v = w[j]; for(i_=0; i_<=m-1;i_++) { ft[j,i_] = v*fmatrix[j,i_]; } b[j] = w[j]*y[j]; } ortfac.rmatrixqr(ref ft, n, m, ref tau); ortfac.rmatrixqrunpackq(ft, n, m, tau, m, ref q); ortfac.rmatrixqrunpackr(ft, n, m, ref r); tmp = new double[m]; for(i=0; i<=m-1; i++) { tmp[i] = 0; } for(i=0; i<=n-1; i++) { v = b[i]; for(i_=0; i_<=m-1;i_++) { tmp[i_] = tmp[i_] + v*q[i,i_]; } } b = new double[m]; for(i_=0; i_<=m-1;i_++) { b[i_] = tmp[i_]; } // // R contains reduced MxM design upper triangular matrix, // B contains reduced Mx1 right part. // // Determine system condition number and decide // should we use triangular solver (faster) or // SVD-based solver (more stable). // // We can use LU-based RCond estimator for this task. // rep.taskrcond = rcond.rmatrixlurcondinf(r, m); if( (double)(rep.taskrcond)>(double)(threshold) ) { // // use QR-based solver // c = new double[m]; c[m-1] = b[m-1]/r[m-1,m-1]; for(i=m-2; i>=0; i--) { v = 0.0; for(i_=i+1; i_<=m-1;i_++) { v += r[i,i_]*c[i_]; } c[i] = (b[i]-v)/r[i,i]; } } else { // // use SVD-based solver // if( !svd.rmatrixsvd(r, m, m, 1, 1, 2, ref sv, ref u, ref vt) ) { info = -4; return; } utb = new double[m]; sutb = new double[m]; for(i=0; i<=m-1; i++) { utb[i] = 0; } for(i=0; i<=m-1; i++) { v = b[i]; for(i_=0; i_<=m-1;i_++) { utb[i_] = utb[i_] + v*u[i,i_]; } } if( (double)(sv[0])>(double)(0) ) { rep.taskrcond = sv[m-1]/sv[0]; for(i=0; i<=m-1; i++) { if( (double)(sv[i])>(double)(threshold*sv[0]) ) { sutb[i] = utb[i]/sv[i]; } else { sutb[i] = 0; } } } else { rep.taskrcond = 0; for(i=0; i<=m-1; i++) { sutb[i] = 0; } } c = new double[m]; for(i=0; i<=m-1; i++) { c[i] = 0; } for(i=0; i<=m-1; i++) { v = sutb[i]; for(i_=0; i_<=m-1;i_++) { c[i_] = c[i_] + v*vt[i,i_]; } } } // // calculate errors // rep.rmserror = 0; rep.avgerror = 0; rep.avgrelerror = 0; rep.maxerror = 0; relcnt = 0; for(i=0; i<=n-1; i++) { v = 0.0; for(i_=0; i_<=m-1;i_++) { v += fmatrix[i,i_]*c[i_]; } rep.rmserror = rep.rmserror+math.sqr(v-y[i]); rep.avgerror = rep.avgerror+Math.Abs(v-y[i]); if( (double)(y[i])!=(double)(0) ) { rep.avgrelerror = rep.avgrelerror+Math.Abs(v-y[i])/Math.Abs(y[i]); relcnt = relcnt+1; } rep.maxerror = Math.Max(rep.maxerror, Math.Abs(v-y[i])); } rep.rmserror = Math.Sqrt(rep.rmserror/n); rep.avgerror = rep.avgerror/n; if( relcnt!=0 ) { rep.avgrelerror = rep.avgrelerror/relcnt; } } /************************************************************************* Internal subroutine *************************************************************************/ private static void lsfitclearrequestfields(lsfitstate state) { state.needf = false; state.needfg = false; state.needfgh = false; state.xupdated = false; } /************************************************************************* Internal subroutine, calculates barycentric basis functions. Used for efficient simultaneous calculation of N basis functions. -- ALGLIB -- Copyright 17.08.2009 by Bochkanov Sergey *************************************************************************/ private static void barycentriccalcbasis(ratint.barycentricinterpolant b, double t, ref double[] y) { double s2 = 0; double s = 0; double v = 0; int i = 0; int j = 0; int i_ = 0; // // special case: N=1 // if( b.n==1 ) { y[0] = 1; return; } // // Here we assume that task is normalized, i.e.: // 1. abs(Y[i])<=1 // 2. abs(W[i])<=1 // 3. X[] is ordered // // First, we decide: should we use "safe" formula (guarded // against overflow) or fast one? // s = Math.Abs(t-b.x[0]); for(i=0; i<=b.n-1; i++) { v = b.x[i]; if( (double)(v)==(double)(t) ) { for(j=0; j<=b.n-1; j++) { y[j] = 0; } y[i] = 1; return; } v = Math.Abs(t-v); if( (double)(v)<(double)(s) ) { s = v; } } s2 = 0; for(i=0; i<=b.n-1; i++) { v = s/(t-b.x[i]); v = v*b.w[i]; y[i] = v; s2 = s2+v; } v = 1/s2; for(i_=0; i_<=b.n-1;i_++) { y[i_] = v*y[i_]; } } /************************************************************************* This is internal function for Chebyshev fitting. It assumes that input data are normalized: * X/XC belong to [-1,+1], * mean(Y)=0, stddev(Y)=1. It does not checks inputs for errors. This function is used to fit general (shifted) Chebyshev models, power basis models or barycentric models. INPUT PARAMETERS: X - points, array[0..N-1]. Y - function values, array[0..N-1]. W - weights, array[0..N-1] N - number of points, N>0. XC - points where polynomial values/derivatives are constrained, array[0..K-1]. YC - values of constraints, array[0..K-1] DC - array[0..K-1], types of constraints: * DC[i]=0 means that P(XC[i])=YC[i] * DC[i]=1 means that P'(XC[i])=YC[i] K - number of constraints, 0<=K=1 OUTPUT PARAMETERS: Info- same format as in LSFitLinearW() subroutine: * Info>0 task is solved * Info<=0 an error occured: -4 means inconvergence of internal SVD -3 means inconsistent constraints C - interpolant in Chebyshev form; [-1,+1] is used as base interval Rep - report, same format as in LSFitLinearW() subroutine. Following fields are set: * RMSError rms error on the (X,Y). * AvgError average error on the (X,Y). * AvgRelError average relative error on the non-zero Y * MaxError maximum error NON-WEIGHTED ERRORS ARE CALCULATED IMPORTANT: this subroitine doesn't calculate task's condition number for K<>0. -- ALGLIB PROJECT -- Copyright 10.12.2009 by Bochkanov Sergey *************************************************************************/ private static void internalchebyshevfit(double[] x, double[] y, double[] w, int n, double[] xc, double[] yc, int[] dc, int k, int m, ref int info, ref double[] c, lsfitreport rep) { double[] y2 = new double[0]; double[] w2 = new double[0]; double[] tmp = new double[0]; double[] tmp2 = new double[0]; double[] tmpdiff = new double[0]; double[] bx = new double[0]; double[] by = new double[0]; double[] bw = new double[0]; double[,] fmatrix = new double[0,0]; double[,] cmatrix = new double[0,0]; int i = 0; int j = 0; double mx = 0; double decay = 0; int i_ = 0; xc = (double[])xc.Clone(); yc = (double[])yc.Clone(); info = 0; c = new double[0]; // // weight decay for correct handling of task which becomes // degenerate after constraints are applied // decay = 10000*math.machineepsilon; // // allocate space, initialize/fill: // * FMatrix- values of basis functions at X[] // * CMatrix- values (derivatives) of basis functions at XC[] // * fill constraints matrix // * fill first N rows of design matrix with values // * fill next M rows of design matrix with regularizing term // * append M zeros to Y // * append M elements, mean(abs(W)) each, to W // y2 = new double[n+m]; w2 = new double[n+m]; tmp = new double[m]; tmpdiff = new double[m]; fmatrix = new double[n+m, m]; if( k>0 ) { cmatrix = new double[k, m+1]; } // // Fill design matrix, Y2, W2: // * first N rows with basis functions for original points // * next M rows with decay terms // for(i=0; i<=n-1; i++) { // // prepare Ith row // use Tmp for calculations to avoid multidimensional arrays overhead // for(j=0; j<=m-1; j++) { if( j==0 ) { tmp[j] = 1; } else { if( j==1 ) { tmp[j] = x[i]; } else { tmp[j] = 2*x[i]*tmp[j-1]-tmp[j-2]; } } } for(i_=0; i_<=m-1;i_++) { fmatrix[i,i_] = tmp[i_]; } } for(i=0; i<=m-1; i++) { for(j=0; j<=m-1; j++) { if( i==j ) { fmatrix[n+i,j] = decay; } else { fmatrix[n+i,j] = 0; } } } for(i_=0; i_<=n-1;i_++) { y2[i_] = y[i_]; } for(i_=0; i_<=n-1;i_++) { w2[i_] = w[i_]; } mx = 0; for(i=0; i<=n-1; i++) { mx = mx+Math.Abs(w[i]); } mx = mx/n; for(i=0; i<=m-1; i++) { y2[n+i] = 0; w2[n+i] = mx; } // // fill constraints matrix // for(i=0; i<=k-1; i++) { // // prepare Ith row // use