/************************************************************************* ALGLIB 3.17.0 (source code generated 2020-12-27) 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 164 #pragma warning disable 219 using System; public partial class alglib { /************************************************************************* Gamma function Input parameters: X - argument Domain: 0 < X < 171.6 -170 < X < 0, X is not an integer. Relative error: arithmetic domain # trials peak rms IEEE -170,-33 20000 2.3e-15 3.3e-16 IEEE -33, 33 20000 9.4e-16 2.2e-16 IEEE 33, 171.6 20000 2.3e-15 3.2e-16 Cephes Math Library Release 2.8: June, 2000 Original copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier Translated to AlgoPascal by Bochkanov Sergey (2005, 2006, 2007). *************************************************************************/ public static double gammafunction(double x) { return gammafunc.gammafunction(x, null); } public static double gammafunction(double x, alglib.xparams _params) { return gammafunc.gammafunction(x, _params); } /************************************************************************* Natural logarithm of gamma function Input parameters: X - argument Result: logarithm of the absolute value of the Gamma(X). Output parameters: SgnGam - sign(Gamma(X)) Domain: 0 < X < 2.55e305 -2.55e305 < X < 0, X is not an integer. ACCURACY: arithmetic domain # trials peak rms IEEE 0, 3 28000 5.4e-16 1.1e-16 IEEE 2.718, 2.556e305 40000 3.5e-16 8.3e-17 The error criterion was relative when the function magnitude was greater than one but absolute when it was less than one. The following test used the relative error criterion, though at certain points the relative error could be much higher than indicated. IEEE -200, -4 10000 4.8e-16 1.3e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier Translated to AlgoPascal by Bochkanov Sergey (2005, 2006, 2007). *************************************************************************/ public static double lngamma(double x, out double sgngam) { sgngam = 0; return gammafunc.lngamma(x, ref sgngam, null); } public static double lngamma(double x, out double sgngam, alglib.xparams _params) { sgngam = 0; return gammafunc.lngamma(x, ref sgngam, _params); } } public partial class alglib { /************************************************************************* Error function The integral is x - 2 | | 2 erf(x) = -------- | exp( - t ) dt. sqrt(pi) | | - 0 For 0 <= |x| < 1, erf(x) = x * P4(x**2)/Q5(x**2); otherwise erf(x) = 1 - erfc(x). ACCURACY: Relative error: arithmetic domain # trials peak rms IEEE 0,1 30000 3.7e-16 1.0e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier *************************************************************************/ public static double errorfunction(double x) { return normaldistr.errorfunction(x, null); } public static double errorfunction(double x, alglib.xparams _params) { return normaldistr.errorfunction(x, _params); } /************************************************************************* Complementary error function 1 - erf(x) = inf. - 2 | | 2 erfc(x) = -------- | exp( - t ) dt sqrt(pi) | | - x For small x, erfc(x) = 1 - erf(x); otherwise rational approximations are computed. ACCURACY: Relative error: arithmetic domain # trials peak rms IEEE 0,26.6417 30000 5.7e-14 1.5e-14 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier *************************************************************************/ public static double errorfunctionc(double x) { return normaldistr.errorfunctionc(x, null); } public static double errorfunctionc(double x, alglib.xparams _params) { return normaldistr.errorfunctionc(x, _params); } /************************************************************************* Same as normalcdf(), obsolete name. *************************************************************************/ public static double normaldistribution(double x) { return normaldistr.normaldistribution(x, null); } public static double normaldistribution(double x, alglib.xparams _params) { return normaldistr.normaldistribution(x, _params); } /************************************************************************* Normal distribution PDF Returns Gaussian probability density function: 1 f(x) = --------- * exp(-x^2/2) sqrt(2pi) Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier *************************************************************************/ public static double normalpdf(double x) { return normaldistr.normalpdf(x, null); } public static double normalpdf(double x, alglib.xparams _params) { return normaldistr.normalpdf(x, _params); } /************************************************************************* Normal distribution CDF Returns the area under the Gaussian probability density function, integrated from minus infinity to x: x - 1 | | 2 ndtr(x) = --------- | exp( - t /2 ) dt sqrt(2pi) | | - -inf. = ( 1 + erf(z) ) / 2 = erfc(z) / 2 where z = x/sqrt(2). Computation is via the functions erf and erfc. ACCURACY: Relative error: arithmetic domain # trials peak rms IEEE -13,0 30000 3.4e-14 6.7e-15 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier *************************************************************************/ public static double normalcdf(double x) { return normaldistr.normalcdf(x, null); } public static double normalcdf(double x, alglib.xparams _params) { return normaldistr.normalcdf(x, _params); } /************************************************************************* Inverse of the error function Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier *************************************************************************/ public static double inverf(double e) { return normaldistr.inverf(e, null); } public static double inverf(double e, alglib.xparams _params) { return normaldistr.inverf(e, _params); } /************************************************************************* Same as invnormalcdf(), deprecated name *************************************************************************/ public static double invnormaldistribution(double y0) { return normaldistr.invnormaldistribution(y0, null); } public static double invnormaldistribution(double y0, alglib.xparams _params) { return normaldistr.invnormaldistribution(y0, _params); } /************************************************************************* Inverse of Normal CDF Returns the argument, x, for which the area under the Gaussian probability density function (integrated from minus infinity to x) is equal to y. For small arguments 0 < y < exp(-2), the program computes z = sqrt( -2.0 * log(y) ); then the approximation is x = z - log(z)/z - (1/z) P(1/z) / Q(1/z). There are two rational functions P/Q, one for 0 < y < exp(-32) and the other for y up to exp(-2). For larger arguments, w = y - 0.5, and x/sqrt(2pi) = w + w**3 R(w**2)/S(w**2)). ACCURACY: Relative error: arithmetic domain # trials peak rms IEEE 0.125, 1 20000 7.2e-16 1.3e-16 IEEE 3e-308, 0.135 50000 4.6e-16 9.8e-17 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier *************************************************************************/ public static double invnormalcdf(double y0) { return normaldistr.invnormalcdf(y0, null); } public static double invnormalcdf(double y0, alglib.xparams _params) { return normaldistr.invnormalcdf(y0, _params); } /************************************************************************* Bivariate normal PDF Returns probability density function of the bivariate Gaussian with correlation parameter equal to Rho: 1 ( x^2 - 2*rho*x*y + y^2 ) f(x,y,rho) = ----------------- * exp( - ----------------------- ) 2pi*sqrt(1-rho^2) ( 2*(1-rho^2) ) with -1=0 x - argument Result: the value of the Hermite polynomial Hn at x *************************************************************************/ public static double hermitecalculate(int n, double x) { return hermite.hermitecalculate(n, x, null); } public static double hermitecalculate(int n, double x, alglib.xparams _params) { return hermite.hermitecalculate(n, x, _params); } /************************************************************************* Summation of Hermite polynomials using Clenshaw's recurrence formula. This routine calculates c[0]*H0(x) + c[1]*H1(x) + ... + c[N]*HN(x) Parameters: n - degree, n>=0 x - argument Result: the value of the Hermite polynomial at x *************************************************************************/ public static double hermitesum(double[] c, int n, double x) { return hermite.hermitesum(c, n, x, null); } public static double hermitesum(double[] c, int n, double x, alglib.xparams _params) { return hermite.hermitesum(c, n, x, _params); } /************************************************************************* Representation of Hn as C[0] + C[1]*X + ... + C[N]*X^N Input parameters: N - polynomial degree, n>=0 Output parameters: C - coefficients *************************************************************************/ public static void hermitecoefficients(int n, out double[] c) { c = new double[0]; hermite.hermitecoefficients(n, ref c, null); } public static void hermitecoefficients(int n, out double[] c, alglib.xparams _params) { c = new double[0]; hermite.hermitecoefficients(n, ref c, _params); } } public partial class alglib { /************************************************************************* Dawson's Integral Approximates the integral x - 2 | | 2 dawsn(x) = exp( -x ) | exp( t ) dt | | - 0 Three different rational approximations are employed, for the intervals 0 to 3.25; 3.25 to 6.25; and 6.25 up. ACCURACY: Relative error: arithmetic domain # trials peak rms IEEE 0,10 10000 6.9e-16 1.0e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier *************************************************************************/ public static double dawsonintegral(double x) { return dawson.dawsonintegral(x, null); } public static double dawsonintegral(double x, alglib.xparams _params) { return dawson.dawsonintegral(x, _params); } } public partial class alglib { /************************************************************************* Sine and cosine integrals Evaluates the integrals x - | cos t - 1 Ci(x) = eul + ln x + | --------- dt, | t - 0 x - | sin t Si(x) = | ----- dt | t - 0 where eul = 0.57721566490153286061 is Euler's constant. The integrals are approximated by rational functions. For x > 8 auxiliary functions f(x) and g(x) are employed such that Ci(x) = f(x) sin(x) - g(x) cos(x) Si(x) = pi/2 - f(x) cos(x) - g(x) sin(x) ACCURACY: Test interval = [0,50]. Absolute error, except relative when > 1: arithmetic function # trials peak rms IEEE Si 30000 4.4e-16 7.3e-17 IEEE Ci 30000 6.9e-16 5.1e-17 Cephes Math Library Release 2.1: January, 1989 Copyright 1984, 1987, 1989 by Stephen L. Moshier *************************************************************************/ public static void sinecosineintegrals(double x, out double si, out double ci) { si = 0; ci = 0; trigintegrals.sinecosineintegrals(x, ref si, ref ci, null); } public static void sinecosineintegrals(double x, out double si, out double ci, alglib.xparams _params) { si = 0; ci = 0; trigintegrals.sinecosineintegrals(x, ref si, ref ci, _params); } /************************************************************************* Hyperbolic sine and cosine integrals Approximates the integrals x - | | cosh t - 1 Chi(x) = eul + ln x + | ----------- dt, | | t - 0 x - | | sinh t Shi(x) = | ------ dt | | t - 0 where eul = 0.57721566490153286061 is Euler's constant. The integrals are evaluated by power series for x < 8 and by Chebyshev expansions for x between 8 and 88. For large x, both functions approach exp(x)/2x. Arguments greater than 88 in magnitude return MAXNUM. ACCURACY: Test interval 0 to 88. Relative error: arithmetic function # trials peak rms IEEE Shi 30000 6.9e-16 1.6e-16 Absolute error, except relative when |Chi| > 1: IEEE Chi 30000 8.4e-16 1.4e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 2000 by Stephen L. Moshier *************************************************************************/ public static void hyperbolicsinecosineintegrals(double x, out double shi, out double chi) { shi = 0; chi = 0; trigintegrals.hyperbolicsinecosineintegrals(x, ref shi, ref chi, null); } public static void hyperbolicsinecosineintegrals(double x, out double shi, out double chi, alglib.xparams _params) { shi = 0; chi = 0; trigintegrals.hyperbolicsinecosineintegrals(x, ref shi, ref chi, _params); } } public partial class alglib { /************************************************************************* Poisson distribution Returns the sum of the first k+1 terms of the Poisson distribution: k j -- -m m > e -- -- j! j=0 The terms are not summed directly; instead the incomplete gamma integral is employed, according to the relation y = pdtr( k, m ) = igamc( k+1, m ). The arguments must both be positive. ACCURACY: See incomplete gamma function Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier *************************************************************************/ public static double poissondistribution(int k, double m) { return poissondistr.poissondistribution(k, m, null); } public static double poissondistribution(int k, double m, alglib.xparams _params) { return poissondistr.poissondistribution(k, m, _params); } /************************************************************************* Complemented Poisson distribution Returns the sum of the terms k+1 to infinity of the Poisson distribution: inf. j -- -m m > e -- -- j! j=k+1 The terms are not summed directly; instead the incomplete gamma integral is employed, according to the formula y = pdtrc( k, m ) = igam( k+1, m ). The arguments must both be positive. ACCURACY: See incomplete gamma function Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier *************************************************************************/ public static double poissoncdistribution(int k, double m) { return poissondistr.poissoncdistribution(k, m, null); } public static double poissoncdistribution(int k, double m, alglib.xparams _params) { return poissondistr.poissoncdistribution(k, m, _params); } /************************************************************************* Inverse Poisson distribution Finds the Poisson variable x such that the integral from 0 to x of the Poisson density is equal to the given probability y. This is accomplished using the inverse gamma integral function and the relation m = igami( k+1, y ). ACCURACY: See inverse incomplete gamma function Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier *************************************************************************/ public static double invpoissondistribution(int k, double y) { return poissondistr.invpoissondistribution(k, y, null); } public static double invpoissondistribution(int k, double y, alglib.xparams _params) { return poissondistr.invpoissondistribution(k, y, _params); } } public partial class alglib { /************************************************************************* Bessel function of order zero Returns Bessel function of order zero of the argument. The domain is divided into the intervals [0, 5] and (5, infinity). In the first interval the following rational approximation is used: 2 2 (w - r ) (w - r ) P (w) / Q (w) 1 2 3 8 2 where w = x and the two r's are zeros of the function. In the second interval, the Hankel asymptotic expansion is employed with two rational functions of degree 6/6 and 7/7. ACCURACY: Absolute error: arithmetic domain # trials peak rms IEEE 0, 30 60000 4.2e-16 1.1e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier *************************************************************************/ public static double besselj0(double x) { return bessel.besselj0(x, null); } public static double besselj0(double x, alglib.xparams _params) { return bessel.besselj0(x, _params); } /************************************************************************* Bessel function of order one Returns Bessel function of order one of the argument. The domain is divided into the intervals [0, 8] and (8, infinity). In the first interval a 24 term Chebyshev expansion is used. In the second, the asymptotic trigonometric representation is employed using two rational functions of degree 5/5. ACCURACY: Absolute error: arithmetic domain # trials peak rms IEEE 0, 30 30000 2.6e-16 1.1e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier *************************************************************************/ public static double besselj1(double x) { return bessel.besselj1(x, null); } public static double besselj1(double x, alglib.xparams _params) { return bessel.besselj1(x, _params); } /************************************************************************* Bessel function of integer order Returns Bessel function of order n, where n is a (possibly negative) integer. The ratio of jn(x) to j0(x) is computed by backward recurrence. First the ratio jn/jn-1 is found by a continued fraction expansion. Then the recurrence relating successive orders is applied until j0 or j1 is reached. If n = 0 or 1 the routine for j0 or j1 is called directly. ACCURACY: Absolute error: arithmetic range # trials peak rms IEEE 0, 30 5000 4.4e-16 7.9e-17 Not suitable for large n or x. Use jv() (fractional order) instead. Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 2000 by Stephen L. Moshier *************************************************************************/ public static double besseljn(int n, double x) { return bessel.besseljn(n, x, null); } public static double besseljn(int n, double x, alglib.xparams _params) { return bessel.besseljn(n, x, _params); } /************************************************************************* Bessel function of the second kind, order zero Returns Bessel function of the second kind, of order zero, of the argument. The domain is divided into the intervals [0, 5] and (5, infinity). In the first interval a rational approximation R(x) is employed to compute y0(x) = R(x) + 2 * log(x) * j0(x) / PI. Thus a call to j0() is required. In the second interval, the Hankel asymptotic expansion is employed with two rational functions of degree 6/6 and 7/7. ACCURACY: Absolute error, when y0(x) < 1; else relative error: arithmetic domain # trials peak rms IEEE 0, 30 30000 1.3e-15 1.6e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier *************************************************************************/ public static double bessely0(double x) { return bessel.bessely0(x, null); } public static double bessely0(double x, alglib.xparams _params) { return bessel.bessely0(x, _params); } /************************************************************************* Bessel function of second kind of order one Returns Bessel function of the second kind of order one of the argument. The domain is divided into the intervals [0, 8] and (8, infinity). In the first interval a 25 term Chebyshev expansion is used, and a call to j1() is required. In the second, the asymptotic trigonometric representation is employed using two rational functions of degree 5/5. ACCURACY: Absolute error: arithmetic domain # trials peak rms IEEE 0, 30 30000 1.0e-15 1.3e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier *************************************************************************/ public static double bessely1(double x) { return bessel.bessely1(x, null); } public static double bessely1(double x, alglib.xparams _params) { return bessel.bessely1(x, _params); } /************************************************************************* Bessel function of second kind of integer order Returns Bessel function of order n, where n is a (possibly negative) integer. The function is evaluated by forward recurrence on n, starting with values computed by the routines y0() and y1(). If n = 0 or 1 the routine for y0 or y1 is called directly. ACCURACY: Absolute error, except relative when y > 1: arithmetic domain # trials peak rms IEEE 0, 30 30000 3.4e-15 4.3e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 2000 by Stephen L. Moshier *************************************************************************/ public static double besselyn(int n, double x) { return bessel.besselyn(n, x, null); } public static double besselyn(int n, double x, alglib.xparams _params) { return bessel.besselyn(n, x, _params); } /************************************************************************* Modified Bessel function of order zero Returns modified Bessel function of order zero of the argument. The function is defined as i0(x) = j0( ix ). The range is partitioned into the two intervals [0,8] and (8, infinity). Chebyshev polynomial expansions are employed in each interval. ACCURACY: Relative error: arithmetic domain # trials peak rms IEEE 0,30 30000 5.8e-16 1.4e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 2000 by Stephen L. Moshier *************************************************************************/ public static double besseli0(double x) { return bessel.besseli0(x, null); } public static double besseli0(double x, alglib.xparams _params) { return bessel.besseli0(x, _params); } /************************************************************************* Modified Bessel function of order one Returns modified Bessel function of order one of the argument. The function is defined as i1(x) = -i j1( ix ). The range is partitioned into the two intervals [0,8] and (8, infinity). Chebyshev polynomial expansions are employed in each interval. ACCURACY: Relative error: arithmetic domain # trials peak rms IEEE 0, 30 30000 1.9e-15 2.1e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1985, 1987, 2000 by Stephen L. Moshier *************************************************************************/ public static double besseli1(double x) { return bessel.besseli1(x, null); } public static double besseli1(double x, alglib.xparams _params) { return bessel.besseli1(x, _params); } /************************************************************************* Modified Bessel function, second kind, order zero Returns modified Bessel function of the second kind of order zero of the argument. The range is partitioned into the two intervals [0,8] and (8, infinity). Chebyshev polynomial expansions are employed in each interval. ACCURACY: Tested at 2000 random points between 0 and 8. Peak absolute error (relative when K0 > 1) was 1.46e-14; rms, 4.26e-15. Relative error: arithmetic domain # trials peak rms IEEE 0, 30 30000 1.2e-15 1.6e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 2000 by Stephen L. Moshier *************************************************************************/ public static double besselk0(double x) { return bessel.besselk0(x, null); } public static double besselk0(double x, alglib.xparams _params) { return bessel.besselk0(x, _params); } /************************************************************************* Modified Bessel function, second kind, order one Computes the modified Bessel function of the second kind of order one of the argument. The range is partitioned into the two intervals [0,2] and (2, infinity). Chebyshev polynomial expansions are employed in each interval. ACCURACY: Relative error: arithmetic domain # trials peak rms IEEE 0, 30 30000 1.2e-15 1.6e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 2000 by Stephen L. Moshier *************************************************************************/ public static double besselk1(double x) { return bessel.besselk1(x, null); } public static double besselk1(double x, alglib.xparams _params) { return bessel.besselk1(x, _params); } /************************************************************************* Modified Bessel function, second kind, integer order Returns modified Bessel function of the second kind of order n of the argument. The range is partitioned into the two intervals [0,9.55] and (9.55, infinity). An ascending power series is used in the low range, and an asymptotic expansion in the high range. ACCURACY: Relative error: arithmetic domain # trials peak rms IEEE 0,30 90000 1.8e-8 3.0e-10 Error is high only near the crossover point x = 9.55 between the two expansions used. Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1988, 2000 by Stephen L. Moshier *************************************************************************/ public static double besselkn(int nn, double x) { return bessel.besselkn(nn, x, null); } public static double besselkn(int nn, double x, alglib.xparams _params) { return bessel.besselkn(nn, x, _params); } } public partial class alglib { /************************************************************************* Incomplete beta integral Returns incomplete beta integral of the arguments, evaluated from zero to x. The function is defined as x - - | (a+b) | | a-1 b-1 ----------- | t (1-t) dt. - - | | | (a) | (b) - 0 The domain of definition is 0 <= x <= 1. In this implementation a and b are restricted to positive values. The integral from x to 1 may be obtained by the symmetry relation 1 - incbet( a, b, x ) = incbet( b, a, 1-x ). The integral is evaluated by a continued fraction expansion or, when b*x is small, by a power series. ACCURACY: Tested at uniformly distributed random points (a,b,x) with a and b in "domain" and x between 0 and 1. Relative error arithmetic domain # trials peak rms IEEE 0,5 10000 6.9e-15 4.5e-16 IEEE 0,85 250000 2.2e-13 1.7e-14 IEEE 0,1000 30000 5.3e-12 6.3e-13 IEEE 0,10000 250000 9.3e-11 7.1e-12 IEEE 0,100000 10000 8.7e-10 4.8e-11 Outputs smaller than the IEEE gradual underflow threshold were excluded from these statistics. Cephes Math Library, Release 2.8: June, 2000 Copyright 1984, 1995, 2000 by Stephen L. Moshier *************************************************************************/ public static double incompletebeta(double a, double b, double x) { return ibetaf.incompletebeta(a, b, x, null); } public static double incompletebeta(double a, double b, double x, alglib.xparams _params) { return ibetaf.incompletebeta(a, b, x, _params); } /************************************************************************* Inverse of imcomplete beta integral Given y, the function finds x such that incbet( a, b, x ) = y . The routine performs interval halving or Newton iterations to find the root of incbet(a,b,x) - y = 0. ACCURACY: Relative error: x a,b arithmetic domain domain # trials peak rms IEEE 0,1 .5,10000 50000 5.8e-12 1.3e-13 IEEE 0,1 .25,100 100000 1.8e-13 3.9e-15 IEEE 0,1 0,5 50000 1.1e-12 5.5e-15 With a and b constrained to half-integer or integer values: IEEE 0,1 .5,10000 50000 5.8e-12 1.1e-13 IEEE 0,1 .5,100 100000 1.7e-14 7.9e-16 With a = .5, b constrained to half-integer or integer values: IEEE 0,1 .5,10000 10000 8.3e-11 1.0e-11 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1996, 2000 by Stephen L. Moshier *************************************************************************/ public static double invincompletebeta(double a, double b, double y) { return ibetaf.invincompletebeta(a, b, y, null); } public static double invincompletebeta(double a, double b, double y, alglib.xparams _params) { return ibetaf.invincompletebeta(a, b, y, _params); } } public partial class alglib { /************************************************************************* F distribution Returns the area from zero to x under the F density function (also known as Snedcor's density or the variance ratio density). This is the density of x = (u1/df1)/(u2/df2), where u1 and u2 are random variables having Chi square distributions with df1 and df2 degrees of freedom, respectively. The incomplete beta integral is used, according to the formula P(x) = incbet( df1/2, df2/2, (df1*x/(df2 + df1*x) ). The arguments a and b are greater than zero, and x is nonnegative. ACCURACY: Tested at random points (a,b,x). x a,b Relative error: arithmetic domain domain # trials peak rms IEEE 0,1 0,100 100000 9.8e-15 1.7e-15 IEEE 1,5 0,100 100000 6.5e-15 3.5e-16 IEEE 0,1 1,10000 100000 2.2e-11 3.3e-12 IEEE 1,5 1,10000 100000 1.1e-11 1.7e-13 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier *************************************************************************/ public static double fdistribution(int a, int b, double x) { return fdistr.fdistribution(a, b, x, null); } public static double fdistribution(int a, int b, double x, alglib.xparams _params) { return fdistr.fdistribution(a, b, x, _params); } /************************************************************************* Complemented F distribution Returns the area from x to infinity under the F density function (also known as Snedcor's density or the variance ratio density). inf. - 1 | | a-1 b-1 1-P(x) = ------ | t (1-t) dt B(a,b) | | - x The incomplete beta integral is used, according to the formula P(x) = incbet( df2/2, df1/2, (df2/(df2 + df1*x) ). ACCURACY: Tested at random points (a,b,x) in the indicated intervals. x a,b Relative error: arithmetic domain domain # trials peak rms IEEE 0,1 1,100 100000 3.7e-14 5.9e-16 IEEE 1,5 1,100 100000 8.0e-15 1.6e-15 IEEE 0,1 1,10000 100000 1.8e-11 3.5e-13 IEEE 1,5 1,10000 100000 2.0e-11 3.0e-12 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier *************************************************************************/ public static double fcdistribution(int a, int b, double x) { return fdistr.fcdistribution(a, b, x, null); } public static double fcdistribution(int a, int b, double x, alglib.xparams _params) { return fdistr.fcdistribution(a, b, x, _params); } /************************************************************************* Inverse of complemented F distribution Finds the F density argument x such that the integral from x to infinity of the F density is equal to the given probability p. This is accomplished using the inverse beta integral function and the relations z = incbi( df2/2, df1/2, p ) x = df2 (1-z) / (df1 z). Note: the following relations hold for the inverse of the uncomplemented F distribution: z = incbi( df1/2, df2/2, p ) x = df2 z / (df1 (1-z)). ACCURACY: Tested at random points (a,b,p). a,b Relative error: arithmetic domain # trials peak rms For p between .001 and 1: IEEE 1,100 100000 8.3e-15 4.7e-16 IEEE 1,10000 100000 2.1e-11 1.4e-13 For p between 10^-6 and 10^-3: IEEE 1,100 50000 1.3e-12 8.4e-15 IEEE 1,10000 50000 3.0e-12 4.8e-14 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier *************************************************************************/ public static double invfdistribution(int a, int b, double y) { return fdistr.invfdistribution(a, b, y, null); } public static double invfdistribution(int a, int b, double y, alglib.xparams _params) { return fdistr.invfdistribution(a, b, y, _params); } } public partial class alglib { /************************************************************************* Fresnel integral Evaluates the Fresnel integrals x - | | C(x) = | cos(pi/2 t**2) dt, | | - 0 x - | | S(x) = | sin(pi/2 t**2) dt. | | - 0 The integrals are evaluated by a power series for x < 1. For x >= 1 auxiliary functions f(x) and g(x) are employed such that C(x) = 0.5 + f(x) sin( pi/2 x**2 ) - g(x) cos( pi/2 x**2 ) S(x) = 0.5 - f(x) cos( pi/2 x**2 ) - g(x) sin( pi/2 x**2 ) ACCURACY: Relative error. Arithmetic function domain # trials peak rms IEEE S(x) 0, 10 10000 2.0e-15 3.2e-16 IEEE C(x) 0, 10 10000 1.8e-15 3.3e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier *************************************************************************/ public static void fresnelintegral(double x, ref double c, ref double s) { fresnel.fresnelintegral(x, ref c, ref s, null); } public static void fresnelintegral(double x, ref double c, ref double s, alglib.xparams _params) { fresnel.fresnelintegral(x, ref c, ref s, _params); } } public partial class alglib { /************************************************************************* Jacobian Elliptic Functions Evaluates the Jacobian elliptic functions sn(u|m), cn(u|m), and dn(u|m) of parameter m between 0 and 1, and real argument u. These functions are periodic, with quarter-period on the real axis equal to the complete elliptic integral ellpk(1.0-m). Relation to incomplete elliptic integral: If u = ellik(phi,m), then sn(u|m) = sin(phi), and cn(u|m) = cos(phi). Phi is called the amplitude of u. Computation is by means of the arithmetic-geometric mean algorithm, except when m is within 1e-9 of 0 or 1. In the latter case with m close to 1, the approximation applies only for phi < pi/2. ACCURACY: Tested at random points with u between 0 and 10, m between 0 and 1. Absolute error (* = relative error): arithmetic function # trials peak rms IEEE phi 10000 9.2e-16* 1.4e-16* IEEE sn 50000 4.1e-15 4.6e-16 IEEE cn 40000 3.6e-15 4.4e-16 IEEE dn 10000 1.3e-12 1.8e-14 Peak error observed in consistency check using addition theorem for sn(u+v) was 4e-16 (absolute). Also tested by the above relation to the incomplete elliptic integral. Accuracy deteriorates when u is large. Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 2000 by Stephen L. Moshier *************************************************************************/ public static void jacobianellipticfunctions(double u, double m, out double sn, out double cn, out double dn, out double ph) { sn = 0; cn = 0; dn = 0; ph = 0; jacobianelliptic.jacobianellipticfunctions(u, m, ref sn, ref cn, ref dn, ref ph, null); } public static void jacobianellipticfunctions(double u, double m, out double sn, out double cn, out double dn, out double ph, alglib.xparams _params) { sn = 0; cn = 0; dn = 0; ph = 0; jacobianelliptic.jacobianellipticfunctions(u, m, ref sn, ref cn, ref dn, ref ph, _params); } } public partial class alglib { /************************************************************************* Psi (digamma) function d - psi(x) = -- ln | (x) dx is the logarithmic derivative of the gamma function. For integer x, n-1 - psi(n) = -EUL + > 1/k. - k=1 This formula is used for 0 < n <= 10. If x is negative, it is transformed to a positive argument by the reflection formula psi(1-x) = psi(x) + pi cot(pi x). For general positive x, the argument is made greater than 10 using the recurrence psi(x+1) = psi(x) + 1/x. Then the following asymptotic expansion is applied: inf. B - 2k psi(x) = log(x) - 1/2x - > ------- - 2k k=1 2k x where the B2k are Bernoulli numbers. ACCURACY: Relative error (except absolute when |psi| < 1): arithmetic domain # trials peak rms IEEE 0,30 30000 1.3e-15 1.4e-16 IEEE -30,0 40000 1.5e-15 2.2e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1992, 2000 by Stephen L. Moshier *************************************************************************/ public static double psi(double x) { return psif.psi(x, null); } public static double psi(double x, alglib.xparams _params) { return psif.psi(x, _params); } } public partial class alglib { /************************************************************************* Exponential integral Ei(x) x - t | | e Ei(x) = -|- --- dt . | | t - -inf Not defined for x <= 0. See also expn.c. ACCURACY: Relative error: arithmetic domain # trials peak rms IEEE 0,100 50000 8.6e-16 1.3e-16 Cephes Math Library Release 2.8: May, 1999 Copyright 1999 by Stephen L. Moshier *************************************************************************/ public static double exponentialintegralei(double x) { return expintegrals.exponentialintegralei(x, null); } public static double exponentialintegralei(double x, alglib.xparams _params) { return expintegrals.exponentialintegralei(x, _params); } /************************************************************************* Exponential integral En(x) Evaluates the exponential integral inf. - | | -xt | e E (x) = | ---- dt. n | n | | t - 1 Both n and x must be nonnegative. The routine employs either a power series, a continued fraction, or an asymptotic formula depending on the relative values of n and x. ACCURACY: Relative error: arithmetic domain # trials peak rms IEEE 0, 30 10000 1.7e-15 3.6e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1985, 2000 by Stephen L. Moshier *************************************************************************/ public static double exponentialintegralen(double x, int n) { return expintegrals.exponentialintegralen(x, n, null); } public static double exponentialintegralen(double x, int n, alglib.xparams _params) { return expintegrals.exponentialintegralen(x, n, _params); } } public partial class alglib { /************************************************************************* Calculation of the value of the Laguerre polynomial. Parameters: n - degree, n>=0 x - argument Result: the value of the Laguerre polynomial Ln at x *************************************************************************/ public static double laguerrecalculate(int n, double x) { return laguerre.laguerrecalculate(n, x, null); } public static double laguerrecalculate(int n, double x, alglib.xparams _params) { return laguerre.laguerrecalculate(n, x, _params); } /************************************************************************* Summation of Laguerre polynomials using Clenshaw's recurrence formula. This routine calculates c[0]*L0(x) + c[1]*L1(x) + ... + c[N]*LN(x) Parameters: n - degree, n>=0 x - argument Result: the value of the Laguerre polynomial at x *************************************************************************/ public static double laguerresum(double[] c, int n, double x) { return laguerre.laguerresum(c, n, x, null); } public static double laguerresum(double[] c, int n, double x, alglib.xparams _params) { return laguerre.laguerresum(c, n, x, _params); } /************************************************************************* Representation of Ln as C[0] + C[1]*X + ... + C[N]*X^N Input parameters: N - polynomial degree, n>=0 Output parameters: C - coefficients *************************************************************************/ public static void laguerrecoefficients(int n, out double[] c) { c = new double[0]; laguerre.laguerrecoefficients(n, ref c, null); } public static void laguerrecoefficients(int n, out double[] c, alglib.xparams _params) { c = new double[0]; laguerre.laguerrecoefficients(n, ref c, _params); } } public partial class alglib { /************************************************************************* Chi-square distribution Returns the area under the left hand tail (from 0 to x) of the Chi square probability density function with v degrees of freedom. x - 1 | | v/2-1 -t/2 P( x | v ) = ----------- | t e dt v/2 - | | 2 | (v/2) - 0 where x is the Chi-square variable. The incomplete gamma integral is used, according to the formula y = chdtr( v, x ) = igam( v/2.0, x/2.0 ). The arguments must both be positive. ACCURACY: See incomplete gamma function Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 2000 by Stephen L. Moshier *************************************************************************/ public static double chisquaredistribution(double v, double x) { return chisquaredistr.chisquaredistribution(v, x, null); } public static double chisquaredistribution(double v, double x, alglib.xparams _params) { return chisquaredistr.chisquaredistribution(v, x, _params); } /************************************************************************* Complemented Chi-square distribution Returns the area under the right hand tail (from x to infinity) of the Chi square probability density function with v degrees of freedom: inf. - 1 | | v/2-1 -t/2 P( x | v ) = ----------- | t e dt v/2 - | | 2 | (v/2) - x where x is the Chi-square variable. The incomplete gamma integral is used, according to the formula y = chdtr( v, x ) = igamc( v/2.0, x/2.0 ). The arguments must both be positive. ACCURACY: See incomplete gamma function Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 2000 by Stephen L. Moshier *************************************************************************/ public static double chisquarecdistribution(double v, double x) { return chisquaredistr.chisquarecdistribution(v, x, null); } public static double chisquarecdistribution(double v, double x, alglib.xparams _params) { return chisquaredistr.chisquarecdistribution(v, x, _params); } /************************************************************************* Inverse of complemented Chi-square distribution Finds the Chi-square argument x such that the integral from x to infinity of the Chi-square density is equal to the given cumulative probability y. This is accomplished using the inverse gamma integral function and the relation x/2 = igami( df/2, y ); ACCURACY: See inverse incomplete gamma function Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 2000 by Stephen L. Moshier *************************************************************************/ public static double invchisquaredistribution(double v, double y) { return chisquaredistr.invchisquaredistribution(v, y, null); } public static double invchisquaredistribution(double v, double y, alglib.xparams _params) { return chisquaredistr.invchisquaredistribution(v, y, _params); } } public partial class alglib { /************************************************************************* Calculation of the value of the Legendre polynomial Pn. Parameters: n - degree, n>=0 x - argument Result: the value of the Legendre polynomial Pn at x *************************************************************************/ public static double legendrecalculate(int n, double x) { return legendre.legendrecalculate(n, x, null); } public static double legendrecalculate(int n, double x, alglib.xparams _params) { return legendre.legendrecalculate(n, x, _params); } /************************************************************************* Summation of Legendre polynomials using Clenshaw's recurrence formula. This routine calculates c[0]*P0(x) + c[1]*P1(x) + ... + c[N]*PN(x) Parameters: n - degree, n>=0 x - argument Result: the value of the Legendre polynomial at x *************************************************************************/ public static double legendresum(double[] c, int n, double x) { return legendre.legendresum(c, n, x, null); } public static double legendresum(double[] c, int n, double x, alglib.xparams _params) { return legendre.legendresum(c, n, x, _params); } /************************************************************************* Representation of Pn as C[0] + C[1]*X + ... + C[N]*X^N Input parameters: N - polynomial degree, n>=0 Output parameters: C - coefficients *************************************************************************/ public static void legendrecoefficients(int n, out double[] c) { c = new double[0]; legendre.legendrecoefficients(n, ref c, null); } public static void legendrecoefficients(int n, out double[] c, alglib.xparams _params) { c = new double[0]; legendre.legendrecoefficients(n, ref c, _params); } } public partial class alglib { /************************************************************************* Beta function - - | (a) | (b) beta( a, b ) = -----------. - | (a+b) For large arguments the logarithm of the function is evaluated using lgam(), then exponentiated. ACCURACY: Relative error: arithmetic domain # trials peak rms IEEE 0,30 30000 8.1e-14 1.1e-14 Cephes Math Library Release 2.0: April, 1987 Copyright 1984, 1987 by Stephen L. Moshier *************************************************************************/ public static double beta(double a, double b) { return betaf.beta(a, b, null); } public static double beta(double a, double b, alglib.xparams _params) { return betaf.beta(a, b, _params); } } public partial class alglib { /************************************************************************* Calculation of the value of the Chebyshev polynomials of the first and second kinds. Parameters: r - polynomial kind, either 1 or 2. n - degree, n>=0 x - argument, -1 <= x <= 1 Result: the value of the Chebyshev polynomial at x *************************************************************************/ public static double chebyshevcalculate(int r, int n, double x) { return chebyshev.chebyshevcalculate(r, n, x, null); } public static double chebyshevcalculate(int r, int n, double x, alglib.xparams _params) { return chebyshev.chebyshevcalculate(r, n, x, _params); } /************************************************************************* Summation of Chebyshev polynomials using Clenshaw's recurrence formula. This routine calculates c[0]*T0(x) + c[1]*T1(x) + ... + c[N]*TN(x) or c[0]*U0(x) + c[1]*U1(x) + ... + c[N]*UN(x) depending on the R. Parameters: r - polynomial kind, either 1 or 2. n - degree, n>=0 x - argument Result: the value of the Chebyshev polynomial at x *************************************************************************/ public static double chebyshevsum(double[] c, int r, int n, double x) { return chebyshev.chebyshevsum(c, r, n, x, null); } public static double chebyshevsum(double[] c, int r, int n, double x, alglib.xparams _params) { return chebyshev.chebyshevsum(c, r, n, x, _params); } /************************************************************************* Representation of Tn as C[0] + C[1]*X + ... + C[N]*X^N Input parameters: N - polynomial degree, n>=0 Output parameters: C - coefficients *************************************************************************/ public static void chebyshevcoefficients(int n, out double[] c) { c = new double[0]; chebyshev.chebyshevcoefficients(n, ref c, null); } public static void chebyshevcoefficients(int n, out double[] c, alglib.xparams _params) { c = new double[0]; chebyshev.chebyshevcoefficients(n, ref c, _params); } /************************************************************************* Conversion of a series of Chebyshev polynomials to a power series. Represents A[0]*T0(x) + A[1]*T1(x) + ... + A[N]*Tn(x) as B[0] + B[1]*X + ... + B[N]*X^N. Input parameters: A - Chebyshev series coefficients N - degree, N>=0 Output parameters B - power series coefficients *************************************************************************/ public static void fromchebyshev(double[] a, int n, out double[] b) { b = new double[0]; chebyshev.fromchebyshev(a, n, ref b, null); } public static void fromchebyshev(double[] a, int n, out double[] b, alglib.xparams _params) { b = new double[0]; chebyshev.fromchebyshev(a, n, ref b, _params); } } public partial class alglib { /************************************************************************* Student's t distribution Computes the integral from minus infinity to t of the Student t distribution with integer k > 0 degrees of freedom: t - | | - | 2 -(k+1)/2 | ( (k+1)/2 ) | ( x ) ---------------------- | ( 1 + --- ) dx - | ( k ) sqrt( k pi ) | ( k/2 ) | | | - -inf. Relation to incomplete beta integral: 1 - stdtr(k,t) = 0.5 * incbet( k/2, 1/2, z ) where z = k/(k + t**2). For t < -2, this is the method of computation. For higher t, a direct method is derived from integration by parts. Since the function is symmetric about t=0, the area under the right tail of the density is found by calling the function with -t instead of t. ACCURACY: Tested at random 1 <= k <= 25. The "domain" refers to t. Relative error: arithmetic domain # trials peak rms IEEE -100,-2 50000 5.9e-15 1.4e-15 IEEE -2,100 500000 2.7e-15 4.9e-17 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier *************************************************************************/ public static double studenttdistribution(int k, double t) { return studenttdistr.studenttdistribution(k, t, null); } public static double studenttdistribution(int k, double t, alglib.xparams _params) { return studenttdistr.studenttdistribution(k, t, _params); } /************************************************************************* Functional inverse of Student's t distribution Given probability p, finds the argument t such that stdtr(k,t) is equal to p. ACCURACY: Tested at random 1 <= k <= 100. The "domain" refers to p: Relative error: arithmetic domain # trials peak rms IEEE .001,.999 25000 5.7e-15 8.0e-16 IEEE 10^-6,.001 25000 2.0e-12 2.9e-14 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier *************************************************************************/ public static double invstudenttdistribution(int k, double p) { return studenttdistr.invstudenttdistribution(k, p, null); } public static double invstudenttdistribution(int k, double p, alglib.xparams _params) { return studenttdistr.invstudenttdistribution(k, p, _params); } } public partial class alglib { /************************************************************************* Binomial distribution Returns the sum of the terms 0 through k of the Binomial probability density: k -- ( n ) j n-j > ( ) p (1-p) -- ( j ) j=0 The terms are not summed directly; instead the incomplete beta integral is employed, according to the formula y = bdtr( k, n, p ) = incbet( n-k, k+1, 1-p ). The arguments must be positive, with p ranging from 0 to 1. ACCURACY: Tested at random points (a,b,p), with p between 0 and 1. a,b Relative error: arithmetic domain # trials peak rms For p between 0.001 and 1: IEEE 0,100 100000 4.3e-15 2.6e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier *************************************************************************/ public static double binomialdistribution(int k, int n, double p) { return binomialdistr.binomialdistribution(k, n, p, null); } public static double binomialdistribution(int k, int n, double p, alglib.xparams _params) { return binomialdistr.binomialdistribution(k, n, p, _params); } /************************************************************************* Complemented binomial distribution Returns the sum of the terms k+1 through n of the Binomial probability density: n -- ( n ) j n-j > ( ) p (1-p) -- ( j ) j=k+1 The terms are not summed directly; instead the incomplete beta integral is employed, according to the formula y = bdtrc( k, n, p ) = incbet( k+1, n-k, p ). The arguments must be positive, with p ranging from 0 to 1. ACCURACY: Tested at random points (a,b,p). a,b Relative error: arithmetic domain # trials peak rms For p between 0.001 and 1: IEEE 0,100 100000 6.7e-15 8.2e-16 For p between 0 and .001: IEEE 0,100 100000 1.5e-13 2.7e-15 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier *************************************************************************/ public static double binomialcdistribution(int k, int n, double p) { return binomialdistr.binomialcdistribution(k, n, p, null); } public static double binomialcdistribution(int k, int n, double p, alglib.xparams _params) { return binomialdistr.binomialcdistribution(k, n, p, _params); } /************************************************************************* Inverse binomial distribution Finds the event probability p such that the sum of the terms 0 through k of the Binomial probability density is equal to the given cumulative probability y. This is accomplished using the inverse beta integral function and the relation 1 - p = incbi( n-k, k+1, y ). ACCURACY: Tested at random points (a,b,p). a,b Relative error: arithmetic domain # trials peak rms For p between 0.001 and 1: IEEE 0,100 100000 2.3e-14 6.4e-16 IEEE 0,10000 100000 6.6e-12 1.2e-13 For p between 10^-6 and 0.001: IEEE 0,100 100000 2.0e-12 1.3e-14 IEEE 0,10000 100000 1.5e-12 3.2e-14 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier *************************************************************************/ public static double invbinomialdistribution(int k, int n, double y) { return binomialdistr.invbinomialdistribution(k, n, y, null); } public static double invbinomialdistribution(int k, int n, double y, alglib.xparams _params) { return binomialdistr.invbinomialdistribution(k, n, y, _params); } } public partial class alglib { /************************************************************************* Airy function Solution of the differential equation y"(x) = xy. The function returns the two independent solutions Ai, Bi and their first derivatives Ai'(x), Bi'(x). Evaluation is by power series summation for small x, by rational minimax approximations for large x. ACCURACY: Error criterion is absolute when function <= 1, relative when function > 1, except * denotes relative error criterion. For large negative x, the absolute error increases as x^1.5. For large positive x, the relative error increases as x^1.5. Arithmetic domain function # trials peak rms IEEE -10, 0 Ai 10000 1.6e-15 2.7e-16 IEEE 0, 10 Ai 10000 2.3e-14* 1.8e-15* IEEE -10, 0 Ai' 10000 4.6e-15 7.6e-16 IEEE 0, 10 Ai' 10000 1.8e-14* 1.5e-15* IEEE -10, 10 Bi 30000 4.2e-15 5.3e-16 IEEE -10, 10 Bi' 30000 4.9e-15 7.3e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier *************************************************************************/ public static void airy(double x, out double ai, out double aip, out double bi, out double bip) { ai = 0; aip = 0; bi = 0; bip = 0; airyf.airy(x, ref ai, ref aip, ref bi, ref bip, null); } public static void airy(double x, out double ai, out double aip, out double bi, out double bip, alglib.xparams _params) { ai = 0; aip = 0; bi = 0; bip = 0; airyf.airy(x, ref ai, ref