[3839] | 1 | /*************************************************************************
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| 2 | Copyright (c) 2006-2009, Sergey Bochkanov (ALGLIB project).
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| 3 |
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| 4 | >>> SOURCE LICENSE >>>
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| 5 | This program is free software; you can redistribute it and/or modify
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| 6 | it under the terms of the GNU General Public License as published by
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| 7 | the Free Software Foundation (www.fsf.org); either version 2 of the
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| 8 | License, or (at your option) any later version.
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| 9 |
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| 10 | This program is distributed in the hope that it will be useful,
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| 11 | but WITHOUT ANY WARRANTY; without even the implied warranty of
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| 12 | MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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| 13 | GNU General Public License for more details.
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| 14 |
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| 15 | A copy of the GNU General Public License is available at
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| 16 | http://www.fsf.org/licensing/licenses
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| 17 |
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| 18 | >>> END OF LICENSE >>>
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| 19 | *************************************************************************/
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| 20 |
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| 21 | using System;
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| 22 |
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| 23 | namespace alglib
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| 24 | {
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| 25 | public class spline1d
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| 26 | {
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| 27 | /*************************************************************************
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| 28 | 1-dimensional spline inteprolant
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| 29 | *************************************************************************/
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| 30 | public struct spline1dinterpolant
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| 31 | {
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| 32 | public int n;
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| 33 | public int k;
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| 34 | public double[] x;
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| 35 | public double[] c;
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| 36 | };
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| 37 |
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| 38 |
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| 39 | /*************************************************************************
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| 40 | Spline fitting report:
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| 41 | TaskRCond reciprocal of task's condition number
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| 42 | RMSError RMS error
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| 43 | AvgError average error
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| 44 | AvgRelError average relative error (for non-zero Y[I])
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| 45 | MaxError maximum error
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| 46 | *************************************************************************/
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| 47 | public struct spline1dfitreport
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| 48 | {
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| 49 | public double taskrcond;
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| 50 | public double rmserror;
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| 51 | public double avgerror;
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| 52 | public double avgrelerror;
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| 53 | public double maxerror;
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| 54 | };
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| 55 |
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| 56 |
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| 57 |
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| 58 |
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| 59 | public const int spline1dvnum = 11;
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| 60 |
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| 61 |
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| 62 | /*************************************************************************
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| 63 | This subroutine builds linear spline interpolant
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| 64 |
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| 65 | INPUT PARAMETERS:
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| 66 | X - spline nodes, array[0..N-1]
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| 67 | Y - function values, array[0..N-1]
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| 68 | N - points count, N>=2
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| 69 |
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| 70 | OUTPUT PARAMETERS:
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| 71 | C - spline interpolant
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| 72 |
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| 73 | -- ALGLIB PROJECT --
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| 74 | Copyright 24.06.2007 by Bochkanov Sergey
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| 75 | *************************************************************************/
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| 76 | public static void spline1dbuildlinear(double[] x,
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| 77 | double[] y,
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| 78 | int n,
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| 79 | ref spline1dinterpolant c)
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| 80 | {
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| 81 | int i = 0;
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| 82 |
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| 83 | x = (double[])x.Clone();
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| 84 | y = (double[])y.Clone();
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| 85 |
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| 86 | System.Diagnostics.Debug.Assert(n>1, "Spline1DBuildLinear: N<2!");
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| 87 |
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| 88 | //
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| 89 | // Sort points
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| 90 | //
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| 91 | heapsortpoints(ref x, ref y, n);
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| 92 |
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| 93 | //
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| 94 | // Build
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| 95 | //
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| 96 | c.n = n;
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| 97 | c.k = 3;
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| 98 | c.x = new double[n];
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| 99 | c.c = new double[4*(n-1)];
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| 100 | for(i=0; i<=n-1; i++)
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| 101 | {
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| 102 | c.x[i] = x[i];
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| 103 | }
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| 104 | for(i=0; i<=n-2; i++)
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| 105 | {
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| 106 | c.c[4*i+0] = y[i];
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| 107 | c.c[4*i+1] = (y[i+1]-y[i])/(x[i+1]-x[i]);
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| 108 | c.c[4*i+2] = 0;
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| 109 | c.c[4*i+3] = 0;
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| 110 | }
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| 111 | }
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| 112 |
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| 113 |
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| 114 | /*************************************************************************
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| 115 | This subroutine builds cubic spline interpolant.
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| 116 |
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| 117 | INPUT PARAMETERS:
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| 118 | X - spline nodes, array[0..N-1]
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| 119 | Y - function values, array[0..N-1]
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| 120 | N - points count, N>=2
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| 121 | BoundLType - boundary condition type for the left boundary
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| 122 | BoundL - left boundary condition (first or second derivative,
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| 123 | depending on the BoundLType)
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| 124 | BoundRType - boundary condition type for the right boundary
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| 125 | BoundR - right boundary condition (first or second derivative,
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| 126 | depending on the BoundRType)
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| 127 |
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| 128 | OUTPUT PARAMETERS:
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| 129 | C - spline interpolant
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| 130 |
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| 131 | The BoundLType/BoundRType parameters can have the following values:
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| 132 | * 0, which corresponds to the parabolically terminated spline
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| 133 | (BoundL/BoundR are ignored).
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| 134 | * 1, which corresponds to the first derivative boundary condition
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| 135 | * 2, which corresponds to the second derivative boundary condition
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| 136 |
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| 137 | -- ALGLIB PROJECT --
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| 138 | Copyright 23.06.2007 by Bochkanov Sergey
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| 139 | *************************************************************************/
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| 140 | public static void spline1dbuildcubic(double[] x,
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| 141 | double[] y,
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| 142 | int n,
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| 143 | int boundltype,
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| 144 | double boundl,
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| 145 | int boundrtype,
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| 146 | double boundr,
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| 147 | ref spline1dinterpolant c)
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| 148 | {
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| 149 | double[] a1 = new double[0];
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| 150 | double[] a2 = new double[0];
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| 151 | double[] a3 = new double[0];
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| 152 | double[] b = new double[0];
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| 153 | double[] d = new double[0];
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| 154 | int i = 0;
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| 155 |
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| 156 | x = (double[])x.Clone();
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| 157 | y = (double[])y.Clone();
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| 158 |
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| 159 | System.Diagnostics.Debug.Assert(n>=2, "BuildCubicSpline: N<2!");
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| 160 | System.Diagnostics.Debug.Assert(boundltype==0 | boundltype==1 | boundltype==2, "BuildCubicSpline: incorrect BoundLType!");
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| 161 | System.Diagnostics.Debug.Assert(boundrtype==0 | boundrtype==1 | boundrtype==2, "BuildCubicSpline: incorrect BoundRType!");
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| 162 | a1 = new double[n];
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| 163 | a2 = new double[n];
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| 164 | a3 = new double[n];
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| 165 | b = new double[n];
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| 166 |
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| 167 | //
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| 168 | // Special case:
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| 169 | // * N=2
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| 170 | // * parabolic terminated boundary condition on both ends
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| 171 | //
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| 172 | if( n==2 & boundltype==0 & boundrtype==0 )
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| 173 | {
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| 174 |
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| 175 | //
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| 176 | // Change task type
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| 177 | //
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| 178 | boundltype = 2;
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| 179 | boundl = 0;
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| 180 | boundrtype = 2;
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| 181 | boundr = 0;
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| 182 | }
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| 183 |
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| 184 | //
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| 185 | //
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| 186 | // Sort points
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| 187 | //
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| 188 | heapsortpoints(ref x, ref y, n);
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| 189 |
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| 190 | //
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| 191 | // Left boundary conditions
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| 192 | //
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| 193 | if( boundltype==0 )
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| 194 | {
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| 195 | a1[0] = 0;
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| 196 | a2[0] = 1;
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| 197 | a3[0] = 1;
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| 198 | b[0] = 2*(y[1]-y[0])/(x[1]-x[0]);
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| 199 | }
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| 200 | if( boundltype==1 )
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| 201 | {
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| 202 | a1[0] = 0;
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| 203 | a2[0] = 1;
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| 204 | a3[0] = 0;
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| 205 | b[0] = boundl;
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| 206 | }
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| 207 | if( boundltype==2 )
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| 208 | {
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| 209 | a1[0] = 0;
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| 210 | a2[0] = 2;
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| 211 | a3[0] = 1;
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| 212 | b[0] = 3*(y[1]-y[0])/(x[1]-x[0])-0.5*boundl*(x[1]-x[0]);
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| 213 | }
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| 214 |
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| 215 | //
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| 216 | // Central conditions
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| 217 | //
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| 218 | for(i=1; i<=n-2; i++)
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| 219 | {
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| 220 | a1[i] = x[i+1]-x[i];
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| 221 | a2[i] = 2*(x[i+1]-x[i-1]);
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| 222 | a3[i] = x[i]-x[i-1];
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| 223 | b[i] = 3*(y[i]-y[i-1])/(x[i]-x[i-1])*(x[i+1]-x[i])+3*(y[i+1]-y[i])/(x[i+1]-x[i])*(x[i]-x[i-1]);
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| 224 | }
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| 225 |
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| 226 | //
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| 227 | // Right boundary conditions
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| 228 | //
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| 229 | if( boundrtype==0 )
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| 230 | {
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| 231 | a1[n-1] = 1;
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| 232 | a2[n-1] = 1;
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| 233 | a3[n-1] = 0;
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| 234 | b[n-1] = 2*(y[n-1]-y[n-2])/(x[n-1]-x[n-2]);
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| 235 | }
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| 236 | if( boundrtype==1 )
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| 237 | {
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| 238 | a1[n-1] = 0;
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| 239 | a2[n-1] = 1;
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| 240 | a3[n-1] = 0;
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| 241 | b[n-1] = boundr;
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| 242 | }
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| 243 | if( boundrtype==2 )
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| 244 | {
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| 245 | a1[n-1] = 1;
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| 246 | a2[n-1] = 2;
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| 247 | a3[n-1] = 0;
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| 248 | b[n-1] = 3*(y[n-1]-y[n-2])/(x[n-1]-x[n-2])+0.5*boundr*(x[n-1]-x[n-2]);
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| 249 | }
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| 250 |
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| 251 | //
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| 252 | // Solve
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| 253 | //
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| 254 | solvetridiagonal(a1, a2, a3, b, n, ref d);
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| 255 |
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| 256 | //
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| 257 | // Now problem is reduced to the cubic Hermite spline
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| 258 | //
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| 259 | spline1dbuildhermite(x, y, d, n, ref c);
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| 260 | }
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| 261 |
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| 262 |
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| 263 | /*************************************************************************
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| 264 | This subroutine builds Hermite spline interpolant.
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| 265 |
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| 266 | INPUT PARAMETERS:
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| 267 | X - spline nodes, array[0..N-1]
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| 268 | Y - function values, array[0..N-1]
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| 269 | D - derivatives, array[0..N-1]
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| 270 | N - points count, N>=2
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| 271 |
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| 272 | OUTPUT PARAMETERS:
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| 273 | C - spline interpolant.