Tmp for basis function values, // TmpDiff for basos function derivatives // for(j=0; j<=m-1; j++) { if( j==0 ) { tmp[j] = 1; tmpdiff[j] = 0; } else { if( j==1 ) { tmp[j] = xc[i]; tmpdiff[j] = 1; } else { tmp[j] = 2*xc[i]*tmp[j-1]-tmp[j-2]; tmpdiff[j] = 2*(tmp[j-1]+xc[i]*tmpdiff[j-1])-tmpdiff[j-2]; } } } if( dc[i]==0 ) { for(i_=0; i_<=m-1;i_++) { cmatrix[i,i_] = tmp[i_]; } } if( dc[i]==1 ) { for(i_=0; i_<=m-1;i_++) { cmatrix[i,i_] = tmpdiff[i_]; } } cmatrix[i,m] = yc[i]; } // // Solve constrained task // if( k>0 ) { // // solve using regularization // lsfitlinearwc(y2, w2, fmatrix, cmatrix, n+m, m, k, ref info, ref c, rep); } else { // // no constraints, no regularization needed // lsfitlinearwc(y, w, fmatrix, cmatrix, n, m, 0, ref info, ref c, rep); } if( info<0 ) { return; } } /************************************************************************* Internal Floater-Hormann fitting subroutine for fixed D *************************************************************************/ private static void barycentricfitwcfixedd(double[] x, double[] y, double[] w, int n, double[] xc, double[] yc, int[] dc, int k, int m, int d, ref int info, ratint.barycentricinterpolant b, barycentricfitreport rep) { double[,] fmatrix = new double[0,0]; double[,] cmatrix = new double[0,0]; double[] y2 = new double[0]; double[] w2 = new double[0]; double[] sx = new double[0]; double[] sy = new double[0]; double[] sbf = new double[0]; double[] xoriginal = new double[0]; double[] yoriginal = new double[0]; double[] tmp = new double[0]; lsfitreport lrep = new lsfitreport(); double v0 = 0; double v1 = 0; double mx = 0; ratint.barycentricinterpolant b2 = new ratint.barycentricinterpolant(); int i = 0; int j = 0; int relcnt = 0; double xa = 0; double xb = 0; double sa = 0; double sb = 0; double decay = 0; int i_ = 0; x = (double[])x.Clone(); y = (double[])y.Clone(); w = (double[])w.Clone(); xc = (double[])xc.Clone(); yc = (double[])yc.Clone(); info = 0; if( ((n<1 | m<2) | k<0) | k>=m ) { info = -1; return; } for(i=0; i<=k-1; i++) { info = 0; if( dc[i]<0 ) { info = -1; } if( dc[i]>1 ) { info = -1; } if( info<0 ) { return; } } // // weight decay for correct handling of task which becomes // degenerate after constraints are applied // decay = 10000*math.machineepsilon; // // Scale X, Y, XC, YC // lsfitscalexy(ref x, ref y, ref w, n, ref xc, ref yc, dc, k, ref xa, ref xb, ref sa, ref sb, ref xoriginal, ref yoriginal); // // allocate space, initialize: // * FMatrix- values of basis functions at X[] // * CMatrix- values (derivatives) of basis functions at XC[] // y2 = new double[n+m]; w2 = new double[n+m]; fmatrix = new double[n+m, m]; if( k>0 ) { cmatrix = new double[k, m+1]; } y2 = new double[n+m]; w2 = new double[n+m]; // // Prepare design and constraints matrices: // * fill constraints matrix // * fill first N rows of design matrix with values // * fill next M rows of design matrix with regularizing term // * append M zeros to Y // * append M elements, mean(abs(W)) each, to W // sx = new double[m]; sy = new double[m]; sbf = new double[m]; for(j=0; j<=m-1; j++) { sx[j] = (double)(2*j)/(double)(m-1)-1; } for(i=0; i<=m-1; i++) { sy[i] = 1; } ratint.barycentricbuildfloaterhormann(sx, sy, m, d, b2); mx = 0; for(i=0; i<=n-1; i++) { barycentriccalcbasis(b2, x[i], ref sbf); for(i_=0; i_<=m-1;i_++) { fmatrix[i,i_] = sbf[i_]; } y2[i] = y[i]; w2[i] = w[i]; mx = mx+Math.Abs(w[i])/n; } for(i=0; i<=m-1; i++) { for(j=0; j<=m-1; j++) { if( i==j ) { fmatrix[n+i,j] = decay; } else { fmatrix[n+i,j] = 0; } } y2[n+i] = 0; w2[n+i] = mx; } if( k>0 ) { for(j=0; j<=m-1; j++) { for(i=0; i<=m-1; i++) { sy[i] = 0; } sy[j] = 1; ratint.barycentricbuildfloaterhormann(sx, sy, m, d, b2); for(i=0; i<=k-1; i++) { ap.assert(dc[i]>=0 & dc[i]<=1, "BarycentricFit: internal error!"); ratint.barycentricdiff1(b2, xc[i], ref v0, ref v1); if( dc[i]==0 ) { cmatrix[i,j] = v0; } if( dc[i]==1 ) { cmatrix[i,j] = v1; } } } for(i=0; i<=k-1; i++) { cmatrix[i,m] = yc[i]; } } // // Solve constrained task // if( k>0 ) { // // solve using regularization // lsfitlinearwc(y2, w2, fmatrix, cmatrix, n+m, m, k, ref info, ref tmp, lrep); } else { // // no constraints, no regularization needed // lsfitlinearwc(y, w, fmatrix, cmatrix, n, m, k, ref info, ref tmp, lrep); } if( info<0 ) { return; } // // Generate interpolant and scale it // for(i_=0; i_<=m-1;i_++) { sy[i_] = tmp[i_]; } ratint.barycentricbuildfloaterhormann(sx, sy, m, d, b); ratint.barycentriclintransx(b, 2/(xb-xa), -((xa+xb)/(xb-xa))); ratint.barycentriclintransy(b, sb-sa, sa); // // Scale absolute errors obtained from LSFitLinearW. // Relative error should be calculated separately // (because of shifting/scaling of the task) // rep.taskrcond = lrep.taskrcond; rep.rmserror = lrep.rmserror*(sb-sa); rep.avgerror = lrep.avgerror*(sb-sa); rep.maxerror = lrep.maxerror*(sb-sa); rep.avgrelerror = 0; relcnt = 0; for(i=0; i<=n-1; i++) { if( (double)(yoriginal[i])!=(double)(0) ) { rep.avgrelerror = rep.avgrelerror+Math.Abs(ratint.barycentriccalc(b, xoriginal[i])-yoriginal[i])/Math.Abs(yoriginal[i]); relcnt = relcnt+1; } } if( relcnt!=0 ) { rep.avgrelerror = rep.avgrelerror/relcnt; } } } public class pspline { /************************************************************************* Parametric spline inteprolant: 2-dimensional curve. You should not try to access its members directly - use PSpline2XXXXXXXX() functions instead. *************************************************************************/ public class pspline2interpolant { public int n; public bool periodic; public double[] p; public spline1d.spline1dinterpolant x; public spline1d.spline1dinterpolant y; public pspline2interpolant() { p = new double[0]; x = new spline1d.spline1dinterpolant(); y = new spline1d.spline1dinterpolant(); } }; /************************************************************************* Parametric spline inteprolant: 3-dimensional curve. You should not try to access its members directly - use PSpline3XXXXXXXX() functions instead. *************************************************************************/ public class pspline3interpolant { public int n; public bool periodic; public double[] p; public spline1d.spline1dinterpolant x; public spline1d.spline1dinterpolant y; public spline1d.spline1dinterpolant z; public pspline3interpolant() { p = new double[0]; x = new spline1d.spline1dinterpolant(); y = new spline1d.spline1dinterpolant(); z = new spline1d.spline1dinterpolant(); } }; /************************************************************************* This function builds non-periodic 2-dimensional parametric spline which starts at (X[0],Y[0]) and ends at (X[N-1],Y[N-1]). INPUT PARAMETERS: XY - points, array[0..N-1,0..1]. XY[I,0:1] corresponds to the Ith point. Order of points is important! N - points count, N>=5 for Akima splines, N>=2 for other types of splines. ST - spline type: * 0 Akima spline * 1 parabolically terminated Catmull-Rom spline (Tension=0) * 2 parabolically terminated cubic spline PT - parameterization type: * 0 uniform * 1 chord length * 2 centripetal OUTPUT PARAMETERS: P - parametric spline interpolant NOTES: * this function assumes that there all consequent points are distinct. I.e. (x0,y0)<>(x1,y1), (x1,y1)<>(x2,y2), (x2,y2)<>(x3,y3) and so on. However, non-consequent points may coincide, i.e. we can have (x0,y0)= =(x2,y2). -- ALGLIB PROJECT -- Copyright 28.05.2010 by Bochkanov Sergey *************************************************************************/ public static void pspline2build(double[,] xy, int