aip, ref bi, ref bip, _params); } } public partial class alglib { public class gammafunc { /************************************************************************* Gamma function Input parameters: X - argument Domain: 0 < X < 171.6 -170 < X < 0, X is not an integer. Relative error: arithmetic domain # trials peak rms IEEE -170,-33 20000 2.3e-15 3.3e-16 IEEE -33, 33 20000 9.4e-16 2.2e-16 IEEE 33, 171.6 20000 2.3e-15 3.2e-16 Cephes Math Library Release 2.8: June, 2000 Original copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier Translated to AlgoPascal by Bochkanov Sergey (2005, 2006, 2007). *************************************************************************/ public static double gammafunction(double x, alglib.xparams _params) { double result = 0; double p = 0; double pp = 0; double q = 0; double qq = 0; double z = 0; int i = 0; double sgngam = 0; sgngam = 1; q = Math.Abs(x); if( (double)(q)>(double)(33.0) ) { if( (double)(x)<(double)(0.0) ) { p = (int)Math.Floor(q); i = (int)Math.Round(p); if( i%2==0 ) { sgngam = -1; } z = q-p; if( (double)(z)>(double)(0.5) ) { p = p+1; z = q-p; } z = q*Math.Sin(Math.PI*z); z = Math.Abs(z); z = Math.PI/(z*gammastirf(q, _params)); } else { z = gammastirf(x, _params); } result = sgngam*z; return result; } z = 1; while( (double)(x)>=(double)(3) ) { x = x-1; z = z*x; } while( (double)(x)<(double)(0) ) { if( (double)(x)>(double)(-0.000000001) ) { result = z/((1+0.5772156649015329*x)*x); return result; } z = z/x; x = x+1; } while( (double)(x)<(double)(2) ) { if( (double)(x)<(double)(0.000000001) ) { result = z/((1+0.5772156649015329*x)*x); return result; } z = z/x; x = x+1.0; } if( (double)(x)==(double)(2) ) { result = z; return result; } x = x-2.0; pp = 1.60119522476751861407E-4; pp = 1.19135147006586384913E-3+x*pp; pp = 1.04213797561761569935E-2+x*pp; pp = 4.76367800457137231464E-2+x*pp; pp = 2.07448227648435975150E-1+x*pp; pp = 4.94214826801497100753E-1+x*pp; pp = 9.99999999999999996796E-1+x*pp; qq = -2.31581873324120129819E-5; qq = 5.39605580493303397842E-4+x*qq; qq = -4.45641913851797240494E-3+x*qq; qq = 1.18139785222060435552E-2+x*qq; qq = 3.58236398605498653373E-2+x*qq; qq = -2.34591795718243348568E-1+x*qq; qq = 7.14304917030273074085E-2+x*qq; qq = 1.00000000000000000320+x*qq; result = z*pp/qq; return result; } /************************************************************************* Natural logarithm of gamma function Input parameters: X - argument Result: logarithm of the absolute value of the Gamma(X). Output parameters: SgnGam - sign(Gamma(X)) Domain: 0 < X < 2.55e305 -2.55e305 < X < 0, X is not an integer. ACCURACY: arithmetic domain # trials peak rms IEEE 0, 3 28000 5.4e-16 1.1e-16 IEEE 2.718, 2.556e305 40000 3.5e-16 8.3e-17 The error criterion was relative when the function magnitude was greater than one but absolute when it was less than one. The following test used the relative error criterion, though at certain points the relative error could be much higher than indicated. IEEE -200, -4 10000 4.8e-16 1.3e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier Translated to AlgoPascal by Bochkanov Sergey (2005, 2006, 2007). *************************************************************************/ public static double lngamma(double x, ref double sgngam, alglib.xparams _params) { double result = 0; double a = 0; double b = 0; double c = 0; double p = 0; double q = 0; double u = 0; double w = 0; double z = 0; int i = 0; double logpi = 0; double ls2pi = 0; double tmp = 0; sgngam = 0; sgngam = 1; logpi = 1.14472988584940017414; ls2pi = 0.91893853320467274178; if( (double)(x)<(double)(-34.0) ) { q = -x; w = lngamma(q, ref tmp, _params); p = (int)Math.Floor(q); i = (int)Math.Round(p); if( i%2==0 ) { sgngam = -1; } else { sgngam = 1; } z = q-p; if( (double)(z)>(double)(0.5) ) { p = p+1; z = p-q; } z = q*Math.Sin(Math.PI*z); result = logpi-Math.Log(z)-w; return result; } if( (double)(x)<(double)(13) ) { z = 1; p = 0; u = x; while( (double)(u)>=(double)(3) ) { p = p-1; u = x+p; z = z*u; } while( (double)(u)<(double)(2) ) { z = z/u; p = p+1; u = x+p; } if( (double)(z)<(double)(0) ) { sgngam = -1; z = -z; } else { sgngam = 1; } if( (double)(u)==(double)(2) ) { result = Math.Log(z); return result; } p = p-2; x = x+p; b = -1378.25152569120859100; b = -38801.6315134637840924+x*b; b = -331612.992738871184744+x*b; b = -1162370.97492762307383+x*b; b = -1721737.00820839662146+x*b; b = -853555.664245765465627+x*b; c = 1; c = -351.815701436523470549+x*c; c = -17064.2106651881159223+x*c; c = -220528.590553854454839+x*c; c = -1139334.44367982507207+x*c; c = -2532523.07177582951285+x*c; c = -2018891.41433532773231+x*c; p = x*b/c; result = Math.Log(z)+p; return result; } q = (x-0.5)*Math.Log(x)-x+ls2pi; if( (double)(x)>(double)(100000000) ) { result = q; return result; } p = 1/(x*x); if( (double)(x)>=(double)(1000.0) ) { q = q+((7.9365079365079365079365*0.0001*p-2.7777777777777777777778*0.001)*p+0.0833333333333333333333)/x; } else { a = 8.11614167470508450300*0.0001; a = -(5.95061904284301438324*0.0001)+p*a; a = 7.93650340457716943945*0.0001+p*a; a = -(2.77777777730099687205*0.001)+p*a; a = 8.33333333333331927722*0.01+p*a; q = q+a/x; } result = q; return result; } private static double gammastirf(double x, alglib.xparams _params) { double result = 0; double y = 0; double w = 0; double v = 0; double stir = 0; w = 1/x; stir = 7.87311395793093628397E-4; stir = -2.29549961613378126380E-4+w*stir; stir = -2.68132617805781232825E-3+w*stir; stir = 3.47222221605458667310E-3+w*stir; stir = 8.33333333333482257126E-2+w*stir; w = 1+w*stir; y = Math.Exp(x); if( (double)(x)>(double)(143.01608) ) { v = Math.Pow(x, 0.5*x-0.25); y = v*(v/y); } else { y = Math.Pow(x, x-0.5)/y; } result = 2.50662827463100050242*y*w; return result; } } public class normaldistr { /************************************************************************* Error function The integral is x - 2 | | 2 erf(x) = -------- | exp( - t ) dt. sqrt(pi) | | - 0 For 0 <= |x| < 1, erf(x) = x * P4(x**2)/Q5(x**2); otherwise erf(x) = 1 - erfc(x). ACCURACY: Relative error: arithmetic domain # trials peak rms IEEE 0,1 30000 3.7e-16 1.0e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier *************************************************************************/ public static double errorfunction(double x, alglib.xparams _params) { double result = 0; double xsq = 0; double s = 0; double p = 0; double q = 0; s = Math.Sign(x); x = Math.Abs(x); if( (double)(x)<(double)(0.5) ) { xsq = x*x; p = 0.007547728033418631287834; p = -0.288805137207594084924010+xsq*p; p = 14.3383842191748205576712+xsq*p; p = 38.0140318123903008244444+xsq*p; p = 3017.82788536507577809226+xsq*p; p = 7404.07142710151470082064+xsq*p; p = 80437.3630960840172832162+xsq*p; q = 0.0; q = 1.00000000000000000000000+xsq*q; q = 38.0190713951939403753468+xsq*q; q = 658.070155459240506326937+xsq*q; q = 6379.60017324428279487120+xsq*q; q = 34216.5257924628539769006+xsq*q; q = 80437.3630960840172826266+xsq*q; result = s*1.1283791670955125738961589031*x*p/q; return result; } if( (double)(x)>=(double)(10) ) { result = s; return result; } result = s*(1-errorfunctionc(x, _params)); return result; } /************************************************************************* Complementary error function 1 - erf(x) = inf. - 2 | | 2 erfc(x) = -------- | exp( - t ) dt sqrt(pi) | | - x For small x, erfc(x) = 1 - erf(x); otherwise rational approximations are computed. ACCURACY: Relative error: arithmetic domain # trials peak rms IEEE 0,26.6417 30000 5.7e-14 1.5e-14 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier *************************************************************************/ public static double errorfunctionc(double x, alglib.xparams _params) { double result = 0; double p = 0; double q = 0; if( (double)(x)<(double)(0) ) { result = 2-errorfunctionc(-x, _params); return result; } if( (double)(x)<(double)(0.5) ) { result = 1.0-errorfunction(x, _params); return result; } if( (double)(x)>=(double)(10) ) { result = 0; return result; } p = 0.0; p = 0.5641877825507397413087057563+x*p; p = 9.675807882987265400604202961+x*p; p = 77.08161730368428609781633646+x*p; p = 368.5196154710010637133875746+x*p; p = 1143.262070703886173606073338+x*p; p = 2320.439590251635247384768711+x*p; p = 2898.0293292167655611275846+x*p; p = 1826.3348842295112592168999+x*p; q = 1.0; q = 17.14980943627607849376131193+x*q; q = 137.1255960500622202878443578+x*q; q = 661.7361207107653469211984771+x*q; q = 2094.384367789539593790281779+x*q; q = 4429.612803883682726711528526+x*q; q = 6089.5424232724435504633068+x*q; q = 4958.82756472114071495438422+x*q; q = 1826.3348842295112595576438+x*q; result = Math.Exp(-math.sqr(x))*p/q; return result; } /************************************************************************* Same as normalcdf(), obsolete name. *************************************************************************/ public static double normaldistribution(double x, alglib.xparams _params) { double result = 0; result = 0.5*(errorfunction(x/1.41421356237309504880, _params)+1); return result; } /************************************************************************* Normal distribution PDF Returns Gaussian probability density function: 1 f(x) = --------- * exp(-x^2/2) sqrt(2pi) Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier *************************************************************************/ public static double normalpdf(double x, alglib.xparams _params) { double result = 0; alglib.ap.assert(math.isfinite(x), "NormalPDF: X is infinite"); result = Math.Exp(-(x*x/2))/Math.Sqrt(2*Math.PI); return result; } /************************************************************************* Normal distribution CDF Returns the area under the Gaussian probability density function, integrated from minus infinity to x: x - 1 | | 2 ndtr(x) = --------- | exp( - t /2 ) dt sqrt(2pi) | | - -inf. = ( 1 + erf(z) ) / 2 = erfc(z) / 2 where z = x/sqrt(2). Computation is via the functions erf and erfc. ACCURACY: Relative error: arithmetic domain # trials peak rms IEEE -13,0 30000 3.4e-14 6.7e-15 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier *************************************************************************/ public static double normalcdf(double x, alglib.xparams _params) { double result = 0; result = 0.5*(errorfunction(x/1.41421356237309504880, _params)+1); return result; } /************************************************************************* Inverse of the error function Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier *************************************************************************/ public static double inverf(double e, alglib.xparams _params) { double result = 0; result = invnormaldistribution(0.5*(e+1), _params)/Math.Sqrt(2); return result; } /************************************************************************* Same as invnormalcdf(), deprecated name *************************************************************************/ public static double invnormaldistribution(double y0, alglib.xparams _params) { double result = 0; result = invnormalcdf(y0, _params); return result; } /************************************************************************* Inverse of Normal CDF Returns the argument, x, for which the area under the Gaussian probability density function (integrated from minus infinity to x) is equal to y. For small arguments 0 < y < exp(-2), the program computes z = sqrt( -2.0 * log(y) ); then the approximation is x = z - log(z)/z - (1/z) P(1/z) / Q(1/z). There are two rational functions P/Q, one for 0 < y < exp(-32) and the other for y up to exp(-2). For larger arguments, w = y - 0.5, and x/sqrt(2pi) = w + w**3 R(w**2)/S(w**2)). ACCURACY: Relative error: arithmetic domain # trials peak rms IEEE 0.125, 1 20000 7.2e-16 1.3e-16 IEEE 3e-308, 0.135 50000 4.6e-16 9.8e-17 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier *************************************************************************/ public static double invnormalcdf(double y0, alglib.xparams _params) { double result = 0; double expm2 = 0; double s2pi = 0; double x = 0; double y = 0; double z = 0; double y2 = 0; double x0 = 0; double x1 = 0; int code = 0; double p0 = 0; double q0 = 0; double p1 = 0; double q1 = 0; double p2 = 0; double q2 = 0; expm2 = 0.13533528323661269189; s2pi = 2.50662827463100050242; if( (double)(y0)<=(double)(0) ) { result = -math.maxrealnumber; return result; } if( (double)(y0)>=(double)(1) ) { result = math.maxrealnumber; return result; } code = 1; y = y0; if( (double)(y)>(double)(1.0-expm2) ) { y = 1.0-y; code = 0; } if( (double)(y)>(double)(expm2) ) { y = y-0.5; y2 = y*y; p0 = -59.9633501014107895267; p0 = 98.0010754185999661536+y2*p0; p0 = -56.6762857469070293439+y2*p0; p0 = 13.9312609387279679503+y2*p0; p0 = -1.23916583867381258016+y2*p0; q0 = 1; q0 = 1.95448858338141759834+y2*q0; q0 = 4.67627912898881538453+y2*q0; q0 = 86.3602421390890590575+y2*q0; q0 = -225.462687854119370527+y2*q0; q0 = 200.260212380060660359+y2*q0; q0 = -82.0372256168333339912+y2*q0; q0 = 15.9056225126211695515+y2*q0; q0 = -1.18331621121330003142+y2*q0; x = y+y*y2*p0/q0; x = x*s2pi; result = x; return result; } x = Math.Sqrt(-(2.0*Math.Log(y))); x0 = x-Math.Log(x)/x; z = 1.0/x; if( (double)(x)<(double)(8.0) ) { p1 = 4.05544892305962419923; p1 = 31.5251094599893866154+z*p1; p1 = 57.1628192246421288162+z*p1; p1 = 44.0805073893200834700+z*p1; p1 = 14.6849561928858024014+z*p1; p1 = 2.18663306850790267539+z*p1; p1 = -(1.40256079171354495875*0.1)+z*p1; p1 = -(3.50424626827848203418*0.01)+z*p1; p1 = -(8.57456785154685413611*0.0001)+z*p1; q1 = 1; q1 = 15.7799883256466749731+z*q1; q1 = 45.3907635128879210584+z*q1; q1 = 41.3172038254672030440+z*q1; q1 = 15.0425385692907503408+z*q1; q1 = 2.50464946208309415979+z*q1; q1 = -(1.42182922854787788574*0.1)+z*q1; q1 = -(3.80806407691578277194*0.01)+z*q1; q1 = -(9.33259480895457427372*0.0001)+z*q1; x1 = z*p1/q1; } else { p2 = 3.23774891776946035970; p2 = 6.91522889068984211695+z*p2; p2 = 3.93881025292474443415+z*p2; p2 = 1.33303460815807542389+z*p2; p2 = 2.01485389549179081538*0.1+z*p2; p2 = 1.23716634817820021358*0.01+z*p2; p2 = 3.01581553508235416007*0.0001+z*p2; p2 = 2.65806974686737550832*0.000001+z*p2; p2 = 6.23974539184983293730*0.000000001+z*p2; q2 = 1; q2 = 6.02427039364742014255+z*q2; q2 = 3.67983563856160859403+z*q2; q2 = 1.37702099489081330271+z*q2; q2 = 2.16236993594496635890*0.1+z*q2; q2 = 1.34204006088543189037*0.01+z*q2; q2 = 3.28014464682127739104*0.0001+z*q2; q2 = 2.89247864745380683936*0.000001+z*q2; q2 = 6.79019408009981274425*0.000000001+z*q2; x1 = z*p2/q2; } x = x0-x1; if( code!=0 ) { x = -x; } result = x; return result; } /************************************************************************* Bivariate normal PDF Returns probability density function of the bivariate Gaussian with correlation parameter equal to Rho: 1 ( x^2 - 2*rho*x*y + y^2 ) f(x,y,rho) = ----------------- * exp( - ----------------------- ) 2pi*sqrt(1-rho^2) ( 2*(1-rho^2) ) with -1(double)(0) ) { fxys = normalcdf(-Math.Max(x, y), _params); } else { fxys = Math.Max(0.0, normalcdf(-x, _params)-normalcdf(y, _params)); } rangea = 0; rangeb = Math.Sqrt((1-rho)*(1+rho)); // // Compute first term (analytic integral) from formula (6) // ta = rangeb; tb = Math.Abs(x-s*y); tc = (4-s*x*y)/8; v0 = ta*(1-tc*(tb*tb-ta*ta)/3)*Math.Exp(-(tb*tb/(2*ta*ta)))-tb*(1-tc*tb*tb/3)*Math.Sqrt(2*Math.PI)*normalcdf(-(tb/ta), _params); v0 = v0*Math.Exp(-(s*x*y/2))/(2*Math.PI); // // Compute second term (numerical integral, 20-point Gauss-Legendre rule) from formula (6) // v1 = 0; v1 = v1+bvnintegrate6(rangea, rangeb, x, y, s, 0.1527533871307258, -0.0765265211334973, _params); v1 = v1+bvnintegrate6(rangea, rangeb, x, y, s, 0.1527533871307258, 0.0765265211334973, _params); v1 = v1+bvnintegrate6(rangea, rangeb, x, y, s, 0.1491729864726037, -0.2277858511416451, _params); v1 = v1+bvnintegrate6(rangea, rangeb, x, y, s, 0.1491729864726037, 0.2277858511416451, _params); v1 = v1+bvnintegrate6(rangea, rangeb, x, y, s, 0.1420961093183820, -0.3737060887154195, _params); v1 = v1+bvnintegrate6(rangea, rangeb, x, y, s, 0.1420961093183820, 0.3737060887154195, _params); v1 = v1+bvnintegrate6(rangea, rangeb, x, y, s, 0.1316886384491766, -0.5108670019508271, _params); v1 = v1+bvnintegrate6(rangea, rangeb, x, y, s, 0.1316886384491766, 0.5108670019508271, _params); v1 = v1+bvnintegrate6(rangea, rangeb, x, y, s, 0.1181945319615184, -0.6360536807265150, _params); v1 = v1+bvnintegrate6(rangea, rangeb, x, y, s, 0.1181945319615184, 0.6360536807265150, _params); v1 = v1+bvnintegrate6(rangea, rangeb, x, y, s, 0.1019301198172404, -0.7463319064601508, _params); v1 = v1+bvnintegrate6(rangea, rangeb, x, y, s, 0.1019301198172404, 0.7463319064601508, _params); v1 = v1+bvnintegrate6(rangea, rangeb, x, y, s, 0.0832767415767048, -0.8391169718222188, _params); v1 = v1+bvnintegrate6(rangea, rangeb, x, y, s, 0.0832767415767048, 0.8391169718222188, _params); v1 = v1+bvnintegrate6(rangea, rangeb, x, y, s, 0.0626720483341091, -0.9122344282513259, _params); v1 = v1+bvnintegrate6(rangea, rangeb, x, y, s, 0.0626720483341091, 0.9122344282513259, _params); v1 = v1+bvnintegrate6(rangea, rangeb, x, y, s, 0.0406014298003869, -0.9639719272779138, _params); v1 = v1+bvnintegrate6(rangea, rangeb, x, y, s, 0.0406014298003869, 0.9639719272779138, _params); v1 = v1+bvnintegrate6(rangea, rangeb, x, y, s, 0.0176140071391521, -0.9931285991850949, _params); v1 = v1+bvnintegrate6(rangea, rangeb, x, y, s, 0.0176140071391521, 0.9931285991850949, _params); v1 = v1*0.5*(rangeb-rangea)/(2*Math.PI); result = fxys-s*(v0+v1); } result = Math.Max(result, 0); result = Math.Min(result, 1); return result; } /************************************************************************* Internal function which computes integrand of formula (3) by Alan Genz times Gaussian weights (passed by user). -- ALGLIB -- Copyright 15.11.2019 by Bochkanov Sergey *************************************************************************/ private static double bvnintegrate3(double rangea, double rangeb, double x, double y, double gw, double gx, alglib.xparams _params) { double result = 0; double r = 0; double t2 = 0; double dd = 0; double sinr = 0; double cosr = 0; r = (rangeb-rangea)*0.5*gx+(rangeb+rangea)*0.5; t2 = Math.Tan(0.5*r); dd = 1/(1+t2*t2); sinr = 2*t2*dd; cosr = (1-t2*t2)*dd; result = gw*Math.Exp(-((x*x+y*y-2*x*y*sinr)/(2*cosr*cosr))); return result; } /************************************************************************* Internal function which computes integrand of formula (6) by Alan Genz times Gaussian weights (passed by user). -- ALGLIB -- Copyright 15.11.2019 by Bochkanov Sergey *************************************************************************/ private static double bvnintegrate6(double rangea, double rangeb, double x, double y, double s, double gw, double gx, alglib.xparams _params) { double result = 0; double r = 0; double exphsk22x2 = 0; double exphsk2 = 0; double sqrt1x2 = 0; double exphsk1sqrt1x2 = 0; r = (rangeb-rangea)*0.5*gx+(rangeb+rangea)*0.5; exphsk22x2 = Math.Exp(-((x-s*y)*(x-s*y)/(2*r*r))); exphsk2 = Math.Exp(-(x*s*y/2)); sqrt1x2 = Math.Sqrt((1-r)*(1+r)); exphsk1sqrt1x2 = Math.Exp(-(x*s*y/(1+sqrt1x2))); result = gw*exphsk22x2*(exphsk1sqrt1x2/sqrt1x2-exphsk2*(1+(4-x*y*s)*r*r/8)); return result; } } public class igammaf { /************************************************************************* Incomplete gamma integral The function is defined by x - 1 | | -t a-1 igam(a,x) = ----- | e t dt. - | | | (a) - 0 In this implementation both arguments must be positive. The integral is evaluated by either a power series or continued fraction expansion, depending on the relative values of a and x. ACCURACY: Relative error: arithmetic domain # trials peak rms IEEE 0,30 200000 3.6e-14 2.9e-15 IEEE 0,100 300000 9.9e-14 1.5e-14 Cephes Math Library Release 2.8: June, 2000 Copyright 1985, 1987, 2000 by Stephen L. Moshier *************************************************************************/ public static double incompletegamma(double a, double x, alglib.xparams _params) { double result = 0; double igammaepsilon = 0; double ans = 0; double ax = 0; double c = 0; double r = 0; double tmp = 0; igammaepsilon = 0.000000000000001; if( (double)(x)<=(double)(0) || (double)(a)<=(double)(0) ) { result = 0; return result; } if( (double)(x)>(double)(1) && (double)(x)>(double)(a) ) { result = 1-incompletegammac(a, x, _params); return result; } ax = a*Math.Log(x)-x-gammafunc.lngamma(a, ref tmp, _params); if( (double)(ax)<(double)(-709.78271289338399) ) { result = 0; return result; } ax = Math.Exp(ax); r = a; c = 1; ans = 1; do { r = r+1; c = c*x/r; ans = ans+c; } while( (double)(c/ans)>(double)(igammaepsilon) ); result = ans*ax/a; return result; } /************************************************************************* Complemented incomplete gamma integral The function is defined by igamc(a,x) = 1 - igam(a,x) inf. - 1 | | -t a-1 = ----- | e t dt. - | | | (a) - x In this implementation both arguments must be positive. The integral is evaluated by either a power series or continued fraction expansion, depending on the relative values of a and x. ACCURACY: Tested at random a, x. a x Relative error: arithmetic domain domain # trials peak rms IEEE 0.5,100 0,100 200000 1.9e-14 1.7e-15 IEEE 0.01,0.5 0,100 200000 1.4e-13 1.6e-15 Cephes Math Library Release 2.8: June, 2000 Copyright 1985, 1987, 2000 by Stephen L. Moshier *************************************************************************/ public static double incompletegammac(double a, double x, alglib.xparams _params) { double result = 0; double igammaepsilon = 0; double igammabignumber = 0; double igammabignumberinv = 0; double ans = 0; double ax = 0; double c = 0; double yc = 0; double r = 0; double t = 0; double y = 0; double z = 0; double pk = 0; double pkm1 = 0; double pkm2 = 0; double qk = 0; double qkm1 = 0; double qkm2 = 0; double tmp = 0; igammaepsilon = 0.000000000000001; igammabignumber = 4503599627370496.0; igammabignumberinv = 2.22044604925031308085*0.0000000000000001; if( (double)(x)<=(double)(0) || (double)(a)<=(double)(0) ) { result = 1; return result; } if( (double)(x)<(double)(1) || (double)(x)<(double)(a) ) { result = 1-incompletegamma(a, x, _params); return result; } ax = a*Math.Log(x)-x-gammafunc.lngamma(a, ref tmp, _params); if( (double)(ax)<(double)(-709.78271289338399) ) { result = 0; return result; } ax = Math.Exp(ax); y = 1-a; z = x+y+1; c = 0; pkm2 = 1; qkm2 = x; pkm1 = x+1; qkm1 = z*x; ans = pkm1/qkm1; do { c = c+1; y = y+1; z = z+2; yc = y*c; pk = pkm1*z-pkm2*yc; qk = qkm1*z-qkm2*yc; if( (double)(qk)!=(double)(0) ) { r = pk/qk; t = Math.Abs((ans-r)/r); ans = r; } else { t = 1; } pkm2 = pkm1; pkm1 = pk; qkm2 = qkm1; qkm1 = qk; if( (double)(Math.Abs(pk))>(double)(igammabignumber) ) { pkm2 = pkm2*igammabignumberinv; pkm1 = pkm1*igammabignumberinv; qkm2 = qkm2*igammabignumberinv; qkm1 = qkm1*igammabignumberinv; } } while( (double)(t)>(double)(igammaepsilon) ); result = ans*ax; return result; } /************************************************************************* Inverse of complemented imcomplete gamma integral Given p, the function finds x such that igamc( a, x ) = p. Starting with the approximate value 3 x = a t where t = 1 - d - ndtri(p) sqrt(d) and d = 1/9a, the routine performs up to 10 Newton iterations to find the root of igamc(a,x) - p = 0. ACCURACY: Tested at random a, p in the intervals indicated. a p Relative error: arithmetic domain domain # trials peak rms IEEE 0.5,100 0,0.5 100000 1.0e-14 1.7e-15 IEEE 0.01,0.5 0,0.5 100000 9.0e-14 3.4e-15 IEEE 0.5,10000 0,0.5 20000 2.3e-13 3.8e-14 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier *************************************************************************/ public static double invincompletegammac(double a, double y0, alglib.xparams _params) { double result = 0; double igammaepsilon = 0; double iinvgammabignumber = 0; double x0 = 0; double x1 = 0; double x = 0; double yl = 0; double yh = 0; double y = 0; double d = 0; double lgm = 0; double dithresh = 0; int i = 0; int dir = 0; double tmp = 0; igammaepsilon = 0.000000000000001; iinvgammabignumber = 4503599627370496.0; x0 = iinvgammabignumber; yl = 0; x1 = 0; yh = 1; dithresh = 5*igammaepsilon; d = 1/(9*a); y = 1-d-normaldistr.invnormaldistribution(y0, _params)*Math.Sqrt(d); x = a*y*y*y; lgm = gammafunc.lngamma(a, ref tmp, _params); i = 0; while( i<10 ) { if( (double)(x)>(double)(x0) || (double)(x)<(double)(x1) ) { d = 0.0625; break; } y = incompletegammac(a, x, _params); if( (double)(y)<(double)(yl) || (double)(y)>(double)(yh) ) { d = 0.0625; break; } if( (double)(y)<(double)(y0) ) { x0 = x; yl = y; } else { x1 = x; yh = y; } d = (a-1)*Math.Log(x)-x-lgm; if( (double)(d)<(double)(-709.78271289338399) ) { d = 0.0625; break; } d = -Math.Exp(d); d = (y-y0)/d; if( (double)(Math.Abs(d/x))<(double)(igammaepsilon) ) { result = x; return result; } x = x-d; i = i+1; } if( (double)(x0)==(double)(iinvgammabignumber) ) { if( (double)(x)<=(double)(0) ) { x = 1; } while( (double)(x0)==(double)(iinvgammabignumber) ) { x = (1+d)*x; y = incompletegammac(a, x, _params); if( (double)(y)<(double)(y0) ) { x0 = x; yl = y; break; } d = d+d; } } d = 0.5; dir = 0; i = 0; while( i<400 ) { x = x1+d*(x0-x1); y = incompletegammac(a, x, _params); lgm = (x0-x1)/(x1+x0); if( (double)(Math.Abs(lgm))<(double)(dithresh) ) { break; } lgm = (y-y0)/y0; if( (double)(Math.Abs(lgm))<(double)(dithresh) ) { break; } if( (double)(x)<=(double)(0.0) ) { break; } if( (double)(y)>=(double)(y0) ) { x1 = x; yh = y; if( dir<0 ) { dir = 0; d = 0.5; } else { if( dir>1 ) { d = 0.5*d+0.5; } else { d = (y0-yl)/(yh-yl); } } dir = dir+1; } else { x0 = x; yl = y; if( dir>0 ) { dir = 0; d = 0.5; } else { if( dir<-1 ) { d = 0.5*d; } else { d = (y0-yl)/(yh-yl); } } dir = dir-1; } i = i+1; } result = x; return result; } } public class elliptic { /************************************************************************* Complete elliptic integral of the first kind Approximates the integral pi/2 - | | | dt K(m) = | ------------------ | 2 | | sqrt( 1 - m sin t ) - 0 using the approximation P(x) - log x Q(x). ACCURACY: Relative error: arithmetic domain # trials peak rms IEEE 0,1 30000 2.5e-16 6.8e-17 Cephes Math Library, Release 2.8: June, 2000 Copyright 1984, 1987, 2000 by Stephen L. Moshier *************************************************************************/ public static double ellipticintegralk(double m, alglib.xparams _params) { double result = 0; result = ellipticintegralkhighprecision(1.0-m, _params); return result; } /************************************************************************* Complete elliptic integral of the first kind Approximates the integral pi/2 - | | | dt K(m) = | ------------------ | 2 | | sqrt( 1 - m sin t ) - 0 where m = 1 - m1, using the approximation P(x) - log x Q(x). The argument m1 is used rather than m so that the logarithmic singularity at m = 1 will be shifted to the origin; this preserves maximum accuracy. K(0) = pi/2. ACCURACY: Relative error: arithmetic domain # trials peak rms IEEE 0,1 30000 2.5e-16 6.8e-17 Cephes Math Library, Release 2.8: June, 2000 Copyright 1984, 1987, 2000 by Stephen L. Moshier *************************************************************************/ public static double ellipticintegralkhighprecision(double m1, alglib.xparams _params) { double result = 0; double p = 0; double q = 0; if( (double)(m1)<=(double)(math.machineepsilon) ) { result = 1.3862943611198906188E0-0.5*Math.Log(m1); } else { p = 1.37982864606273237150E-4; p = p*m1+2.28025724005875567385E-3; p = p*m1+7.97404013220415179367E-3; p = p*m1+9.85821379021226008714E-3; p = p*m1+6.87489687449949877925E-3; p = p*m1+6.18901033637687613229E-3; p = p*m1+8.79078273952743772254E-3; p = p*m1+1.49380448916805252718E-2; p = p*m1+3.08851465246711995998E-2; p = p*m1+9.65735902811690126535E-2; p = p*m1+1.38629436111989062502E0; q = 2.94078955048598507511E-5; q = q*m1+9.14184723865917226571E-4; q = q*m1+5.94058303753167793257E-3; q = q*m1+1.54850516649762399335E-2; q = q*m1+2.39089602715924892727E-2; q = q*m1+3.01204715227604046988E-2; q = q*m1+3.73774314173823228969E-2; q = q*m1+4.88280347570998239232E-2; q = q*m1+7.03124996963957469739E-2; q = q*m1+1.24999999999870820058E-1; q = q*m1+4.99999999999999999821E-1; result = p-q*Math.Log(m1); } return result; } /************************************************************************* Incomplete elliptic integral of the first kind F(phi|m) Approximates the integral phi - | | | dt F(phi_\m) = | ------------------ | 2 | | sqrt( 1 - m sin t ) - 0 of amplitude phi and modulus m, using the arithmetic - geometric mean algorithm. ACCURACY: Tested at random points with m in [0, 1] and phi as indicated. Relative error: arithmetic domain # trials peak rms IEEE -10,10 200000 7.4e-16 1.0e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 2000 by Stephen L. Moshier *************************************************************************/ public static double incompleteellipticintegralk(double phi, double m, alglib.xparams _params) { double result = 0; double a = 0; double b = 0; double c = 0; double e = 0; double temp = 0; double pio2 = 0; double t = 0; double k = 0; int d = 0; int md = 0; int s = 0; int npio2 = 0; pio2 = 1.57079632679489661923; if( (double)(m)==(double)(0) ) { result = phi; return result; } a = 1-m; if( (double)(a)==(double)(0) ) { result = Math.Log(Math.Tan(0.5*(pio2+phi))); return result; } npio2 = (int)Math.Floor(phi/pio2); if( npio2%2!