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| 274 |
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| 275 | -- ALGLIB PROJECT --
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| 276 | Copyright 23.06.2007 by Bochkanov Sergey
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| 277 | *************************************************************************/
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| 278 | public static void spline1dbuildhermite(double[] x,
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| 279 | double[] y,
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| 280 | double[] d,
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| 281 | int n,
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| 282 | ref spline1dinterpolant c)
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| 283 | {
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| 284 | int i = 0;
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| 285 | double delta = 0;
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| 286 | double delta2 = 0;
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| 287 | double delta3 = 0;
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| 288 |
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| 289 | x = (double[])x.Clone();
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| 290 | y = (double[])y.Clone();
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| 291 | d = (double[])d.Clone();
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| 292 |
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| 293 | System.Diagnostics.Debug.Assert(n>=2, "BuildHermiteSpline: N<2!");
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| 294 |
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| 295 | //
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| 296 | // Sort points
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| 297 | //
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| 298 | heapsortdpoints(ref x, ref y, ref d, n);
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| 299 |
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| 300 | //
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| 301 | // Build
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| 302 | //
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| 303 | c.x = new double[n];
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| 304 | c.c = new double[4*(n-1)];
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| 305 | c.k = 3;
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| 306 | c.n = n;
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| 307 | for(i=0; i<=n-1; i++)
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| 308 | {
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| 309 | c.x[i] = x[i];
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| 310 | }
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| 311 | for(i=0; i<=n-2; i++)
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| 312 | {
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| 313 | delta = x[i+1]-x[i];
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| 314 | delta2 = AP.Math.Sqr(delta);
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| 315 | delta3 = delta*delta2;
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| 316 | c.c[4*i+0] = y[i];
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| 317 | c.c[4*i+1] = d[i];
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| 318 | c.c[4*i+2] = (3*(y[i+1]-y[i])-2*d[i]*delta-d[i+1]*delta)/delta2;
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| 319 | c.c[4*i+3] = (2*(y[i]-y[i+1])+d[i]*delta+d[i+1]*delta)/delta3;
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| 320 | }
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| 321 | }
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| 322 |
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| 323 |
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| 324 | /*************************************************************************
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| 325 | This subroutine builds Akima spline interpolant
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| 326 |
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| 327 | INPUT PARAMETERS:
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| 328 | X - spline nodes, array[0..N-1]
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| 329 | Y - function values, array[0..N-1]
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| 330 | N - points count, N>=5
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| 331 |
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| 332 | OUTPUT PARAMETERS:
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| 333 | C - spline interpolant
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| 334 |
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| 335 | -- ALGLIB PROJECT --
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| 336 | Copyright 24.06.2007 by Bochkanov Sergey
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| 337 | *************************************************************************/
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| 338 | public static void spline1dbuildakima(double[] x,
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| 339 | double[] y,
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| 340 | int n,
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| 341 | ref spline1dinterpolant c)
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| 342 | {
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| 343 | int i = 0;
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| 344 | double[] d = new double[0];
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| 345 | double[] w = new double[0];
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| 346 | double[] diff = new double[0];
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| 347 |
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| 348 | x = (double[])x.Clone();
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| 349 | y = (double[])y.Clone();
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| 350 |
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| 351 | System.Diagnostics.Debug.Assert(n>=5, "BuildAkimaSpline: N<5!");
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| 352 |
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| 353 | //
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| 354 | // Sort points
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| 355 | //
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| 356 | heapsortpoints(ref x, ref y, n);
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| 357 |
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| 358 | //
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| 359 | // Prepare W (weights), Diff (divided differences)
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| 360 | //
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| 361 | w = new double[n-1];
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| 362 | diff = new double[n-1];
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| 363 | for(i=0; i<=n-2; i++)
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| 364 | {
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| 365 | diff[i] = (y[i+1]-y[i])/(x[i+1]-x[i]);
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| 366 | }
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| 367 | for(i=1; i<=n-2; i++)
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| 368 | {
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| 369 | w[i] = Math.Abs(diff[i]-diff[i-1]);
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| 370 | }
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| 371 |
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| 372 | //
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| 373 | // Prepare Hermite interpolation scheme
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| 374 | //
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| 375 | d = new double[n];
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| 376 | for(i=2; i<=n-3; i++)
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| 377 | {
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| 378 | if( (double)(Math.Abs(w[i-1])+Math.Abs(w[i+1]))!=(double)(0) )
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| 379 | {
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| 380 | d[i] = (w[i+1]*diff[i-1]+w[i-1]*diff[i])/(w[i+1]+w[i-1]);
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| 381 | }
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| 382 | else
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| 383 | {
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| 384 | d[i] = ((x[i+1]-x[i])*diff[i-1]+(x[i]-x[i-1])*diff[i])/(x[i+1]-x[i-1]);
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| 385 | }
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| 386 | }
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| 387 | d[0] = diffthreepoint(x[0], x[0], y[0], x[1], y[1], x[2], y[2]);
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| 388 | d[1] = diffthreepoint(x[1], x[0], y[0], x[1], y[1], x[2], y[2]);
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| 389 | d[n-2] = diffthreepoint(x[n-2], x[n-3], y[n-3], x[n-2], y[n-2], x[n-1], y[n-1]);
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| 390 | d[n-1] = diffthreepoint(x[n-1], x[n-3], y[n-3], x[n-2], y[n-2], x[n-1], y[n-1]);
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| 391 |
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| 392 | //
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| 393 | // Build Akima spline using Hermite interpolation scheme
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| 394 | //
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| 395 | spline1dbuildhermite(x, y, d, n, ref c);
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| 396 | }
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| 397 |
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| 398 |
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| 399 | /*************************************************************************
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| 400 | Weighted fitting by cubic spline, with constraints on function values or
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| 401 | derivatives.
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| 402 |
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| 403 | Equidistant grid with M-2 nodes on [min(x,xc),max(x,xc)] is used to build
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| 404 | basis functions. Basis functions are cubic splines with continuous second
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| 405 | derivatives and non-fixed first derivatives at interval ends. Small
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| 406 | regularizing term is used when solving constrained tasks (to improve
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| 407 | stability).
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| 408 |
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| 409 | Task is linear, so linear least squares solver is used. Complexity of this
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| 410 | computational scheme is O(N*M^2), mostly dominated by least squares solver
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| 411 |
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| 412 | SEE ALSO
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| 413 | Spline1DFitHermiteWC() - fitting by Hermite splines (more flexible,
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| 414 | less smooth)
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| 415 | Spline1DFitCubic() - "lightweight" fitting by cubic splines,
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| 416 | without invididual weights and constraints
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| 417 |
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| 418 | INPUT PARAMETERS:
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| 419 | X - points, array[0..N-1].
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| 420 | Y - function values, array[0..N-1].
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| 421 | W - weights, array[0..N-1]
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| 422 | Each summand in square sum of approximation deviations from
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| 423 | given values is multiplied by the square of corresponding
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| 424 | weight. Fill it by 1's if you don't want to solve weighted
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| 425 | task.
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| 426 | N - number of points, N>0.
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| 427 | XC - points where spline values/derivatives are constrained,
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| 428 | array[0..K-1].
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| 429 | YC - values of constraints, array[0..K-1]
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| 430 | DC - array[0..K-1], types of constraints:
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| 431 | * DC[i]=0 means that S(XC[i])=YC[i]
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| 432 | * DC[i]=1 means that S'(XC[i])=YC[i]
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| 433 | SEE BELOW FOR IMPORTANT INFORMATION ON CONSTRAINTS
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| 434 | K - number of constraints, 0<=K<M.
|
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| 435 | K=0 means no constraints (XC/YC/DC are not used in such cases)
|
---|
| 436 | M - number of basis functions ( = number_of_nodes+2), M>=4.
|
---|
| 437 |
|
---|
| 438 | OUTPUT PARAMETERS:
|
---|
| 439 | Info- same format as in LSFitLinearWC() subroutine.
|
---|
| 440 | * Info>0 task is solved
|
---|
| 441 | * Info<=0 an error occured:
|
---|
| 442 | -4 means inconvergence of internal SVD
|
---|
| 443 | -3 means inconsistent constraints
|
---|
| 444 | -1 means another errors in parameters passed
|
---|
| 445 | (N<=0, for example)
|
---|
| 446 | S - spline interpolant.
|
---|
| 447 | Rep - report, same format as in LSFitLinearWC() subroutine.
|
---|
| 448 | Following fields are set:
|
---|
| 449 | * RMSError rms error on the (X,Y).
|
---|
| 450 | * AvgError average error on the (X,Y).
|
---|
| 451 | * AvgRelError average relative error on the non-zero Y
|
---|
| 452 | * MaxError maximum error
|
---|
| 453 | NON-WEIGHTED ERRORS ARE CALCULATED
|
---|
| 454 |
|
---|
| 455 | IMPORTANT:
|
---|
| 456 | this subroitine doesn't calculate task's condition number for K<>0.
|
---|
| 457 |
|
---|
| 458 | SETTING CONSTRAINTS - DANGERS AND OPPORTUNITIES:
|
---|
| 459 |
|
---|
| 460 | Setting constraints can lead to undesired results, like ill-conditioned
|
---|
| 461 | behavior, or inconsistency being detected. From the other side, it allows
|
---|
| 462 | us to improve quality of the fit. Here we summarize our experience with
|
---|
| 463 | constrained regression splines:
|
---|
| 464 | * excessive constraints can be inconsistent. Splines are piecewise cubic
|
---|
| 465 | functions, and it is easy to create an example, where large number of
|
---|
| 466 | constraints concentrated in small area will result in inconsistency.
|
---|
| 467 | Just because spline is not flexible enough to satisfy all of them. And
|
---|
| 468 | same constraints spread across the [min(x),max(x)] will be perfectly
|
---|
| 469 | consistent.
|
---|
| 470 | * the more evenly constraints are spread across [min(x),max(x)], the more
|
---|
| 471 | chances that they will be consistent
|
---|
| 472 | * the greater is M (given fixed constraints), the more chances that
|
---|
| 473 | constraints will be consistent
|
---|
| 474 | * in the general case, consistency of constraints IS NOT GUARANTEED.
|
---|
| 475 | * in the several special cases, however, we CAN guarantee consistency.
|
---|
| 476 | * one of this cases is constraints on the function values AND/OR its
|
---|
| 477 | derivatives at the interval boundaries.
|
---|
| 478 | * another special case is ONE constraint on the function value (OR, but
|
---|
| 479 | not AND, derivative) anywhere in the interval
|
---|
| 480 |
|
---|
| 481 | Our final recommendation is to use constraints WHEN AND ONLY WHEN you
|
---|
| 482 | can't solve your task without them. Anything beyond special cases given
|
---|
| 483 | above is not guaranteed and may result in inconsistency.