n, int st, int pt, pspline2interpolant p) { double[] tmp = new double[0]; int i_ = 0; xy = (double[,])xy.Clone(); ap.assert(st>=0 & st<=2, "PSpline2Build: incorrect spline type!"); ap.assert(pt>=0 & pt<=2, "PSpline2Build: incorrect parameterization type!"); if( st==0 ) { ap.assert(n>=5, "PSpline2Build: N<5 (minimum value for Akima splines)!"); } else { ap.assert(n>=2, "PSpline2Build: N<2!"); } // // Prepare // p.n = n; p.periodic = false; tmp = new double[n]; // // Build parameterization, check that all parameters are distinct // pspline2par(xy, n, pt, ref p.p); ap.assert(apserv.aredistinct(p.p, n), "PSpline2Build: consequent points are too close!"); // // Build splines // if( st==0 ) { for(i_=0; i_<=n-1;i_++) { tmp[i_] = xy[i_,0]; } spline1d.spline1dbuildakima(p.p, tmp, n, p.x); for(i_=0; i_<=n-1;i_++) { tmp[i_] = xy[i_,1]; } spline1d.spline1dbuildakima(p.p, tmp, n, p.y); } if( st==1 ) { for(i_=0; i_<=n-1;i_++) { tmp[i_] = xy[i_,0]; } spline1d.spline1dbuildcatmullrom(p.p, tmp, n, 0, 0.0, p.x); for(i_=0; i_<=n-1;i_++) { tmp[i_] = xy[i_,1]; } spline1d.spline1dbuildcatmullrom(p.p, tmp, n, 0, 0.0, p.y); } if( st==2 ) { for(i_=0; i_<=n-1;i_++) { tmp[i_] = xy[i_,0]; } spline1d.spline1dbuildcubic(p.p, tmp, n, 0, 0.0, 0, 0.0, p.x); for(i_=0; i_<=n-1;i_++) { tmp[i_] = xy[i_,1]; } spline1d.spline1dbuildcubic(p.p, tmp, n, 0, 0.0, 0, 0.0, p.y); } } /************************************************************************* This function builds non-periodic 3-dimensional parametric spline which starts at (X[0],Y[0],Z[0]) and ends at (X[N-1],Y[N-1],Z[N-1]). Same as PSpline2Build() function, but for 3D, so we won't duplicate its description here. -- ALGLIB PROJECT -- Copyright 28.05.2010 by Bochkanov Sergey *************************************************************************/ public static void pspline3build(double[,] xy, int n, int st, int pt, pspline3interpolant p) { double[] tmp = new double[0]; int i_ = 0; xy = (double[,])xy.Clone(); ap.assert(st>=0 & st<=2, "PSpline3Build: incorrect spline type!"); ap.assert(pt>=0 & pt<=2, "PSpline3Build: incorrect parameterization type!"); if( st==0 ) { ap.assert(n>=5, "PSpline3Build: N<5 (minimum value for Akima splines)!"); } else { ap.assert(n>=2, "PSpline3Build: N<2!"); } // // Prepare // p.n = n; p.periodic = false; tmp = new double[n]; // // Build parameterization, check that all parameters are distinct // pspline3par(xy, n, pt, ref p.p); ap.assert(apserv.aredistinct(p.p, n), "PSpline3Build: consequent points are too close!"); // // Build splines // if( st==0 ) { for(i_=0; i_<=n-1;i_++) { tmp[i_] = xy[i_,0]; } spline1d.spline1dbuildakima(p.p, tmp, n, p.x); for(i_=0; i_<=n-1;i_++) { tmp[i_] = xy[i_,1]; } spline1d.spline1dbuildakima(p.p, tmp, n, p.y); for(i_=0; i_<=n-1;i_++) { tmp[i_] = xy[i_,2]; } spline1d.spline1dbuildakima(p.p, tmp, n, p.z); } if( st==1 ) { for(i_=0; i_<=n-1;i_++) { tmp[i_] = xy[i_,0]; } spline1d.spline1dbuildcatmullrom(p.p, tmp, n, 0, 0.0, p.x); for(i_=0; i_<=n-1;i_++) { tmp[i_] = xy[i_,1]; } spline1d.spline1dbuildcatmullrom(p.p, tmp, n, 0, 0.0, p.y); for(i_=0; i_<=n-1;i_++) { tmp[i_] = xy[i_,2]; } spline1d.spline1dbuildcatmullrom(p.p, tmp, n, 0, 0.0, p.z); } if( st==2 ) { for(i_=0; i_<=n-1;i_++) { tmp[i_] = xy[i_,0]; } spline1d.spline1dbuildcubic(p.p, tmp, n, 0, 0.0, 0, 0.0, p.x); for(i_=0; i_<=n-1;i_++) { tmp[i_] = xy[i_,1]; } spline1d.spline1dbuildcubic(p.p, tmp, n, 0, 0.0, 0, 0.0, p.y); for(i_=0; i_<=n-1;i_++) { tmp[i_] = xy[i_,2]; } spline1d.spline1dbuildcubic(p.p, tmp, n, 0, 0.0, 0, 0.0, p.z); } } /************************************************************************* This function builds periodic 2-dimensional parametric spline which starts at (X[0],Y[0]), goes through all points to (X[N-1],Y[N-1]) and then back to (X[0],Y[0]). INPUT PARAMETERS: XY - points, array[0..N-1,0..1]. XY[I,0:1] corresponds to the Ith point. XY[N-1,0:1] must be different from XY[0,0:1]. Order of points is important! N - points count, N>=3 for other types of splines. ST - spline type: * 1 Catmull-Rom spline (Tension=0) with cyclic boundary conditions * 2 cubic spline with cyclic boundary conditions PT - parameterization type: * 0 uniform * 1 chord length * 2 centripetal OUTPUT PARAMETERS: P - parametric spline interpolant NOTES: * this function assumes that there all consequent points are distinct. I.e. (x0,y0)<>(x1,y1), (x1,y1)<>(x2,y2), (x2,y2)<>(x3,y3) and so on. However, non-consequent points may coincide, i.e. we can have (x0,y0)= =(x2,y2). * last point of sequence is NOT equal to the first point. You shouldn't make curve "explicitly periodic" by making them equal. -- ALGLIB PROJECT -- Copyright 28.05.2010 by Bochkanov Sergey *************************************************************************/ public static void pspline2buildperiodic(double[,] xy, int n, int st, int pt, pspline2interpolant p) { double[,] xyp = new double[0,0]; double[] tmp = new double[0]; int i_ = 0; xy = (double[,])xy.Clone(); ap.assert(st>=1 & st<=2, "PSpline2BuildPeriodic: incorrect spline type!"); ap.assert(pt>=0 & pt<=2, "PSpline2BuildPeriodic: incorrect parameterization type!"); ap.assert(n>=3, "PSpline2BuildPeriodic: N<3!"); // // Prepare // p.n = n; p.periodic = true; tmp = new double[n+1]; xyp = new double[n+1, 2]; for(i_=0; i_<=n-1;i_++) { xyp[i_,0] = xy[i_,0]; } for(i_=0; i_<=n-1;i_++) { xyp[i_,1] = xy[i_,1]; } for(i_=0; i_<=1;i_++) { xyp[n,i_] = xy[0,i_]; } // // Build parameterization, check that all parameters are distinct // pspline2par(xyp, n+1, pt, ref p.p); ap.assert(apserv.aredistinct(p.p, n+1), "PSpline2BuildPeriodic: consequent (or first and last) points are too close!"); // // Build splines // if( st==1 ) { for(i_=0; i_<=n;i_++) { tmp[i_] = xyp[i_,0]; } spline1d.spline1dbuildcatmullrom(p.p, tmp, n+1, -1, 0.0, p.x); for(i_=0; i_<=n;i_++) { tmp[i_] = xyp[i_,1]; } spline1d.spline1dbuildcatmullrom(p.p, tmp, n+1, -1, 0.0, p.y); } if( st==2 ) { for(i_=0; i_<=n;i_++) { tmp[i_] = xyp[i_,0]; } spline1d.spline1dbuildcubic(p.p, tmp, n+1, -1, 0.0, -1, 0.0, p.x); for(i_=0; i_<=n;i_++) { tmp[i_] = xyp[i_,1]; } spline1d.spline1dbuildcubic(p.p, tmp, n+1, -1, 0.0, -1, 0.0, p.y); } } /************************************************************************* This function builds periodic 3-dimensional parametric spline which starts at (X[0],Y[0],Z[0]), goes through all points to (X[N-1],Y[N-1],Z[N-1]) and then back to (X[0],Y[0],Z[0]). Same as PSpline2Build() function, but for 3D, so we won't duplicate its description here. -- ALGLIB PROJECT -- Copyright 28.05.2010 by Bochkanov Sergey *************************************************************************/ public static void pspline3buildperiodic(double[,] xy, int n, int st, int pt, pspline3interpolant p) { double[,] xyp = new double[0,0]; double[] tmp = new double[0]; int i_ = 0; xy = (double[,])xy.Clone(); ap.assert(st>=1 & st<=2, "PSpline3BuildPeriodic: incorrect spline type!"); ap.assert(pt>=0 & pt<=2, "PSpline3BuildPeriodic: incorrect parameterization type!"); ap.assert(n>=3, "PSpline3BuildPeriodic: N<3!"); // // Prepare // p.n = n; p.periodic = true; tmp = new double[n+1]; xyp = new double[n+1, 3]; for(i_=0; i_<=n-1;i_++) { xyp[i_,0] = xy[i_,0]; } for(i_=0; i_<=n-1;i_++) { xyp[i_,1] = xy[i_,1]; } for(i_=0; i_<=n-1;i_++) { xyp[i_,2] = xy[i_,2]; } for(i_=0; i_<=2;i_++) { xyp[n,i_] = xy[0,i_]; } // // Build parameterization, check that all parameters are distinct // pspline3par(xyp, n+1, pt, ref p.p); ap.assert(apserv.aredistinct(p.p, n+1), "PSplineBuild2Periodic: consequent (or first and last) points are too close!"); // // Build splines // if( st==1 ) { for(i_=0; i_<=n;i_++) { tmp[i_] = xyp[i_,0]; } spline1d.spline1dbuildcatmullrom(p.p, tmp, n+1, -1, 0.0, p.x); for(i_=0; i_<=n;i_++) { tmp[i_] = xyp[i_,1]; } spline1d.spline1dbuildcatmullrom(p.p, tmp, n+1, -1, 0.0, p.y); for(i_=0; i_<=n;i_++) { tmp[i_] = xyp[i_,2]; } spline1d.spline1dbuildcatmullrom(p.p, tmp, n+1, -1, 0.0, p.z); } if( st==2 ) { for(i_=0; i_<=n;i_++) { tmp[i_] = xyp[i_,0]; } spline1d.spline1dbuildcubic(p.p, tmp, n+1, -1, 0.0, -1, 0.0, p.x); for(i_=0; i_<=n;i_++) { tmp[i_] = xyp[i_,1]; } spline1d.spline1dbuildcubic(p.p, tmp, n+1, -1, 0.0, -1, 0.0, p.y); for(i_=0; i_<=n;i_++) { tmp[i_] = xyp[i_,2]; } spline1d.spline1dbuildcubic(p.p, tmp, n+1, -1, 0.0, -1, 0.0, p.z); } } /************************************************************************* This function returns vector of parameter values correspoding to points. I.e. for P created from (X[0],Y[0])...