=0 ) { npio2 = npio2+1; } if( npio2!=0 ) { k = ellipticintegralk(1-a, _params); phi = phi-npio2*pio2; } else { k = 0; } if( (double)(phi)<(double)(0) ) { phi = -phi; s = -1; } else { s = 0; } b = Math.Sqrt(a); t = Math.Tan(phi); if( (double)(Math.Abs(t))>(double)(10) ) { e = 1.0/(b*t); if( (double)(Math.Abs(e))<(double)(10) ) { e = Math.Atan(e); if( npio2==0 ) { k = ellipticintegralk(1-a, _params); } temp = k-incompleteellipticintegralk(e, m, _params); if( s<0 ) { temp = -temp; } result = temp+npio2*k; return result; } } a = 1.0; c = Math.Sqrt(m); d = 1; md = 0; while( (double)(Math.Abs(c/a))>(double)(math.machineepsilon) ) { temp = b/a; phi = phi+Math.Atan(t*temp)+md*Math.PI; md = (int)((phi+pio2)/Math.PI); t = t*(1.0+temp)/(1.0-temp*t*t); c = 0.5*(a-b); temp = Math.Sqrt(a*b); a = 0.5*(a+b); b = temp; d = d+d; } temp = (Math.Atan(t)+md*Math.PI)/(d*a); if( s<0 ) { temp = -temp; } result = temp+npio2*k; return result; } /************************************************************************* Complete elliptic integral of the second kind Approximates the integral pi/2 - | | 2 E(m) = | sqrt( 1 - m sin t ) dt | | - 0 using the approximation P(x) - x log x Q(x). ACCURACY: Relative error: arithmetic domain # trials peak rms IEEE 0, 1 10000 2.1e-16 7.3e-17 Cephes Math Library, Release 2.8: June, 2000 Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier *************************************************************************/ public static double ellipticintegrale(double m, alglib.xparams _params) { double result = 0; double p = 0; double q = 0; alglib.ap.assert((double)(m)>=(double)(0) && (double)(m)<=(double)(1), "Domain error in EllipticIntegralE: m<0 or m>1"); m = 1-m; if( (double)(m)==(double)(0) ) { result = 1; return result; } p = 1.53552577301013293365E-4; p = p*m+2.50888492163602060990E-3; p = p*m+8.68786816565889628429E-3; p = p*m+1.07350949056076193403E-2; p = p*m+7.77395492516787092951E-3; p = p*m+7.58395289413514708519E-3; p = p*m+1.15688436810574127319E-2; p = p*m+2.18317996015557253103E-2; p = p*m+5.68051945617860553470E-2; p = p*m+4.43147180560990850618E-1; p = p*m+1.00000000000000000299E0; q = 3.27954898576485872656E-5; q = q*m+1.00962792679356715133E-3; q = q*m+6.50609489976927491433E-3; q = q*m+1.68862163993311317300E-2; q = q*m+2.61769742454493659583E-2; q = q*m+3.34833904888224918614E-2; q = q*m+4.27180926518931511717E-2; q = q*m+5.85936634471101055642E-2; q = q*m+9.37499997197644278445E-2; q = q*m+2.49999999999888314361E-1; result = p-q*m*Math.Log(m); return result; } /************************************************************************* Incomplete elliptic integral of the second kind Approximates the integral phi - | | | 2 E(phi_\m) = | sqrt( 1 - m sin t ) dt | | | - 0 of amplitude phi and modulus m, using the arithmetic - geometric mean algorithm. ACCURACY: Tested at random arguments with phi in [-10, 10] and m in [0, 1]. Relative error: arithmetic domain # trials peak rms IEEE -10,10 150000 3.3e-15 1.4e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1993, 2000 by Stephen L. Moshier *************************************************************************/ public static double incompleteellipticintegrale(double phi, double m, alglib.xparams _params) { double result = 0; double pio2 = 0; double a = 0; double b = 0; double c = 0; double e = 0; double temp = 0; double lphi = 0; double t = 0; double ebig = 0; int d = 0; int md = 0; int npio2 = 0; int s = 0; pio2 = 1.57079632679489661923; if( (double)(m)==(double)(0) ) { result = phi; return result; } lphi = phi; npio2 = (int)Math.Floor(lphi/pio2); if( npio2%2!=0 ) { npio2 = npio2+1; } lphi = lphi-npio2*pio2; if( (double)(lphi)<(double)(0) ) { lphi = -lphi; s = -1; } else { s = 1; } a = 1.0-m; ebig = ellipticintegrale(m, _params); if( (double)(a)==(double)(0) ) { temp = Math.Sin(lphi); if( s<0 ) { temp = -temp; } result = temp+npio2*ebig; return result; } t = Math.Tan(lphi); b = Math.Sqrt(a); // // Thanks to Brian Fitzgerald // for pointing out an instability near odd multiples of pi/2 // if( (double)(Math.Abs(t))>(double)(10) ) { // // Transform the amplitude // e = 1.0/(b*t); // // ... but avoid multiple recursions. // if( (double)(Math.Abs(e))<(double)(10) ) { e = Math.Atan(e); temp = ebig+m*Math.Sin(lphi)*Math.Sin(e)-incompleteellipticintegrale(e, m, _params); if( s<0 ) { temp = -temp; } result = temp+npio2*ebig; return result; } } c = Math.Sqrt(m); a = 1.0; d = 1; e = 0.0; md = 0; while( (double)(Math.Abs(c/a))>(double)(math.machineepsilon) ) { temp = b/a; lphi = lphi+Math.Atan(t*temp)+md*Math.PI; md = (int)((lphi+pio2)/Math.PI); t = t*(1.0+temp)/(1.0-temp*t*t); c = 0.5*(a-b); temp = Math.Sqrt(a*b); a = 0.5*(a+b); b = temp; d = d+d; e = e+c*Math.Sin(lphi); } temp = ebig/ellipticintegralk(m, _params); temp = temp*((Math.Atan(t)+md*Math.PI)/(d*a)); temp = temp+e; if( s<0 ) { temp = -temp; } result = temp+npio2*ebig; return result; } } public class hermite { /************************************************************************* Calculation of the value of the Hermite polynomial. Parameters: n - degree, n>=0 x - argument Result: the value of the Hermite polynomial Hn at x *************************************************************************/ public static double hermitecalculate(int n, double x, alglib.xparams _params) { double result = 0; int i = 0; double a = 0; double b = 0; result = 0; // // Prepare A and B // a = 1; b = 2*x; // // Special cases: N=0 or N=1 // if( n==0 ) { result = a; return result; } if( n==1 ) { result = b; return result; } // // General case: N>=2 // for(i=2; i<=n; i++) { result = 2*x*b-2*(i-1)*a; a = b; b = result; } return result; } /************************************************************************* Summation of Hermite polynomials using Clenshaw's recurrence formula. This routine calculates c[0]*H0(x) + c[1]*H1(x) + ... + c[N]*HN(x) Parameters: n - degree, n>=0 x - argument Result: the value of the Hermite polynomial at x *************************************************************************/ public static double hermitesum(double[] c, int n, double x, alglib.xparams _params) { double result = 0; double b1 = 0; double b2 = 0; int i = 0; b1 = 0; b2 = 0; result = 0; for(i=n; i>=0; i--) { result = 2*(x*b1-(i+1)*b2)+c[i]; b2 = b1; b1 = result; } return result; } /************************************************************************* Representation of Hn as C[0] + C[1]*X + ... + C[N]*X^N Input parameters: N - polynomial degree, n>=0 Output parameters: C - coefficients *************************************************************************/ public static void hermitecoefficients(int n, ref double[] c, alglib.xparams _params) { int i = 0; c = new double[0]; c = new double[n+1]; for(i=0; i<=n; i++) { c[i] = 0; } c[n] = Math.Exp(n*Math.Log(2)); for(i=0; i<=n/2-1; i++) { c[n-2*(i+1)] = -(c[n-2*i]*(n-2*i)*(n-2*i-1)/4/(i+1)); } } } public class dawson { /************************************************************************* Dawson's Integral Approximates the integral x - 2 | | 2 dawsn(x) = exp( -x ) | exp( t ) dt | | - 0 Three different rational approximations are employed, for the intervals 0 to 3.25; 3.25 to 6.25; and 6.25 up. ACCURACY: Relative error: arithmetic domain # trials peak rms IEEE 0,10 10000 6.9e-16 1.0e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier *************************************************************************/ public static double dawsonintegral(double x, alglib.xparams _params) { double result = 0; double x2 = 0; double y = 0; int sg = 0; double an = 0; double ad = 0; double bn = 0; double bd = 0; double cn = 0; double cd = 0; sg = 1; if( (double)(x)<(double)(0) ) { sg = -1; x = -x; } if( (double)(x)<(double)(3.25) ) { x2 = x*x; an = 1.13681498971755972054E-11; an = an*x2+8.49262267667473811108E-10; an = an*x2+1.94434204175553054283E-8; an = an*x2+9.53151741254484363489E-7; an = an*x2+3.07828309874913200438E-6; an = an*x2+3.52513368520288738649E-4; an = an*x2+-8.50149846724410912031E-4; an = an*x2+4.22618223005546594270E-2; an = an*x2+-9.17480371773452345351E-2; an = an*x2+9.99999999999999994612E-1; ad = 2.40372073066762605484E-11; ad = ad*x2+1.48864681368493396752E-9; ad = ad*x2+5.21265281010541664570E-8; ad = ad*x2+1.27258478273186970203E-6; ad = ad*x2+2.32490249820789513991E-5; ad = ad*x2+3.25524741826057911661E-4; ad = ad*x2+3.48805814657162590916E-3; ad = ad*x2+2.79448531198828973716E-2; ad = ad*x2+1.58874241960120565368E-1; ad = ad*x2+5.74918629489320327824E-1; ad = ad*x2+1.00000000000000000539E0; y = x*an/ad; result = sg*y; return result; } x2 = 1.0/(x*x); if( (double)(x)<(double)(6.25) ) { bn = 5.08955156417900903354E-1; bn = bn*x2-2.44754418142697847934E-1; bn = bn*x2+9.41512335303534411857E-2; bn = bn*x2-2.18711255142039025206E-2; bn = bn*x2+3.66207612329569181322E-3; bn = bn*x2-4.23209114460388756528E-4; bn = bn*x2+3.59641304793896631888E-5; bn = bn*x2-2.14640351719968974225E-6; bn = bn*x2+9.10010780076391431042E-8; bn = bn*x2-2.40274520828250956942E-9; bn = bn*x2+3.59233385440928410398E-11; bd = 1.00000000000000000000E0; bd = bd*x2-6.31839869873368190192E-1; bd = bd*x2+2.36706788228248691528E-1; bd = bd*x2-5.31806367003223277662E-2; bd = bd*x2+8.48041718586295374409E-3; bd = bd*x2-9.47996768486665330168E-4; bd = bd*x2+7.81025592944552338085E-5; bd = bd*x2-4.55875153252442634831E-6; bd = bd*x2+1.89100358111421846170E-7; bd = bd*x2-4.91324691331920606875E-9; bd = bd*x2+7.18466403235734541950E-11; y = 1.0/x+x2*bn/(bd*x); result = sg*0.5*y; return result; } if( (double)(x)>(double)(1.0E9) ) { result = sg*0.5/x; return result; } cn = -5.90592860534773254987E-1; cn = cn*x2+6.29235242724368800674E-1; cn = cn*x2-1.72858975380388136411E-1; cn = cn*x2+1.64837047825189632310E-2; cn = cn*x2-4.86827613020462700845E-4; cd = 1.00000000000000000000E0; cd = cd*x2-2.69820057197544900361E0; cd = cd*x2+1.73270799045947845857E0; cd = cd*x2-3.93708582281939493482E-1; cd = cd*x2+3.44278924041233391079E-2; cd = cd*x2-9.73655226040941223894E-4; y = 1.0/x+x2*cn/(cd*x); result = sg*0.5*y; return result; } } public class trigintegrals { /************************************************************************* Sine and cosine integrals Evaluates the integrals x - | cos t - 1 Ci(x) = eul + ln x + | --------- dt, | t - 0 x - | sin t Si(x) = | ----- dt | t - 0 where eul = 0.57721566490153286061 is Euler's constant. The integrals are approximated by rational functions. For x > 8 auxiliary functions f(x) and g(x) are employed such that Ci(x) = f(x) sin(x) - g(x) cos(x) Si(x) = pi/2 - f(x) cos(x) - g(x) sin(x) ACCURACY: Test interval = [0,50]. Absolute error, except relative when > 1: arithmetic function # trials peak rms IEEE Si 30000 4.4e-16 7.3e-17 IEEE Ci 30000 6.9e-16 5.1e-17 Cephes Math Library Release 2.1: January, 1989 Copyright 1984, 1987, 1989 by Stephen L. Moshier *************************************************************************/ public static void sinecosineintegrals(double x, ref double si, ref double ci, alglib.xparams _params) { double z = 0; double c = 0; double s = 0; double f = 0; double g = 0; int sg = 0; double sn = 0; double sd = 0; double cn = 0; double cd = 0; double fn = 0; double fd = 0; double gn = 0; double gd = 0; si = 0; ci = 0; if( (double)(x)<(double)(0) ) { sg = -1; x = -x; } else { sg = 0; } if( (double)(x)==(double)(0) ) { si = 0; ci = -math.maxrealnumber; return; } if( (double)(x)>(double)(1.0E9) ) { si = 1.570796326794896619-Math.Cos(x)/x; ci = Math.Sin(x)/x; return; } if( (double)(x)<=(double)(4) ) { z = x*x; sn = -8.39167827910303881427E-11; sn = sn*z+4.62591714427012837309E-8; sn = sn*z-9.75759303843632795789E-6; sn = sn*z+9.76945438170435310816E-4; sn = sn*z-4.13470316229406538752E-2; sn = sn*z+1.00000000000000000302E0; sd = 2.03269266195951942049E-12; sd = sd*z+1.27997891179943299903E-9; sd = sd*z+4.41827842801218905784E-7; sd = sd*z+9.96412122043875552487E-5; sd = sd*z+1.42085239326149893930E-2; sd = sd*z+9.99999999999999996984E-1; s = x*sn/sd; cn = 2.02524002389102268789E-11; cn = cn*z-1.35249504915790756375E-8; cn = cn*z+3.59325051419993077021E-6; cn = cn*z-4.74007206873407909465E-4; cn = cn*z+2.89159652607555242092E-2; cn = cn*z-1.00000000000000000080E0; cd = 4.07746040061880559506E-12; cd = cd*z+3.06780997581887812692E-9; cd = cd*z+1.23210355685883423679E-6; cd = cd*z+3.17442024775032769882E-4; cd = cd*z+5.10028056236446052392E-2; cd = cd*z+4.00000000000000000080E0; c = z*cn/cd; if( sg!=0 ) { s = -s; } si = s; ci = 0.57721566490153286061+Math.Log(x)+c; return; } s = Math.Sin(x); c = Math.Cos(x); z = 1.0/(x*x); if( (double)(x)<(double)(8) ) { fn = 4.23612862892216586994E0; fn = fn*z+5.45937717161812843388E0; fn = fn*z+1.62083287701538329132E0; fn = fn*z+1.67006611831323023771E-1; fn = fn*z+6.81020132472518137426E-3; fn = fn*z+1.08936580650328664411E-4; fn = fn*z+5.48900223421373614008E-7; fd = 1.00000000000000000000E0; fd = fd*z+8.16496634205391016773E0; fd = fd*z+7.30828822505564552187E0; fd = fd*z+1.86792257950184183883E0; fd = fd*z+1.78792052963149907262E-1; fd = fd*z+7.01710668322789753610E-3; fd = fd*z+1.10034357153915731354E-4; fd = fd*z+5.48900252756255700982E-7; f = fn/(x*fd); gn = 8.71001698973114191777E-2; gn = gn*z+6.11379109952219284151E-1; gn = gn*z+3.97180296392337498885E-1; gn = gn*z+7.48527737628469092119E-2; gn = gn*z+5.38868681462177273157E-3; gn = gn*z+1.61999794598934024525E-4; gn = gn*z+1.97963874140963632189E-6; gn = gn*z+7.82579040744090311069E-9; gd = 1.00000000000000000000E0; gd = gd*z+1.64402202413355338886E0; gd = gd*z+6.66296701268987968381E-1; gd = gd*z+9.88771761277688796203E-2; gd = gd*z+6.22396345441768420760E-3; gd = gd*z+1.73221081474177119497E-4; gd = gd*z+2.02659182086343991969E-6; gd = gd*z+7.82579218933534490868E-9; g = z*gn/gd; } else { fn = 4.55880873470465315206E-1; fn = fn*z+7.13715274100146711374E-1; fn = fn*z+1.60300158222319456320E-1; fn = fn*z+1.16064229408124407915E-2; fn = fn*z+3.49556442447859055605E-4; fn = fn*z+4.86215430826454749482E-6; fn = fn*z+3.20092790091004902806E-8; fn = fn*z+9.41779576128512936592E-11; fn = fn*z+9.70507110881952024631E-14; fd = 1.00000000000000000000E0; fd = fd*z+9.17463611873684053703E-1; fd = fd*z+1.78685545332074536321E-1; fd = fd*z+1.22253594771971293032E-2; fd = fd*z+3.58696481881851580297E-4; fd = fd*z+4.92435064317881464393E-6; fd = fd*z+3.21956939101046018377E-8; fd = fd*z+9.43720590350276732376E-11; fd = fd*z+9.70507110881952025725E-14; f = fn/(x*fd); gn = 6.97359953443276214934E-1; gn = gn*z+3.30410979305632063225E-1; gn = gn*z+3.84878767649974295920E-2; gn = gn*z+1.71718239052347903558E-3; gn = gn*z+3.48941165502279436777E-5; gn = gn*z+3.47131167084116673800E-7; gn = gn*z+1.70404452782044526189E-9; gn = gn*z+3.85945925430276600453E-12; gn = gn*z+3.14040098946363334640E-15; gd = 1.00000000000000000000E0; gd = gd*z+1.68548898811011640017E0; gd = gd*z+4.87852258695304967486E-1; gd = gd*z+4.67913194259625806320E-2; gd = gd*z+1.90284426674399523638E-3; gd = gd*z+3.68475504442561108162E-5; gd = gd*z+3.57043223443740838771E-7; gd = gd*z+1.72693748966316146736E-9; gd = gd*z+3.87830166023954706752E-12; gd = gd*z+3.14040098946363335242E-15; g = z*gn/gd; } si = 1.570796326794896619-f*c-g*s; if( sg!=0 ) { si = -si; } ci = f*s-g*c; } /************************************************************************* Hyperbolic sine and cosine integrals Approximates the integrals x - | | cosh t - 1 Chi(x) = eul + ln x + | ----------- dt, | | t - 0 x - | | sinh t Shi(x) = | ------ dt | | t - 0 where eul = 0.57721566490153286061 is Euler's constant. The integrals are evaluated by power series for x < 8 and by Chebyshev expansions for x between 8 and 88. For large x, both functions approach exp(x)/2x. Arguments greater than 88 in magnitude return MAXNUM. ACCURACY: Test interval 0 to 88. Relative error: arithmetic function # trials peak rms IEEE Shi 30000 6.9e-16 1.6e-16 Absolute error, except relative when |Chi| > 1: IEEE Chi 30000 8.4e-16 1.4e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 2000 by Stephen L. Moshier *************************************************************************/ public static void hyperbolicsinecosineintegrals(double x, ref double shi, ref double chi, alglib.xparams _params) { double k = 0; double z = 0; double c = 0; double s = 0; double a = 0; int sg = 0; double b0 = 0; double b1 = 0; double b2 = 0; shi = 0; chi = 0; if( (double)(x)<(double)(0) ) { sg = -1; x = -x; } else { sg = 0; } if( (double)(x)==(double)(0) ) { shi = 0; chi = -math.maxrealnumber; return; } if( (double)(x)<(double)(8.0) ) { z = x*x; a = 1.0; s = 1.0; c = 0.0; k = 2.0; do { a = a*z/k; c = c+a/k; k = k+1.0; a = a/k; s = s+a/k; k = k+1.0; } while( (double)(Math.Abs(a/s))>=(double)(math.machineepsilon) ); s = s*x; } else { if( (double)(x)<(double)(18.0) ) { a = (576.0/x-52.0)/10.0; k = Math.Exp(x)/x; b0 = 1.83889230173399459482E-17; b1 = 0.0; chebiterationshichi(a, -9.55485532279655569575E-17, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 2.04326105980879882648E-16, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 1.09896949074905343022E-15, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, -1.31313534344092599234E-14, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 5.93976226264314278932E-14, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, -3.47197010497749154755E-14, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, -1.40059764613117131000E-12, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 9.49044626224223543299E-12, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, -1.61596181145435454033E-11, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, -1.77899784436430310321E-10, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 1.35455469767246947469E-9, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, -1.03257121792819495123E-9, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, -3.56699611114982536845E-8, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 1.44818877384267342057E-7, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 7.82018215184051295296E-7, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, -5.39919118403805073710E-6, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, -3.12458202168959833422E-5, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 8.90136741950727517826E-5, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 2.02558474743846862168E-3, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 2.96064440855633256972E-2, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 1.11847751047257036625E0, ref b0, ref b1, ref b2, _params); s = k*0.5*(b0-b2); b0 = -8.12435385225864036372E-18; b1 = 0.0; chebiterationshichi(a, 2.17586413290339214377E-17, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 5.22624394924072204667E-17, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, -9.48812110591690559363E-16, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 5.35546311647465209166E-15, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, -1.21009970113732918701E-14, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, -6.00865178553447437951E-14, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 7.16339649156028587775E-13, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, -2.93496072607599856104E-12, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, -1.40359438136491256904E-12, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 8.76302288609054966081E-11, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, -4.40092476213282340617E-10, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, -1.87992075640569295479E-10, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 1.31458150989474594064E-8, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, -4.75513930924765465590E-8, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, -2.21775018801848880741E-7, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 1.94635531373272490962E-6, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 4.33505889257316408893E-6, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, -6.13387001076494349496E-5, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, -3.13085477492997465138E-4, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 4.97164789823116062801E-4, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 2.64347496031374526641E-2, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 1.11446150876699213025E0, ref b0, ref b1, ref b2, _params); c = k*0.5*(b0-b2); } else { if( (double)(x)<=(double)(88.0) ) { a = (6336.0/x-212.0)/70.0; k = Math.Exp(x)/x; b0 = -1.05311574154850938805E-17; b1 = 0.0; chebiterationshichi(a, 2.62446095596355225821E-17, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 8.82090135625368160657E-17, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, -3.38459811878103047136E-16, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, -8.30608026366935789136E-16, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 3.93397875437050071776E-15, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 1.01765565969729044505E-14, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, -4.21128170307640802703E-14, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, -1.60818204519802480035E-13, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 3.34714954175994481761E-13, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 2.72600352129153073807E-12, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 1.66894954752839083608E-12, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, -3.49278141024730899554E-11, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, -1.58580661666482709598E-10, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, -1.79289437183355633342E-10, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 1.76281629144264523277E-9, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 1.69050228879421288846E-8, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 1.25391771228487041649E-7, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 1.16229947068677338732E-6, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 1.61038260117376323993E-5, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 3.49810375601053973070E-4, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 1.28478065259647610779E-2, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 1.03665722588798326712E0, ref b0, ref b1, ref b2, _params); s = k*0.5*(b0-b2); b0 = 8.06913408255155572081E-18; b1 = 0.0; chebiterationshichi(a, -2.08074168180148170312E-17, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, -5.98111329658272336816E-17, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 2.68533951085945765591E-16, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 4.52313941698904694774E-16, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, -3.10734917335299464535E-15, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, -4.42823207332531972288E-15, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 3.49639695410806959872E-14, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 6.63406731718911586609E-14, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, -3.71902448093119218395E-13, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, -1.27135418132338309016E-12, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 2.74851141935315395333E-12, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 2.33781843985453438400E-11, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 2.71436006377612442764E-11, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, -2.56600180000355990529E-10, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, -1.61021375163803438552E-9, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, -4.72543064876271773512E-9, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, -3.00095178028681682282E-9, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 7.79387474390914922337E-8, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 1.06942765566401507066E-6, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 1.59503164802313196374E-5, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 3.49592575153777996871E-4, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 1.28475387530065247392E-2, ref b0, ref b1, ref b2, _params); chebiterationshichi(a, 1.03665693917934275131E0, ref b0, ref b1, ref b2, _params); c = k*0.5*(b0-b2); } else { if( sg!