|
---|
| 484 |
|
---|
| 485 |
|
---|
| 486 | -- ALGLIB PROJECT --
|
---|
| 487 | Copyright 18.08.2009 by Bochkanov Sergey
|
---|
| 488 | *************************************************************************/
|
---|
| 489 | public static void spline1dfitcubicwc(ref double[] x,
|
---|
| 490 | ref double[] y,
|
---|
| 491 | ref double[] w,
|
---|
| 492 | int n,
|
---|
| 493 | ref double[] xc,
|
---|
| 494 | ref double[] yc,
|
---|
| 495 | ref int[] dc,
|
---|
| 496 | int k,
|
---|
| 497 | int m,
|
---|
| 498 | ref int info,
|
---|
| 499 | ref spline1dinterpolant s,
|
---|
| 500 | ref spline1dfitreport rep)
|
---|
| 501 | {
|
---|
| 502 | spline1dfitinternal(0, x, y, ref w, n, xc, yc, ref dc, k, m, ref info, ref s, ref rep);
|
---|
| 503 | }
|
---|
| 504 |
|
---|
| 505 |
|
---|
| 506 | /*************************************************************************
|
---|
| 507 | Weighted fitting by Hermite spline, with constraints on function values
|
---|
| 508 | or first derivatives.
|
---|
| 509 |
|
---|
| 510 | Equidistant grid with M nodes on [min(x,xc),max(x,xc)] is used to build
|
---|
| 511 | basis functions. Basis functions are Hermite splines. Small regularizing
|
---|
| 512 | term is used when solving constrained tasks (to improve stability).
|
---|
| 513 |
|
---|
| 514 | Task is linear, so linear least squares solver is used. Complexity of this
|
---|
| 515 | computational scheme is O(N*M^2), mostly dominated by least squares solver
|
---|
| 516 |
|
---|
| 517 | SEE ALSO
|
---|
| 518 | Spline1DFitCubicWC() - fitting by Cubic splines (less flexible,
|
---|
| 519 | more smooth)
|
---|
| 520 | Spline1DFitHermite() - "lightweight" Hermite fitting, without
|
---|
| 521 | invididual weights and constraints
|
---|
| 522 |
|
---|
| 523 | INPUT PARAMETERS:
|
---|
| 524 | X - points, array[0..N-1].
|
---|
| 525 | Y - function values, array[0..N-1].
|
---|
| 526 | W - weights, array[0..N-1]
|
---|
| 527 | Each summand in square sum of approximation deviations from
|
---|
| 528 | given values is multiplied by the square of corresponding
|
---|
| 529 | weight. Fill it by 1's if you don't want to solve weighted
|
---|
| 530 | task.
|
---|
| 531 | N - number of points, N>0.
|
---|
| 532 | XC - points where spline values/derivatives are constrained,
|
---|
| 533 | array[0..K-1].
|
---|
| 534 | YC - values of constraints, array[0..K-1]
|
---|
| 535 | DC - array[0..K-1], types of constraints:
|
---|
| 536 | * DC[i]=0 means that S(XC[i])=YC[i]
|
---|
| 537 | * DC[i]=1 means that S'(XC[i])=YC[i]
|
---|
| 538 | SEE BELOW FOR IMPORTANT INFORMATION ON CONSTRAINTS
|
---|
| 539 | K - number of constraints, 0<=K<M.
|
---|
| 540 | K=0 means no constraints (XC/YC/DC are not used in such cases)
|
---|
| 541 | M - number of basis functions (= 2 * number of nodes),
|
---|
| 542 | M>=4,
|
---|
| 543 | M IS EVEN!
|
---|
| 544 |
|
---|
| 545 | OUTPUT PARAMETERS:
|
---|
| 546 | Info- same format as in LSFitLinearW() subroutine:
|
---|
| 547 | * Info>0 task is solved
|
---|
| 548 | * Info<=0 an error occured:
|
---|
| 549 | -4 means inconvergence of internal SVD
|
---|
| 550 | -3 means inconsistent constraints
|
---|
| 551 | -2 means odd M was passed (which is not supported)
|
---|
| 552 | -1 means another errors in parameters passed
|
---|
| 553 | (N<=0, for example)
|
---|
| 554 | S - spline interpolant.
|
---|
| 555 | Rep - report, same format as in LSFitLinearW() subroutine.
|
---|
| 556 | Following fields are set:
|
---|
| 557 | * RMSError rms error on the (X,Y).
|
---|
| 558 | * AvgError average error on the (X,Y).
|
---|
| 559 | * AvgRelError average relative error on the non-zero Y
|
---|
| 560 | * MaxError maximum error
|
---|
| 561 | NON-WEIGHTED ERRORS ARE CALCULATED
|
---|
| 562 |
|
---|
| 563 | IMPORTANT:
|
---|
| 564 | this subroitine doesn't calculate task's condition number for K<>0.
|
---|
| 565 |
|
---|
| 566 | IMPORTANT:
|
---|
| 567 | this subroitine supports only even M's
|
---|
| 568 |
|
---|
| 569 | SETTING CONSTRAINTS - DANGERS AND OPPORTUNITIES:
|
---|
| 570 |
|
---|
| 571 | Setting constraints can lead to undesired results, like ill-conditioned
|
---|
| 572 | behavior, or inconsistency being detected. From the other side, it allows
|
---|
| 573 | us to improve quality of the fit. Here we summarize our experience with
|
---|
| 574 | constrained regression splines:
|
---|
| 575 | * excessive constraints can be inconsistent. Splines are piecewise cubic
|
---|
| 576 | functions, and it is easy to create an example, where large number of
|
---|
| 577 | constraints concentrated in small area will result in inconsistency.
|
---|
| 578 | Just because spline is not flexible enough to satisfy all of them. And
|
---|
| 579 | same constraints spread across the [min(x),max(x)] will be perfectly
|
---|
| 580 | consistent.
|
---|
| 581 | * the more evenly constraints are spread across [min(x),max(x)], the more
|
---|
| 582 | chances that they will be consistent
|
---|
| 583 | * the greater is M (given fixed constraints), the more chances that
|
---|
| 584 | constraints will be consistent
|
---|
| 585 | * in the general case, consistency of constraints is NOT GUARANTEED.
|
---|
| 586 | * in the several special cases, however, we can guarantee consistency.
|
---|
| 587 | * one of this cases is M>=4 and constraints on the function value
|
---|
| 588 | (AND/OR its derivative) at the interval boundaries.
|
---|
| 589 | * another special case is M>=4 and ONE constraint on the function value
|
---|
| 590 | (OR, BUT NOT AND, derivative) anywhere in [min(x),max(x)]
|
---|
| 591 |
|
---|
| 592 | Our final recommendation is to use constraints WHEN AND ONLY when you
|
---|
| 593 | can't solve your task without them. Anything beyond special cases given
|
---|
| 594 | above is not guaranteed and may result in inconsistency.
|
---|
| 595 |
|
---|
| 596 | -- ALGLIB PROJECT --
|
---|
| 597 | Copyright 18.08.2009 by Bochkanov Sergey
|
---|
| 598 | *************************************************************************/
|
---|
| 599 | public static void spline1dfithermitewc(ref double[] x,
|
---|
| 600 | ref double[] y,
|
---|
| 601 | ref double[] w,
|
---|
| 602 | int n,
|
---|
| 603 | ref double[] xc,
|
---|
| 604 | ref double[] yc,
|
---|
| 605 | ref int[] dc,
|
---|
| 606 | int k,
|
---|
| 607 | int m,
|
---|
| 608 | ref int info,
|
---|
| 609 | ref spline1dinterpolant s,
|
---|
| 610 | ref spline1dfitreport rep)
|
---|
| 611 | {
|
---|
| 612 | spline1dfitinternal(1, x, y, ref w, n, xc, yc, ref dc, k, m, ref info, ref s, ref rep);
|
---|
| 613 | }
|
---|
| 614 |
|
---|
| 615 |
|
---|
| 616 | /*************************************************************************
|
---|
| 617 | Least squares fitting by cubic spline.
|
---|
| 618 |
|
---|
| 619 | This subroutine is "lightweight" alternative for more complex and feature-
|
---|
| 620 | rich Spline1DFitCubicWC(). See Spline1DFitCubicWC() for more information
|
---|
| 621 | about subroutine parameters (we don't duplicate it here because of length)
|
---|
| 622 |
|
---|
| 623 | -- ALGLIB PROJECT --
|
---|
| 624 | Copyright 18.08.2009 by Bochkanov Sergey
|
---|
| 625 | *************************************************************************/
|
---|
| 626 | public static void spline1dfitcubic(ref double[] x,
|
---|
| 627 | ref double[] y,
|
---|
| 628 | int n,
|
---|
| 629 | int m,
|
---|
| 630 | ref int info,
|
---|
| 631 | ref spline1dinterpolant s,
|
---|
| 632 | ref spline1dfitreport rep)
|
---|
| 633 | {
|
---|
| 634 | int i = 0;
|
---|
| 635 | double[] w = new double[0];
|
---|
| 636 | double[] xc = new double[0];
|
---|
| 637 | double[] yc = new double[0];
|
---|
| 638 | int[] dc = new int[0];
|
---|
| 639 |
|
---|
| 640 | if( n>0 )
|
---|
| 641 | {
|
---|
| 642 | w = new double[n];
|
---|
| 643 | for(i=0; i<=n-1; i++)
|
---|
| 644 | {
|
---|
| 645 | w[i] = 1;
|
---|
| 646 | }
|
---|
| 647 | }
|
---|
| 648 | spline1dfitcubicwc(ref x, ref y, ref w, n, ref xc, ref yc, ref dc, 0, m, ref info, ref s, ref rep);
|
---|
| 649 | }
|
---|
| 650 |
|
---|
| 651 |
|
---|
| 652 | /*************************************************************************
|
---|
| 653 | Least squares fitting by Hermite spline.
|
---|
| 654 |
|
---|
| 655 | This subroutine is "lightweight" alternative for more complex and feature-
|
---|
| 656 | rich Spline1DFitHermiteWC(). See Spline1DFitHermiteWC() description for
|
---|
| 657 | more information about subroutine parameters (we don't duplicate it here
|
---|
| 658 | because of length).
|
---|
| 659 |
|
---|
| 660 | -- ALGLIB PROJECT --
|
---|
| 661 | Copyright 18.08.2009 by Bochkanov Sergey
|
---|
| 662 | *************************************************************************/
|
---|
| 663 | public static void spline1dfithermite(ref double[] x,
|
---|
| 664 | ref double[] y,
|
---|
| 665 | int n,
|
---|
| 666 | int m,
|
---|
| 667 | ref int info,
|
---|
| 668 | ref spline1dinterpolant s,
|
---|
| 669 | ref spline1dfitreport rep)
|
---|
| 670 | {
|
---|
| 671 | int i = 0;
|
---|
| 672 | double[] w = new double[0];
|
---|
| 673 | double[] xc = new double[0];
|
---|
| 674 | double[] yc = new double[0];
|
---|
| 675 | int[] dc = new int[0];
|
---|
| 676 |
|
---|
| 677 | if( n>0 )
|
---|
| 678 | {
|
---|
| 679 | w = new double[n];
|
---|
| 680 | for(i=0; i<=n-1; i++)
|
---|
| 681 | {
|
---|
| 682 | w[i] = 1;
|
---|
| 683 | }
|
---|
| 684 | }
|
---|
| 685 | spline1dfithermitewc(ref x, ref y, ref w, n, ref xc, ref yc, ref dc, 0, m, ref info, ref s, ref rep);
|
---|
| 686 | }
|
---|
| 687 |
|
---|
| 688 |
|
---|
| 689 | /*************************************************************************
|
---|
| 690 | This subroutine calculates the value of the spline at the given point X.