(X[N-1],Y[N-1]) and U=TValues(P) we have (X[0],Y[0]) = PSpline2Calc(P,U[0]), (X[1],Y[1]) = PSpline2Calc(P,U[1]), (X[2],Y[2]) = PSpline2Calc(P,U[2]), ... INPUT PARAMETERS: P - parametric spline interpolant OUTPUT PARAMETERS: N - array size T - array[0..N-1] NOTES: * for non-periodic splines U[0]=0, U[0]=2, "PSpline2ParameterValues: internal error!"); n = p.n; t = new double[n]; for(i_=0; i_<=n-1;i_++) { t[i_] = p.p[i_]; } t[0] = 0; if( !p.periodic ) { t[n-1] = 1; } } /************************************************************************* This function returns vector of parameter values correspoding to points. Same as PSpline2ParameterValues(), but for 3D. -- ALGLIB PROJECT -- Copyright 28.05.2010 by Bochkanov Sergey *************************************************************************/ public static void pspline3parametervalues(pspline3interpolant p, ref int n, ref double[] t) { int i_ = 0; n = 0; t = new double[0]; ap.assert(p.n>=2, "PSpline3ParameterValues: internal error!"); n = p.n; t = new double[n]; for(i_=0; i_<=n-1;i_++) { t[i_] = p.p[i_]; } t[0] = 0; if( !p.periodic ) { t[n-1] = 1; } } /************************************************************************* This function calculates the value of the parametric spline for a given value of parameter T INPUT PARAMETERS: P - parametric spline interpolant T - point: * T in [0,1] corresponds to interval spanned by points * for non-periodic splines T<0 (or T>1) correspond to parts of the curve before the first (after the last) point * for periodic splines T<0 (or T>1) are projected into [0,1] by making T=T-floor(T). OUTPUT PARAMETERS: X - X-position Y - Y-position -- ALGLIB PROJECT -- Copyright 28.05.2010 by Bochkanov Sergey *************************************************************************/ public static void pspline2calc(pspline2interpolant p, double t, ref double x, ref double y) { x = 0; y = 0; if( p.periodic ) { t = t-(int)Math.Floor(t); } x = spline1d.spline1dcalc(p.x, t); y = spline1d.spline1dcalc(p.y, t); } /************************************************************************* This function calculates the value of the parametric spline for a given value of parameter T. INPUT PARAMETERS: P - parametric spline interpolant T - point: * T in [0,1] corresponds to interval spanned by points * for non-periodic splines T<0 (or T>1) correspond to parts of the curve before the first (after the last) point * for periodic splines T<0 (or T>1) are projected into [0,1] by making T=T-floor(T). OUTPUT PARAMETERS: X - X-position Y - Y-position Z - Z-position -- ALGLIB PROJECT -- Copyright 28.05.2010 by Bochkanov Sergey *************************************************************************/ public static void pspline3calc(pspline3interpolant p, double t, ref double x, ref double y, ref double z) { x = 0; y = 0; z = 0; if( p.periodic ) { t = t-(int)Math.Floor(t); } x = spline1d.spline1dcalc(p.x, t); y = spline1d.spline1dcalc(p.y, t); z = spline1d.spline1dcalc(p.z, t); } /************************************************************************* This function calculates tangent vector for a given value of parameter T INPUT PARAMETERS: P - parametric spline interpolant T - point: * T in [0,1] corresponds to interval spanned by points * for non-periodic splines T<0 (or T>1) correspond to parts of the curve before the first (after the last) point * for periodic splines T<0 (or T>1) are projected into [0,1] by making T=T-floor(T). OUTPUT PARAMETERS: X - X-component of tangent vector (normalized) Y - Y-component of tangent vector (normalized) NOTE: X^2+Y^2 is either 1 (for non-zero tangent vector) or 0. -- ALGLIB PROJECT -- Copyright 28.05.2010 by Bochkanov Sergey *************************************************************************/ public static void pspline2tangent(pspline2interpolant p, double t, ref double x, ref double y) { double v = 0; double v0 = 0; double v1 = 0; x = 0; y = 0; if( p.periodic ) { t = t-(int)Math.Floor(t); } pspline2diff(p, t, ref v0, ref x, ref v1, ref y); if( (double)(x)!=(double)(0) | (double)(y)!=(double)(0) ) { // // this code is a bit more complex than X^2+Y^2 to avoid // overflow for large values of X and Y. // v = apserv.safepythag2(x, y); x = x/v; y = y/v; } } /************************************************************************* This function calculates tangent vector for a given value of parameter T INPUT PARAMETERS: P - parametric spline interpolant T - point: * T in [0,1] corresponds to interval spanned by points * for non-periodic splines T<0 (or T>1) correspond to parts of the curve before the first (after the last) point * for periodic splines T<0 (or T>1) are projected into [0,1] by making T=T-floor(T). OUTPUT PARAMETERS: X - X-component of tangent vector (normalized) Y - Y-component of tangent vector (normalized) Z - Z-component of tangent vector (normalized) NOTE: X^2+Y^2+Z^2 is either 1 (for non-zero tangent vector) or 0. -- ALGLIB PROJECT -- Copyright 28.05.2010 by Bochkanov Sergey *************************************************************************/ public static void pspline3tangent(pspline3interpolant p, double t, ref double x, ref double y, ref double z) { double v = 0; double v0 = 0; double v1 = 0; double v2 = 0; x = 0; y = 0; z = 0; if( p.periodic ) { t = t-(int)Math.Floor(t); } pspline3diff(p, t, ref v0, ref x, ref v1, ref y, ref v2, ref z); if( ((double)(x)!=(double)(0) | (double)(y)!=(double)(0)) | (double)(z)!=(double)(0) ) { v = apserv.safepythag3(x, y, z); x = x/v; y = y/v; z = z/v; } } /************************************************************************* This function calculates derivative, i.e. it returns (dX/dT,dY/dT). INPUT PARAMETERS: P - parametric spline interpolant T - point: * T in [0,1] corresponds to interval spanned by points * for non-periodic splines T<0 (or T>1) correspond to parts of the curve before the first (after the last) point * for periodic splines T<0 (or T>1) are projected into [0,1] by making T=T-floor(T). OUTPUT PARAMETERS: X - X-value DX - X-derivative Y - Y-value DY - Y-derivative -- ALGLIB PROJECT -- Copyright 28.05.2010 by Bochkanov Sergey *************************************************************************/ public static void pspline2diff(pspline2interpolant p, double t, ref double x, ref double dx, ref double y, ref double dy) { double d2s = 0; x = 0; dx = 0; y = 0; dy = 0; if( p.periodic ) { t = t-(int)Math.Floor(t); } spline1d.spline1ddiff(p.x, t, ref x, ref dx, ref d2s); spline1d.spline1ddiff(p.y, t, ref y, ref dy, ref d2s); } /************************************************************************* This