=0 ) { shi = -math.maxrealnumber; } else { shi = math.maxrealnumber; } chi = math.maxrealnumber; return; } } } if( sg!=0 ) { s = -s; } shi = s; chi = 0.57721566490153286061+Math.Log(x)+c; } private static void chebiterationshichi(double x, double c, ref double b0, ref double b1, ref double b2, alglib.xparams _params) { b2 = b1; b1 = b0; b0 = x*b1-b2+c; } } public class poissondistr { /************************************************************************* Poisson distribution Returns the sum of the first k+1 terms of the Poisson distribution: k j -- -m m > e -- -- j! j=0 The terms are not summed directly; instead the incomplete gamma integral is employed, according to the relation y = pdtr( k, m ) = igamc( k+1, m ). The arguments must both be positive. ACCURACY: See incomplete gamma function Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier *************************************************************************/ public static double poissondistribution(int k, double m, alglib.xparams _params) { double result = 0; alglib.ap.assert(k>=0 && (double)(m)>(double)(0), "Domain error in PoissonDistribution"); result = igammaf.incompletegammac(k+1, m, _params); return result; } /************************************************************************* Complemented Poisson distribution Returns the sum of the terms k+1 to infinity of the Poisson distribution: inf. j -- -m m > e -- -- j! j=k+1 The terms are not summed directly; instead the incomplete gamma integral is employed, according to the formula y = pdtrc( k, m ) = igam( k+1, m ). The arguments must both be positive. ACCURACY: See incomplete gamma function Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier *************************************************************************/ public static double poissoncdistribution(int k, double m, alglib.xparams _params) { double result = 0; alglib.ap.assert(k>=0 && (double)(m)>(double)(0), "Domain error in PoissonDistributionC"); result = igammaf.incompletegamma(k+1, m, _params); return result; } /************************************************************************* Inverse Poisson distribution Finds the Poisson variable x such that the integral from 0 to x of the Poisson density is equal to the given probability y. This is accomplished using the inverse gamma integral function and the relation m = igami( k+1, y ). ACCURACY: See inverse incomplete gamma function Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier *************************************************************************/ public static double invpoissondistribution(int k, double y, alglib.xparams _params) { double result = 0; alglib.ap.assert((k>=0 && (double)(y)>=(double)(0)) && (double)(y)<(double)(1), "Domain error in InvPoissonDistribution"); result = igammaf.invincompletegammac(k+1, y, _params); return result; } } public class bessel { /************************************************************************* Bessel function of order zero Returns Bessel function of order zero of the argument. The domain is divided into the intervals [0, 5] and (5, infinity). In the first interval the following rational approximation is used: 2 2 (w - r ) (w - r ) P (w) / Q (w) 1 2 3 8 2 where w = x and the two r's are zeros of the function. In the second interval, the Hankel asymptotic expansion is employed with two rational functions of degree 6/6 and 7/7. ACCURACY: Absolute error: arithmetic domain # trials peak rms IEEE 0, 30 60000 4.2e-16 1.1e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier *************************************************************************/ public static double besselj0(double x, alglib.xparams _params) { double result = 0; double xsq = 0; double nn = 0; double pzero = 0; double qzero = 0; double p1 = 0; double q1 = 0; if( (double)(x)<(double)(0) ) { x = -x; } if( (double)(x)>(double)(8.0) ) { besselasympt0(x, ref pzero, ref qzero, _params); nn = x-Math.PI/4; result = Math.Sqrt(2/Math.PI/x)*(pzero*Math.Cos(nn)-qzero*Math.Sin(nn)); return result; } xsq = math.sqr(x); p1 = 26857.86856980014981415848441; p1 = -40504123.71833132706360663322+xsq*p1; p1 = 25071582855.36881945555156435+xsq*p1; p1 = -8085222034853.793871199468171+xsq*p1; p1 = 1434354939140344.111664316553+xsq*p1; p1 = -136762035308817138.6865416609+xsq*p1; p1 = 6382059341072356562.289432465+xsq*p1; p1 = -117915762910761053603.8440800+xsq*p1; p1 = 493378725179413356181.6813446+xsq*p1; q1 = 1.0; q1 = 1363.063652328970604442810507+xsq*q1; q1 = 1114636.098462985378182402543+xsq*q1; q1 = 669998767.2982239671814028660+xsq*q1; q1 = 312304311494.1213172572469442+xsq*q1; q1 = 112775673967979.8507056031594+xsq*q1; q1 = 30246356167094626.98627330784+xsq*q1; q1 = 5428918384092285160.200195092+xsq*q1; q1 = 493378725179413356211.3278438+xsq*q1; result = p1/q1; return result; } /************************************************************************* Bessel function of order one Returns Bessel function of order one of the argument. The domain is divided into the intervals [0, 8] and (8, infinity). In the first interval a 24 term Chebyshev expansion is used. In the second, the asymptotic trigonometric representation is employed using two rational functions of degree 5/5. ACCURACY: Absolute error: arithmetic domain # trials peak rms IEEE 0, 30 30000 2.6e-16 1.1e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier *************************************************************************/ public static double besselj1(double x, alglib.xparams _params) { double result = 0; double s = 0; double xsq = 0; double nn = 0; double pzero = 0; double qzero = 0; double p1 = 0; double q1 = 0; s = Math.Sign(x); if( (double)(x)<(double)(0) ) { x = -x; } if( (double)(x)>(double)(8.0) ) { besselasympt1(x, ref pzero, ref qzero, _params); nn = x-3*Math.PI/4; result = Math.Sqrt(2/Math.PI/x)*(pzero*Math.Cos(nn)-qzero*Math.Sin(nn)); if( (double)(s)<(double)(0) ) { result = -result; } return result; } xsq = math.sqr(x); p1 = 2701.122710892323414856790990; p1 = -4695753.530642995859767162166+xsq*p1; p1 = 3413234182.301700539091292655+xsq*p1; p1 = -1322983480332.126453125473247+xsq*p1; p1 = 290879526383477.5409737601689+xsq*p1; p1 = -35888175699101060.50743641413+xsq*p1; p1 = 2316433580634002297.931815435+xsq*p1; p1 = -66721065689249162980.20941484+xsq*p1; p1 = 581199354001606143928.050809+xsq*p1; q1 = 1.0; q1 = 1606.931573481487801970916749+xsq*q1; q1 = 1501793.594998585505921097578+xsq*q1; q1 = 1013863514.358673989967045588+xsq*q1; q1 = 524371026216.7649715406728642+xsq*q1; q1 = 208166122130760.7351240184229+xsq*q1; q1 = 60920613989175217.46105196863+xsq*q1; q1 = 11857707121903209998.37113348+xsq*q1; q1 = 1162398708003212287858.529400+xsq*q1; result = s*x*p1/q1; return result; } /************************************************************************* Bessel function of integer order Returns Bessel function of order n, where n is a (possibly negative) integer. The ratio of jn(x) to j0(x) is computed by backward recurrence. First the ratio jn/jn-1 is found by a continued fraction expansion. Then the recurrence relating successive orders is applied until j0 or j1 is reached. If n = 0 or 1 the routine for j0 or j1 is called directly. ACCURACY: Absolute error: arithmetic range # trials peak rms IEEE 0, 30 5000 4.4e-16 7.9e-17 Not suitable for large n or x. Use jv() (fractional order) instead. Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 2000 by Stephen L. Moshier *************************************************************************/ public static double besseljn(int n, double x, alglib.xparams _params) { double result = 0; double pkm2 = 0; double pkm1 = 0; double pk = 0; double xk = 0; double r = 0; double ans = 0; int k = 0; int sg = 0; if( n<0 ) { n = -n; if( n%2==0 ) { sg = 1; } else { sg = -1; } } else { sg = 1; } if( (double)(x)<(double)(0) ) { if( n%2!=0 ) { sg = -sg; } x = -x; } if( n==0 ) { result = sg*besselj0(x, _params); return result; } if( n==1 ) { result = sg*besselj1(x, _params); return result; } if( n==2 ) { if( (double)(x)==(double)(0) ) { result = 0; } else { result = sg*(2.0*besselj1(x, _params)/x-besselj0(x, _params)); } return result; } if( (double)(x)<(double)(math.machineepsilon) ) { result = 0; return result; } k = 53; pk = 2*(n+k); ans = pk; xk = x*x; do { pk = pk-2.0; ans = pk-xk/ans; k = k-1; } while( k!=0 ); ans = x/ans; pk = 1.0; pkm1 = 1.0/ans; k = n-1; r = 2*k; do { pkm2 = (pkm1*r-pk*x)/x; pk = pkm1; pkm1 = pkm2; r = r-2.0; k = k-1; } while( k!=0 ); if( (double)(Math.Abs(pk))>(double)(Math.Abs(pkm1)) ) { ans = besselj1(x, _params)/pk; } else { ans = besselj0(x, _params)/pkm1; } result = sg*ans; return result; } /************************************************************************* Bessel function of the second kind, order zero Returns Bessel function of the second kind, of order zero, of the argument. The domain is divided into the intervals [0, 5] and (5, infinity). In the first interval a rational approximation R(x) is employed to compute y0(x) = R(x) + 2 * log(x) * j0(x) / PI. Thus a call to j0() is required. In the second interval, the Hankel asymptotic expansion is employed with two rational functions of degree 6/6 and 7/7. ACCURACY: Absolute error, when y0(x) < 1; else relative error: arithmetic domain # trials peak rms IEEE 0, 30 30000 1.3e-15 1.6e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier *************************************************************************/ public static double bessely0(double x, alglib.xparams _params) { double result = 0; double nn = 0; double xsq = 0; double pzero = 0; double qzero = 0; double p4 = 0; double q4 = 0; if( (double)(x)>(double)(8.0) ) { besselasympt0(x, ref pzero, ref qzero, _params); nn = x-Math.PI/4; result = Math.Sqrt(2/Math.PI/x)*(pzero*Math.Sin(nn)+qzero*Math.Cos(nn)); return result; } xsq = math.sqr(x); p4 = -41370.35497933148554125235152; p4 = 59152134.65686889654273830069+xsq*p4; p4 = -34363712229.79040378171030138+xsq*p4; p4 = 10255208596863.94284509167421+xsq*p4; p4 = -1648605817185729.473122082537+xsq*p4; p4 = 137562431639934407.8571335453+xsq*p4; p4 = -5247065581112764941.297350814+xsq*p4; p4 = 65874732757195549259.99402049+xsq*p4; p4 = -27502866786291095837.01933175+xsq*p4; q4 = 1.0; q4 = 1282.452772478993804176329391+xsq*q4; q4 = 1001702.641288906265666651753+xsq*q4; q4 = 579512264.0700729537480087915+xsq*q4; q4 = 261306575504.1081249568482092+xsq*q4; q4 = 91620380340751.85262489147968+xsq*q4; q4 = 23928830434997818.57439356652+xsq*q4; q4 = 4192417043410839973.904769661+xsq*q4; q4 = 372645883898616588198.9980+xsq*q4; result = p4/q4+2/Math.PI*besselj0(x, _params)*Math.Log(x); return result; } /************************************************************************* Bessel function of second kind of order one Returns Bessel function of the second kind of order one of the argument. The domain is divided into the intervals [0, 8] and (8, infinity). In the first interval a 25 term Chebyshev expansion is used, and a call to j1() is required. In the second, the asymptotic trigonometric representation is employed using two rational functions of degree 5/5. ACCURACY: Absolute error: arithmetic domain # trials peak rms IEEE 0, 30 30000 1.0e-15 1.3e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier *************************************************************************/ public static double bessely1(double x, alglib.xparams _params) { double result = 0; double nn = 0; double xsq = 0; double pzero = 0; double qzero = 0; double p4 = 0; double q4 = 0; if( (double)(x)>(double)(8.0) ) { besselasympt1(x, ref pzero, ref qzero, _params); nn = x-3*Math.PI/4; result = Math.Sqrt(2/Math.PI/x)*(pzero*Math.Sin(nn)+qzero*Math.Cos(nn)); return result; } xsq = math.sqr(x); p4 = -2108847.540133123652824139923; p4 = 3639488548.124002058278999428+xsq*p4; p4 = -2580681702194.450950541426399+xsq*p4; p4 = 956993023992168.3481121552788+xsq*p4; p4 = -196588746272214065.8820322248+xsq*p4; p4 = 21931073399177975921.11427556+xsq*p4; p4 = -1212297555414509577913.561535+xsq*p4; p4 = 26554738314348543268942.48968+xsq*p4; p4 = -99637534243069222259967.44354+xsq*p4; q4 = 1.0; q4 = 1612.361029677000859332072312+xsq*q4; q4 = 1563282.754899580604737366452+xsq*q4; q4 = 1128686837.169442121732366891+xsq*q4; q4 = 646534088126.5275571961681500+xsq*q4; q4 = 297663212564727.6729292742282+xsq*q4; q4 = 108225825940881955.2553850180+xsq*q4; q4 = 29549879358971486742.90758119+xsq*q4; q4 = 5435310377188854170800.653097+xsq*q4; q4 = 508206736694124324531442.4152+xsq*q4; result = x*p4/q4+2/Math.PI*(besselj1(x, _params)*Math.Log(x)-1/x); return result; } /************************************************************************* Bessel function of second kind of integer order Returns Bessel function of order n, where n is a (possibly negative) integer. The function is evaluated by forward recurrence on n, starting with values computed by the routines y0() and y1(). If n = 0 or 1 the routine for y0 or y1 is called directly. ACCURACY: Absolute error, except relative when y > 1: arithmetic domain # trials peak rms IEEE 0, 30 30000 3.4e-15 4.3e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 2000 by Stephen L. Moshier *************************************************************************/ public static double besselyn(int n, double x, alglib.xparams _params) { double result = 0; int i = 0; double a = 0; double b = 0; double tmp = 0; double s = 0; s = 1; if( n<0 ) { n = -n; if( n%2!=0 ) { s = -1; } } if( n==0 ) { result = bessely0(x, _params); return result; } if( n==1 ) { result = s*bessely1(x, _params); return result; } a = bessely0(x, _params); b = bessely1(x, _params); for(i=1; i<=n-1; i++) { tmp = b; b = 2*i/x*b-a; a = tmp; } result = s*b; return result; } /************************************************************************* Modified Bessel function of order zero Returns modified Bessel function of order zero of the argument. The function is defined as i0(x) = j0( ix ). The range is partitioned into the two intervals [0,8] and (8, infinity). Chebyshev polynomial expansions are employed in each interval. ACCURACY: Relative error: arithmetic domain # trials peak rms IEEE 0,30 30000 5.8e-16 1.4e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 2000 by Stephen L. Moshier *************************************************************************/ public static double besseli0(double x, alglib.xparams _params) { double result = 0; double y = 0; double v = 0; double z = 0; double b0 = 0; double b1 = 0; double b2 = 0; if( (double)(x)<(double)(0) ) { x = -x; } if( (double)(x)<=(double)(8.0) ) { y = x/2.0-2.0; besselmfirstcheb(-4.41534164647933937950E-18, ref b0, ref b1, ref b2, _params); besselmnextcheb(y, 3.33079451882223809783E-17, ref b0, ref b1, ref b2, _params); besselmnextcheb(y, -2.43127984654795469359E-16, ref b0, ref b1, ref b2, _params); besselmnextcheb(y, 1.71539128555513303061E-15, ref b0, ref b1, ref b2, _params); besselmnextcheb(y, -1.16853328779934516808E-14, ref b0, ref b1, ref b2, _params); besselmnextcheb(y, 7.67618549860493561688E-14, ref b0, ref b1, ref b2, _params); besselmnextcheb(y, -4.85644678311192946090E-13, ref b0, ref b1, ref b2, _params); besselmnextcheb(y, 2.95505266312963983461E-12, ref b0, ref b1, ref b2, _params); besselmnextcheb(y, -1.72682629144155570723E-11, ref b0, ref b1, ref b2, _params); besselmnextcheb(y, 9.67580903537323691224E-11, ref b0, ref b1, ref b2, _params); besselmnextcheb(y, -5.18979560163526290666E-10, ref b0, ref b1, ref b2, _params); besselmnextcheb(y, 2.65982372468238665035E-9, ref b0, ref b1, ref b2, _params); besselmnextcheb(y, -1.30002500998624804212E-8, ref b0, ref b1, ref b2, _params); besselmnextcheb(y, 6.04699502254191894932E-8, ref b0, ref b1, ref b2, _params); besselmnextcheb(y, -2.67079385394061173391E-7, ref b0, ref b1, ref b2, _params); besselmnextcheb(y, 1.11738753912010371815E-6, ref b0, ref b1, ref b2, _params); besselmnextcheb(y, -4.41673835845875056359E-6, ref b0, ref b1, ref b2, _params); besselmnextcheb(y, 1.64484480707288970893E-5, ref b0, ref b1, ref b2, _params); besselmnextcheb(y, -5.75419501008210370398E-5, ref b0, ref b1, ref b2, _params); besselmnextcheb(y, 1.88502885095841655729E-4, ref b0, ref b1, ref b2, _params); besselmnextcheb(y, -5.76375574538582365885E-4, ref b0, ref b1, ref b2, _params); besselmnextcheb(y, 1.63947561694133579842E-3, ref b0, ref b1, ref b2, _params); besselmnextcheb(y, -4.32430999505057594430E-3, ref b0, ref b1, ref b2, _params); besselmnextcheb(y, 1.05464603945949983183E-2, ref b0, ref b1, ref b2, _params); besselmnextcheb(y, -2.37374148058994688156E-2, ref b0, ref b1, ref b2, _params); besselmnextcheb(y, 4.93052842396707084878E-2, ref b0, ref b1, ref b2, _params); besselmnextcheb(y, -9.49010970480476444210E-2, ref b0, ref b1, ref b2, _params); besselmnextcheb(y, 1.71620901522208775349E-1, ref b0, ref b1, ref b2, _params); besselmnextcheb(y, -3.04682672343198398683E-1, ref b0, ref b1, ref b2, _params); besselmnextcheb(y, 6.76795274409476084995E-1, ref b0, ref b1, ref b2, _params); v = 0.5*(b0-b2); result = Math.Exp(x)*v; return result; } z = 32.0/x-2.0; besselmfirstcheb(-7.23318048787475395456E-18, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, -4.83050448594418207126E-18, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, 4.46562142029675999901E-17, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, 3.46122286769746109310E-17, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, -2.82762398051658348494E-16, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, -3.42548561967721913462E-16, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, 1.77256013305652638360E-15, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, 3.81168066935262242075E-15, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, -9.55484669882830764870E-15, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, -4.15056934728722208663E-14, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, 1.54008621752140982691E-14, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, 3.85277838274214270114E-13, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, 7.18012445138366623367E-13, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, -1.79417853150680611778E-12, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, -1.32158118404477131188E-11, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, -3.14991652796324136454E-11, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, 1.18891471078464383424E-11, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, 4.94060238822496958910E-10, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, 3.39623202570838634515E-9, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, 2.26666899049817806459E-8, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, 2.04891858946906374183E-7, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, 2.89137052083475648297E-6, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, 6.88975834691682398426E-5, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, 3.36911647825569408990E-3, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, 8.04490411014108831608E-1, ref b0, ref b1, ref b2, _params); v = 0.5*(b0-b2); result = Math.Exp(x)*v/Math.Sqrt(x); return result; } /************************************************************************* Modified Bessel function of order one Returns modified Bessel function of order one of the argument. The function is defined as i1(x) = -i j1( ix ). The range is partitioned into the two intervals [0,8] and (8, infinity). Chebyshev polynomial expansions are employed in each interval. ACCURACY: Relative error: arithmetic domain # trials peak rms IEEE 0, 30 30000 1.9e-15 2.1e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1985, 1987, 2000 by Stephen L. Moshier *************************************************************************/ public static double besseli1(double x, alglib.xparams _params) { double result = 0; double y = 0; double z = 0; double v = 0; double b0 = 0; double b1 = 0; double b2 = 0; z = Math.Abs(x); if( (double)(z)<=(double)(8.0) ) { y = z/2.0-2.0; besselm1firstcheb(2.77791411276104639959E-18, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -2.11142121435816608115E-17, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, 1.55363195773620046921E-16, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -1.10559694773538630805E-15, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, 7.60068429473540693410E-15, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -5.04218550472791168711E-14, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, 3.22379336594557470981E-13, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -1.98397439776494371520E-12, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, 1.17361862988909016308E-11, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -6.66348972350202774223E-11, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, 3.62559028155211703701E-10, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -1.88724975172282928790E-9, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, 9.38153738649577178388E-9, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -4.44505912879632808065E-8, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, 2.00329475355213526229E-7, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -8.56872026469545474066E-7, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, 3.47025130813767847674E-6, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -1.32731636560394358279E-5, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, 4.78156510755005422638E-5, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -1.61760815825896745588E-4, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, 5.12285956168575772895E-4, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -1.51357245063125314899E-3, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, 4.15642294431288815669E-3, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -1.05640848946261981558E-2, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, 2.47264490306265168283E-2, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -5.29459812080949914269E-2, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, 1.02643658689847095384E-1, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -1.76416518357834055153E-1, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, 2.52587186443633654823E-1, ref b0, ref b1, ref b2, _params); v = 0.5*(b0-b2); z = v*z*Math.Exp(z); } else { y = 32.0/z-2.0; besselm1firstcheb(7.51729631084210481353E-18, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, 4.41434832307170791151E-18, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -4.65030536848935832153E-17, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -3.20952592199342395980E-17, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, 2.96262899764595013876E-16, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, 3.30820231092092828324E-16, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -1.88035477551078244854E-15, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -3.81440307243700780478E-15, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, 1.04202769841288027642E-14, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, 4.27244001671195135429E-14, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -2.10154184277266431302E-14, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -4.08355111109219731823E-13, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -7.19855177624590851209E-13, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, 2.03562854414708950722E-12, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, 1.41258074366137813316E-11, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, 3.25260358301548823856E-11, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -1.89749581235054123450E-11, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -5.58974346219658380687E-10, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -3.83538038596423702205E-9, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -2.63146884688951950684E-8, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -2.51223623787020892529E-7, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -3.88256480887769039346E-6, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -1.10588938762623716291E-4, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -9.76109749136146840777E-3, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, 7.78576235018280120474E-1, ref b0, ref b1, ref b2, _params); v = 0.5*(b0-b2); z = v*Math.Exp(z)/Math.Sqrt(z); } if( (double)(x)<(double)(0) ) { z = -z; } result = z; return result; } /************************************************************************* Modified Bessel function, second kind, order zero Returns modified Bessel function of the second kind of order zero of the argument. The range is partitioned into the two intervals [0,8] and (8, infinity). Chebyshev polynomial expansions are employed in each interval. ACCURACY: Tested at 2000 random points between 0 and 8. Peak absolute error (relative when K0 > 1) was 1.46e-14; rms, 4.26e-15. Relative error: arithmetic domain # trials peak rms IEEE 0, 30 30000 1.2e-15 1.6e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 2000 by Stephen L. Moshier *************************************************************************/ public static double besselk0(double x, alglib.xparams _params) { double result = 0; double y = 0; double z = 0; double v = 0; double b0 = 0; double b1 = 0; double b2 = 