|
---|
| 691 |
|
---|
| 692 | INPUT PARAMETERS:
|
---|
| 693 | C - spline interpolant
|
---|
| 694 | X - point
|
---|
| 695 |
|
---|
| 696 | Result:
|
---|
| 697 | S(x)
|
---|
| 698 |
|
---|
| 699 | -- ALGLIB PROJECT --
|
---|
| 700 | Copyright 23.06.2007 by Bochkanov Sergey
|
---|
| 701 | *************************************************************************/
|
---|
| 702 | public static double spline1dcalc(ref spline1dinterpolant c,
|
---|
| 703 | double x)
|
---|
| 704 | {
|
---|
| 705 | double result = 0;
|
---|
| 706 | int l = 0;
|
---|
| 707 | int r = 0;
|
---|
| 708 | int m = 0;
|
---|
| 709 |
|
---|
| 710 | System.Diagnostics.Debug.Assert(c.k==3, "Spline1DCalc: internal error");
|
---|
| 711 |
|
---|
| 712 | //
|
---|
| 713 | // Binary search in the [ x[0], ..., x[n-2] ] (x[n-1] is not included)
|
---|
| 714 | //
|
---|
| 715 | l = 0;
|
---|
| 716 | r = c.n-2+1;
|
---|
| 717 | while( l!=r-1 )
|
---|
| 718 | {
|
---|
| 719 | m = (l+r)/2;
|
---|
| 720 | if( (double)(c.x[m])>=(double)(x) )
|
---|
| 721 | {
|
---|
| 722 | r = m;
|
---|
| 723 | }
|
---|
| 724 | else
|
---|
| 725 | {
|
---|
| 726 | l = m;
|
---|
| 727 | }
|
---|
| 728 | }
|
---|
| 729 |
|
---|
| 730 | //
|
---|
| 731 | // Interpolation
|
---|
| 732 | //
|
---|
| 733 | x = x-c.x[l];
|
---|
| 734 | m = 4*l;
|
---|
| 735 | result = c.c[m]+x*(c.c[m+1]+x*(c.c[m+2]+x*c.c[m+3]));
|
---|
| 736 | return result;
|
---|
| 737 | }
|
---|
| 738 |
|
---|
| 739 |
|
---|
| 740 | /*************************************************************************
|
---|
| 741 | This subroutine differentiates the spline.
|
---|
| 742 |
|
---|
| 743 | INPUT PARAMETERS:
|
---|
| 744 | C - spline interpolant.
|
---|
| 745 | X - point
|
---|
| 746 |
|
---|
| 747 | Result:
|
---|
| 748 | S - S(x)
|
---|
| 749 | DS - S'(x)
|
---|
| 750 | D2S - S''(x)
|
---|
| 751 |
|
---|
| 752 | -- ALGLIB PROJECT --
|
---|
| 753 | Copyright 24.06.2007 by Bochkanov Sergey
|
---|
| 754 | *************************************************************************/
|
---|
| 755 | public static void spline1ddiff(ref spline1dinterpolant c,
|
---|
| 756 | double x,
|
---|
| 757 | ref double s,
|
---|
| 758 | ref double ds,
|
---|
| 759 | ref double d2s)
|
---|
| 760 | {
|
---|
| 761 | int l = 0;
|
---|
| 762 | int r = 0;
|
---|
| 763 | int m = 0;
|
---|
| 764 |
|
---|
| 765 | System.Diagnostics.Debug.Assert(c.k==3, "Spline1DCalc: internal error");
|
---|
| 766 |
|
---|
| 767 | //
|
---|
| 768 | // Binary search
|
---|
| 769 | //
|
---|
| 770 | l = 0;
|
---|
| 771 | r = c.n-2+1;
|
---|
| 772 | while( l!=r-1 )
|
---|
| 773 | {
|
---|
| 774 | m = (l+r)/2;
|
---|
| 775 | if( (double)(c.x[m])>=(double)(x) )
|
---|
| 776 | {
|
---|
| 777 | r = m;
|
---|
| 778 | }
|
---|
| 779 | else
|
---|
| 780 | {
|
---|
| 781 | l = m;
|
---|
| 782 | }
|
---|
| 783 | }
|
---|
| 784 |
|
---|
| 785 | //
|
---|
| 786 | // Differentiation
|
---|
| 787 | //
|
---|
| 788 | x = x-c.x[l];
|
---|
| 789 | m = 4*l;
|
---|
| 790 | s = c.c[m]+x*(c.c[m+1]+x*(c.c[m+2]+x*c.c[m+3]));
|
---|
| 791 | ds = c.c[m+1]+2*x*c.c[m+2]+3*AP.Math.Sqr(x)*c.c[m+3];
|
---|
| 792 | d2s = 2*c.c[m+2]+6*x*c.c[m+3];
|
---|
| 793 | }
|
---|
| 794 |
|
---|
| 795 |
|
---|
| 796 | /*************************************************************************
|
---|
| 797 | This subroutine makes the copy of the spline.
|
---|
| 798 |
|
---|
| 799 | INPUT PARAMETERS:
|
---|
| 800 | C - spline interpolant.
|
---|
| 801 |
|
---|
| 802 | Result:
|
---|
| 803 | CC - spline copy
|
---|
| 804 |
|
---|
| 805 | -- ALGLIB PROJECT --
|
---|
| 806 | Copyright 29.06.2007 by Bochkanov Sergey
|
---|
| 807 | *************************************************************************/
|
---|
| 808 | public static void spline1dcopy(ref spline1dinterpolant c,
|
---|
| 809 | ref spline1dinterpolant cc)
|
---|
| 810 | {
|
---|
| 811 | int i_ = 0;
|
---|
| 812 |
|
---|
| 813 | cc.n = c.n;
|
---|
| 814 | cc.k = c.k;
|
---|
| 815 | cc.x = new double[cc.n];
|
---|
| 816 | for(i_=0; i_<=cc.n-1;i_++)
|
---|
| 817 | {
|
---|
| 818 | cc.x[i_] = c.x[i_];
|
---|
| 819 | }
|
---|
| 820 | cc.c = new double[(cc.k+1)*(cc.n-1)];
|
---|
| 821 | for(i_=0; i_<=(cc.k+1)*(cc.n-1)-1;i_++)
|
---|
| 822 | {
|
---|
| 823 | cc.c[i_] = c.c[i_];
|
---|
| 824 | }
|
---|
| 825 | }
|
---|
| 826 |
|
---|
| 827 |
|
---|
| 828 | /*************************************************************************
|
---|
| 829 | Serialization of the spline interpolant
|
---|
| 830 |
|
---|
| 831 | INPUT PARAMETERS:
|
---|
| 832 | B - spline interpolant
|
---|
| 833 |
|
---|
| 834 | OUTPUT PARAMETERS:
|
---|
| 835 | RA - array of real numbers which contains interpolant,
|
---|
| 836 | array[0..RLen-1]
|
---|
| 837 | RLen - RA lenght
|
---|
| 838 |
|
---|
| 839 | -- ALGLIB --
|
---|
| 840 | Copyright 17.08.2009 by Bochkanov Sergey
|
---|
| 841 | *************************************************************************/
|
---|
| 842 | public static void spline1dserialize(ref spline1dinterpolant c,
|
---|
| 843 | ref double[] ra,
|
---|
| 844 | ref int ralen)
|
---|
| 845 | {
|
---|
| 846 | int i_ = 0;
|
---|
| 847 | int i1_ = 0;
|
---|
| 848 |
|
---|
| 849 | ralen = 2+2+c.n+(c.k+1)*(c.n-1);
|
---|
| 850 | ra = new double[ralen];
|
---|
| 851 | ra[0] = ralen;
|
---|
| 852 | ra[1] = spline1dvnum;
|
---|
| 853 | ra[2] = c.n;
|
---|
| 854 | ra[3] = c.k;
|
---|
| 855 | i1_ = (0) - (4);
|
---|
| 856 | for(i_=4; i_<=4+c.n-1;i_++)
|
---|
| 857 | {
|
---|
| 858 | ra[i_] = c.x[i_+i1_];
|
---|
| 859 | }
|
---|
| 860 | i1_ = (0) - (4+c.n);
|
---|
| 861 | for(i_=4+c.n; i_<=4+c.n+(c.k+1)*(c.n-1)-1;i_++)
|
---|
| 862 | {
|
---|
| 863 | ra[i_] = c.c[i_+i1_];
|
---|
| 864 | }
|
---|
| 865 | }
|
---|
| 866 |
|
---|
| 867 |
|
---|
| 868 | /*************************************************************************
|
---|
| 869 | Unserialization of the spline interpolant
|
---|
| 870 |
|
---|
| 871 | INPUT PARAMETERS:
|
---|
| 872 | RA - array of real numbers which contains interpolant,
|
---|
| 873 |
|
---|
| 874 | OUTPUT PARAMETERS:
|
---|
| 875 | B - spline interpolant
|
---|
| 876 |
|
---|
| 877 | -- ALGLIB --
|
---|
| 878 | Copyright 17.08.2009 by Bochkanov Sergey
|
---|
| 879 | *************************************************************************/
|
---|
| 880 | public static void spline1dunserialize(ref double[] ra,
|
---|
| 881 | ref spline1dinterpolant c)
|
---|
| 882 | {
|
---|
| 883 | int i_ = 0;
|
---|
| 884 | int i1_ = 0;
|
---|
| 885 |
|
---|
| 886 | System.Diagnostics.Debug.Assert((int)Math.Round(ra[1])==spline1dvnum, "Spline1DUnserialize: corrupted array!");
|
---|
| 887 | c.n = (int)Math.Round(ra[2]);
|
---|
| 888 | c.k = (int)Math.Round(ra[3]);
|
---|
| 889 | c.x = new double[c.n];
|
---|
| 890 | c.c = new double[(c.k+1)*(c.n-1)];
|
---|
| 891 | i1_ = (4) - (0);
|
---|
| 892 | for(i_=0; i_<=c.n-1;i_++)
|
---|
| 893 | {
|
---|
| 894 | c.x[i_] = ra[i_+i1_];
|
---|
| 895 | }
|
---|
| 896 | i1_ = (4+c.n) - (0);
|
---|
| 897 | for(i_=0; i_<=(c.k+1)*(c.n-1)-1;i_++)
|
---|
| 898 | {
|
---|
| 899 | c.c[i_] = ra[i_+i1_];
|
---|
| 900 | }
|
---|
| 901 | }
|
---|
| 902 |
|
---|
| 903 |
|
---|
| 904 | /*************************************************************************
|
---|
| 905 | This subroutine unpacks the spline into the coefficients table.