function calculates derivative, i.e. it returns (dX/dT,dY/dT,dZ/dT). INPUT PARAMETERS: P - parametric spline interpolant T - point: * T in [0,1] corresponds to interval spanned by points * for non-periodic splines T<0 (or T>1) correspond to parts of the curve before the first (after the last) point * for periodic splines T<0 (or T>1) are projected into [0,1] by making T=T-floor(T). OUTPUT PARAMETERS: X - X-value DX - X-derivative Y - Y-value DY - Y-derivative Z - Z-value DZ - Z-derivative -- ALGLIB PROJECT -- Copyright 28.05.2010 by Bochkanov Sergey *************************************************************************/ public static void pspline3diff(pspline3interpolant p, double t, ref double x, ref double dx, ref double y, ref double dy, ref double z, ref double dz) { double d2s = 0; x = 0; dx = 0; y = 0; dy = 0; z = 0; dz = 0; if( p.periodic ) { t = t-(int)Math.Floor(t); } spline1d.spline1ddiff(p.x, t, ref x, ref dx, ref d2s); spline1d.spline1ddiff(p.y, t, ref y, ref dy, ref d2s); spline1d.spline1ddiff(p.z, t, ref z, ref dz, ref d2s); } /************************************************************************* This function calculates first and second derivative with respect to T. INPUT PARAMETERS: P - parametric spline interpolant T - point: * T in [0,1] corresponds to interval spanned by points * for non-periodic splines T<0 (or T>1) correspond to parts of the curve before the first (after the last) point * for periodic splines T<0 (or T>1) are projected into [0,1] by making T=T-floor(T). OUTPUT PARAMETERS: X - X-value DX - derivative D2X - second derivative Y - Y-value DY - derivative D2Y - second derivative -- ALGLIB PROJECT -- Copyright 28.05.2010 by Bochkanov Sergey *************************************************************************/ public static void pspline2diff2(pspline2interpolant p, double t, ref double x, ref double dx, ref double d2x, ref double y, ref double dy, ref double d2y) { x = 0; dx = 0; d2x = 0; y = 0; dy = 0; d2y = 0; if( p.periodic ) { t = t-(int)Math.Floor(t); } spline1d.spline1ddiff(p.x, t, ref x, ref dx, ref d2x); spline1d.spline1ddiff(p.y, t, ref y, ref dy, ref d2y); } /************************************************************************* This function calculates first and second derivative with respect to T. INPUT PARAMETERS: P - parametric spline interpolant T - point: * T in [0,1] corresponds to interval spanned by points * for non-periodic splines T<0 (or T>1) correspond to parts of the curve before the first (after the last) point * for periodic splines T<0 (or T>1) are projected into [0,1] by making T=T-floor(T). OUTPUT PARAMETERS: X - X-value DX - derivative D2X - second derivative Y - Y-value DY - derivative D2Y - second derivative Z - Z-value DZ - derivative D2Z - second derivative -- ALGLIB PROJECT -- Copyright 28.05.2010 by Bochkanov Sergey *************************************************************************/ public static void pspline3diff2(pspline3interpolant p, double t, ref double x, ref double dx, ref double d2x, ref double y, ref double dy, ref double d2y, ref double z, ref double dz, ref double d2z) { x = 0; dx = 0; d2x = 0; y = 0; dy = 0; d2y = 0; z = 0; dz = 0; d2z = 0; if( p.periodic ) { t = t-(int)Math.Floor(t); } spline1d.spline1ddiff(p.x, t, ref x, ref dx, ref d2x); spline1d.spline1ddiff(p.y, t, ref y, ref dy, ref d2y); spline1d.spline1ddiff(p.z, t, ref z, ref dz, ref d2z); } /************************************************************************* This function calculates arc length, i.e. length of curve between t=a and t=b. INPUT PARAMETERS: P - parametric spline interpolant A,B - parameter values corresponding to arc ends: * B>A will result in positive length returned * B0, "PSpline2ArcLength: internal error!"); return result; } /************************************************************************* This function calculates arc length, i.e. length of curve between t=a and t=b. INPUT PARAMETERS: P - parametric spline interpolant A,B - parameter values corresponding to arc ends: * B>A will result in positive length returned * B0, "PSpline3ArcLength: internal error!"); return result; } /************************************************************************* Builds non-periodic parameterization for 2-dimensional spline *************************************************************************/ private static void pspline2par(double[,] xy, int n, int pt, ref double[] p) { double v = 0; int i = 0; int i_ = 0; p = new double[0]; ap.assert(pt>=0 & pt<=2, "PSpline2Par: internal error!"); // // Build parameterization: // * fill by non-normalized values // * normalize them so we have P[0]=0, P[N-1]=1. // p = new double[n]; if( pt==0 ) { for(i=0; i<=n-1; i++) { p[i] = i; } } if( pt==1 ) { p[0] = 0; for(i=1; i<=n-1; i++) { p[i] = p[i-1]+apserv.safepythag2(xy[i,0]-xy[i-1,0], xy[i,1]-xy[i-1,1]); } } if( pt==2 ) { p[0] = 0; for(i=1; i<=n-1; i++) { p[i] = p[i-1]+Math.Sqrt(apserv.safepythag2(xy[i,0]-xy[i-1,0], xy[i,1]-xy[i-1,1])); } } v = 1/p[n-1]; for(i_=0; i_<=n-1;i_++) { p[i_] = v*p[i_]; } } /************************************************************************* Builds non-periodic parameterization for 3-dimensional spline *************************************************************************/ private static void pspline3par(double[,] xy, int n, int pt, ref double[] p) { double v = 0; int i = 0; int i_ = 0; p = new double[0]; ap.assert(pt>=0 & pt<=2, "PSpline3Par: internal error!"); // // Build parameterization: // * fill by non-normalized values // * normalize them so we have P[0]=0, P[N-1]=1. // p = new double[n]; if( pt==0 ) { for(i=0; i<=n-1; i++) { p[i] = i; } } if( pt==1 ) { p[0] = 0; for(i=1; i<=n-1; i++) { p[i] = p[i-1]+apserv.safepythag3(xy[i,0]-xy[i-1,0], xy[i,1]-xy[i-1,1], xy[i,2]-xy[i-1,2]); } } if( pt==2 ) { p[0] = 0; for(i=1; i<=n-1; i++) { p[i] = p[i-1]+Math.Sqrt(apserv.safepythag3(xy[i,0]-xy[i-1,0], xy[i,1]-xy[i-1,1], xy[i,2]-xy[i-1,2])); } } v = 1/p[n-1]; for(i_=0; i_<=n-1;i_++) { p[i_] = v*p[i_]; } } } public class spline2d { /************************************************************************* 2-dimensional spline inteprolant *************************************************************************/ public class spline2dinterpolant { public int k; public double[] c; public spline2dinterpolant() { c = new double[0]; } }; /************************************************************************* This subroutine builds bilinear spline coefficients table. Input parameters: X - spline abscissas, array[0..N-1] Y - spline ordinates, array[0..M-1] F - function values, array[0..M-1,0..N-1] M,N - grid size, M>=2, N>=2 Output parameters: C - spline interpolant -- ALGLIB PROJECT -- Copyright 05.07.2007 by Bochkanov Sergey *************************************************************************/ public static void spline2dbuildbilinear(double[] x, double[] y, double[,] f, int m, int n, spline2dinterpolant c) { int i = 0; int j = 0; int k = 0; int tblsize = 0; int shift = 0; double t = 0; double[,] dx = new double[0,0]; double[,] dy = new double[0,0]; double[,] dxy = new double[0,0]; x = (double[])x.Clone(); y = (double[])y.Clone(); f = (double[,])f.Clone(); ap.assert(n>=2 & m>=2, "Spline2DBuildBilinear: N<2 or M<2!"); // // Sort points // for(j=0; j<=n-1; j++) { k = j; for(i=j+1; i<=n-1; i++) { if( (double)(x[i])<(double)(x[k]) ) { k = i; } } if( k!