0; alglib.ap.assert((double)(x)>(double)(0), "Domain error in BesselK0: x<=0"); if( (double)(x)<=(double)(2) ) { y = x*x-2.0; besselmfirstcheb(1.37446543561352307156E-16, ref b0, ref b1, ref b2, _params); besselmnextcheb(y, 4.25981614279661018399E-14, ref b0, ref b1, ref b2, _params); besselmnextcheb(y, 1.03496952576338420167E-11, ref b0, ref b1, ref b2, _params); besselmnextcheb(y, 1.90451637722020886025E-9, ref b0, ref b1, ref b2, _params); besselmnextcheb(y, 2.53479107902614945675E-7, ref b0, ref b1, ref b2, _params); besselmnextcheb(y, 2.28621210311945178607E-5, ref b0, ref b1, ref b2, _params); besselmnextcheb(y, 1.26461541144692592338E-3, ref b0, ref b1, ref b2, _params); besselmnextcheb(y, 3.59799365153615016266E-2, ref b0, ref b1, ref b2, _params); besselmnextcheb(y, 3.44289899924628486886E-1, ref b0, ref b1, ref b2, _params); besselmnextcheb(y, -5.35327393233902768720E-1, ref b0, ref b1, ref b2, _params); v = 0.5*(b0-b2); v = v-Math.Log(0.5*x)*besseli0(x, _params); } else { z = 8.0/x-2.0; besselmfirstcheb(5.30043377268626276149E-18, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, -1.64758043015242134646E-17, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, 5.21039150503902756861E-17, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, -1.67823109680541210385E-16, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, 5.51205597852431940784E-16, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, -1.84859337734377901440E-15, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, 6.34007647740507060557E-15, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, -2.22751332699166985548E-14, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, 8.03289077536357521100E-14, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, -2.98009692317273043925E-13, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, 1.14034058820847496303E-12, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, -4.51459788337394416547E-12, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, 1.85594911495471785253E-11, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, -7.95748924447710747776E-11, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, 3.57739728140030116597E-10, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, -1.69753450938905987466E-9, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, 8.57403401741422608519E-9, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, -4.66048989768794782956E-8, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, 2.76681363944501510342E-7, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, -1.83175552271911948767E-6, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, 1.39498137188764993662E-5, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, -1.28495495816278026384E-4, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, 1.56988388573005337491E-3, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, -3.14481013119645005427E-2, ref b0, ref b1, ref b2, _params); besselmnextcheb(z, 2.44030308206595545468E0, ref b0, ref b1, ref b2, _params); v = 0.5*(b0-b2); v = v*Math.Exp(-x)/Math.Sqrt(x); } result = v; return result; } /************************************************************************* Modified Bessel function, second kind, order one Computes the modified Bessel function of the second kind of order one of the argument. The range is partitioned into the two intervals [0,2] and (2, infinity). Chebyshev polynomial expansions are employed in each interval. ACCURACY: Relative error: arithmetic domain # trials peak rms IEEE 0, 30 30000 1.2e-15 1.6e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 2000 by Stephen L. Moshier *************************************************************************/ public static double besselk1(double x, alglib.xparams _params) { double result = 0; double y = 0; double z = 0; double v = 0; double b0 = 0; double b1 = 0; double b2 = 0; z = 0.5*x; alglib.ap.assert((double)(z)>(double)(0), "Domain error in K1"); if( (double)(x)<=(double)(2) ) { y = x*x-2.0; besselm1firstcheb(-7.02386347938628759343E-18, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -2.42744985051936593393E-15, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -6.66690169419932900609E-13, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -1.41148839263352776110E-10, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -2.21338763073472585583E-8, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -2.43340614156596823496E-6, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -1.73028895751305206302E-4, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -6.97572385963986435018E-3, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -1.22611180822657148235E-1, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -3.53155960776544875667E-1, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, 1.52530022733894777053E0, ref b0, ref b1, ref b2, _params); v = 0.5*(b0-b2); result = Math.Log(z)*besseli1(x, _params)+v/x; } else { y = 8.0/x-2.0; besselm1firstcheb(-5.75674448366501715755E-18, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, 1.79405087314755922667E-17, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -5.68946255844285935196E-17, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, 1.83809354436663880070E-16, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -6.05704724837331885336E-16, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, 2.03870316562433424052E-15, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -7.01983709041831346144E-15, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, 2.47715442448130437068E-14, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -8.97670518232499435011E-14, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, 3.34841966607842919884E-13, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -1.28917396095102890680E-12, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, 5.13963967348173025100E-12, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -2.12996783842756842877E-11, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, 9.21831518760500529508E-11, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -4.19035475934189648750E-10, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, 2.01504975519703286596E-9, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -1.03457624656780970260E-8, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, 5.74108412545004946722E-8, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -3.50196060308781257119E-7, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, 2.40648494783721712015E-6, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -1.93619797416608296024E-5, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, 1.95215518471351631108E-4, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, -2.85781685962277938680E-3, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, 1.03923736576817238437E-1, ref b0, ref b1, ref b2, _params); besselm1nextcheb(y, 2.72062619048444266945E0, ref b0, ref b1, ref b2, _params); v = 0.5*(b0-b2); result = Math.Exp(-x)*v/Math.Sqrt(x); } return result; } /************************************************************************* Modified Bessel function, second kind, integer order Returns modified Bessel function of the second kind of order n of the argument. The range is partitioned into the two intervals [0,9.55] and (9.55, infinity). An ascending power series is used in the low range, and an asymptotic expansion in the high range. ACCURACY: Relative error: arithmetic domain # trials peak rms IEEE 0,30 90000 1.8e-8 3.0e-10 Error is high only near the crossover point x = 9.55 between the two expansions used. Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1988, 2000 by Stephen L. Moshier *************************************************************************/ public static double besselkn(int nn, double x, alglib.xparams _params) { double result = 0; double k = 0; double kf = 0; double nk1f = 0; double nkf = 0; double zn = 0; double t = 0; double s = 0; double z0 = 0; double z = 0; double ans = 0; double fn = 0; double pn = 0; double pk = 0; double zmn = 0; double tlg = 0; double tox = 0; int i = 0; int n = 0; double eul = 0; eul = 5.772156649015328606065e-1; if( nn<0 ) { n = -nn; } else { n = nn; } alglib.ap.assert(n<=31, "Overflow in BesselKN"); alglib.ap.assert((double)(x)>(double)(0), "Domain error in BesselKN"); if( (double)(x)<=(double)(9.55) ) { ans = 0.0; z0 = 0.25*x*x; fn = 1.0; pn = 0.0; zmn = 1.0; tox = 2.0/x; if( n>0 ) { pn = -eul; k = 1.0; for(i=1; i<=n-1; i++) { pn = pn+1.0/k; k = k+1.0; fn = fn*k; } zmn = tox; if( n==1 ) { ans = 1.0/x; } else { nk1f = fn/n; kf = 1.0; s = nk1f; z = -z0; zn = 1.0; for(i=1; i<=n-1; i++) { nk1f = nk1f/(n-i); kf = kf*i; zn = zn*z; t = nk1f*zn/kf; s = s+t; alglib.ap.assert((double)(math.maxrealnumber-Math.Abs(t))>(double)(Math.Abs(s)), "Overflow in BesselKN"); alglib.ap.assert(!((double)(tox)>(double)(1.0) && (double)(math.maxrealnumber/tox)<(double)(zmn)), "Overflow in BesselKN"); zmn = zmn*tox; } s = s*0.5; t = Math.Abs(s); alglib.ap.assert(!((double)(zmn)>(double)(1.0) && (double)(math.maxrealnumber/zmn)<(double)(t)), "Overflow in BesselKN"); alglib.ap.assert(!((double)(t)>(double)(1.0) && (double)(math.maxrealnumber/t)<(double)(zmn)), "Overflow in BesselKN"); ans = s*zmn; } } tlg = 2.0*Math.Log(0.5*x); pk = -eul; if( n==0 ) { pn = pk; t = 1.0; } else { pn = pn+1.0/n; t = 1.0/fn; } s = (pk+pn-tlg)*t; k = 1.0; do { t = t*(z0/(k*(k+n))); pk = pk+1.0/k; pn = pn+1.0/(k+n); s = s+(pk+pn-tlg)*t; k = k+1.0; } while( (double)(Math.Abs(t/s))>(double)(math.machineepsilon) ); s = 0.5*s/zmn; if( n%2!=0 ) { s = -s; } ans = ans+s; result = ans; return result; } if( (double)(x)>(double)(Math.Log(math.maxrealnumber)) ) { result = 0; return result; } k = n; pn = 4.0*k*k; pk = 1.0; z0 = 8.0*x; fn = 1.0; t = 1.0; s = t; nkf = math.maxrealnumber; i = 0; do { z = pn-pk*pk; t = t*z/(fn*z0); nk1f = Math.Abs(t); if( i>=n && (double)(nk1f)>(double)(nkf) ) { break; } nkf = nk1f; s = s+t; fn = fn+1.0; pk = pk+2.0; i = i+1; } while( (double)(Math.Abs(t/s))>(double)(math.machineepsilon) ); result = Math.Exp(-x)*Math.Sqrt(Math.PI/(2.0*x))*s; return result; } /************************************************************************* Internal subroutine Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 2000 by Stephen L. Moshier *************************************************************************/ private static void besselmfirstcheb(double c, ref double b0, ref double b1, ref double b2, alglib.xparams _params) { b0 = c; b1 = 0.0; b2 = 0.0; } /************************************************************************* Internal subroutine Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 2000 by Stephen L. Moshier *************************************************************************/ private static void besselmnextcheb(double x, double c, ref double b0, ref double b1, ref double b2, alglib.xparams _params) { b2 = b1; b1 = b0; b0 = x*b1-b2+c; } /************************************************************************* Internal subroutine Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 2000 by Stephen L. Moshier *************************************************************************/ private static void besselm1firstcheb(double c, ref double b0, ref double b1, ref double b2, alglib.xparams _params) { b0 = c; b1 = 0.0; b2 = 0.0; } /************************************************************************* Internal subroutine Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 2000 by Stephen L. Moshier *************************************************************************/ private static void besselm1nextcheb(double x, double c, ref double b0, ref double b1, ref double b2, alglib.xparams _params) { b2 = b1; b1 = b0; b0 = x*b1-b2+c; } private static void besselasympt0(double x, ref double pzero, ref double qzero, alglib.xparams _params) { double xsq = 0; double p2 = 0; double q2 = 0; double p3 = 0; double q3 = 0; pzero = 0; qzero = 0; xsq = 64.0/(x*x); p2 = 0.0; p2 = 2485.271928957404011288128951+xsq*p2; p2 = 153982.6532623911470917825993+xsq*p2; p2 = 2016135.283049983642487182349+xsq*p2; p2 = 8413041.456550439208464315611+xsq*p2; p2 = 12332384.76817638145232406055+xsq*p2; p2 = 5393485.083869438325262122897+xsq*p2; q2 = 1.0; q2 = 2615.700736920839685159081813+xsq*q2; q2 = 156001.7276940030940592769933+xsq*q2; q2 = 2025066.801570134013891035236+xsq*q2; q2 = 8426449.050629797331554404810+xsq*q2; q2 = 12338310.22786324960844856182+xsq*q2; q2 = 5393485.083869438325560444960+xsq*q2; p3 = -0.0; p3 = -4.887199395841261531199129300+xsq*p3; p3 = -226.2630641933704113967255053+xsq*p3; p3 = -2365.956170779108192723612816+xsq*p3; p3 = -8239.066313485606568803548860+xsq*p3; p3 = -10381.41698748464093880530341+xsq*p3; p3 = -3984.617357595222463506790588+xsq*p3; q3 = 1.0; q3 = 408.7714673983499223402830260+xsq*q3; q3 = 15704.89191515395519392882766+xsq*q3; q3 = 156021.3206679291652539287109+xsq*q3; q3 = 533291.3634216897168722255057+xsq*q3; q3 = 666745.4239319826986004038103+xsq*q3; q3 = 255015.5108860942382983170882+xsq*q3; pzero = p2/q2; qzero = 8*p3/q3/x; } private static void besselasympt1(double x, ref double pzero, ref double qzero, alglib.xparams _params) { double xsq = 0; double p2 = 0; double q2 = 0; double p3 = 0; double q3 = 0; pzero = 0; qzero = 0; xsq = 64.0/(x*x); p2 = -1611.616644324610116477412898; p2 = -109824.0554345934672737413139+xsq*p2; p2 = -1523529.351181137383255105722+xsq*p2; p2 = -6603373.248364939109255245434+xsq*p2; p2 = -9942246.505077641195658377899+xsq*p2; p2 = -4435757.816794127857114720794+xsq*p2; q2 = 1.0; q2 = -1455.009440190496182453565068+xsq*q2; q2 = -107263.8599110382011903063867+xsq*q2; q2 = -1511809.506634160881644546358+xsq*q2; q2 = -6585339.479723087072826915069+xsq*q2; q2 = -9934124.389934585658967556309+xsq*q2; q2 = -4435757.816794127856828016962+xsq*q2; p3 = 35.26513384663603218592175580; p3 = 1706.375429020768002061283546+xsq*p3; p3 = 18494.26287322386679652009819+xsq*p3; p3 = 66178.83658127083517939992166+xsq*p3; p3 = 85145.16067533570196555001171+xsq*p3; p3 = 33220.91340985722351859704442+xsq*p3; q3 = 1.0; q3 = 863.8367769604990967475517183+xsq*q3; q3 = 37890.22974577220264142952256+xsq*q3; q3 = 400294.4358226697511708610813+xsq*q3; q3 = 1419460.669603720892855755253+xsq*q3; q3 = 1819458.042243997298924553839+xsq*q3; q3 = 708712.8194102874357377502472+xsq*q3; pzero = p2/q2; qzero = 8*p3/q3/x; } } public class ibetaf { /************************************************************************* Incomplete beta integral Returns incomplete beta integral of the arguments, evaluated from zero to x. The function is defined as x - - | (a+b) | | a-1 b-1 ----------- | t (1-t) dt. - - | | | (a) | (b) - 0 The domain of definition is 0 <= x <= 1. In this implementation a and b are restricted to positive values. The integral from x to 1 may be obtained by the symmetry relation 1 - incbet( a, b, x ) = incbet( b, a, 1-x ). The integral is evaluated by a continued fraction expansion or, when b*x is small, by a power series. ACCURACY: Tested at uniformly distributed random points (a,b,x) with a and b in "domain" and x between 0 and 1. Relative error arithmetic domain # trials peak rms IEEE 0,5 10000 6.9e-15 4.5e-16 IEEE 0,85 250000 2.2e-13 1.7e-14 IEEE 0,1000 30000 5.3e-12 6.3e-13 IEEE 0,10000 250000 9.3e-11 7.1e-12 IEEE 0,100000 10000 8.7e-10 4.8e-11 Outputs smaller than the IEEE gradual underflow threshold were excluded from these statistics. Cephes Math Library, Release 2.8: June, 2000 Copyright 1984, 1995, 2000 by Stephen L. Moshier *************************************************************************/ public static double incompletebeta(double a, double b, double x, alglib.xparams _params) { double result = 0; double t = 0; double xc = 0; double w = 0; double y = 0; int flag = 0; double sg = 0; double big = 0; double biginv = 0; double maxgam = 0; double minlog = 0; double maxlog = 0; big = 4.503599627370496e15; biginv = 2.22044604925031308085e-16; maxgam = 171.624376956302725; minlog = Math.Log(math.minrealnumber); maxlog = Math.Log(math.maxrealnumber); alglib.ap.assert((double)(a)>(double)(0) && (double)(b)>(double)(0), "Domain error in IncompleteBeta"); alglib.ap.assert((double)(x)>=(double)(0) && (double)(x)<=(double)(1), "Domain error in IncompleteBeta"); if( (double)(x)==(double)(0) ) { result = 0; return result; } if( (double)(x)==(double)(1) ) { result = 1; return result; } flag = 0; if( (double)(b*x)<=(double)(1.0) && (double)(x)<=(double)(0.95) ) { result = incompletebetaps(a, b, x, maxgam, _params); return result; } w = 1.0-x; if( (double)(x)>(double)(a/(a+b)) ) { flag = 1; t = a; a = b; b = t; xc = x; x = w; } else { xc = w; } if( (flag==1 && (double)(b*x)<=(double)(1.0)) && (double)(x)<=(double)(0.95) ) { t = incompletebetaps(a, b, x, maxgam, _params); if( (double)(t)<=(double)(math.machineepsilon) ) { result = 1.0-math.machineepsilon; } else { result = 1.0-t; } return result; } y = x*(a+b-2.0)-(a-1.0); if( (double)(y)<(double)(0.0) ) { w = incompletebetafe(a, b, x, big, biginv, _params); } else { w = incompletebetafe2(a, b, x, big, biginv, _params)/xc; } y = a*Math.Log(x); t = b*Math.Log(xc); if( ((double)(a+b)<(double)(maxgam) && (double)(Math.Abs(y))<(double)(maxlog)) && (double)(Math.Abs(t))<(double)(maxlog) ) { t = Math.Pow(xc, b); t = t*Math.Pow(x, a); t = t/a; t = t*w; t = t*(gammafunc.gammafunction(a+b, _params)/(gammafunc.gammafunction(a, _params)*gammafunc.gammafunction(b, _params))); if( flag==1 ) { if( (double)(t)<=(double)(math.machineepsilon) ) { result = 1.0-math.machineepsilon; } else { result = 1.0-t; } } else { result = t; } return result; } y = y+t+gammafunc.lngamma(a+b, ref sg, _params)-gammafunc.lngamma(a, ref sg, _params)-gammafunc.lngamma(b, ref sg, _params); y = y+Math.Log(w/a); if( (double)(y)<(double)(minlog) ) { t = 0.0; } else { t = Math.Exp(y); } if( flag==1 ) { if( (double)(t)<=(double)(math.machineepsilon) ) { t = 1.0-math.machineepsilon; } else { t = 1.0-t; } } result = t; return result; } /************************************************************************* Inverse of imcomplete beta integral Given y, the function finds x such that incbet( a, b, x ) = y . The routine performs interval halving or Newton iterations to find the root of incbet(a,b,x) - y = 0. ACCURACY: Relative error: x a,b arithmetic domain domain # trials peak rms IEEE 0,1 .5,10000 50000 5.8e-12 1.3e-13 IEEE 0,1 .25,100 100000 1.8e-13 3.9e-15 IEEE 0,1 0,5 50000 1.1e-12 5.5e-15 With a and b constrained to half-integer or integer values: IEEE 0,1 .5,10000 50000 5.8e-12 1.1e-13 IEEE 0,1 .5,100 100000 1.7e-14 7.9e-16 With a = .5, b constrained to half-integer or integer values: IEEE 0,1 .5,10000 10000 8.3e-11 1.0e-11 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1996, 2000 by Stephen L. Moshier *************************************************************************/ public static double invincompletebeta(double a, double b, double y, alglib.xparams _params) { double result = 0; double aaa = 0; double bbb = 0; double y0 = 0; double d = 0; double yyy = 0; double x = 0; double x0 = 0; double x1 = 0; double lgm = 0; double yp = 0; double di = 0; double dithresh = 0; double yl = 0; double yh = 0; double xt = 0; int i = 0; int rflg = 0; int dir = 0; int nflg = 0; double s = 0; int mainlooppos = 0; int ihalve = 0; int ihalvecycle = 0; int newt = 0; int newtcycle = 0; int breaknewtcycle = 0; int breakihalvecycle = 0; i = 0; alglib.ap.assert((double)(y)>=(double)(0) && (double)(y)<=(double)(1), "Domain error in InvIncompleteBeta"); // // special cases // if( (double)(y)==(double)(0) ) { result = 0; return result; } if( (double)(y)==(double)(1.0) ) { result = 1; return result; } // // these initializations are not really necessary, // but without them compiler complains about 'possibly uninitialized variables'. // dithresh = 0; rflg = 0; aaa = 0; bbb = 0; y0 = 0; x = 0; yyy = 0; lgm = 0; dir = 0; di = 0; // // normal initializations // x0 = 0.0; yl = 0.0; x1 = 1.0; yh = 1.0; nflg = 0; mainlooppos = 0; ihalve = 1; ihalvecycle = 2; newt = 3; newtcycle = 4; breaknewtcycle = 5; breakihalvecycle = 6; // // main loop // while( true ) { // // start // if( mainlooppos==0 ) { if( (double)(a)<=(double)(1.0) || (double)(b)<=(double)(1.0) ) { dithresh = 1.0e-6; rflg = 0; aaa = a; bbb = b; y0 = y; x = aaa/(aaa+bbb); yyy = incompletebeta(aaa, bbb, x, _params); mainlooppos = ihalve; continue; } else { dithresh = 1.0e-4; } yp = -normaldistr.invnormaldistribution(y, _params); if( (double)(y)>(double)(0.5) ) { rflg = 1; aaa = b; bbb = a; y0 = 1.0-y; yp = -yp; } else { rflg = 0; aaa = a; bbb = b; y0 = y; } lgm = (yp*yp-3.0)/6.0; x = 2.0/(1.0/(2.0*aaa-1.0)+1.0/(2.0*bbb-1.0)); d = yp*Math.Sqrt(x+lgm)/x-(1.0/(2.0*bbb-1.0)-1.0/(2.0*aaa-1.0))*(lgm+5.0/6.0-2.0/(3.0*x)); d = 2.0*d; if( (double)(d)<(double)(Math.Log(math.minrealnumber)) ) { x = 0; break; } x = aaa/(aaa+bbb*Math.Exp(d)); yyy = incompletebeta(aaa, bbb, x, _params); yp = (yyy-y0)/y0; if( (double)(Math.Abs(yp))<(double)(0.2) ) { mainlooppos = newt; continue; } mainlooppos = ihalve; continue; } // // ihalve // if( mainlooppos==ihalve ) { dir = 0; di = 0.5; i = 0; mainlooppos = ihalvecycle; continue; } // // ihalvecycle // if( mainlooppos==ihalvecycle ) { if( i<=99 ) { if( i!=0 ) { x = x0+di*(x1-x0); if( (double)(x)==(double)(1.0) ) { x = 1.0-math.machineepsilon; } if( (double)(x)==(double)(0.0) ) { di = 0.5; x = x0+di*(x1-x0); if( (double)(x)==(double)(0.0) ) { break; } } yyy = incompletebeta(aaa, bbb, x, _params); yp = (x1-x0)/(x1+x0); if( (double)(Math.Abs(yp))<(double)(dithresh) ) { mainlooppos = newt; continue; } yp = (yyy-y0)/y0; if( (double)(Math.Abs(yp))<(double)(dithresh) ) { mainlooppos = newt; continue; } } if( (double)(yyy)<(double)(y0) ) { x0 = x; yl = yyy; if( dir<0 ) { dir = 0; di = 0.5; } else { if( dir>3 ) { di = 1.0-(1.0-di)*(1.0-di); } else { if( dir>1 ) { di = 0.5*di+0.5; } else { di = (y0-yyy)/(yh-yl); } } } dir = dir+1; if( (double)(x0)>(double)(0.75) ) { if( rflg==1 ) { rflg = 0; aaa = a; bbb = b; y0 = y; } else { rflg = 1; aaa = b; bbb = a; y0 = 1.0-y; } x = 1.0-x; yyy = incompletebeta(aaa, bbb, x, _params); x0 = 0.0; yl = 0.0; x1 = 1.0; yh = 1.0; mainlooppos = ihalve; continue; } } else { x1 = x; if( rflg==1 && (double)(x1)<(double)(math.machineepsilon) ) { x = 0.0; break; } yh = yyy; if( dir>0 ) { dir = 0; di = 0.5; } else { if( dir<-3 ) { di = di*di; } else { if( dir<-1 ) { di = 0.5*di; } else { di = (yyy-y0)/(yh-yl); } } } dir = dir-1; } i = i+1; mainlooppos = ihalvecycle; continue; } else { mainlooppos = breakihalvecycle; continue; } } // // breakihalvecycle // if( mainlooppos==breakihalvecycle ) { if( (double)(x0)>=(double)(1.0) ) { x = 1.0-math.machineepsilon; break; } if( (double)(x)<=(double)(0.0) ) { x = 0.0; break; } mainlooppos = newt; continue; } // // newt // if( mainlooppos==newt ) { if( nflg!=0 ) { break; } nflg = 1; lgm = gammafunc.lngamma(aaa+bbb, ref s, _params)-gammafunc.lngamma(aaa, ref s, _params)-gammafunc.lngamma(bbb, ref s, _params); i = 0; mainlooppos = newtcycle; continue; } // // newtcycle // if( mainlooppos==newtcycle ) { if( i<=7 ) { if( i!=0 ) { yyy = incompletebeta(aaa, bbb, x, _params); } if( (double)(yyy)<(double)(yl) ) { x = x0; yyy = yl; } else { if( (double)(yyy)>(double)(yh) ) { x = x1; yyy = yh; } else { if( (double)(yyy)<(double)(y0) ) { x0 = x; yl = yyy; } else { x1 = x; yh = yyy; } } } if( (double)(x)==(double)(1.0) || (double)(x)==(double)(0.0) ) { mainlooppos = breaknewtcycle; continue; } d = (aaa-1.0)*Math.Log(x)+(bbb-1.0)*Math.Log(1.0-x)+lgm; if( (double)(d)<(double)(Math.Log(math.minrealnumber)) ) { break; } if( (double)(d)>(double)(Math.Log(math.maxrealnumber)) ) { mainlooppos = breaknewtcycle; continue; } d = Math.Exp(d); d = (yyy-y0)/d; xt = x-d; if( (double)(xt)<=(double)(x0) ) { yyy = (x-x0)/(x1-x0); xt = x0+0.5*yyy*(x-x0); if( (double)(xt)<=(double)(0.0) ) { mainlooppos = breaknewtcycle; continue; } } if( (double)(xt)>=(double)(x1) ) { yyy = (x1-x)/(x1-x0); xt = x1-0.5*yyy*(x1-x); if( (double)(xt)>=(double)(1.0) ) { mainlooppos = breaknewtcycle; continue; } } x = xt; if( (double)(Math.Abs(d/x))<(double)(128.0*math.machineepsilon) ) { break; } i = i+1; mainlooppos = newtcycle; continue; } else { mainlooppos = breaknewtcycle; continue; } } // // breaknewtcycle // if( mainlooppos==breaknewtcycle ) { dithresh = 256.0*math.machineepsilon; mainlooppos = ihalve; continue; } } // // done // if( rflg!=0 ) { if( (double)(x)<=(double)(math.machineepsilon) ) { x = 1.0-math.machineepsilon; } else { x = 1.0-x; } } result = x; return result; } /************************************************************************* Continued fraction expansion #1 for incomplete beta integral Cephes Math Library, Release 2.8: June, 2000 Copyright 1984, 1995, 2000 by Stephen L. Moshier *************************************************************************/ private static double incompletebetafe(double a, double b, double x, double big, double biginv, alglib.xparams _params) { double result = 0; double xk = 0; double pk = 0; double pkm1 = 0; double pkm2 = 0; double qk = 0; double qkm1 = 0; double qkm2 = 0; double k1 = 0; double k2 = 0; double k3 = 0; double k4 = 0; double k5 = 0; double k6 = 0; double k7 = 0; double k8 = 0; double r = 0; double t = 0; double ans = 0; double thresh = 0; int n = 0; k1 = a; k2 = a+b; k3 = a; k4 = a+1.0; k5 = 1.0; k6 = b-1.0; k7 = k4; k8 = a+2.0; pkm2 = 0.0; qkm2 = 1.0; pkm1 = 1.0; qkm1 = 1.0; ans = 1.0; r = 1.0; n = 0; thresh = 3.0*math.machineepsilon; do { xk = -(x*k1*k2/(k3*k4)); pk = pkm1+pkm2*xk; qk = qkm1+qkm2*xk; pkm2 = pkm1; pkm1 = pk; qkm2 = qkm1; qkm1 = qk; xk = x*k5*k6/(k7*k8); pk = pkm1+pkm2*xk; qk = qkm1+qkm2*xk; pkm2 = pkm1; pkm1 = pk; qkm2 = qkm1; qkm1 = qk; if( (double)(qk)!=(double)(0) ) { r = pk/qk; } if( (double)(r)!=(double)(0) ) { t = Math.Abs((ans-r)/r); ans = r; } else { t = 1.0; } if( (double)(t)<(double)(thresh) ) { break; } k1 = k1+1.0; k2 = k2+1.0; k3 = k3+2.0; k4 = k4+2.0; k5 = k5+1.0; k6 = k6-1.0; k7 = k7+2.0; k8 = k8+2.0; if( (double)(Math.Abs(qk)+Math.Abs(pk))>(double)(big) ) { pkm2 = pkm2*biginv; pkm1 = pkm1*biginv; qkm2 = qkm2*biginv; qkm1 = qkm1*biginv; } if( (double)(Math.Abs(qk))<(double)(biginv) || (double)(Math.Abs(pk))<(double)(biginv) ) { pkm2 = pkm2*big; pkm1 = pkm1*big; qkm2 = qkm2*big; qkm1 = qkm1*big; } n = n+1; } while( n!