|
---|
| 906 |
|
---|
| 907 | INPUT PARAMETERS:
|
---|
| 908 | C - spline interpolant.
|
---|
| 909 | X - point
|
---|
| 910 |
|
---|
| 911 | Result:
|
---|
| 912 | Tbl - coefficients table, unpacked format, array[0..N-2, 0..5].
|
---|
| 913 | For I = 0...N-2:
|
---|
| 914 | Tbl[I,0] = X[i]
|
---|
| 915 | Tbl[I,1] = X[i+1]
|
---|
| 916 | Tbl[I,2] = C0
|
---|
| 917 | Tbl[I,3] = C1
|
---|
| 918 | Tbl[I,4] = C2
|
---|
| 919 | Tbl[I,5] = C3
|
---|
| 920 | On [x[i], x[i+1]] spline is equals to:
|
---|
| 921 | S(x) = C0 + C1*t + C2*t^2 + C3*t^3
|
---|
| 922 | t = x-x[i]
|
---|
| 923 |
|
---|
| 924 | -- ALGLIB PROJECT --
|
---|
| 925 | Copyright 29.06.2007 by Bochkanov Sergey
|
---|
| 926 | *************************************************************************/
|
---|
| 927 | public static void spline1dunpack(ref spline1dinterpolant c,
|
---|
| 928 | ref int n,
|
---|
| 929 | ref double[,] tbl)
|
---|
| 930 | {
|
---|
| 931 | int i = 0;
|
---|
| 932 | int j = 0;
|
---|
| 933 |
|
---|
| 934 | tbl = new double[c.n-2+1, 2+c.k+1];
|
---|
| 935 | n = c.n;
|
---|
| 936 |
|
---|
| 937 | //
|
---|
| 938 | // Fill
|
---|
| 939 | //
|
---|
| 940 | for(i=0; i<=n-2; i++)
|
---|
| 941 | {
|
---|
| 942 | tbl[i,0] = c.x[i];
|
---|
| 943 | tbl[i,1] = c.x[i+1];
|
---|
| 944 | for(j=0; j<=c.k; j++)
|
---|
| 945 | {
|
---|
| 946 | tbl[i,2+j] = c.c[(c.k+1)*i+j];
|
---|
| 947 | }
|
---|
| 948 | }
|
---|
| 949 | }
|
---|
| 950 |
|
---|
| 951 |
|
---|
| 952 | /*************************************************************************
|
---|
| 953 | This subroutine performs linear transformation of the spline argument.
|
---|
| 954 |
|
---|
| 955 | INPUT PARAMETERS:
|
---|
| 956 | C - spline interpolant.
|
---|
| 957 | A, B- transformation coefficients: x = A*t + B
|
---|
| 958 | Result:
|
---|
| 959 | C - transformed spline
|
---|
| 960 |
|
---|
| 961 | -- ALGLIB PROJECT --
|
---|
| 962 | Copyright 30.06.2007 by Bochkanov Sergey
|
---|
| 963 | *************************************************************************/
|
---|
| 964 | public static void spline1dlintransx(ref spline1dinterpolant c,
|
---|
| 965 | double a,
|
---|
| 966 | double b)
|
---|
| 967 | {
|
---|
| 968 | int i = 0;
|
---|
| 969 | int j = 0;
|
---|
| 970 | int n = 0;
|
---|
| 971 | double v = 0;
|
---|
| 972 | double dv = 0;
|
---|
| 973 | double d2v = 0;
|
---|
| 974 | double[] x = new double[0];
|
---|
| 975 | double[] y = new double[0];
|
---|
| 976 | double[] d = new double[0];
|
---|
| 977 |
|
---|
| 978 | n = c.n;
|
---|
| 979 |
|
---|
| 980 | //
|
---|
| 981 | // Special case: A=0
|
---|
| 982 | //
|
---|
| 983 | if( (double)(a)==(double)(0) )
|
---|
| 984 | {
|
---|
| 985 | v = spline1dcalc(ref c, b);
|
---|
| 986 | for(i=0; i<=n-2; i++)
|
---|
| 987 | {
|
---|
| 988 | c.c[(c.k+1)*i] = v;
|
---|
| 989 | for(j=1; j<=c.k; j++)
|
---|
| 990 | {
|
---|
| 991 | c.c[(c.k+1)*i+j] = 0;
|
---|
| 992 | }
|
---|
| 993 | }
|
---|
| 994 | return;
|
---|
| 995 | }
|
---|
| 996 |
|
---|
| 997 | //
|
---|
| 998 | // General case: A<>0.
|
---|
| 999 | // Unpack, X, Y, dY/dX.
|
---|
| 1000 | // Scale and pack again.
|
---|
| 1001 | //
|
---|
| 1002 | System.Diagnostics.Debug.Assert(c.k==3, "Spline1DLinTransX: internal error");
|
---|
| 1003 | x = new double[n-1+1];
|
---|
| 1004 | y = new double[n-1+1];
|
---|
| 1005 | d = new double[n-1+1];
|
---|
| 1006 | for(i=0; i<=n-1; i++)
|
---|
| 1007 | {
|
---|
| 1008 | x[i] = c.x[i];
|
---|
| 1009 | spline1ddiff(ref c, x[i], ref v, ref dv, ref d2v);
|
---|
| 1010 | x[i] = (x[i]-b)/a;
|
---|
| 1011 | y[i] = v;
|
---|
| 1012 | d[i] = a*dv;
|
---|
| 1013 | }
|
---|
| 1014 | spline1dbuildhermite(x, y, d, n, ref c);
|
---|
| 1015 | }
|
---|
| 1016 |
|
---|
| 1017 |
|
---|
| 1018 | /*************************************************************************
|
---|
| 1019 | This subroutine performs linear transformation of the spline.
|
---|
| 1020 |
|
---|
| 1021 | INPUT PARAMETERS:
|
---|
| 1022 | C - spline interpolant.
|
---|
| 1023 | A, B- transformation coefficients: S2(x) = A*S(x) + B
|
---|
| 1024 | Result:
|
---|
| 1025 | C - transformed spline
|
---|
| 1026 |
|
---|
| 1027 | -- ALGLIB PROJECT --
|
---|
| 1028 | Copyright 30.06.2007 by Bochkanov Sergey
|
---|
| 1029 | *************************************************************************/
|
---|
| 1030 | public static void spline1dlintransy(ref spline1dinterpolant c,
|
---|
| 1031 | double a,
|
---|
| 1032 | double b)
|
---|
| 1033 | {
|
---|
| 1034 | int i = 0;
|
---|
| 1035 | int j = 0;
|
---|
| 1036 | int n = 0;
|
---|
| 1037 |
|
---|
| 1038 | n = c.n;
|
---|
| 1039 | for(i=0; i<=n-2; i++)
|
---|
| 1040 | {
|
---|
| 1041 | c.c[(c.k+1)*i] = a*c.c[(c.k+1)*i]+b;
|
---|
| 1042 | for(j=1; j<=c.k; j++)
|
---|
| 1043 | {
|
---|
| 1044 | c.c[(c.k+1)*i+j] = a*c.c[(c.k+1)*i+j];
|
---|
| 1045 | }
|
---|
| 1046 | }
|
---|
| 1047 | }
|
---|
| 1048 |
|
---|
| 1049 |
|
---|
| 1050 | /*************************************************************************
|
---|
| 1051 | This subroutine integrates the spline.
|
---|
| 1052 |
|
---|
| 1053 | INPUT PARAMETERS:
|
---|
| 1054 | C - spline interpolant.
|
---|
| 1055 | X - right bound of the integration interval [a, x]
|
---|
| 1056 | Result:
|
---|
| 1057 | integral(S(t)dt,a,x)
|
---|
| 1058 |
|
---|
| 1059 | -- ALGLIB PROJECT --
|
---|
| 1060 | Copyright 23.06.2007 by Bochkanov Sergey
|
---|
| 1061 | *************************************************************************/
|
---|
| 1062 | public static double spline1dintegrate(ref spline1dinterpolant c,
|
---|
| 1063 | double x)
|
---|
| 1064 | {
|
---|
| 1065 | double result = 0;
|
---|
| 1066 | int n = 0;
|
---|
| 1067 | int i = 0;
|
---|
| 1068 | int j = 0;
|
---|
| 1069 | int l = 0;
|
---|
| 1070 | int r = 0;
|
---|
| 1071 | int m = 0;
|
---|
| 1072 | double w = 0;
|
---|
| 1073 | double v = 0;
|
---|
| 1074 |
|
---|
| 1075 | n = c.n;
|
---|
| 1076 |
|
---|
| 1077 | //
|
---|
| 1078 | // Binary search in the [ x[0], ..., x[n-2] ] (x[n-1] is not included)
|
---|
| 1079 | //
|
---|
| 1080 | l = 0;
|
---|
| 1081 | r = n-2+1;
|
---|
| 1082 | while( l!=r-1 )
|
---|
| 1083 | {
|
---|
| 1084 | m = (l+r)/2;
|
---|
| 1085 | if( (double)(c.x[m])>=(double)(x) )
|
---|
| 1086 | {
|
---|
| 1087 | r = m;
|
---|
| 1088 | }
|
---|
| 1089 | else
|
---|
| 1090 | {
|
---|
| 1091 | l = m;
|
---|
| 1092 | }
|
---|
| 1093 | }
|
---|
| 1094 |
|
---|
| 1095 | //
|
---|
| 1096 | // Integration
|
---|
| 1097 | //
|
---|
| 1098 | result = 0;
|
---|
| 1099 | for(i=0; i<=l-1; i++)
|
---|
| 1100 | {
|
---|
| 1101 | w = c.x[i+1]-c.x[i];
|
---|
| 1102 | m = (c.k+1)*i;
|
---|
| 1103 | result = result+c.c[m]*w;
|
---|
| 1104 | v = w;
|
---|
| 1105 | for(j=1; j<=c.k; j++)
|
---|
| 1106 | {
|
---|
| 1107 | v = v*w;
|
---|
| 1108 | result = result+c.c[m+j]*v/(j+1);
|
---|
| 1109 | }
|
---|
| 1110 | }
|
---|
| 1111 | w = x-c.x[l];
|
---|
| 1112 | m = (c.k+1)*l;
|
---|
| 1113 | v = w;
|
---|
| 1114 | result = result+c.c[m]*w;
|
---|
| 1115 | for(j=1; j<=c.k; j++)
|
---|
| 1116 | {
|
---|
| 1117 | v = v*w;
|
---|
| 1118 | result = result+c.c[m+j]*v/(j+1);
|
---|
| 1119 | }
|
---|
| 1120 | return result;
|
---|
| 1121 | }
|
---|
| 1122 |
|
---|
| 1123 |
|
---|
| 1124 | /*************************************************************************
|
---|
| 1125 | Internal spline fitting subroutine
|
---|
| 1126 |
|
---|
| 1127 | -- ALGLIB PROJECT --
|
---|
| 1128 | Copyright 08.09.2009 by Bochkanov Sergey