=j ) { for(i=0; i<=m-1; i++) { t = f[i,j]; f[i,j] = f[i,k]; f[i,k] = t; } t = x[j]; x[j] = x[k]; x[k] = t; } } for(i=0; i<=m-1; i++) { k = i; for(j=i+1; j<=m-1; j++) { if( (double)(y[j])<(double)(y[k]) ) { k = j; } } if( k!=i ) { for(j=0; j<=n-1; j++) { t = f[i,j]; f[i,j] = f[k,j]; f[k,j] = t; } t = y[i]; y[i] = y[k]; y[k] = t; } } // // Fill C: // C[0] - length(C) // C[1] - type(C): // -1 = bilinear interpolant // -3 = general cubic spline // (see BuildBicubicSpline) // C[2]: // N (x count) // C[3]: // M (y count) // C[4]...C[4+N-1]: // x[i], i = 0...N-1 // C[4+N]...C[4+N+M-1]: // y[i], i = 0...M-1 // C[4+N+M]...C[4+N+M+(N*M-1)]: // f(i,j) table. f(0,0), f(0, 1), f(0,2) and so on... // c.k = 1; tblsize = 4+n+m+n*m; c.c = new double[tblsize-1+1]; c.c[0] = tblsize; c.c[1] = -1; c.c[2] = n; c.c[3] = m; for(i=0; i<=n-1; i++) { c.c[4+i] = x[i]; } for(i=0; i<=m-1; i++) { c.c[4+n+i] = y[i]; } for(i=0; i<=m-1; i++) { for(j=0; j<=n-1; j++) { shift = i*n+j; c.c[4+n+m+shift] = f[i,j]; } } } /************************************************************************* This subroutine builds bicubic spline coefficients table. Input parameters: X - spline abscissas, array[0..N-1] Y - spline ordinates, array[0..M-1] F - function values, array[0..M-1,0..N-1] M,N - grid size, M>=2, N>=2 Output parameters: C - spline interpolant -- ALGLIB PROJECT -- Copyright 05.07.2007 by Bochkanov Sergey *************************************************************************/ public static void spline2dbuildbicubic(double[] x, double[] y, double[,] f, int m, int n, spline2dinterpolant c) { int i = 0; int j = 0; int k = 0; int tblsize = 0; int shift = 0; double t = 0; double[,] dx = new double[0,0]; double[,] dy = new double[0,0]; double[,] dxy = new double[0,0]; x = (double[])x.Clone(); y = (double[])y.Clone(); f = (double[,])f.Clone(); ap.assert(n>=2 & m>=2, "BuildBicubicSpline: N<2 or M<2!"); // // Sort points // for(j=0; j<=n-1; j++) { k = j; for(i=j+1; i<=n-1; i++) { if( (double)(x[i])<(double)(x[k]) ) { k = i; } } if( k!=j ) { for(i=0; i<=m-1; i++) { t = f[i,j]; f[i,j] = f[i,k]; f[i,k] = t; } t = x[j]; x[j] = x[k]; x[k] = t; } } for(i=0; i<=m-1; i++) { k = i; for(j=i+1; j<=m-1; j++) { if( (double)(y[j])<(double)(y[k]) ) { k = j; } } if( k!=i ) { for(j=0; j<=n-1; j++) { t = f[i,j]; f[i,j] = f[k,j]; f[k,j] = t; } t = y[i]; y[i] = y[k]; y[k] = t; } } // // Fill C: // C[0] - length(C) // C[1] - type(C): // -1 = bilinear interpolant // (see BuildBilinearInterpolant) // -3 = general cubic spline // C[2]: // N (x count) // C[3]: // M (y count) // C[4]...C[4+N-1]: // x[i], i = 0...N-1 // C[4+N]...C[4+N+M-1]: // y[i], i = 0...M-1 // C[4+N+M]...C[4+N+M+(N*M-1)]: // f(i,j) table. f(0,0), f(0, 1), f(0,2) and so on... // C[4+N+M+N*M]...C[4+N+M+(2*N*M-1)]: // df(i,j)/dx table. // C[4+N+M+2*N*M]...C[4+N+M+(3*N*M-1)]: // df(i,j)/dy table. // C[4+N+M+3*N*M]...C[4+N+M+(4*N*M-1)]: // d2f(i,j)/dxdy table. // c.k = 3; tblsize = 4+n+m+4*n*m; c.c = new double[tblsize-1+1]; c.c[0] = tblsize; c.c[1] = -3; c.c[2] = n; c.c[3] = m; for(i=0; i<=n-1; i++) { c.c[4+i] = x[i]; } for(i=0; i<=m-1; i++) { c.c[4+n+i] = y[i]; } bicubiccalcderivatives(f, x, y, m, n, ref dx, ref dy, ref dxy); for(i=0; i<=m-1; i++) { for(j=0; j<=n-1; j++) { shift = i*n+j; c.c[4+n+m+shift] = f[i,j]; c.c[4+n+m+n*m+shift] = dx[i,j]; c.c[4+n+m+2*n*m+shift] = dy[i,j]; c.c[4+n+m+3*n*m+shift] = dxy[i,j]; } } } /************************************************************************* This subroutine calculates the value of the bilinear or bicubic spline at the given point X. Input parameters: C - coefficients table. Built by BuildBilinearSpline or BuildBicubicSpline. X, Y- point Result: S(x,y) -- ALGLIB PROJECT -- Copyright 05.07.2007 by Bochkanov Sergey *************************************************************************/ public static double spline2dcalc(spline2dinterpolant c, double x, double y) { double result = 0; double v = 0; double vx = 0; double vy = 0; double vxy = 0; spline2ddiff(c, x, y, ref v, ref vx, ref vy, ref vxy); result = v; return result; } /************************************************************************* This subroutine calculates the value of the bilinear or bicubic spline at the given point X and its derivatives. Input parameters: C - spline interpolant. X, Y- point Output parameters: F - S(x,y) FX - dS(x,y)/dX FY - dS(x,y)/dY FXY - d2S(x,y)/dXdY -- ALGLIB PROJECT -- Copyright 05.07.2007 by Bochkanov Sergey *************************************************************************/ public static void spline2ddiff(spline2dinterpolant c, double x, double y, ref double f, ref double fx, ref double fy, ref double fxy) { int n = 0; int m = 0; double t = 0; double dt = 0; double u = 0; double du = 0; int ix = 0; int iy = 0; int l = 0; int r = 0; int h = 0; int shift1 = 0; int s1 = 0; int s2 = 0; int s3 = 0; int s4 = 0; int sf = 0; int sfx = 0; int sfy = 0; int sfxy = 0; double y1 = 0; double y2 = 0; double y3 = 0; double y4 = 0; double v = 0; double t0 = 0; double t1 = 0; double t2 = 0; double t3 = 0; double u0 = 0; double u1 = 0; double u2 = 0; double u3 = 0; f = 0; fx = 0; fy = 0; fxy = 0; ap.assert((int)Math.Round(c.c[1])==-1 | (int)Math.Round(c.c[1])==-3, "Spline2DDiff: incorrect C!"); n = (int)Math.Round(c.c[2]); m = (int)Math.Round(c.c[3]); // // Binary search in the [ x[0], ..., x[n-2] ] (x[n-1] is not included) // l = 4; r = 4+n-2+1; while( l!=r-1 ) { h = (l+r)/2; if( (double)(c.c[h])>=(double)(x) ) { r = h; } else { l = h; } } t = (x-c.c[l])/(c.c[l+1]-c.c[l]); dt = 1.0/(c.c[l+1]-c.c[l]); ix = l-4; // // Binary search in the [ y[0], ..., y[m-2] ] (y[m-1] is not included) // l = 4+n; r = 4+n+(m-2)+1; while( l!=r-1 ) { h = (l+r)/2; if( (double)(c.c[h])>=(double)(y) ) { r = h; } else { l = h; } } u = (y-c.c[l])/(c.c[l+1]-c.c[l]); du = 1.0/(c.c[l+1]-c.c[l]); iy = l-(4+n); // // Prepare F, dF/dX, dF/dY, d2F/dXdY // f = 0; fx = 0; fy = 0; fxy = 0; // // Bilinear interpolation // if( (int)Math.Round(c.c[1])==-1 ) { shift1 = 4+n+m; y1 = c.c[shift1+n*iy+ix]; y2 = c.c[shift1+n*iy+(ix+1)]; y3 = c.c[shift1+n*(iy+1)+(ix+1)]; y4 = c.c[shift1+n*(iy+1)+ix]; f = (1-t)*(1-u)*y1+t*(1-u)*y2+t*u*y3+(1-t)*u*y4; fx = (-((1-u)*y1)+(1-u)*y2+u*y3-u*y4)*dt; fy = (-((1-t)*y1)-t*y2+t*y3+(1-t)*y4)*du; fxy = (y1-y2+y3-y4)*du*dt; return; } // // Bicubic interpolation // if( (int)Math.Round(c.c[1])==-3 ) { // // Prepare info // t0 = 1; t1 = t; t2 = math.sqr(t); t3 = t*t2; u0 = 1; u1 = u; u2 = math.sqr(u); u3 = u*u2; sf = 4+n+m; sfx = 4+n+m+n*m; sfy = 4+n+m+2*n*m; sfxy = 4+n+m+3*n*m; s1 = n*iy+ix; s2 = n*iy+(ix+1); s3 = n*(iy+1)+(ix+1); s4 = n*(iy+1)+ix; // // Calculate // v = 1*c.c[sf+s1]; f = f+v*t0*u0; v = 1*c.c[sfy+s1]/du; f = f+v*t0*u1; fy = fy+1*v*t0*u0*du; v = -(3*c.c[sf+s1])+3*c.c[sf+s4]-2*c.c[sfy+s1]/du-1*c.c[sfy+s4]/du; f = f+v*t0*u2; fy = fy+2*v*t0*u1*du; v = 2*c.c[sf+s1]-2*c.c[sf+s4]+1*c.c[sfy+s1]/du+1*c.c[sfy+s4]/du; f = f+v*t0*u3; fy = fy+3*v*t0*u2*du; v = 1*c.c[sfx+s1]/dt; f = f+v*t1*u0; fx = fx+1*v*t0*u0*dt; v = 1*c.c[sfxy+s1]/(dt*du); f = f+v*t1*u1; fx = fx+1*v*t0*u1*dt; fy = fy+1*v*t1*u0*du; fxy = fxy+1*v*t0*u0*dt*du; v = -(3*c.c[sfx+s1]/dt)+3*c.c[sfx+s4]/dt-2*c.c[sfxy+s1]/(dt*du)-1*c.c[sfxy+s4]/(dt*du); f = f+v*t1*u2; fx = fx+1*v*t0*u2*dt; fy = fy+2*v*t1*u1*du; fxy = fxy+2*v*t0*u1*dt*du; v = 2*c.c[sfx+s1]/dt-2*c.c[sfx+s4]/dt+1*c.c[sfxy+s1]/(dt*du)+1*c.c[sfxy+s4]/(dt*du); f = f+v*t1*u3; fx = fx+1*v*t0*u3*dt; fy = fy+3*v*t1*u2*du; fxy = fxy+3*v*t0*u2*dt*du; v = -(3*c.c[sf+s1])+3*c.c[sf+s2]-2*c.c[sfx+s1]/dt-1*c.c[sfx+s2]/dt; f = f+v*t2*u0; fx = fx+2*v*t1*u0*dt; v = -(3*c.c[sfy+s1]/du)+3*c.c[sfy+s2]/du-2*c.c[sfxy+s1]/(dt*du)-1*c.c[sfxy+s2]/(dt*du); f = f+v*t2*u1; fx = fx+2*v*t1*u1*dt; fy = fy+1*v*t2*u0*du; fxy = fxy+2*v*t1*u0*dt*du; v = 9*c.c[sf+s1]-9*c.c[sf+s2]+9*c.c[sf+s3]-9*c.c[sf+s4]+6*c.c[sfx+s1]/dt+3*c.c[sfx+s2]/dt-3*c.c[sfx+s3]/dt-6*c.c[sfx+s4]/dt+6*c.c[sfy+s1]/du-6*c.c[sfy+s2]/du-3*c.c[sfy+s3]/du+3*c.c[sfy+s4]/du+4*c.c[sfxy+s1]/(dt*du)+2*c.c[sfxy+s2]/(dt*du)+1*c.c[sfxy+s3]/(dt*du)+2*c.c[sfxy+s4]/(dt*du); f = f+v*t2*u2; fx = fx+2*v*t1*u2*dt; fy = fy+2*v*t2*u1*du; fxy = fxy+4*v*t1*u1*dt*du; v = -(6*c.c[sf+s1])+6*c.c[sf+s2]-6*c.c[sf+s3]+6*c.c[sf+s4]-4*c.c[sfx+s1]/dt-2*c.c[sfx+s2]/dt+2*c.c[sfx+s3]/dt+4*c.c[sfx+s4]/dt-3*c.c[sfy+s1]/du+3*c.c[sfy+s2]/du+3*c.c[sfy+s3]/du-3*c.c[sfy+s4]/du-2*c.c[sfxy+s1]/(dt*du)-1*c.c[sfxy+s2]/(dt*du)-1*c.c[sfxy+s3]/(dt*du)-2*c.c[sfxy+s4]/(dt*du); f = f+v*t2*u3; fx = fx+2*v*t1*u3*dt; fy = fy+3*v*t2*u2*du; fxy = fxy+6*v*t1*u2*dt*du; v = 2*c.c[sf+s1]-2*c.c[sf+s2]+1*c.c[sfx+s1]/dt+1*c.c[sfx+s2]/dt; f = f+v*t3*u0; fx = fx+3*v*t2*u0*dt; v = 2*c.c[sfy+s1]/du-2*c.c[sfy+s2]/du+1*c.c[sfxy+s1]/(dt*du)+1*c.c[sfxy+s2]/(dt*du); f = f+v*t3*u1; fx = fx+3*v*t2*u1*dt; fy = fy+1*v*t3*u0*du; fxy = fxy+3*v*t2*u0*dt*du; v = -(6*c.c[sf+s1])+6*c.c[sf+s2]-6*c.c[sf+s3]+6*c.c[sf+s4]-3*c.c[sfx+s1]/dt-3*c.c[sfx+s2]/dt+3*c.c[sfx+s3]/dt+3*c.c[sfx+s4]/dt-4*c.c[sfy+s1]/du+4*c.c[sfy+s2]/du+2*c.c[sfy+s3]/du-2*c.c[sfy+s4]/du-2*c.c[sfxy+s1]/(dt*du)-2*c.c[sfxy+s2]/(dt*du)-1*c.c[sfxy+s3]/(dt*du)-1*c.c[sfxy+s4]/(dt*du); f = f+v*t3*u2; fx = fx+3*v*t2*u2*dt; fy = fy+2*v*t3*u1*du; fxy = fxy+6*v*t2*u1*dt*du; v = 4*c.c[sf+s1]-4*c.c[sf+s2]+4*c.c[sf+s3]-4*c.c[sf+s4]+2*c.c[sfx+s1]/dt+2*c.c[sfx+s2]/dt-2*c.c[sfx+s3]/dt-2*c.c[sfx+s4]/dt+2*c.c[sfy+s1]/du-2*c.c[sfy+s2]/du-2*c.c[sfy+s3]/du+2*c.c[sfy+s4]/du+1*c.c[sfxy+s1]/(dt*du)+1*c.c[sfxy+s2]/(dt*du)+1*c.c[sfxy+s3]/(dt*du)+1*c.c[sfxy+s4]/(dt*du); f = f+v*t3*u3; fx = fx+3*v*t2*u3*dt; fy = fy+3*v*t3*u2*du; fxy = fxy+9*v*t2*u2*dt*du; return; } } /************************************************************************* This subroutine unpacks two-dimensional spline into the coefficients table Input parameters: C - spline interpolant. Result: M, N- grid size (x-axis and y-axis) Tbl - coefficients table, unpacked format, [0..(N-1)*(M-1)-1, 0..19]. For I = 0...M-2, J=0..N-2: K = I*(N-1)+J Tbl[K,0] = X[j] Tbl[K,1] = X[j+1] Tbl[K,2] = Y[i] Tbl[K,3] = Y[i+1] Tbl[K,4] = C00 Tbl[K,5] = C01 Tbl[K,6] = C02 Tbl[K,7] = C03 Tbl[K,8] = C10 Tbl[K,9] = C11 ... Tbl[K,19] = C33 On each grid square spline is equals to: S(x) = SUM(c[i,j]*(x^i)*(y^j), i=0..3, j=0..3) t = x-x[j] u = y-y[i] -- ALGLIB PROJECT -- Copyright 29.06.2007 by Bochkanov Sergey *************************************************************************/ public static void spline2dunpack(spline2dinterpolant c, ref int m, ref int n, ref double[,] tbl) { int i = 0; int j = 0; int ci = 0; int cj = 0; int k = 0; int p = 0; int shift = 0; int s1 = 0; int s2 = 0; int s3 = 0; int s4 = 0; int sf = 0; int sfx = 0; int sfy = 0; int sfxy = 0; double y1 = 0; double y2 = 0; double y3 = 0; double y4 = 0; double dt = 0; double du = 0; m = 0; n = 0; tbl = new double[0,0]; ap.assert((int)Math.Round(c.c[1])==-3 | (int)Math.Round(c.c[1])==-1, "SplineUnpack2D: incorrect C!"); n = (int)Math.Round(c.c[2]); m = (int)Math.Round(c.c[3]); tbl = new double[(n-1)*(m-1)-1+1, 19+1]; // // Fill // for(i=0; i<=m-2; i++) { for(j=0; j<=n-2; j++) { p = i*(n-1)+j; tbl[p,0] = c.c[4+j]; tbl[p,1] = c.c[4+j+1]; tbl[p,2] = c.c[4+n+i]; tbl[p,3] = c.c[4+n+i+1]; dt = 1/(tbl[p,1]-tbl[p,0]); du = 1/(tbl[p,3]-tbl[p,2]); // // Bilinear interpolation // if( (int)Math.Round(c.c[1])==-1 ) { for(k=4; k<=19; k++) { tbl[p,k] = 0; } shift = 4+n+m; y1 = c.c[shift+n*i+j]; y2 = c.c[shift+n*i+(j+1)]; y3 = c.c[shift+n*(i+1)+(j+1)]; y4 = c.c[shift+n*(i+1)+j]; tbl[p,4] = y1; tbl[p,4+1*4+0] = y2-y1; tbl[p,4+0*4+1] = y4-y1; tbl[p,4+1*4+1] = y3-y2-y4+y1; } // // Bicubic interpolation // if( (int)Math.Round(c.c[1])==-3 ) { sf = 4+n+m; sfx = 4+n+m+n*m; sfy = 4+n+m+2*n*m; sfxy = 4+n+m+3*n*m; s1 = n*i+j; s2 = n*i+(j+1); s3 = n*(i+1)+(j+1); s4 = n*(i+1)+j; tbl[p,4+0*4+0] = 1*c.c[sf+s1]; tbl[p,4+0*4+1] = 1*c.c[sfy+s1]/du; tbl[p,4+0*4+2] = -(3*c.c[sf+s1])+3*c.c[sf+s4]-2*c.c[sfy+s1]/du-1*c.c[sfy+s4]/du; tbl[p,4+0*4+3] = 2*c.c[sf+s1]-2*c.c[sf+s4]+1*c.c[sfy+s1]/du+1*c.c[sfy+s4]/du; tbl[p,4+1*4+0] = 1*c.c[sfx+s1]/dt; tbl[p,4+1*4+1] = 1*c.c[sfxy+s1]/(dt*du); tbl[p,4+1*4+2] = -(3*c.c[sfx+s1]/dt)+3*c.c[sfx+s4]/dt-2*c.c[sfxy+s1]/(dt*du)-1*c.c[sfxy+s4]/(dt*du); tbl[p,4+1*4+3] = 2*c.c[sfx+s1]/dt-2*c.c[sfx+s4]/dt+1*c.c[sfxy+s1]/(dt*du)+1*c.c[sfxy+s4]/(dt*du); tbl[p,4+2*4+0] = -(3*c.c[sf+s1])+3*c.c[sf+s2]-2*c.c[sfx+s1]/dt-1*c.c[sfx+s2]/dt; tbl[p,4+2*4+1] = -(3*c.c[sfy+s1]/du)+3*c.c[sfy+s2]/du-2*c.c[sfxy+s1]/(dt*du)-1*c.c[sfxy+s2]/(dt*du); tbl[p,4+2*4+2] = 9*c.c[sf+s1]-9*c.c[sf+s2]+9*c.c[sf+s3]-9*c.c[sf+s4]+6*c.c[sfx+s1]/dt+3*c.c[sfx+s2]/dt-3*c.c[sfx+s3]/dt-6*c.c[sfx+s4]/dt+6*c.c[sfy+s1]/du-6*c.c[sfy+s2]/du-3*c.c[sfy+s3]/du+3*c.c[sfy+s4]/du+4*c.c[sfxy+s1]/(dt*du)+2*c.c[sfxy+s2]/(dt*du)+1*c.c[sfxy+s3]/(dt*du)+2*c.c[sfxy+s4]/(dt*du); tbl[p,4+2*4+3] = -(6*c.c[sf+s1])+6*c.c[sf+s2]-6*c.c[sf+s3]+6*c.c[sf+s4]-4*c.c[sfx+s1]/dt-2*c.c[sfx+s2]/dt+2*c.c[sfx+s3]/dt+4*c.c[sfx+s4]/dt-3*c.c[sfy+s1]/du+3*c.c[sfy+s2]/du+3*c.c[sfy+s3]/du-3*c.c[sfy+s4]/du-2*c.c[sfxy+s1]/(dt*du)-1*c.c[sfxy+s2]/(dt*du)-1*c.c[sfxy+s3]/(dt*du)-2*c.c[sfxy+s4]/(dt*du); tbl[p,4+3*4+0] = 2*c.c[sf+s1]-2*c.c[sf+s2]+1*c.c[sfx+s1]/dt+1*c.c[sfx+s2]/dt; tbl[p,4+3*4+1] = 2*c.c[sfy+s1]/du-2*c.c[sfy+s2]/du+1*c.c[sfxy+s1]/(dt*du)+1*c.c[sfxy+s2]/(dt*du); tbl[p,4+3*4+2] = -(6*c.c[sf+s1])+6*c.c[sf+s2]-6*c.c[sf+s3]+6*c.c[sf+s4]-3*c.c[sfx+s1]/dt-3*c.c[sfx+s2]/dt+3*c.c[sfx+s3]/dt+3*c.c[sfx+s4]/dt-4*c.c[sfy+s1]/du+4*c.c[sfy+s2]/du+2*c.c[sfy+s3]/du-2*c.c[sfy+s4]/du-2*c.c[sfxy+s1]/(dt*du)-2*c.c[sfxy+s2]/(dt*du)-1*c.c[sfxy+s3]/(dt*du)-1*c.c[sfxy+s4]/(dt*du); tbl[p,4+3*4+3] = 4*c.c[sf+s1]-4*c.c[sf+s2]+4*c.c[sf+s3]-4*c.c[sf+s4]+2*c.c[sfx+s1]/dt+2*c.c[sfx+s2]/dt-2*c.c[sfx+s3]/dt-2*c.c[sfx+s4]/dt+2*c.c[sfy+s1]/du-2*c.c[sfy+s2]/du-2*c.c[sfy+s3]/du+2*c.c[sfy+s4]/du+1*c.c[sfxy+s1]/(dt*du)+1*c.c[sfxy+s2]/(dt*du)+1*c.c[sfxy+s3]/(dt*du)+1*c.c[sfxy+s4]/(dt*du); } // // Rescale Cij // for(ci=0; ci<=3; ci++) { for(cj=0; cj<=3; cj++) { tbl[p,4+ci*4+cj] = tbl[p,4+ci*4+cj]*Math.Pow(dt, ci)*Math.Pow(du, cj); } } } } } /************************************************************************* This subroutine performs linear transformation of the spline argument. Input parameters: C - spline interpolant AX, BX - transformation coefficients: x = A*t + B AY, BY - transformation coefficients: y = A*u + B Result: C - transformed spline -- ALGLIB PROJECT -- Copyright 30.06.2007 by Bochkanov Sergey *************************************************************************/ public static void spline2dlintransxy(spline2dinterpolant c, double ax, double bx, double ay, double by) { int i = 0; int j = 0; int n = 0; int m = 0; double v = 0; double[] x = new double[0]; double[] y = new double[0]; double[,] f = new double[0,0]; int typec = 0; typec = (int)Math.Round(c.c[1]); ap.assert(typec==-3 | typec==-1, "Spline2DLinTransXY: incorrect C!"); n = (int)Math.Round(c.c[2]); m = (int)Math.Round(c.c[3]); x = new double[n-1+1]; y = new double[m-1+1]; f = new double[m-1+1, n-1+1]; for(j=0; j<=n-1; j++) { x[j] = c.c[4+j]; } for(i=0; i<=m-1; i++) { y[i] = c.c[4+n+i]; } for(i=0; i<=m-1; i++) { for(j=0; j<=n-1; j++) { f[i,j] = c.c[4+n+m+i*n+j]; } } // // Special case: AX=0 or AY=0 // if( (double)(ax)==(double)(0) ) { for(i=0; i<=m-1; i++) { v = spline2dcalc(c, bx, y[i]); for(j=0; j<=n-1; j++) { f[i,j] = v; } } if( typec==-3 ) { spline2dbuildbicubic(x, y, f, m, n, c); } if( typec==-1 ) { spline2dbuildbilinear(x, y, f, m, n, c); } ax = 1; bx = 0; } if( (double)(ay)==(double)(0) ) { for(j=0; j<=n-1; j++) { v = spline2dcalc(c, x[j], by); for(i=0; i<=m-1; i++) { f[i,j] = v; } } if( typec==-3 ) { spline2dbuildbicubic(x, y, f, m, n, c); } if( typec==-1 ) { spline2dbuildbilinear(x, y, f, m, n, c); } ay = 1; by = 0; } // // General case: AX<>0, AY<>0 // Unpack, scale and pack again. // for(j=0; j<=n-1; j++) { x[j] = (x[j]-bx)/ax; } for(i=0; i<=m-1; i++) { y[i] = (y[i]-by)/ay; } if( typec==-3 ) { spline2dbuildbicubic(x, y, f, m, n, c); } if( typec==-1 ) { spline2dbuildbilinear(x, y, f, m, n, c); } } /************************************************************************* This subroutine performs linear transformation of the spline. Input parameters: C - spline interpolant. A, B- transformation coefficients: S2(x,y) = A*S(x,y) + B Output parameters: C - transformed spline -- ALGLIB PROJECT -- Copyright 30.06.2007 by Bochkanov Sergey *************************************************************************/ public static void spline2dlintransf(spline2dinterpolant c, double a, double b) { int i = 0; int j = 0; int n = 0; int m = 0; double[] x = new double[0]; double[] y = new double[0]; double[,] f = new double[0,0]; int typec = 0; typec = (int)Math.Round(c.c[1]); ap.assert(typec==-3 | typec==-1, "Spline2DLinTransXY: incorrect C!"); n = (int)Math.Round(c.c[2]); m = (int)Math.Round(c.c[3]); x = new double[n-1+1]; y = new double[m-1+1]; f = new double[m-1+1, n-1+1]; for(j=0; j<=n-1; j++) { x[j] = c.c[4+j]; } for(i=0; i<=m-1; i++) { y[i] = c.c[4+n+i]; } for(i=0; i<=m-1; i++) { for(j=0; j<=n-1; j++) { f[i,j] = a*c.c[4+n+m+i*n+j]+b; } } if( typec==-3 ) { spline2dbuildbicubic(x, y, f, m, n, c); } if( typec==-1 ) { spline2dbuildbilinear(x, y, f, m, n, c); } } /************************************************************************* This subroutine makes the copy of the spline model. Input parameters: C - spline interpolant Output parameters: CC - spline copy -- ALGLIB PROJECT -- Copyright 29.06.2007 by Bochkanov Sergey *************************************************************************/ public static void spline2dcopy(spline2dinterpolant c, spline2dinterpolant cc) { int n = 0; int i_ = 0; ap.assert(c.k==1 | c.k==3, "Spline2DCopy: incorrect C!"); cc.k = c.k; n = (int)Math.Round(c.c[0]); cc.c = new double[n]; for(i_=0; i_<=n-1;i_++) { cc.c[i_] = c.c[i_]; } } /************************************************************************* Bicubic spline resampling Input parameters: A - function values at the old grid, array[0..OldHeight-1, 0..OldWidth-1] OldHeight - old grid height, OldHeight>1 OldWidth - old grid width, OldWidth>1 NewHeight - new grid height, NewHeight>1 NewWidth - new grid width, NewWidth>1 Output parameters: B - function values at the new grid, array[0..NewHeight-1, 0..NewWidth-1] -- ALGLIB routine -- 15 May, 2007 Copyright by Bochkanov Sergey *************************************************************************/ public static void spline2dresamplebicubic(double[,] a, int oldheight, int oldwidth, ref double[,] b, int newheight, int newwidth) { double[,] buf = new double[0,0]; double[] x = new double[0]; double[] y = new double[0]; spline1d.spline1dinterpolant c = new spline1d.spline1dinterpolant(); int i = 0; int j = 0; int mw = 0; int mh = 0; b = new double[0,0]; ap.assert(oldwidth>1 & oldheight>1, "Spline2DResampleBicubic: width/height less than 1"); ap.assert(newwidth>1 & newheight>1, "Spline2DResampleBicubic: width/height less than 1"); // // Prepare // mw = Math.Max(oldwidth, newwidth); mh = Math.Max(oldheight, newheight); b = new double[newheight-1+1, newwidth-1+1]; buf = new double[oldheight-1+1, newwidth-1+1]; x = new double[Math.Max(mw, mh)-1+1]; y = new double[Math.Max(mw, mh)-1+1]; // // Horizontal interpolation // for(i=0; i<=oldheight-1; i++) { // // Fill X, Y // for(j=0; j<=oldwidth-1; j++) { x[j] = (double)j/(double)(oldwidth-1); y[j] = a[i,j]; } // // Interpolate and place result into temporary matrix // spline1d.spline1dbuildcubic(x, y, oldwidth, 0, 0.0, 0, 0.0, c); for(j=0; j<=newwidth-1; j++) { buf[i,j] = spline1d.spline1dcalc(c, (double)j/(double)(newwidth-1)); } } // // Vertical interpolation // for(j=0; j<=newwidth-1; j++) { // // Fill X, Y // for(i=0; i<=oldheight-1; i++) { x[i] = (double)i/(double)(oldheight-1); y[i] = buf[i,j]; } // // Interpolate and place result into B // spline1d.spline1dbuildcubic(x, y, oldheight, 0, 0.0, 0, 0.0, c); for(i=0; i<=newheight-1; i++) { b[i,j] = spline1d.spline1dcalc(c, (double)i/(double)(newheight-1)); } } } /************************************************************************* Bilinear spline resampling Input parameters: A - function values at the old grid, array[0..OldHeight-1, 0..OldWidth-1] OldHeight - old grid height, OldHeight>1 OldWidth - old grid width, OldWidth>1 NewHeight - new grid height, NewHeight>1 NewWidth - new grid width, NewWidth>1 Output parameters: B - function values at the new grid, array[0..NewHeight-1, 0..NewWidth-1] -- ALGLIB routine -- 09.07.2007 Copyright by Bochkanov Sergey *************************************************************************/ public static void spline2dresamplebilinear(double[,] a, int oldheight, int oldwidth, ref double[,] b, int newheight, int newwidth) { int i = 0; int j = 0; int l = 0; int c = 0; double t = 0; double u = 0; b = new double[0,0]; b = new double[newheight-1+1, newwidth-1+1]; for(i=0; i<=newheight-1; i++) { for(j=0; j<=newwidth-1; j++) { l = i*(oldheight-1)/(newheight-1); if( l==oldheight-1 ) { l = oldheight-2; } u = (double)i/(double)(newheight-1)*(oldheight-1)-l; c = j*(oldwidth-1)/(newwidth-1); if( c==oldwidth-1 ) { c = oldwidth-2; } t = (double)(j*(oldwidth-1))/(double)(newwidth-1)-c; b[i,j] = (1-t)*(1-u)*a[l,c]+t*(1-u)*a[l,c+1]+t*u*a[l+1,c+1]+(1-t)*u*a[l+1,c]; } } } /************************************************************************* Internal subroutine. Calculation of the first derivatives and the cross-derivative. *************************************************************************/ private static void bicubiccalcderivatives(double[,] a, double[] x, double[] y, int m, int n, ref double[,] dx, ref double[,] dy, ref double[,] dxy) { int i = 0; int j = 0; double[] xt = new double[0]; double[] ft = new double[0]; double s = 0; double ds = 0; double d2s = 0; spline1d.spline1dinterpolant c = new spline1d.spline1dinterpolant(); dx = new double[0,0]; dy = new double[0,0]; dxy = new double[0,0]; dx = new double[m, n]; dy = new double[m, n]; dxy = new double[m, n]; // // dF/dX // xt = new double[n]; ft = new double[n]; for(i=0; i<=m-1; i++) { for(j=0; j<=n-1; j++) { xt[j] = x[j]; ft[j] = a[i,j]; } spline1d.spline1dbuildcubic(xt, ft, n, 0, 0.0, 0, 0.0, c); for(j=0; j<=n-1; j++) { spline1d.spline1ddiff(c, x[j], ref s, ref ds, ref d2s); dx[i,j] = ds; } } // // dF/dY // xt = new double[m]; ft = new double[m]; for(j=0; j<=n-1; j++) { for(i=0; i<=m-1; i++) { xt[i] = y[i]; ft[i] = a[i,j]; } spline1d.spline1dbuildcubic(xt, ft, m, 0, 0.0, 0, 0.0, c); for(i=0; i<=m-1; i++) { spline1d.spline1ddiff(c, y[i], ref s, ref ds, ref d2s); dy[i,j] = ds; } } // // d2F/dXdY // xt = new double[n]; ft = new double[n]; for(i=0; i<=m-1; i++) { for(j=0; j<=n-1; j++) { xt[j] = x[j]; ft[j] = dy[i,j]; } spline1d.spline1dbuildcubic(xt, ft, n, 0, 0.0, 0, 0.0, c); for(j=0; j<=n-1; j++) { spline1d.spline1ddiff(c, x[j], ref s, ref ds, ref d2s); dxy[i,j] = ds; } } } } }