=300 ); result = ans; return result; } /************************************************************************* Continued fraction expansion #2 for incomplete beta integral Cephes Math Library, Release 2.8: June, 2000 Copyright 1984, 1995, 2000 by Stephen L. Moshier *************************************************************************/ private static double incompletebetafe2(double a, double b, double x, double big, double biginv, alglib.xparams _params) { double result = 0; double xk = 0; double pk = 0; double pkm1 = 0; double pkm2 = 0; double qk = 0; double qkm1 = 0; double qkm2 = 0; double k1 = 0; double k2 = 0; double k3 = 0; double k4 = 0; double k5 = 0; double k6 = 0; double k7 = 0; double k8 = 0; double r = 0; double t = 0; double ans = 0; double z = 0; double thresh = 0; int n = 0; k1 = a; k2 = b-1.0; k3 = a; k4 = a+1.0; k5 = 1.0; k6 = a+b; k7 = a+1.0; k8 = a+2.0; pkm2 = 0.0; qkm2 = 1.0; pkm1 = 1.0; qkm1 = 1.0; z = x/(1.0-x); ans = 1.0; r = 1.0; n = 0; thresh = 3.0*math.machineepsilon; do { xk = -(z*k1*k2/(k3*k4)); pk = pkm1+pkm2*xk; qk = qkm1+qkm2*xk; pkm2 = pkm1; pkm1 = pk; qkm2 = qkm1; qkm1 = qk; xk = z*k5*k6/(k7*k8); pk = pkm1+pkm2*xk; qk = qkm1+qkm2*xk; pkm2 = pkm1; pkm1 = pk; qkm2 = qkm1; qkm1 = qk; if( (double)(qk)!=(double)(0) ) { r = pk/qk; } if( (double)(r)!=(double)(0) ) { t = Math.Abs((ans-r)/r); ans = r; } else { t = 1.0; } if( (double)(t)<(double)(thresh) ) { break; } k1 = k1+1.0; k2 = k2-1.0; k3 = k3+2.0; k4 = k4+2.0; k5 = k5+1.0; k6 = k6+1.0; k7 = k7+2.0; k8 = k8+2.0; if( (double)(Math.Abs(qk)+Math.Abs(pk))>(double)(big) ) { pkm2 = pkm2*biginv; pkm1 = pkm1*biginv; qkm2 = qkm2*biginv; qkm1 = qkm1*biginv; } if( (double)(Math.Abs(qk))<(double)(biginv) || (double)(Math.Abs(pk))<(double)(biginv) ) { pkm2 = pkm2*big; pkm1 = pkm1*big; qkm2 = qkm2*big; qkm1 = qkm1*big; } n = n+1; } while( n!=300 ); result = ans; return result; } /************************************************************************* Power series for incomplete beta integral. Use when b*x is small and x not too close to 1. Cephes Math Library, Release 2.8: June, 2000 Copyright 1984, 1995, 2000 by Stephen L. Moshier *************************************************************************/ private static double incompletebetaps(double a, double b, double x, double maxgam, alglib.xparams _params) { double result = 0; double s = 0; double t = 0; double u = 0; double v = 0; double n = 0; double t1 = 0; double z = 0; double ai = 0; double sg = 0; ai = 1.0/a; u = (1.0-b)*x; v = u/(a+1.0); t1 = v; t = u; n = 2.0; s = 0.0; z = math.machineepsilon*ai; while( (double)(Math.Abs(v))>(double)(z) ) { u = (n-b)*x/n; t = t*u; v = t/(a+n); s = s+v; n = n+1.0; } s = s+t1; s = s+ai; u = a*Math.Log(x); if( (double)(a+b)<(double)(maxgam) && (double)(Math.Abs(u))<(double)(Math.Log(math.maxrealnumber)) ) { t = gammafunc.gammafunction(a+b, _params)/(gammafunc.gammafunction(a, _params)*gammafunc.gammafunction(b, _params)); s = s*t*Math.Pow(x, a); } else { t = gammafunc.lngamma(a+b, ref sg, _params)-gammafunc.lngamma(a, ref sg, _params)-gammafunc.lngamma(b, ref sg, _params)+u+Math.Log(s); if( (double)(t)<(double)(Math.Log(math.minrealnumber)) ) { s = 0.0; } else { s = Math.Exp(t); } } result = s; return result; } } public class fdistr { /************************************************************************* F distribution Returns the area from zero to x under the F density function (also known as Snedcor's density or the variance ratio density). This is the density of x = (u1/df1)/(u2/df2), where u1 and u2 are random variables having Chi square distributions with df1 and df2 degrees of freedom, respectively. The incomplete beta integral is used, according to the formula P(x) = incbet( df1/2, df2/2, (df1*x/(df2 + df1*x) ). The arguments a and b are greater than zero, and x is nonnegative. ACCURACY: Tested at random points (a,b,x). x a,b Relative error: arithmetic domain domain # trials peak rms IEEE 0,1 0,100 100000 9.8e-15 1.7e-15 IEEE 1,5 0,100 100000 6.5e-15 3.5e-16 IEEE 0,1 1,10000 100000 2.2e-11 3.3e-12 IEEE 1,5 1,10000 100000 1.1e-11 1.7e-13 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier *************************************************************************/ public static double fdistribution(int a, int b, double x, alglib.xparams _params) { double result = 0; double w = 0; alglib.ap.assert((a>=1 && b>=1) && (double)(x)>=(double)(0), "Domain error in FDistribution"); w = a*x; w = w/(b+w); result = ibetaf.incompletebeta(0.5*a, 0.5*b, w, _params); return result; } /************************************************************************* Complemented F distribution Returns the area from x to infinity under the F density function (also known as Snedcor's density or the variance ratio density). inf. - 1 | | a-1 b-1 1-P(x) = ------ | t (1-t) dt B(a,b) | | - x The incomplete beta integral is used, according to the formula P(x) = incbet( df2/2, df1/2, (df2/(df2 + df1*x) ). ACCURACY: Tested at random points (a,b,x) in the indicated intervals. x a,b Relative error: arithmetic domain domain # trials peak rms IEEE 0,1 1,100 100000 3.7e-14 5.9e-16 IEEE 1,5 1,100 100000 8.0e-15 1.6e-15 IEEE 0,1 1,10000 100000 1.8e-11 3.5e-13 IEEE 1,5 1,10000 100000 2.0e-11 3.0e-12 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier *************************************************************************/ public static double fcdistribution(int a, int b, double x, alglib.xparams _params) { double result = 0; double w = 0; alglib.ap.assert((a>=1 && b>=1) && (double)(x)>=(double)(0), "Domain error in FCDistribution"); w = b/(b+a*x); result = ibetaf.incompletebeta(0.5*b, 0.5*a, w, _params); return result; } /************************************************************************* Inverse of complemented F distribution Finds the F density argument x such that the integral from x to infinity of the F density is equal to the given probability p. This is accomplished using the inverse beta integral function and the relations z = incbi( df2/2, df1/2, p ) x = df2 (1-z) / (df1 z). Note: the following relations hold for the inverse of the uncomplemented F distribution: z = incbi( df1/2, df2/2, p ) x = df2 z / (df1 (1-z)). ACCURACY: Tested at random points (a,b,p). a,b Relative error: arithmetic domain # trials peak rms For p between .001 and 1: IEEE 1,100 100000 8.3e-15 4.7e-16 IEEE 1,10000 100000 2.1e-11 1.4e-13 For p between 10^-6 and 10^-3: IEEE 1,100 50000 1.3e-12 8.4e-15 IEEE 1,10000 50000 3.0e-12 4.8e-14 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier *************************************************************************/ public static double invfdistribution(int a, int b, double y, alglib.xparams _params) { double result = 0; double w = 0; alglib.ap.assert(((a>=1 && b>=1) && (double)(y)>(double)(0)) && (double)(y)<=(double)(1), "Domain error in InvFDistribution"); // // Compute probability for x = 0.5 // w = ibetaf.incompletebeta(0.5*b, 0.5*a, 0.5, _params); // // If that is greater than y, then the solution w < .5 // Otherwise, solve at 1-y to remove cancellation in (b - b*w) // if( (double)(w)>(double)(y) || (double)(y)<(double)(0.001) ) { w = ibetaf.invincompletebeta(0.5*b, 0.5*a, y, _params); result = (b-b*w)/(a*w); } else { w = ibetaf.invincompletebeta(0.5*a, 0.5*b, 1.0-y, _params); result = b*w/(a*(1.0-w)); } return result; } } public class fresnel { /************************************************************************* Fresnel integral Evaluates the Fresnel integrals x - | | C(x) = | cos(pi/2 t**2) dt, | | - 0 x - | | S(x) = | sin(pi/2 t**2) dt. | | - 0 The integrals are evaluated by a power series for x < 1. For x >= 1 auxiliary functions f(x) and g(x) are employed such that C(x) = 0.5 + f(x) sin( pi/2 x**2 ) - g(x) cos( pi/2 x**2 ) S(x) = 0.5 - f(x) cos( pi/2 x**2 ) - g(x) sin( pi/2 x**2 ) ACCURACY: Relative error. Arithmetic function domain # trials peak rms IEEE S(x) 0, 10 10000 2.0e-15 3.2e-16 IEEE C(x) 0, 10 10000 1.8e-15 3.3e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier *************************************************************************/ public static void fresnelintegral(double x, ref double c, ref double s, alglib.xparams _params) { double xxa = 0; double f = 0; double g = 0; double cc = 0; double ss = 0; double t = 0; double u = 0; double x2 = 0; double sn = 0; double sd = 0; double cn = 0; double cd = 0; double fn = 0; double fd = 0; double gn = 0; double gd = 0; double mpi = 0; double mpio2 = 0; mpi = 3.14159265358979323846; mpio2 = 1.57079632679489661923; xxa = x; x = Math.Abs(xxa); x2 = x*x; if( (double)(x2)<(double)(2.5625) ) { t = x2*x2; sn = -2.99181919401019853726E3; sn = sn*t+7.08840045257738576863E5; sn = sn*t-6.29741486205862506537E7; sn = sn*t+2.54890880573376359104E9; sn = sn*t-4.42979518059697779103E10; sn = sn*t+3.18016297876567817986E11; sd = 1.00000000000000000000E0; sd = sd*t+2.81376268889994315696E2; sd = sd*t+4.55847810806532581675E4; sd = sd*t+5.17343888770096400730E6; sd = sd*t+4.19320245898111231129E8; sd = sd*t+2.24411795645340920940E10; sd = sd*t+6.07366389490084639049E11; cn = -4.98843114573573548651E-8; cn = cn*t+9.50428062829859605134E-6; cn = cn*t-6.45191435683965050962E-4; cn = cn*t+1.88843319396703850064E-2; cn = cn*t-2.05525900955013891793E-1; cn = cn*t+9.99999999999999998822E-1; cd = 3.99982968972495980367E-12; cd = cd*t+9.15439215774657478799E-10; cd = cd*t+1.25001862479598821474E-7; cd = cd*t+1.22262789024179030997E-5; cd = cd*t+8.68029542941784300606E-4; cd = cd*t+4.12142090722199792936E-2; cd = cd*t+1.00000000000000000118E0; s = Math.Sign(xxa)*x*x2*sn/sd; c = Math.Sign(xxa)*x*cn/cd; return; } if( (double)(x)>(double)(36974.0) ) { c = Math.Sign(xxa)*0.5; s = Math.Sign(xxa)*0.5; return; } x2 = x*x; t = mpi*x2; u = 1/(t*t); t = 1/t; fn = 4.21543555043677546506E-1; fn = fn*u+1.43407919780758885261E-1; fn = fn*u+1.15220955073585758835E-2; fn = fn*u+3.45017939782574027900E-4; fn = fn*u+4.63613749287867322088E-6; fn = fn*u+3.05568983790257605827E-8; fn = fn*u+1.02304514164907233465E-10; fn = fn*u+1.72010743268161828879E-13; fn = fn*u+1.34283276233062758925E-16; fn = fn*u+3.76329711269987889006E-20; fd = 1.00000000000000000000E0; fd = fd*u+7.51586398353378947175E-1; fd = fd*u+1.16888925859191382142E-1; fd = fd*u+6.44051526508858611005E-3; fd = fd*u+1.55934409164153020873E-4; fd = fd*u+1.84627567348930545870E-6; fd = fd*u+1.12699224763999035261E-8; fd = fd*u+3.60140029589371370404E-11; fd = fd*u+5.88754533621578410010E-14; fd = fd*u+4.52001434074129701496E-17; fd = fd*u+1.25443237090011264384E-20; gn = 5.04442073643383265887E-1; gn = gn*u+1.97102833525523411709E-1; gn = gn*u+1.87648584092575249293E-2; gn = gn*u+6.84079380915393090172E-4; gn = gn*u+1.15138826111884280931E-5; gn = gn*u+9.82852443688422223854E-8; gn = gn*u+4.45344415861750144738E-10; gn = gn*u+1.08268041139020870318E-12; gn = gn*u+1.37555460633261799868E-15; gn = gn*u+8.36354435630677421531E-19; gn = gn*u+1.86958710162783235106E-22; gd = 1.00000000000000000000E0; gd = gd*u+1.47495759925128324529E0; gd = gd*u+3.37748989120019970451E-1; gd = gd*u+2.53603741420338795122E-2; gd = gd*u+8.14679107184306179049E-4; gd = gd*u+1.27545075667729118702E-5; gd = gd*u+1.04314589657571990585E-7; gd = gd*u+4.60680728146520428211E-10; gd = gd*u+1.10273215066240270757E-12; gd = gd*u+1.38796531259578871258E-15; gd = gd*u+8.39158816283118707363E-19; gd = gd*u+1.86958710162783236342E-22; f = 1-u*fn/fd; g = t*gn/gd; t = mpio2*x2; cc = Math.Cos(t); ss = Math.Sin(t); t = mpi*x; c = 0.5+(f*ss-g*cc)/t; s = 0.5-(f*cc+g*ss)/t; c = c*Math.Sign(xxa); s = s*Math.Sign(xxa); } } public class jacobianelliptic { /************************************************************************* Jacobian Elliptic Functions Evaluates the Jacobian elliptic functions sn(u|m), cn(u|m), and dn(u|m) of parameter m between 0 and 1, and real argument u. These functions are periodic, with quarter-period on the real axis equal to the complete elliptic integral ellpk(1.0-m). Relation to incomplete elliptic integral: If u = ellik(phi,m), then sn(u|m) = sin(phi), and cn(u|m) = cos(phi). Phi is called the amplitude of u. Computation is by means of the arithmetic-geometric mean algorithm, except when m is within 1e-9 of 0 or 1. In the latter case with m close to 1, the approximation applies only for phi < pi/2. ACCURACY: Tested at random points with u between 0 and 10, m between 0 and 1. Absolute error (* = relative error): arithmetic function # trials peak rms IEEE phi 10000 9.2e-16* 1.4e-16* IEEE sn 50000 4.1e-15 4.6e-16 IEEE cn 40000 3.6e-15 4.4e-16 IEEE dn 10000 1.3e-12 1.8e-14 Peak error observed in consistency check using addition theorem for sn(u+v) was 4e-16 (absolute). Also tested by the above relation to the incomplete elliptic integral. Accuracy deteriorates when u is large. Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 2000 by Stephen L. Moshier *************************************************************************/ public static void jacobianellipticfunctions(double u, double m, ref double sn, ref double cn, ref double dn, ref double ph, alglib.xparams _params) { double ai = 0; double b = 0; double phi = 0; double t = 0; double twon = 0; double[] a = new double[0]; double[] c = new double[0]; int i = 0; sn = 0; cn = 0; dn = 0; ph = 0; alglib.ap.assert((double)(m)>=(double)(0) && (double)(m)<=(double)(1), "Domain error in JacobianEllipticFunctions: m<0 or m>1"); a = new double[8+1]; c = new double[8+1]; if( (double)(m)<(double)(1.0e-9) ) { t = Math.Sin(u); b = Math.Cos(u); ai = 0.25*m*(u-t*b); sn = t-ai*b; cn = b+ai*t; ph = u-ai; dn = 1.0-0.5*m*t*t; return; } if( (double)(m)>=(double)(0.9999999999) ) { ai = 0.25*(1.0-m); b = Math.Cosh(u); t = Math.Tanh(u); phi = 1.0/b; twon = b*Math.Sinh(u); sn = t+ai*(twon-u)/(b*b); ph = 2.0*Math.Atan(Math.Exp(u))-1.57079632679489661923+ai*(twon-u)/b; ai = ai*t*phi; cn = phi-ai*(twon-u); dn = phi+ai*(twon+u); return; } a[0] = 1.0; b = Math.Sqrt(1.0-m); c[0] = Math.Sqrt(m); twon = 1.0; i = 0; while( (double)(Math.Abs(c[i]/a[i]))>(double)(math.machineepsilon) ) { if( i>7 ) { alglib.ap.assert(false, "Overflow in JacobianEllipticFunctions"); break; } ai = a[i]; i = i+1; c[i] = 0.5*(ai-b); t = Math.Sqrt(ai*b); a[i] = 0.5*(ai+b); b = t; twon = twon*2.0; } phi = twon*a[i]*u; do { t = c[i]*Math.Sin(phi)/a[i]; b = phi; phi = (Math.Asin(t)+phi)/2.0; i = i-1; } while( i!=0 ); sn = Math.Sin(phi); t = Math.Cos(phi); cn = t; dn = t/Math.Cos(phi-b); ph = phi; } } public class psif { /************************************************************************* Psi (digamma) function d - psi(x) = -- ln | (x) dx is the logarithmic derivative of the gamma function. For integer x, n-1 - psi(n) = -EUL + > 1/k. - k=1 This formula is used for 0 < n <= 10. If x is negative, it is transformed to a positive argument by the reflection formula psi(1-x) = psi(x) + pi cot(pi x). For general positive x, the argument is made greater than 10 using the recurrence psi(x+1) = psi(x) + 1/x. Then the following asymptotic expansion is applied: inf. B - 2k psi(x) = log(x) - 1/2x - > ------- - 2k k=1 2k x where the B2k are Bernoulli numbers. ACCURACY: Relative error (except absolute when |psi| < 1): arithmetic domain # trials peak rms IEEE 0,30 30000 1.3e-15 1.4e-16 IEEE -30,0 40000 1.5e-15 2.2e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1992, 2000 by Stephen L. Moshier *************************************************************************/ public static double psi(double x, alglib.xparams _params) { double result = 0; double p = 0; double q = 0; double nz = 0; double s = 0; double w = 0; double y = 0; double z = 0; double polv = 0; int i = 0; int n = 0; int negative = 0; negative = 0; nz = 0.0; if( (double)(x)<=(double)(0) ) { negative = 1; q = x; p = (int)Math.Floor(q); if( (double)(p)==(double)(q) ) { alglib.ap.assert(false, "Singularity in Psi(x)"); result = math.maxrealnumber; return result; } nz = q-p; if( (double)(nz)!=(double)(0.5) ) { if( (double)(nz)>(double)(0.5) ) { p = p+1.0; nz = q-p; } nz = Math.PI/Math.Tan(Math.PI*nz); } else { nz = 0.0; } x = 1.0-x; } if( (double)(x)<=(double)(10.0) && (double)(x)==(double)((int)Math.Floor(x)) ) { y = 0.0; n = (int)Math.Floor(x); for(i=1; i<=n-1; i++) { w = i; y = y+1.0/w; } y = y-0.57721566490153286061; } else { s = x; w = 0.0; while( (double)(s)<(double)(10.0) ) { w = w+1.0/s; s = s+1.0; } if( (double)(s)<(double)(1.0E17) ) { z = 1.0/(s*s); polv = 8.33333333333333333333E-2; polv = polv*z-2.10927960927960927961E-2; polv = polv*z+7.57575757575757575758E-3; polv = polv*z-4.16666666666666666667E-3; polv = polv*z+3.96825396825396825397E-3; polv = polv*z-8.33333333333333333333E-3; polv = polv*z+8.33333333333333333333E-2; y = z*polv; } else { y = 0.0; } y = Math.Log(s)-0.5/s-y-w; } if( negative!=0 ) { y = y-nz; } result = y; return result; } } public class expintegrals { /************************************************************************* Exponential integral Ei(x) x - t | | e Ei(x) = -|- --- dt . | | t - -inf Not defined for x <= 0. See also expn.c. ACCURACY: Relative error: arithmetic domain # trials peak rms IEEE 0,100 50000 8.6e-16 1.3e-16 Cephes Math Library Release 2.8: May, 1999 Copyright 1999 by Stephen L. Moshier *************************************************************************/ public static double exponentialintegralei(double x, alglib.xparams _params) { double result = 0; double eul = 0; double f = 0; double f1 = 0; double f2 = 0; double w = 0; eul = 0.5772156649015328606065; if( (double)(x)<=(double)(0) ) { result = 0; return result; } if( (double)(x)<(double)(2) ) { f1 = -5.350447357812542947283; f1 = f1*x+218.5049168816613393830; f1 = f1*x-4176.572384826693777058; f1 = f1*x+55411.76756393557601232; f1 = f1*x-331338.1331178144034309; f1 = f1*x+1592627.163384945414220; f2 = 1.000000000000000000000; f2 = f2*x-52.50547959112862969197; f2 = f2*x+1259.616186786790571525; f2 = f2*x-17565.49581973534652631; f2 = f2*x+149306.2117002725991967; f2 = f2*x-729494.9239640527645655; f2 = f2*x+1592627.163384945429726; f = f1/f2; result = eul+Math.Log(x)+x*f; return result; } if( (double)(x)<(double)(4) ) { w = 1/x; f1 = 1.981808503259689673238E-2; f1 = f1*w-1.271645625984917501326; f1 = f1*w-2.088160335681228318920; f1 = f1*w+2.755544509187936721172; f1 = f1*w-4.409507048701600257171E-1; f1 = f1*w+4.665623805935891391017E-2; f1 = f1*w-1.545042679673485262580E-3; f1 = f1*w+7.059980605299617478514E-5; f2 = 1.000000000000000000000; f2 = f2*w+1.476498670914921440652; f2 = f2*w+5.629177174822436244827E-1; f2 = f2*w+1.699017897879307263248E-1; f2 = f2*w+2.291647179034212017463E-2; f2 = f2*w+4.450150439728752875043E-3; f2 = f2*w+1.727439612206521482874E-4; f2 = f2*w+3.953167195549672482304E-5; f = f1/f2; result = Math.Exp(x)*w*(1+w*f); return result; } if( (double)(x)<(double)(8) ) { w = 1/x; f1 = -1.373215375871208729803; f1 = f1*w-7.084559133740838761406E-1; f1 = f1*w+1.580806855547941010501; f1 = f1*w-2.601500427425622944234E-1; f1 = f1*w+2.994674694113713763365E-2; f1 = f1*w-1.038086040188744005513E-3; f1 = f1*w+4.371064420753005429514E-5; f1 = f1*w+2.141783679522602903795E-6; f2 = 1.000000000000000000000; f2 = f2*w+8.585231423622028380768E-1; f2 = f2*w+4.483285822873995129957E-1; f2 = f2*w+7.687932158124475434091E-2; f2 = f2*w+2.449868241021887685904E-2; f2 = f2*w+8.832165941927796567926E-4; f2 = f2*w+4.590952299511353531215E-4; f2 = f2*w+-4.729848351866523044863E-6; f2 = f2*w+2.665195537390710170105E-6; f = f1/f2; result = Math.Exp(x)*w*(1+w*f); return result; } if( (double)(x)<(double)(16) ) { w = 1/x; f1 = -2.106934601691916512584; f1 = f1*w+1.732733869664688041885; f1 = f1*w-2.423619178935841904839E-1; f1 = f1*w+2.322724180937565842585E-2; f1 = f1*w+2.372880440493179832059E-4; f1 = f1*w-8.343219561192552752335E-5; f1 = f1*w+1.363408795605250394881E-5; f1 = f1*w-3.655412321999253963714E-7; f1 = f1*w+1.464941733975961318456E-8; f1 = f1*w+6.176407863710360207074E-10; f2 = 1.000000000000000000000; f2 = f2*w-2.298062239901678075778E-1; f2 = f2*w+1.105077041474037862347E-1; f2 = f2*w-1.566542966630792353556E-2; f2 = f2*w+2.761106850817352773874E-3; f2 = f2*w-2.089148012284048449115E-4; f2 = f2*w+1.708528938807675304186E-5; f2 = f2*w-4.459311796356686423199E-7; f2 = f2*w+1.394634930353847498145E-8; f2 = f2*w+6.150865933977338354138E-10; f = f1/f2; result = Math.Exp(x)*w*(1+w*f); return result; } if( (double)(x)<(double)(32) ) { w = 1/x; f1 = -2.458119367674020323359E-1; f1 = f1*w-1.483382253322077687183E-1; f1 = f1*w+7.248291795735551591813E-2; f1 = f1*w-1.348315687380940523823E-2; f1 = f1*w+1.342775069788636972294E-3; f1 = f1*w-7.942465637159712264564E-5; f1 = f1*w+2.644179518984235952241E-6; f1 = f1*w-4.239473659313765177195E-8; f2 = 1.000000000000000000000; f2 = f2*w-1.044225908443871106315E-1; f2 = f2*w-2.676453128101402655055E-1; f2 = f2*w+9.695000254621984627876E-2; f2 = f2*w-1.601745692712991078208E-2; f2 = f2*w+1.496414899205908021882E-3; f2 = f2*w-8.462452563778485013756E-5; f2 = f2*w+2.728938403476726394024E-6; f2 = f2*w-4.239462431819542051337E-8; f = f1/f2; result = Math.Exp(x)*w*(1+w*f); return result; } if( (double)(x)<(double)(64) ) { w = 1/x; f1 = 1.212561118105456670844E-1; f1 = f1*w-5.823133179043894485122E-1; f1 = f1*w+2.348887314557016779211E-1; f1 = f1*w-3.040034318113248237280E-2; f1 = f1*w+1.510082146865190661777E-3; f1 = f1*w-2.523137095499571377122E-5; f2 = 1.000000000000000000000; f2 = f2*w-1.002252150365854016662; f2 = f2*w+2.928709694872224144953E-1; f2 = f2*w-3.337004338674007801307E-2; f2 = f2*w+1.560544881127388842819E-3; f2 = f2*w-2.523137093603234562648E-5; f = f1/f2; result = Math.Exp(x)*w*(1+w*f); return result; } w = 1/x; f1 = -7.657847078286127362028E-1; f1 = f1*w+6.886192415566705051750E-1; f1 = f1*w-2.132598113545206124553E-1; f1 = f1*w+3.346107552384193813594E-2; f1 = f1*w-3.076541477344756050249E-3; f1 = f1*w+1.747119316454907477380E-4; f1 = f1*w-6.103711682274170530369E-6; f1 = f1*w+1.218032765428652199087E-7; f1 = f1*w-1.086076102793290233007E-9; f2 = 1.000000000000000000000; f2 = f2*w-1.888802868662308731041; f2 = f2*w+1.066691687211408896850; f2 = f2*w-2.751915982306380647738E-1; f2 = f2*w+3.930852688233823569726E-2; f2 = f2*w-3.414684558602365085394E-3; f2 = f2*w+1.866844370703555398195E-4; f2 = f2*w-6.345146083130515357861E-6; f2 = f2*w+1.239754287483206878024E-7; f2 = f2*w-1.086076102793126632978E-9; f = f1/f2; result = Math.Exp(x)*w*(1+w*f); return result; } /************************************************************************* Exponential integral En(x) Evaluates the exponential integral inf. - | | -xt | e E (x) = | ---- dt. n | n | | t - 1 Both n and x must be nonnegative. The routine employs either a power series, a continued fraction, or an asymptotic formula depending on the relative values of n and x. ACCURACY: Relative error: arithmetic domain # trials peak rms IEEE 0, 30 10000 1.7e-15 3.6e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1985, 2000 by Stephen L. Moshier *************************************************************************/ public static double exponentialintegralen(double x, int n, alglib.xparams _params) { double result = 0; double r = 0; double t = 0; double yk = 0; double xk = 0; double pk = 0; double pkm1 = 0; double pkm2 = 0; double qk = 0; double qkm1 = 0; double qkm2 = 0; double psi = 0; double z = 0; int i = 0; int k = 0; double big = 0; double eul = 0; eul = 0.57721566490153286060; big = 1.44115188075855872*Math.Pow(10, 17); if( ((n<0 || (double)(x)<(double)(0)) || (double)(x)>(double)(170)) || ((double)(x)==(double)(0) && n<2) ) { result = -1; return result; } if( (double)(x)==(double)(0) ) { result = (double)1/(double)(n-1); return result; } if( n==0 ) { result = Math.Exp(-x)/x; return result; } if( n>5000 ) { xk = x+n; yk = 1/(xk*xk); t = n; result = yk*t*(6*x*x-8*t*x+t*t); result = yk*(result+t*(t-2.0*x)); result = yk*(result+t); result = (result+1)*Math.Exp(-x)/xk; return result; } if( (double)(x)<=(double)(1) ) { psi = -eul-Math.Log(x); for(i=1; i<=n-1; i++) { psi = psi+(double)1/(double)i; } z = -x; xk = 0; yk = 1; pk = 1-n; if( n==1 ) { result = 0.0; } else { result = 1.0/pk; } do { xk = xk+1; yk = yk*z/xk; pk = pk+1; if( (double)(pk)!=(double)(0) ) { result = result+yk/pk; } if( (double)(result)!=(double)(0) ) { t = Math.Abs(yk/result); } else { t = 1; } } while( (double)(t)>=(double)(math.machineepsilon) ); t = 1; for(i=1; i<=n-1; i++) { t = t*z/i; } result = psi*t-result; return result; } else { k = 1; pkm2 = 1; qkm2 = x; pkm1 = 1.0; qkm1 = x+n; result = pkm1/qkm1; do { k = k+1; if( k%2==1 ) { yk = 1; xk = n+(double)(k-1)/(double)2; } else { yk = x; xk = (double)k/(double)2; } pk = pkm1*yk+pkm2*xk; qk = qkm1*yk+qkm2*xk; if( (double)(qk)!