|
---|
| 1129 | *************************************************************************/
|
---|
| 1130 | private static void spline1dfitinternal(int st,
|
---|
| 1131 | double[] x,
|
---|
| 1132 | double[] y,
|
---|
| 1133 | ref double[] w,
|
---|
| 1134 | int n,
|
---|
| 1135 | double[] xc,
|
---|
| 1136 | double[] yc,
|
---|
| 1137 | ref int[] dc,
|
---|
| 1138 | int k,
|
---|
| 1139 | int m,
|
---|
| 1140 | ref int info,
|
---|
| 1141 | ref spline1dinterpolant s,
|
---|
| 1142 | ref spline1dfitreport rep)
|
---|
| 1143 | {
|
---|
| 1144 | double[,] fmatrix = new double[0,0];
|
---|
| 1145 | double[,] cmatrix = new double[0,0];
|
---|
| 1146 | double[] y2 = new double[0];
|
---|
| 1147 | double[] w2 = new double[0];
|
---|
| 1148 | double[] sx = new double[0];
|
---|
| 1149 | double[] sy = new double[0];
|
---|
| 1150 | double[] sd = new double[0];
|
---|
| 1151 | double[] tmp = new double[0];
|
---|
| 1152 | double[] xoriginal = new double[0];
|
---|
| 1153 | double[] yoriginal = new double[0];
|
---|
| 1154 | lsfit.lsfitreport lrep = new lsfit.lsfitreport();
|
---|
| 1155 | double v0 = 0;
|
---|
| 1156 | double v1 = 0;
|
---|
| 1157 | double v2 = 0;
|
---|
| 1158 | double mx = 0;
|
---|
| 1159 | spline1dinterpolant s2 = new spline1dinterpolant();
|
---|
| 1160 | int i = 0;
|
---|
| 1161 | int j = 0;
|
---|
| 1162 | int relcnt = 0;
|
---|
| 1163 | double xa = 0;
|
---|
| 1164 | double xb = 0;
|
---|
| 1165 | double sa = 0;
|
---|
| 1166 | double sb = 0;
|
---|
| 1167 | double bl = 0;
|
---|
| 1168 | double br = 0;
|
---|
| 1169 | double decay = 0;
|
---|
| 1170 | int i_ = 0;
|
---|
| 1171 |
|
---|
| 1172 | x = (double[])x.Clone();
|
---|
| 1173 | y = (double[])y.Clone();
|
---|
| 1174 | xc = (double[])xc.Clone();
|
---|
| 1175 | yc = (double[])yc.Clone();
|
---|
| 1176 |
|
---|
| 1177 | System.Diagnostics.Debug.Assert(st==0 | st==1, "Spline1DFit: internal error!");
|
---|
| 1178 | if( st==0 & m<4 )
|
---|
| 1179 | {
|
---|
| 1180 | info = -1;
|
---|
| 1181 | return;
|
---|
| 1182 | }
|
---|
| 1183 | if( st==1 & m<4 )
|
---|
| 1184 | {
|
---|
| 1185 | info = -1;
|
---|
| 1186 | return;
|
---|
| 1187 | }
|
---|
| 1188 | if( n<1 | k<0 | k>=m )
|
---|
| 1189 | {
|
---|
| 1190 | info = -1;
|
---|
| 1191 | return;
|
---|
| 1192 | }
|
---|
| 1193 | for(i=0; i<=k-1; i++)
|
---|
| 1194 | {
|
---|
| 1195 | info = 0;
|
---|
| 1196 | if( dc[i]<0 )
|
---|
| 1197 | {
|
---|
| 1198 | info = -1;
|
---|
| 1199 | }
|
---|
| 1200 | if( dc[i]>1 )
|
---|
| 1201 | {
|
---|
| 1202 | info = -1;
|
---|
| 1203 | }
|
---|
| 1204 | if( info<0 )
|
---|
| 1205 | {
|
---|
| 1206 | return;
|
---|
| 1207 | }
|
---|
| 1208 | }
|
---|
| 1209 | if( st==1 & m%2!=0 )
|
---|
| 1210 | {
|
---|
| 1211 |
|
---|
| 1212 | //
|
---|
| 1213 | // Hermite fitter must have even number of basis functions
|
---|
| 1214 | //
|
---|
| 1215 | info = -2;
|
---|
| 1216 | return;
|
---|
| 1217 | }
|
---|
| 1218 |
|
---|
| 1219 | //
|
---|
| 1220 | // weight decay for correct handling of task which becomes
|
---|
| 1221 | // degenerate after constraints are applied
|
---|
| 1222 | //
|
---|
| 1223 | decay = 10000*AP.Math.MachineEpsilon;
|
---|
| 1224 |
|
---|
| 1225 | //
|
---|
| 1226 | // Scale X, Y, XC, YC
|
---|
| 1227 | //
|
---|
| 1228 | lsfit.lsfitscalexy(ref x, ref y, n, ref xc, ref yc, ref dc, k, ref xa, ref xb, ref sa, ref sb, ref xoriginal, ref yoriginal);
|
---|
| 1229 |
|
---|
| 1230 | //
|
---|
| 1231 | // allocate space, initialize:
|
---|
| 1232 | // * SX - grid for basis functions
|
---|
| 1233 | // * SY - values of basis functions at grid points
|
---|
| 1234 | // * FMatrix- values of basis functions at X[]
|
---|
| 1235 | // * CMatrix- values (derivatives) of basis functions at XC[]
|
---|
| 1236 | //
|
---|
| 1237 | y2 = new double[n+m];
|
---|
| 1238 | w2 = new double[n+m];
|
---|
| 1239 | fmatrix = new double[n+m, m];
|
---|
| 1240 | if( k>0 )
|
---|
| 1241 | {
|
---|
| 1242 | cmatrix = new double[k, m+1];
|
---|
| 1243 | }
|
---|
| 1244 | if( st==0 )
|
---|
| 1245 | {
|
---|
| 1246 |
|
---|
| 1247 | //
|
---|
| 1248 | // allocate space for cubic spline
|
---|
| 1249 | //
|
---|
| 1250 | sx = new double[m-2];
|
---|
| 1251 | sy = new double[m-2];
|
---|
| 1252 | for(j=0; j<=m-2-1; j++)
|
---|
| 1253 | {
|
---|
| 1254 | sx[j] = (double)(2*j)/((double)(m-2-1))-1;
|
---|
| 1255 | }
|
---|
| 1256 | }
|
---|
| 1257 | if( st==1 )
|
---|
| 1258 | {
|
---|
| 1259 |
|
---|
| 1260 | //
|
---|
| 1261 | // allocate space for Hermite spline
|
---|
| 1262 | //
|
---|
| 1263 | sx = new double[m/2];
|
---|
| 1264 | sy = new double[m/2];
|
---|
| 1265 | sd = new double[m/2];
|
---|
| 1266 | for(j=0; j<=m/2-1; j++)
|
---|
| 1267 | {
|
---|
| 1268 | sx[j] = (double)(2*j)/((double)(m/2-1))-1;
|
---|
| 1269 | }
|
---|
| 1270 | }
|
---|
| 1271 |
|
---|
| 1272 | //
|
---|
| 1273 | // Prepare design and constraints matrices:
|
---|
| 1274 | // * fill constraints matrix
|
---|
| 1275 | // * fill first N rows of design matrix with values
|
---|
| 1276 | // * fill next M rows of design matrix with regularizing term
|
---|
| 1277 | // * append M zeros to Y
|
---|
| 1278 | // * append M elements, mean(abs(W)) each, to W
|
---|
| 1279 | //
|
---|
| 1280 | for(j=0; j<=m-1; j++)
|
---|
| 1281 | {
|
---|
| 1282 |
|
---|
| 1283 | //
|
---|
| 1284 | // prepare Jth basis function
|
---|
| 1285 | //
|
---|
| 1286 | if( st==0 )
|
---|
| 1287 | {
|
---|
| 1288 |
|
---|
| 1289 | //
|
---|
| 1290 | // cubic spline basis
|
---|
| 1291 | //
|
---|
| 1292 | for(i=0; i<=m-2-1; i++)
|
---|
| 1293 | {
|
---|
| 1294 | sy[i] = 0;
|
---|
| 1295 | }
|
---|
| 1296 | bl = 0;
|
---|
| 1297 | br = 0;
|
---|
| 1298 | if( j<m-2 )
|
---|
| 1299 | {
|
---|
| 1300 | sy[j] = 1;
|
---|
| 1301 | }
|
---|
| 1302 | if( j==m-2 )
|
---|
| 1303 | {
|
---|
| 1304 | bl = 1;
|
---|
| 1305 | }
|
---|
| 1306 | if( j==m-1 )
|
---|
| 1307 | {
|
---|
| 1308 | br = 1;
|
---|
| 1309 | }
|
---|
| 1310 | spline1dbuildcubic(sx, sy, m-2, 1, bl, 1, br, ref s2);
|
---|
| 1311 | }
|
---|
| 1312 | if( st==1 )
|
---|
| 1313 | {
|
---|
| 1314 |
|
---|
| 1315 | //
|
---|
| 1316 | // Hermite basis
|
---|
| 1317 | //
|
---|
| 1318 | for(i=0; i<=m/2-1; i++)
|
---|
| 1319 | {
|
---|
| 1320 | sy[i] = 0;
|
---|
| 1321 | sd[i] = 0;
|
---|
| 1322 | }
|
---|
| 1323 | if( j%2==0 )
|
---|
| 1324 | {
|
---|
| 1325 | sy[j/2] = 1;
|
---|
| 1326 | }
|
---|
| 1327 | else
|
---|
| 1328 | {
|
---|
| 1329 | sd[j/2] = 1;
|
---|
| 1330 | }
|
---|
| 1331 | spline1dbuildhermite(sx, sy, sd, m/2, ref s2);
|
---|
| 1332 | }
|
---|
| 1333 |
|
---|
| 1334 | //
|
---|
| 1335 | // values at X[], XC[]
|
---|
| 1336 | //
|
---|
| 1337 | for(i=0; i<=n-1; i++)
|
---|
| 1338 | {
|
---|
| 1339 | fmatrix[i,j] = spline1dcalc(ref s2, x[i]);
|
---|
| 1340 | }
|
---|
| 1341 | for(i=0; i<=k-1; i++)
|
---|
| 1342 | {
|
---|
| 1343 | System.Diagnostics.Debug.Assert(dc[i]>=0 & dc[i]<=2, "Spline1DFit: internal error!");
|
---|
| 1344 | spline1ddiff(ref s2, xc[i], ref v0, ref v1, ref v2);
|
---|
| 1345 | if( dc[i]==0 )
|
---|
| 1346 | {
|
---|
| 1347 | cmatrix[i,j] = v0;
|
---|
| 1348 | }
|
---|
| 1349 | if( dc[i]==1 )
|
---|
| 1350 | {
|
---|
| 1351 | cmatrix[i,j] = v1;
|
---|
| 1352 | }
|
---|
| 1353 | if( dc[i]==2 )
|
---|
| 1354 | {
|
---|
| 1355 | cmatrix[i,j] = v2;
|
---|
| 1356 | }
|
---|
| 1357 | }
|
---|
| 1358 | }
|
---|
| 1359 | for(i=0; i<=k-1; i++)
|
---|
| 1360 | {
|
---|
| 1361 | cmatrix[i,m] = yc[i];
|
---|
| 1362 | }
|
---|
| 1363 | for(i=0; i<=m-1; i++)
|
---|
| 1364 | {
|
---|
| 1365 | for(j=0; j<=m-1; j++)
|
---|
| 1366 | {
|
---|
| 1367 | if( i==j )
|
---|
| 1368 | {
|
---|
| 1369 | fmatrix[n+i,j] = decay;
|
---|
| 1370 | }
|
---|
| 1371 | else
|
---|
| 1372 | {
|
---|
| 1373 | fmatrix[n+i,j] = 0;
|
---|
| 1374 | }
|
---|
| 1375 | }
|
---|
| 1376 | }
|
---|
| 1377 | y2 = new double[n+m];
|
---|
| 1378 | w2 = new double[n+m];
|
---|
| 1379 | for(i_=0; i_<=n-1;i_++)
|
---|
| 1380 | {
|
---|
| 1381 | y2[i_] = y[i_];
|
---|
| 1382 | }
|
---|
| 1383 | for(i_=0; i_<=n-1;i_++)
|
---|
| 1384 | {
|
---|
| 1385 | w2[i_] = w[i_];
|
---|
| 1386 | }
|
---|
| 1387 | mx = 0;
|
---|
| 1388 | for(i=0; i<=n-1; i++)
|
---|
| 1389 | {
|
---|
| 1390 | mx = mx+Math.Abs(w[i]);
|
---|
| 1391 | }
|
---|
| 1392 | mx = mx/n;
|
---|
| 1393 | for(i=0; i<=m-1; i++)
|
---|
| 1394 | {
|
---|
| 1395 | y2[n+i] = 0;
|
---|
| 1396 | w2[n+i] = mx;
|
---|
| 1397 | }
|
---|
| 1398 |
|
---|
| 1399 | //
|
---|
| 1400 | // Solve constrained task
|
---|
| 1401 | //
|
---|
| 1402 | if( k>0 )
|
---|
| 1403 | {
|
---|
| 1404 |
|
---|
| 1405 | //
|
---|
| 1406 | // solve using regularization
|
---|
| 1407 | //
|
---|
| 1408 | lsfit.lsfitlinearwc(y2, ref w2, ref fmatrix, cmatrix, n+m, m, k, ref info, ref tmp, ref lrep);
|
---|
| 1409 | }
|
---|
| 1410 | else
|
---|
| 1411 | {
|
---|
| 1412 |
|
---|
| 1413 | //
|
---|
| 1414 | // no constraints, no regularization needed
|
---|
| 1415 | //
|
---|
| 1416 | lsfit.lsfitlinearwc(y, ref w, ref fmatrix, cmatrix, n, m, k, ref info, ref tmp, ref lrep);
|
---|
| 1417 | }
|
---|
| 1418 | if( info<0 )
|
---|
| 1419 | {
|
---|
| 1420 | return;
|
---|
| 1421 | }
|
---|
| 1422 |
|
---|
| 1423 | //
|
---|
| 1424 | // Generate spline and scale it
|
---|
| 1425 | //
|
---|
| 1426 | if( st==0 )
|
---|
| 1427 | {
|
---|
| 1428 |
|
---|
| 1429 | //
|
---|
| 1430 | // cubic spline basis
|
---|
| 1431 | //
|
---|
| 1432 | for(i_=0; i_<=m-2-1;i_++)
|
---|
| 1433 | {
|
---|
| 1434 | sy[i_] = tmp[i_];
|
---|
| 1435 | }
|
---|
| 1436 | spline1dbuildcubic(sx, sy, m-2, 1, tmp[m-2], 1, tmp[m-1], ref s);
|
---|
| 1437 | }
|
---|
| 1438 | if( st==1 )
|
---|
| 1439 | {
|
---|
| 1440 |
|
---|
| 1441 | //
|
---|
| 1442 | // Hermite basis
|
---|
| 1443 | //
|
---|
| 1444 | for(i=0; i<=m/2-1; i++)
|
---|
| 1445 | {
|
---|
| 1446 | sy[i] = tmp[2*i];
|
---|
| 1447 | sd[i] = tmp[2*i+1];
|
---|
| 1448 | }
|
---|
| 1449 | spline1dbuildhermite(sx, sy, sd, m/2, ref s);
|
---|
| 1450 | }
|
---|
| 1451 | spline1dlintransx(ref s, 2/(xb-xa), -((xa+xb)/(xb-xa)));
|
---|
| 1452 | spline1dlintransy(ref s, sb-sa, sa);
|
---|
| 1453 |
|
---|
| 1454 | //
|
---|
| 1455 | // Scale absolute errors obtained from LSFitLinearW.
|
---|
| 1456 | // Relative error should be calculated separately
|
---|
| 1457 | // (because of shifting/scaling of the task)
|
---|
| 1458 | //
|
---|
| 1459 | rep.taskrcond = lrep.taskrcond;
|
---|
| 1460 | rep.rmserror = lrep.rmserror*(sb-sa);
|
---|
| 1461 | rep.avgerror = lrep.avgerror*(sb-sa);
|
---|
| 1462 | rep.maxerror = lrep.maxerror*(sb-sa);
|
---|
| 1463 | rep.avgrelerror = 0;
|
---|
| 1464 | relcnt = 0;
|
---|
| 1465 | for(i=0; i<=n-1; i++)
|
---|
| 1466 | {
|
---|
| 1467 | if( (double)(yoriginal[i])!=(double)(0) )
|
---|
| 1468 | {
|
---|
| 1469 | rep.avgrelerror = rep.avgrelerror+Math.Abs(spline1dcalc(ref s, xoriginal[i])-yoriginal[i])/Math.Abs(yoriginal[i]);
|
---|
| 1470 | relcnt = relcnt+1;
|
---|
| 1471 | }
|
---|
| 1472 | }
|
---|
| 1473 | if( relcnt!=0 )
|
---|
| 1474 | {
|
---|
| 1475 | rep.avgrelerror = rep.avgrelerror/relcnt;
|
---|
| 1476 | }
|
---|
| 1477 | }
|
---|
| 1478 |
|
---|
| 1479 |
|
---|
| 1480 | /*************************************************************************
|
---|
| 1481 | Internal subroutine. Heap sort.
|
---|
| 1482 | *************************************************************************/
|
---|
| 1483 | private static void heapsortpoints(ref double[] x,
|
---|
| 1484 | ref double[] y,
|
---|
| 1485 | int n)
|
---|
| 1486 | {
|
---|
| 1487 | int i = 0;
|
---|
| 1488 | int j = 0;
|
---|
| 1489 | int k = 0;
|
---|
| 1490 | int t = 0;
|
---|
| 1491 | double tmp = 0;
|
---|
| 1492 | bool isascending = new bool();
|
---|
| 1493 | bool isdescending = new bool();
|
---|
| 1494 |
|
---|
| 1495 |
|
---|
| 1496 | //
|
---|
| 1497 | // Test for already sorted set
|
---|
| 1498 | //
|
---|
| 1499 | isascending = true;
|
---|
| 1500 | isdescending = true;
|
---|
| 1501 | for(i=1; i<=n-1; i++)
|
---|
| 1502 | {
|
---|
| 1503 | isascending = isascending & (double)(x[i])>(double)(x[i-1]);
|
---|
| 1504 | isdescending = isdescending & (double)(x[i])<(double)(x[i-1]);
|
---|
| 1505 | }
|
---|
| 1506 | if( isascending )
|
---|
| 1507 | {
|
---|
| 1508 | return;
|
---|
| 1509 | }
|
---|
| 1510 | if( isdescending )
|
---|
| 1511 | {
|
---|
| 1512 | for(i=0; i<=n-1; i++)
|
---|
| 1513 | {
|
---|
| 1514 | j = n-1-i;
|
---|
| 1515 | if( j<=i )
|
---|
| 1516 | {
|
---|
| 1517 | break;
|
---|
| 1518 | }
|
---|
| 1519 | tmp = x[i];
|
---|
| 1520 | x[i] = x[j];
|
---|
| 1521 | x[j] = tmp;
|
---|
| 1522 | tmp = y[i];
|
---|
| 1523 | y[i] = y[j];
|
---|
| 1524 | y[j] = tmp;
|
---|
| 1525 | }
|
---|
| 1526 | return;
|
---|
| 1527 | }
|
---|
| 1528 |
|
---|
| 1529 | //
|
---|
| 1530 | // Special case: N=1
|
---|
| 1531 | //
|
---|
| 1532 | if( n==1 )
|
---|
| 1533 | {
|
---|
| 1534 | return;
|
---|
| 1535 | }
|
---|
| 1536 |
|
---|
| 1537 | //
|
---|
| 1538 | // General case
|
---|
| 1539 | //
|
---|
| 1540 | i = 2;
|
---|
| 1541 | do
|
---|
| 1542 | {
|
---|
| 1543 | t = i;
|
---|
| 1544 | while( t!=1 )
|
---|
| 1545 | {
|
---|
| 1546 | k = t/2;
|
---|
| 1547 | if( (double)(x[k-1])>=(double)(x[t-1]) )
|
---|
| 1548 | {
|
---|
| 1549 | t = 1;
|
---|
| 1550 | }
|
---|
| 1551 | else
|
---|
| 1552 | {
|
---|
| 1553 | tmp = x[k-1];
|
---|
| 1554 | x[k-1] = x[t-1];
|
---|
| 1555 | x[t-1] = tmp;
|
---|
| 1556 | tmp = y[k-1];
|
---|
| 1557 | y[k-1] = y[t-1];
|
---|
| 1558 | y[t-1] = tmp;
|
---|
| 1559 | t = k;
|
---|
| 1560 | }
|
---|
| 1561 | }
|
---|
| 1562 | i = i+1;
|
---|
| 1563 | }
|
---|
| 1564 | while( i<=n );
|
---|
| 1565 | i = n-1;
|
---|
| 1566 | do
|
---|
| 1567 | {
|
---|
| 1568 | tmp = x[i];
|
---|
| 1569 | x[i] = x[0];
|
---|
| 1570 | x[0] = tmp;
|
---|
| 1571 | tmp = y[i];
|
---|
| 1572 | y[i] = y[0];
|
---|
| 1573 | y[0] = tmp;
|
---|
| 1574 | t = 1;
|
---|
| 1575 | while( t!=0 )
|
---|
| 1576 | {
|
---|
| 1577 | k = 2*t;
|
---|
| 1578 | if( k>i )
|
---|
| 1579 | {
|
---|
| 1580 | t = 0;
|
---|
| 1581 | }
|
---|
| 1582 | else
|
---|
| 1583 | {
|
---|
| 1584 | if( k<i )
|
---|
| 1585 | {
|
---|
| 1586 | if( (double)(x[k])>(double)(x[k-1]) )
|
---|
| 1587 | {
|
---|
| 1588 | k = k+1;
|
---|
| 1589 | }
|
---|
| 1590 | }
|
---|
| 1591 | if( (double)(x[t-1])>=(double)(x[k-1]) )
|
---|
| 1592 | {
|
---|
| 1593 | t = 0;
|
---|
| 1594 | }
|
---|
| 1595 | else
|
---|
| 1596 | {
|
---|
| 1597 | tmp = x[k-1];
|
---|
| 1598 | x[k-1] = x[t-1];
|
---|
| 1599 | x[t-1] = tmp;
|
---|
| 1600 | tmp = y[k-1];
|
---|
| 1601 | y[k-1] = y[t-1];
|
---|
| 1602 | y[t-1] = tmp;
|
---|
| 1603 | t = k;
|
---|
| 1604 | }
|
---|
| 1605 | }
|
---|
| 1606 | }
|
---|
| 1607 | i = i-1;
|
---|
| 1608 | }
|
---|
| 1609 | while( i>=1 );
|
---|
| 1610 | }
|
---|
| 1611 |
|
---|
| 1612 |
|
---|
| 1613 | /*************************************************************************
|
---|
| 1614 | Internal subroutine. Heap sort.