=(double)(0) ) { r = pk/qk; t = Math.Abs((result-r)/r); result = r; } else { t = 1; } pkm2 = pkm1; pkm1 = pk; qkm2 = qkm1; qkm1 = qk; if( (double)(Math.Abs(pk))>(double)(big) ) { pkm2 = pkm2/big; pkm1 = pkm1/big; qkm2 = qkm2/big; qkm1 = qkm1/big; } } while( (double)(t)>=(double)(math.machineepsilon) ); result = result*Math.Exp(-x); } return result; } } public class laguerre { /************************************************************************* Calculation of the value of the Laguerre polynomial. Parameters: n - degree, n>=0 x - argument Result: the value of the Laguerre polynomial Ln at x *************************************************************************/ public static double laguerrecalculate(int n, double x, alglib.xparams _params) { double result = 0; double a = 0; double b = 0; double i = 0; result = 1; a = 1; b = 1-x; if( n==1 ) { result = b; } i = 2; while( (double)(i)<=(double)(n) ) { result = ((2*i-1-x)*b-(i-1)*a)/i; a = b; b = result; i = i+1; } return result; } /************************************************************************* Summation of Laguerre polynomials using Clenshaw's recurrence formula. This routine calculates c[0]*L0(x) + c[1]*L1(x) + ... + c[N]*LN(x) Parameters: n - degree, n>=0 x - argument Result: the value of the Laguerre polynomial at x *************************************************************************/ public static double laguerresum(double[] c, int n, double x, alglib.xparams _params) { double result = 0; double b1 = 0; double b2 = 0; int i = 0; b1 = 0; b2 = 0; result = 0; for(i=n; i>=0; i--) { result = (2*i+1-x)*b1/(i+1)-(i+1)*b2/(i+2)+c[i]; b2 = b1; b1 = result; } return result; } /************************************************************************* Representation of Ln as C[0] + C[1]*X + ... + C[N]*X^N Input parameters: N - polynomial degree, n>=0 Output parameters: C - coefficients *************************************************************************/ public static void laguerrecoefficients(int n, ref double[] c, alglib.xparams _params) { int i = 0; c = new double[0]; c = new double[n+1]; c[0] = 1; for(i=0; i<=n-1; i++) { c[i+1] = -(c[i]*(n-i)/(i+1)/(i+1)); } } } public class chisquaredistr { /************************************************************************* Chi-square distribution Returns the area under the left hand tail (from 0 to x) of the Chi square probability density function with v degrees of freedom. x - 1 | | v/2-1 -t/2 P( x | v ) = ----------- | t e dt v/2 - | | 2 | (v/2) - 0 where x is the Chi-square variable. The incomplete gamma integral is used, according to the formula y = chdtr( v, x ) = igam( v/2.0, x/2.0 ). The arguments must both be positive. ACCURACY: See incomplete gamma function Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 2000 by Stephen L. Moshier *************************************************************************/ public static double chisquaredistribution(double v, double x, alglib.xparams _params) { double result = 0; alglib.ap.assert((double)(x)>=(double)(0) && (double)(v)>=(double)(1), "Domain error in ChiSquareDistribution"); result = igammaf.incompletegamma(v/2.0, x/2.0, _params); return result; } /************************************************************************* Complemented Chi-square distribution Returns the area under the right hand tail (from x to infinity) of the Chi square probability density function with v degrees of freedom: inf. - 1 | | v/2-1 -t/2 P( x | v ) = ----------- | t e dt v/2 - | | 2 | (v/2) - x where x is the Chi-square variable. The incomplete gamma integral is used, according to the formula y = chdtr( v, x ) = igamc( v/2.0, x/2.0 ). The arguments must both be positive. ACCURACY: See incomplete gamma function Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 2000 by Stephen L. Moshier *************************************************************************/ public static double chisquarecdistribution(double v, double x, alglib.xparams _params) { double result = 0; alglib.ap.assert((double)(x)>=(double)(0) && (double)(v)>=(double)(1), "Domain error in ChiSquareDistributionC"); result = igammaf.incompletegammac(v/2.0, x/2.0, _params); return result; } /************************************************************************* Inverse of complemented Chi-square distribution Finds the Chi-square argument x such that the integral from x to infinity of the Chi-square density is equal to the given cumulative probability y. This is accomplished using the inverse gamma integral function and the relation x/2 = igami( df/2, y ); ACCURACY: See inverse incomplete gamma function Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 2000 by Stephen L. Moshier *************************************************************************/ public static double invchisquaredistribution(double v, double y, alglib.xparams _params) { double result = 0; alglib.ap.assert(((double)(y)>=(double)(0) && (double)(y)<=(double)(1)) && (double)(v)>=(double)(1), "Domain error in InvChiSquareDistribution"); result = 2*igammaf.invincompletegammac(0.5*v, y, _params); return result; } } public class legendre { /************************************************************************* Calculation of the value of the Legendre polynomial Pn. Parameters: n - degree, n>=0 x - argument Result: the value of the Legendre polynomial Pn at x *************************************************************************/ public static double legendrecalculate(int n, double x, alglib.xparams _params) { double result = 0; double a = 0; double b = 0; int i = 0; result = 1; a = 1; b = x; if( n==0 ) { result = a; return result; } if( n==1 ) { result = b; return result; } for(i=2; i<=n; i++) { result = ((2*i-1)*x*b-(i-1)*a)/i; a = b; b = result; } return result; } /************************************************************************* Summation of Legendre polynomials using Clenshaw's recurrence formula. This routine calculates c[0]*P0(x) + c[1]*P1(x) + ... + c[N]*PN(x) Parameters: n - degree, n>=0 x - argument Result: the value of the Legendre polynomial at x *************************************************************************/ public static double legendresum(double[] c, int n, double x, alglib.xparams _params) { double result = 0; double b1 = 0; double b2 = 0; int i = 0; b1 = 0; b2 = 0; result = 0; for(i=n; i>=0; i--) { result = (2*i+1)*x*b1/(i+1)-(i+1)*b2/(i+2)+c[i]; b2 = b1; b1 = result; } return result; } /************************************************************************* Representation of Pn as C[0] + C[1]*X + ... + C[N]*X^N Input parameters: N - polynomial degree, n>=0 Output parameters: C - coefficients *************************************************************************/ public static void legendrecoefficients(int n, ref double[] c, alglib.xparams _params) { int i = 0; c = new double[0]; c = new double[n+1]; for(i=0; i<=n; i++) { c[i] = 0; } c[n] = 1; for(i=1; i<=n; i++) { c[n] = c[n]*(n+i)/2/i; } for(i=0; i<=n/2-1; i++) { c[n-2*(i+1)] = -(c[n-2*i]*(n-2*i)*(n-2*i-1)/2/(i+1)/(2*(n-i)-1)); } } } public class betaf { /************************************************************************* Beta function - - | (a) | (b) beta( a, b ) = -----------. - | (a+b) For large arguments the logarithm of the function is evaluated using lgam(), then exponentiated. ACCURACY: Relative error: arithmetic domain # trials peak rms IEEE 0,30 30000 8.1e-14 1.1e-14 Cephes Math Library Release 2.0: April, 1987 Copyright 1984, 1987 by Stephen L. Moshier *************************************************************************/ public static double beta(double a, double b, alglib.xparams _params) { double result = 0; double y = 0; double sg = 0; double s = 0; sg = 1; alglib.ap.assert((double)(a)>(double)(0) || (double)(a)!=(double)((int)Math.Floor(a)), "Overflow in Beta"); alglib.ap.assert((double)(b)>(double)(0) || (double)(b)!=(double)((int)Math.Floor(b)), "Overflow in Beta"); y = a+b; if( (double)(Math.Abs(y))>(double)(171.624376956302725) ) { y = gammafunc.lngamma(y, ref s, _params); sg = sg*s; y = gammafunc.lngamma(b, ref s, _params)-y; sg = sg*s; y = gammafunc.lngamma(a, ref s, _params)+y; sg = sg*s; alglib.ap.assert((double)(y)<=(double)(Math.Log(math.maxrealnumber)), "Overflow in Beta"); result = sg*Math.Exp(y); return result; } y = gammafunc.gammafunction(y, _params); alglib.ap.assert((double)(y)!=(double)(0), "Overflow in Beta"); if( (double)(a)>(double)(b) ) { y = gammafunc.gammafunction(a, _params)/y; y = y*gammafunc.gammafunction(b, _params); } else { y = gammafunc.gammafunction(b, _params)/y; y = y*gammafunc.gammafunction(a, _params); } result = y; return result; } } public class chebyshev { /************************************************************************* Calculation of the value of the Chebyshev polynomials of the first and second kinds. Parameters: r - polynomial kind, either 1 or 2. n - degree, n>=0 x - argument, -1 <= x <= 1 Result: the value of the Chebyshev polynomial at x *************************************************************************/ public static double chebyshevcalculate(int r, int n, double x, alglib.xparams _params) { double result = 0; int i = 0; double a = 0; double b = 0; result = 0; // // Prepare A and B // if( r==1 ) { a = 1; b = x; } else { a = 1; b = 2*x; } // // Special cases: N=0 or N=1 // if( n==0 ) { result = a; return result; } if( n==1 ) { result = b; return result; } // // General case: N>=2 // for(i=2; i<=n; i++) { result = 2*x*b-a; a = b; b = result; } return result; } /************************************************************************* Summation of Chebyshev polynomials using Clenshaw's recurrence formula. This routine calculates c[0]*T0(x) + c[1]*T1(x) + ... + c[N]*TN(x) or c[0]*U0(x) + c[1]*U1(x) + ... + c[N]*UN(x) depending on the R. Parameters: r - polynomial kind, either 1 or 2. n - degree, n>=0 x - argument Result: the value of the Chebyshev polynomial at x *************************************************************************/ public static double chebyshevsum(double[] c, int r, int n, double x, alglib.xparams _params) { double result = 0; double b1 = 0; double b2 = 0; int i = 0; b1 = 0; b2 = 0; for(i=n; i>=1; i--) { result = 2*x*b1-b2+c[i]; b2 = b1; b1 = result; } if( r==1 ) { result = -b2+x*b1+c[0]; } else { result = -b2+2*x*b1+c[0]; } return result; } /************************************************************************* Representation of Tn as C[0] + C[1]*X + ... + C[N]*X^N Input parameters: N - polynomial degree, n>=0 Output parameters: C - coefficients *************************************************************************/ public static void chebyshevcoefficients(int n, ref double[] c, alglib.xparams _params) { int i = 0; c = new double[0]; c = new double[n+1]; for(i=0; i<=n; i++) { c[i] = 0; } if( n==0 || n==1 ) { c[n] = 1; } else { c[n] = Math.Exp((n-1)*Math.Log(2)); for(i=0; i<=n/2-1; i++) { c[n-2*(i+1)] = -(c[n-2*i]*(n-2*i)*(n-2*i-1)/4/(i+1)/(n-i-1)); } } } /************************************************************************* Conversion of a series of Chebyshev polynomials to a power series. Represents A[0]*T0(x) + A[1]*T1(x) + ... + A[N]*Tn(x) as B[0] + B[1]*X + ... + B[N]*X^N. Input parameters: A - Chebyshev series coefficients N - degree, N>=0 Output parameters B - power series coefficients *************************************************************************/ public static void fromchebyshev(double[] a, int n, ref double[] b, alglib.xparams _params) { int i = 0; int k = 0; double e = 0; double d = 0; b = new double[0]; b = new double[n+1]; for(i=0; i<=n; i++) { b[i] = 0; } d = 0; i = 0; do { k = i; do { e = b[k]; b[k] = 0; if( i<=1 && k==i ) { b[k] = 1; } else { if( i!=0 ) { b[k] = 2*d; } if( k>i+1 ) { b[k] = b[k]-b[k-2]; } } d = e; k = k+1; } while( k<=n ); d = b[i]; e = 0; k = i; while( k<=n ) { e = e+b[k]*a[k]; k = k+2; } b[i] = e; i = i+1; } while( i<=n ); } } public class studenttdistr { /************************************************************************* Student's t distribution Computes the integral from minus infinity to t of the Student t distribution with integer k > 0 degrees of freedom: t - | | - | 2 -(k+1)/2 | ( (k+1)/2 ) | ( x ) ---------------------- | ( 1 + --- ) dx - | ( k ) sqrt( k pi ) | ( k/2 ) | | | - -inf. Relation to incomplete beta integral: 1 - stdtr(k,t) = 0.5 * incbet( k/2, 1/2, z ) where z = k/(k + t**2). For t < -2, this is the method of computation. For higher t, a direct method is derived from integration by parts. Since the function is symmetric about t=0, the area under the right tail of the density is found by calling the function with -t instead of t. ACCURACY: Tested at random 1 <= k <= 25. The "domain" refers to t. Relative error: arithmetic domain # trials peak rms IEEE -100,-2 50000 5.9e-15 1.4e-15 IEEE -2,100 500000 2.7e-15 4.9e-17 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier *************************************************************************/ public static double studenttdistribution(int k, double t, alglib.xparams _params) { double result = 0; double x = 0; double rk = 0; double z = 0; double f = 0; double tz = 0; double p = 0; double xsqk = 0; int j = 0; alglib.ap.assert(k>0, "Domain error in StudentTDistribution"); if( (double)(t)==(double)(0) ) { result = 0.5; return result; } if( (double)(t)<(double)(-2.0) ) { rk = k; z = rk/(rk+t*t); result = 0.5*ibetaf.incompletebeta(0.5*rk, 0.5, z, _params); return result; } if( (double)(t)<(double)(0) ) { x = -t; } else { x = t; } rk = k; z = 1.0+x*x/rk; if( k%2!=0 ) { xsqk = x/Math.Sqrt(rk); p = Math.Atan(xsqk); if( k>1 ) { f = 1.0; tz = 1.0; j = 3; while( j<=k-2 && (double)(tz/f)>(double)(math.machineepsilon) ) { tz = tz*((j-1)/(z*j)); f = f+tz; j = j+2; } p = p+f*xsqk/z; } p = p*2.0/Math.PI; } else { f = 1.0; tz = 1.0; j = 2; while( j<=k-2 && (double)(tz/f)>(double)(math.machineepsilon) ) { tz = tz*((j-1)/(z*j)); f = f+tz; j = j+2; } p = f*x/Math.Sqrt(z*rk); } if( (double)(t)<(double)(0) ) { p = -p; } result = 0.5+0.5*p; return result; } /************************************************************************* Functional inverse of Student's t distribution Given probability p, finds the argument t such that stdtr(k,t) is equal to p. ACCURACY: Tested at random 1 <= k <= 100. The "domain" refers to p: Relative error: arithmetic domain # trials peak rms IEEE .001,.999 25000 5.7e-15 8.0e-16 IEEE 10^-6,.001 25000 2.0e-12 2.9e-14 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier *************************************************************************/ public static double invstudenttdistribution(int k, double p, alglib.xparams _params) { double result = 0; double t = 0; double rk = 0; double z = 0; int rflg = 0; alglib.ap.assert((k>0 && (double)(p)>(double)(0)) && (double)(p)<(double)(1), "Domain error in InvStudentTDistribution"); rk = k; if( (double)(p)>(double)(0.25) && (double)(p)<(double)(0.75) ) { if( (double)(p)==(double)(0.5) ) { result = 0; return result; } z = 1.0-2.0*p; z = ibetaf.invincompletebeta(0.5, 0.5*rk, Math.Abs(z), _params); t = Math.Sqrt(rk*z/(1.0-z)); if( (double)(p)<(double)(0.5) ) { t = -t; } result = t; return result; } rflg = -1; if( (double)(p)>=(double)(0.5) ) { p = 1.0-p; rflg = 1; } z = ibetaf.invincompletebeta(0.5*rk, 0.5, 2.0*p, _params); if( (double)(math.maxrealnumber*z)<(double)(rk) ) { result = rflg*math.maxrealnumber; return result; } t = Math.Sqrt(rk/z-rk); result = rflg*t; return result; } } public class binomialdistr { /************************************************************************* Binomial distribution Returns the sum of the terms 0 through k of the Binomial probability density: k -- ( n ) j n-j > ( ) p (1-p) -- ( j ) j=0 The terms are not summed directly; instead the incomplete beta integral is employed, according to the formula y = bdtr( k, n, p ) = incbet( n-k, k+1, 1-p ). The arguments must be positive, with p ranging from 0 to 1. ACCURACY: Tested at random points (a,b,p), with p between 0 and 1. a,b Relative error: arithmetic domain # trials peak rms For p between 0.001 and 1: IEEE 0,100 100000 4.3e-15 2.6e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier *************************************************************************/ public static double binomialdistribution(int k, int n, double p, alglib.xparams _params) { double result = 0; double dk = 0; double dn = 0; alglib.ap.assert((double)(p)>=(double)(0) && (double)(p)<=(double)(1), "Domain error in BinomialDistribution"); alglib.ap.assert(k>=-1 && k<=n, "Domain error in BinomialDistribution"); if( k==-1 ) { result = 0; return result; } if( k==n ) { result = 1; return result; } dn = n-k; if( k==0 ) { dk = Math.Pow(1.0-p, dn); } else { dk = k+1; dk = ibetaf.incompletebeta(dn, dk, 1.0-p, _params); } result = dk; return result; } /************************************************************************* Complemented binomial distribution Returns the sum of the terms k+1 through n of the Binomial probability density: n -- ( n ) j n-j > ( ) p (1-p) -- ( j ) j=k+1 The terms are not summed directly; instead the incomplete beta integral is employed, according to the formula y = bdtrc( k, n, p ) = incbet( k+1, n-k, p ). The arguments must be positive, with p ranging from 0 to 1. ACCURACY: Tested at random points (a,b,p). a,b Relative error: arithmetic domain # trials peak rms For p between 0.001 and 1: IEEE 0,100 100000 6.7e-15 8.2e-16 For p between 0 and .001: IEEE 0,100 100000 1.5e-13 2.7e-15 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier *************************************************************************/ public static double binomialcdistribution(int k, int n, double p, alglib.xparams _params) { double result = 0; double dk = 0; double dn = 0; alglib.ap.assert((double)(p)>=(double)(0) && (double)(p)<=(double)(1), "Domain error in BinomialDistributionC"); alglib.ap.assert(k>=-1 && k<=n, "Domain error in BinomialDistributionC"); if( k==-1 ) { result = 1; return result; } if( k==n ) { result = 0; return result; } dn = n-k; if( k==0 ) { if( (double)(p)<(double)(0.01) ) { dk = -nearunityunit.nuexpm1(dn*nearunityunit.nulog1p(-p, _params), _params); } else { dk = 1.0-Math.Pow(1.0-p, dn); } } else { dk = k+1; dk = ibetaf.incompletebeta(dk, dn, p, _params); } result = dk; return result; } /************************************************************************* Inverse binomial distribution Finds the event probability p such that the sum of the terms 0 through k of the Binomial probability density is equal to the given cumulative probability y. This is accomplished using the inverse beta integral function and the relation 1 - p = incbi( n-k, k+1, y ). ACCURACY: Tested at random points (a,b,p). a,b Relative error: arithmetic domain # trials peak rms For p between 0.001 and 1: IEEE 0,100 100000 2.3e-14 6.4e-16 IEEE 0,10000 100000 6.6e-12 1.2e-13 For p between 10^-6 and 0.001: IEEE 0,100 100000 2.0e-12 1.3e-14 IEEE 0,10000 100000 1.5e-12 3.2e-14 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier *************************************************************************/ public static double invbinomialdistribution(int k, int n, double y, alglib.xparams _params) { double result = 0; double dk = 0; double dn = 0; double p = 0; alglib.ap.assert(k>=0 && k(double)(0.8) ) { p = -nearunityunit.nuexpm1(nearunityunit.nulog1p(y-1.0, _params)/dn, _params); } else { p = 1.0-Math.Pow(y, 1.0/dn); } } else { dk = k+1; p = ibetaf.incompletebeta(dn, dk, 0.5, _params); if( (double)(p)>(double)(0.5) ) { p = ibetaf.invincompletebeta(dk, dn, 1.0-y, _params); } else { p = 1.0-ibetaf.invincompletebeta(dn, dk, y, _params); } } result = p; return result; } } public class airyf { /************************************************************************* Airy function Solution of the differential equation y"(x) = xy. The function returns the two independent solutions Ai, Bi and their first derivatives Ai'(x), Bi'(x). Evaluation is by power series summation for small x, by rational minimax approximations for large x. ACCURACY: Error criterion is absolute when function <= 1, relative when function > 1, except * denotes relative error criterion. For large negative x, the absolute error increases as x^1.5. For large positive x, the relative error increases as x^1.5. Arithmetic domain function # trials peak rms IEEE -10, 0 Ai 10000 1.6e-15 2.7e-16 IEEE 0, 10 Ai 10000 2.3e-14* 1.8e-15* IEEE -10, 0 Ai' 10000 4.6e-15 7.6e-16 IEEE 0, 10 Ai' 10000 1.8e-14* 1.5e-15* IEEE -10, 10 Bi 30000 4.2e-15 5.3e-16 IEEE -10, 10 Bi' 30000 4.9e-15 7.3e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier *************************************************************************/ public static void airy(double x, ref double ai, ref double aip, ref double bi, ref double bip, alglib.xparams _params) { double z = 0; double zz = 0; double t = 0; double f = 0; double g = 0; double uf = 0; double ug = 0; double k = 0; double zeta = 0; double theta = 0; int domflg = 0; double c1 = 0; double c2 = 0; double sqrt3 = 0; double sqpii = 0; double afn = 0; double afd = 0; double agn = 0; double agd = 0; double apfn = 0; double apfd = 0; double apgn = 0; double apgd = 0; double an = 0; double ad = 0; double apn = 0; double apd = 0; double bn16 = 0; double bd16 = 0; double bppn = 0; double bppd = 0; ai = 0; aip = 0; bi = 0; bip = 0; sqpii = 5.64189583547756286948E-1; c1 = 0.35502805388781723926; c2 = 0.258819403792806798405; sqrt3 = 1.732050807568877293527; domflg = 0; if( (double)(x)>(double)(25.77) ) { ai = 0; aip = 0; bi = math.maxrealnumber; bip = math.maxrealnumber; return; } if( (double)(x)<(double)(-2.09) ) { domflg = 15; t = Math.Sqrt(-x); zeta = -(2.0*x*t/3.0); t = Math.Sqrt(t); k = sqpii/t; z = 1.0/zeta; zz = z*z; afn = -1.31696323418331795333E-1; afn = afn*zz-6.26456544431912369773E-1; afn = afn*zz-6.93158036036933542233E-1; afn = afn*zz-2.79779981545119124951E-1; afn = afn*zz-4.91900132609500318020E-2; afn = afn*zz-4.06265923594885404393E-3; afn = afn*zz-1.59276496239262096340E-4; afn = afn*zz-2.77649108155232920844E-6; afn = afn*zz-1.67787698489114633780E-8; afd = 1.00000000000000000000E0; afd = afd*zz+1.33560420706553243746E1; afd = afd*zz+3.26825032795224613948E1; afd = afd*zz+2.67367040941499554804E1; afd = afd*zz+9.18707402907259625840E0; afd = afd*zz+1.47529146771666414581E0; afd = afd*zz+1.15687173795188044134E-1; afd = afd*zz+4.40291641615211203805E-3; afd = afd*zz+7.54720348287414296618E-5; afd = afd*zz+4.51850092970580378464E-7; uf = 1.0+zz*afn/afd; agn = 1.97339932091685679179E-2; agn = agn*zz+3.91103029615688277255E-1; agn = agn*zz+1.06579897599595591108E0; agn = agn*zz+9.39169229816650230044E-1; agn = agn*zz+3.51465656105547619242E-1; agn = agn*zz+6.33888919628925490927E-2; agn = agn*zz+5.85804113048388458567E-3; agn = agn*zz+2.82851600836737019778E-4; agn = agn*zz+6.98793669997260967291E-6; agn = agn*zz+8.11789239554389293311E-8; agn = agn*zz+3.41551784765923618484E-10; agd = 1.00000000000000000000E0; agd = agd*zz+9.30892908077441974853E0; agd = agd*zz+1.98352928718312140417E1; agd = agd*zz+1.55646628932864612953E1; agd = agd*zz+5.47686069422975497931E0; agd = agd*zz+9.54293611618961883998E-1; agd = agd*zz+8.64580826352392193095E-2; agd = agd*zz+4.12656523824222607191E-3; agd = agd*zz+1.01259085116509135510E-4; agd = agd*zz+1.17166733214413521882E-6; agd = agd*zz+4.91834570062930015649E-9; ug = z*agn/agd; theta = zeta+0.25*Math.PI; f = Math.Sin(theta); g = Math.Cos(theta); ai = k*(f*uf-g*ug); bi = k*(g*uf+f*ug); apfn = 1.85365624022535566142E-1; apfn = apfn*zz+8.86712188052584095637E-1; apfn = apfn*zz+9.87391981747398547272E-1; apfn = apfn*zz+4.01241082318003734092E-1; apfn = apfn*zz+7.10304926289631174579E-2; apfn = apfn*zz+5.90618657995661810071E-3; apfn = apfn*zz+2.33051409401776799569E-4; apfn = apfn*zz+4.08718778289035454598E-6; apfn = apfn*zz+2.48379932900442457853E-8; apfd = 1.00000000000000000000E0; apfd = apfd*zz+1.47345854687502542552E1; apfd = apfd*zz+3.75423933435489594466E1; apfd = apfd*zz+3.14657751203046424330E1; apfd = apfd*zz+1.09969125207298778536E1; apfd = apfd*zz+1.78885054766999417817E0; apfd = apfd*zz+1.41733275753662636873E-1; apfd = apfd*zz+5.44066067017226003627E-3; apfd = apfd*zz+9.39421290654511171663E-5; apfd = apfd*zz+5.65978713036027009243E-7; uf = 1.0+zz*apfn/apfd; apgn = -3.55615429033082288335E-2; apgn = apgn*zz-6.37311518129435504426E-1; apgn = apgn*zz-1.70856738884312371053E0; apgn = apgn*zz-1.50221872117316635393E0; apgn = apgn*zz-5.63606665822102676611E-1; apgn = apgn*zz-1.02101031120216891789E-1; apgn = apgn*zz-9.48396695961445269093E-3; apgn = apgn*zz-4.60325307486780994357E-4; apgn = apgn*zz-1.14300836484517375919E-5; apgn = apgn*zz-1.33415518685547420648E-7; apgn = apgn*zz-5.63803833958893494476E-10; apgd = 1.00000000000000000000E0; apgd = apgd*zz+9.85865801696130355144E0; apgd = apgd*zz+2.16401867356585941885E1; apgd = apgd*zz+1.73130776389749389525E1; apgd = apgd*zz+6.17872175280828766327E0; apgd = apgd*zz+1.08848694396321495475E0; apgd = apgd*zz+9.95005543440888479402E-2; apgd = apgd*zz+4.78468199683886610842E-3; apgd = apgd*zz+1.18159633322838625562E-4; apgd = apgd*zz+1.37480673554219441465E-6; apgd = apgd*zz+5.79912514929147598821E-9; ug = z*apgn/apgd; k = sqpii*t; aip = -(k*(g*uf+f*ug)); bip = k*(f*uf-g*ug); return; } if( (double)(x)>=(double)(2.09) ) { domflg = 5; t = Math.Sqrt(x); zeta = 2.0*x*t/3.0; g = Math.Exp(zeta); t = Math.Sqrt(t); k = 2.0*t*g; z = 1.0/zeta; an = 3.46538101525629032477E-1; an = an*z+1.20075952739645805542E1; an = an*z+7.62796053615234516538E1; an = an*z+1.68089224934630576269E2; an = an*z+1.59756391350164413639E2; an = an*z+7.05360906840444183113E1; an = an*z+1.40264691163389668864E1; an = an*z+9.99999999999999995305E-1; ad = 5.67594532638770212846E-1; ad = ad*z+1.47562562584847203173E1; ad = ad*z+8.45138970141474626562E1; ad = ad*z+1.77318088145400459522E2; ad = ad*z+1.64234692871529701831E2; ad = ad*z+7.14778400825575695274E1; ad = ad*z+1.40959135607834029598E1; ad = ad*z+1.00000000000000000470E0; f = an/ad; ai = sqpii*f/k; k = -(0.5*sqpii*t/g); apn = 6.13759184814035759225E-1; apn = apn*z+1.47454670787755323881E1; apn = apn*z+8.20584123476060982430E1; apn = apn*z+1.71184781360976385540E2; apn = apn*z+1.59317847137141783523E2; apn = apn*z+6.99778599330103016170E1; apn = apn*z+1.39470856980481566958E1; apn = apn*z+1.00000000000000000550E0; apd = 3.34203677749736953049E-1; apd = apd*z+1.11810297306158156705E1; apd = apd*z+7.11727352147859965283E1; apd = apd*z+1.58778084372838313640E2; apd = apd*z+1.53206427475809220834E2; apd = apd*z+6.86752304592780337944E1; apd = apd*z+1.38498634758259442477E1; apd = apd*z+9.99999999999999994502E-1; f = apn/apd; aip = f*k; if( (double)(x)>(double)(8.3203353) ) { bn16 = -2.53240795869364152689E-1; bn16 = bn16*z+5.75285167332467384228E-1; bn16 = bn16*z-3.29907036873225371650E-1; bn16 = bn16*z+6.44404068948199951727E-2; bn16 = bn16*z-3.82519546641336734394E-3; bd16 = 1.00000000000000000000E0; bd16 = bd16*z-7.15685095054035237902E0; bd16 = bd16*z+1.06039580715664694291E1; bd16 = bd16*z-5.23246636471251500874E0; bd16 = bd16*z+9.57395864378383833152E-1; bd16 = bd16*z-5.50828147163549611107E-2; f = z*bn16/bd16; k = sqpii*g; bi = k*(1.0+f)/t; bppn = 4.65461162774651610328E-1; bppn = bppn*z-1.08992173800493920734E0; bppn = bppn*z+6.38800117371827987759E-1; bppn = bppn*z-1.26844349553102907034E-1; bppn = bppn*z+7.62487844342109852105E-3; bppd = 1.00000000000000000000E0; bppd = bppd*z-8.70622787633159124240E0; bppd = bppd*z+1.38993162704553213172E1; bppd = bppd*z-7.14116144616431159572E0; bppd = bppd*z+1.34008595960680518666E0; bppd = bppd*z-7.84273211323341930448E-2; f = z*bppn/bppd; bip = k*t*(1.0+f); return; } } f = 1.0; g = x; t = 1.0; uf = 1.0; ug = x; k = 1.0; z = x*x*x; while( (double)(t)>(double)(math.machineepsilon) ) { uf = uf*z; k = k+1.0; uf = uf/k; ug = ug*z; k = k+1.0; ug = ug/k; uf = uf/k; f = f+uf; k = k+1.0; ug = ug/k; g = g+ug; t = Math.Abs(uf/f); } uf = c1*f; ug = c2*g; if( domflg%2==0 ) { ai = uf-ug; } if( domflg/2%2==0 ) { bi = sqrt3*(uf+ug); } k = 4.0; uf = x*x/2.0; ug = z/3.0; f = uf; g = 1.0+ug; uf = uf/3.0; t = 1.0; while( (double)(t)>(double)(math.machineepsilon) ) { uf = uf*z; ug = ug/k; k = k+1.0; ug = ug*z; uf = uf/k; f = f+uf; k = k+1.0; ug = ug/k; uf = uf/k; g = g+ug; k = k+1.0; t = Math.Abs(ug/g); } uf = c1*f; ug = c2*g; if( domflg/4%2==0 ) { aip = uf-ug; } if( domflg/8%2==0 ) { bip = sqrt3*(uf+ug); } } } }