|
---|
| 1615 | *************************************************************************/
|
---|
| 1616 | private static void heapsortdpoints(ref double[] x,
|
---|
| 1617 | ref double[] y,
|
---|
| 1618 | ref double[] d,
|
---|
| 1619 | int n)
|
---|
| 1620 | {
|
---|
| 1621 | int i = 0;
|
---|
| 1622 | int j = 0;
|
---|
| 1623 | int k = 0;
|
---|
| 1624 | int t = 0;
|
---|
| 1625 | double tmp = 0;
|
---|
| 1626 | bool isascending = new bool();
|
---|
| 1627 | bool isdescending = new bool();
|
---|
| 1628 |
|
---|
| 1629 |
|
---|
| 1630 | //
|
---|
| 1631 | // Test for already sorted set
|
---|
| 1632 | //
|
---|
| 1633 | isascending = true;
|
---|
| 1634 | isdescending = true;
|
---|
| 1635 | for(i=1; i<=n-1; i++)
|
---|
| 1636 | {
|
---|
| 1637 | isascending = isascending & (double)(x[i])>(double)(x[i-1]);
|
---|
| 1638 | isdescending = isdescending & (double)(x[i])<(double)(x[i-1]);
|
---|
| 1639 | }
|
---|
| 1640 | if( isascending )
|
---|
| 1641 | {
|
---|
| 1642 | return;
|
---|
| 1643 | }
|
---|
| 1644 | if( isdescending )
|
---|
| 1645 | {
|
---|
| 1646 | for(i=0; i<=n-1; i++)
|
---|
| 1647 | {
|
---|
| 1648 | j = n-1-i;
|
---|
| 1649 | if( j<=i )
|
---|
| 1650 | {
|
---|
| 1651 | break;
|
---|
| 1652 | }
|
---|
| 1653 | tmp = x[i];
|
---|
| 1654 | x[i] = x[j];
|
---|
| 1655 | x[j] = tmp;
|
---|
| 1656 | tmp = y[i];
|
---|
| 1657 | y[i] = y[j];
|
---|
| 1658 | y[j] = tmp;
|
---|
| 1659 | tmp = d[i];
|
---|
| 1660 | d[i] = d[j];
|
---|
| 1661 | d[j] = tmp;
|
---|
| 1662 | }
|
---|
| 1663 | return;
|
---|
| 1664 | }
|
---|
| 1665 |
|
---|
| 1666 | //
|
---|
| 1667 | // Special case: N=1
|
---|
| 1668 | //
|
---|
| 1669 | if( n==1 )
|
---|
| 1670 | {
|
---|
| 1671 | return;
|
---|
| 1672 | }
|
---|
| 1673 |
|
---|
| 1674 | //
|
---|
| 1675 | // General case
|
---|
| 1676 | //
|
---|
| 1677 | i = 2;
|
---|
| 1678 | do
|
---|
| 1679 | {
|
---|
| 1680 | t = i;
|
---|
| 1681 | while( t!=1 )
|
---|
| 1682 | {
|
---|
| 1683 | k = t/2;
|
---|
| 1684 | if( (double)(x[k-1])>=(double)(x[t-1]) )
|
---|
| 1685 | {
|
---|
| 1686 | t = 1;
|
---|
| 1687 | }
|
---|
| 1688 | else
|
---|
| 1689 | {
|
---|
| 1690 | tmp = x[k-1];
|
---|
| 1691 | x[k-1] = x[t-1];
|
---|
| 1692 | x[t-1] = tmp;
|
---|
| 1693 | tmp = y[k-1];
|
---|
| 1694 | y[k-1] = y[t-1];
|
---|
| 1695 | y[t-1] = tmp;
|
---|
| 1696 | tmp = d[k-1];
|
---|
| 1697 | d[k-1] = d[t-1];
|
---|
| 1698 | d[t-1] = tmp;
|
---|
| 1699 | t = k;
|
---|
| 1700 | }
|
---|
| 1701 | }
|
---|
| 1702 | i = i+1;
|
---|
| 1703 | }
|
---|
| 1704 | while( i<=n );
|
---|
| 1705 | i = n-1;
|
---|
| 1706 | do
|
---|
| 1707 | {
|
---|
| 1708 | tmp = x[i];
|
---|
| 1709 | x[i] = x[0];
|
---|
| 1710 | x[0] = tmp;
|
---|
| 1711 | tmp = y[i];
|
---|
| 1712 | y[i] = y[0];
|
---|
| 1713 | y[0] = tmp;
|
---|
| 1714 | tmp = d[i];
|
---|
| 1715 | d[i] = d[0];
|
---|
| 1716 | d[0] = tmp;
|
---|
| 1717 | t = 1;
|
---|
| 1718 | while( t!=0 )
|
---|
| 1719 | {
|
---|
| 1720 | k = 2*t;
|
---|
| 1721 | if( k>i )
|
---|
| 1722 | {
|
---|
| 1723 | t = 0;
|
---|
| 1724 | }
|
---|
| 1725 | else
|
---|
| 1726 | {
|
---|
| 1727 | if( k<i )
|
---|
| 1728 | {
|
---|
| 1729 | if( (double)(x[k])>(double)(x[k-1]) )
|
---|
| 1730 | {
|
---|
| 1731 | k = k+1;
|
---|
| 1732 | }
|
---|
| 1733 | }
|
---|
| 1734 | if( (double)(x[t-1])>=(double)(x[k-1]) )
|
---|
| 1735 | {
|
---|
| 1736 | t = 0;
|
---|
| 1737 | }
|
---|
| 1738 | else
|
---|
| 1739 | {
|
---|
| 1740 | tmp = x[k-1];
|
---|
| 1741 | x[k-1] = x[t-1];
|
---|
| 1742 | x[t-1] = tmp;
|
---|
| 1743 | tmp = y[k-1];
|
---|
| 1744 | y[k-1] = y[t-1];
|
---|
| 1745 | y[t-1] = tmp;
|
---|
| 1746 | tmp = d[k-1];
|
---|
| 1747 | d[k-1] = d[t-1];
|
---|
| 1748 | d[t-1] = tmp;
|
---|
| 1749 | t = k;
|
---|
| 1750 | }
|
---|
| 1751 | }
|
---|
| 1752 | }
|
---|
| 1753 | i = i-1;
|
---|
| 1754 | }
|
---|
| 1755 | while( i>=1 );
|
---|
| 1756 | }
|
---|
| 1757 |
|
---|
| 1758 |
|
---|
| 1759 | /*************************************************************************
|
---|
| 1760 | Internal subroutine. Tridiagonal solver.
|
---|
| 1761 | *************************************************************************/
|
---|
| 1762 | private static void solvetridiagonal(double[] a,
|
---|
| 1763 | double[] b,
|
---|
| 1764 | double[] c,
|
---|
| 1765 | double[] d,
|
---|
| 1766 | int n,
|
---|
| 1767 | ref double[] x)
|
---|
| 1768 | {
|
---|
| 1769 | int k = 0;
|
---|
| 1770 | double t = 0;
|
---|
| 1771 |
|
---|
| 1772 | a = (double[])a.Clone();
|
---|
| 1773 | b = (double[])b.Clone();
|
---|
| 1774 | c = (double[])c.Clone();
|
---|
| 1775 | d = (double[])d.Clone();
|
---|
| 1776 |
|
---|
| 1777 | x = new double[n-1+1];
|
---|
| 1778 | a[0] = 0;
|
---|
| 1779 | c[n-1] = 0;
|
---|
| 1780 | for(k=1; k<=n-1; k++)
|
---|
| 1781 | {
|
---|
| 1782 | t = a[k]/b[k-1];
|
---|
| 1783 | b[k] = b[k]-t*c[k-1];
|
---|
| 1784 | d[k] = d[k]-t*d[k-1];
|
---|
| 1785 | }
|
---|
| 1786 | x[n-1] = d[n-1]/b[n-1];
|
---|
| 1787 | for(k=n-2; k>=0; k--)
|
---|
| 1788 | {
|
---|
| 1789 | x[k] = (d[k]-c[k]*x[k+1])/b[k];
|
---|
| 1790 | }
|
---|
| 1791 | }
|
---|
| 1792 |
|
---|
| 1793 |
|
---|
| 1794 | /*************************************************************************
|
---|
| 1795 | Internal subroutine. Three-point differentiation
|
---|
| 1796 | *************************************************************************/
|
---|
| 1797 | private static double diffthreepoint(double t,
|
---|
| 1798 | double x0,
|
---|
| 1799 | double f0,
|
---|
| 1800 | double x1,
|
---|
| 1801 | double f1,
|
---|
| 1802 | double x2,
|
---|
| 1803 | double f2)
|
---|
| 1804 | {
|
---|
| 1805 | double result = 0;
|
---|
| 1806 | double a = 0;
|
---|
| 1807 | double b = 0;
|
---|
| 1808 |
|
---|
| 1809 | t = t-x0;
|
---|
| 1810 | x1 = x1-x0;
|
---|
| 1811 | x2 = x2-x0;
|
---|
| 1812 | a = (f2-f0-x2/x1*(f1-f0))/(AP.Math.Sqr(x2)-x1*x2);
|
---|
| 1813 | b = (f1-f0-a*AP.Math.Sqr(x1))/x1;
|
---|
| 1814 | result = 2*a*t+b;
|
---|
| 1815 | return result;
|
---|
| 1816 | }
|
---|
| 1817 | }
|
---|
| 1818 | }
|
---|