/************************************************************************* ALGLIB 3.16.0 (source code generated 2019-12-19) 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 >>> *************************************************************************/ #ifndef _specialfunctions_pkg_h #define _specialfunctions_pkg_h #include "ap.h" #include "alglibinternal.h" #include "alglibmisc.h" ///////////////////////////////////////////////////////////////////////// // // THIS SECTION CONTAINS COMPUTATIONAL CORE DECLARATIONS (DATATYPES) // ///////////////////////////////////////////////////////////////////////// namespace alglib_impl { #if defined(AE_COMPILE_GAMMAFUNC) || !defined(AE_PARTIAL_BUILD) #endif #if defined(AE_COMPILE_NORMALDISTR) || !defined(AE_PARTIAL_BUILD) #endif #if defined(AE_COMPILE_IGAMMAF) || !defined(AE_PARTIAL_BUILD) #endif #if defined(AE_COMPILE_ELLIPTIC) || !defined(AE_PARTIAL_BUILD) #endif #if defined(AE_COMPILE_HERMITE) || !defined(AE_PARTIAL_BUILD) #endif #if defined(AE_COMPILE_DAWSON) || !defined(AE_PARTIAL_BUILD) #endif #if defined(AE_COMPILE_TRIGINTEGRALS) || !defined(AE_PARTIAL_BUILD) #endif #if defined(AE_COMPILE_POISSONDISTR) || !defined(AE_PARTIAL_BUILD) #endif #if defined(AE_COMPILE_BESSEL) || !defined(AE_PARTIAL_BUILD) #endif #if defined(AE_COMPILE_IBETAF) || !defined(AE_PARTIAL_BUILD) #endif #if defined(AE_COMPILE_FDISTR) || !defined(AE_PARTIAL_BUILD) #endif #if defined(AE_COMPILE_FRESNEL) || !defined(AE_PARTIAL_BUILD) #endif #if defined(AE_COMPILE_JACOBIANELLIPTIC) || !defined(AE_PARTIAL_BUILD) #endif #if defined(AE_COMPILE_PSIF) || !defined(AE_PARTIAL_BUILD) #endif #if defined(AE_COMPILE_EXPINTEGRALS) || !defined(AE_PARTIAL_BUILD) #endif #if defined(AE_COMPILE_LAGUERRE) || !defined(AE_PARTIAL_BUILD) #endif #if defined(AE_COMPILE_CHISQUAREDISTR) || !defined(AE_PARTIAL_BUILD) #endif #if defined(AE_COMPILE_LEGENDRE) || !defined(AE_PARTIAL_BUILD) #endif #if defined(AE_COMPILE_BETAF) || !defined(AE_PARTIAL_BUILD) #endif #if defined(AE_COMPILE_CHEBYSHEV) || !defined(AE_PARTIAL_BUILD) #endif #if defined(AE_COMPILE_STUDENTTDISTR) || !defined(AE_PARTIAL_BUILD) #endif #if defined(AE_COMPILE_BINOMIALDISTR) || !defined(AE_PARTIAL_BUILD) #endif #if defined(AE_COMPILE_AIRYF) || !defined(AE_PARTIAL_BUILD) #endif } ///////////////////////////////////////////////////////////////////////// // // THIS SECTION CONTAINS C++ INTERFACE // ///////////////////////////////////////////////////////////////////////// namespace alglib { #if defined(AE_COMPILE_GAMMAFUNC) || !defined(AE_PARTIAL_BUILD) #endif #if defined(AE_COMPILE_NORMALDISTR) || !defined(AE_PARTIAL_BUILD) #endif #if defined(AE_COMPILE_IGAMMAF) || !defined(AE_PARTIAL_BUILD) #endif #if defined(AE_COMPILE_ELLIPTIC) || !defined(AE_PARTIAL_BUILD) #endif #if defined(AE_COMPILE_HERMITE) || !defined(AE_PARTIAL_BUILD) #endif #if defined(AE_COMPILE_DAWSON) || !defined(AE_PARTIAL_BUILD) #endif #if defined(AE_COMPILE_TRIGINTEGRALS) || !defined(AE_PARTIAL_BUILD) #endif #if defined(AE_COMPILE_POISSONDISTR) || !defined(AE_PARTIAL_BUILD) #endif #if defined(AE_COMPILE_BESSEL) || !defined(AE_PARTIAL_BUILD) #endif #if defined(AE_COMPILE_IBETAF) || !defined(AE_PARTIAL_BUILD) #endif #if defined(AE_COMPILE_FDISTR) || !defined(AE_PARTIAL_BUILD) #endif #if defined(AE_COMPILE_FRESNEL) || !defined(AE_PARTIAL_BUILD) #endif #if defined(AE_COMPILE_JACOBIANELLIPTIC) || !defined(AE_PARTIAL_BUILD) #endif #if defined(AE_COMPILE_PSIF) || !defined(AE_PARTIAL_BUILD) #endif #if defined(AE_COMPILE_EXPINTEGRALS) || !defined(AE_PARTIAL_BUILD) #endif #if defined(AE_COMPILE_LAGUERRE) || !defined(AE_PARTIAL_BUILD) #endif #if defined(AE_COMPILE_CHISQUAREDISTR) || !defined(AE_PARTIAL_BUILD) #endif #if defined(AE_COMPILE_LEGENDRE) || !defined(AE_PARTIAL_BUILD) #endif #if defined(AE_COMPILE_BETAF) || !defined(AE_PARTIAL_BUILD) #endif #if defined(AE_COMPILE_CHEBYSHEV) || !defined(AE_PARTIAL_BUILD) #endif #if defined(AE_COMPILE_STUDENTTDISTR) || !defined(AE_PARTIAL_BUILD) #endif #if defined(AE_COMPILE_BINOMIALDISTR) || !defined(AE_PARTIAL_BUILD) #endif #if defined(AE_COMPILE_AIRYF) || !defined(AE_PARTIAL_BUILD) #endif #if defined(AE_COMPILE_GAMMAFUNC) || !defined(AE_PARTIAL_BUILD) /************************************************************************* 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). *************************************************************************/ double gammafunction(const double x, const xparams _xparams = alglib::xdefault); /************************************************************************* 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). *************************************************************************/ double lngamma(const double x, double &sgngam, const xparams _xparams = alglib::xdefault); #endif #if defined(AE_COMPILE_NORMALDISTR) || !defined(AE_PARTIAL_BUILD) /************************************************************************* 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 *************************************************************************/ double errorfunction(const double x, const xparams _xparams = alglib::xdefault); /************************************************************************* 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 *************************************************************************/ double errorfunctionc(const double x, const xparams _xparams = alglib::xdefault); /************************************************************************* Same as normalcdf(), obsolete name. *************************************************************************/ double normaldistribution(const double x, const xparams _xparams = alglib::xdefault); /************************************************************************* 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 *************************************************************************/ double normalpdf(const double x, const xparams _xparams = alglib::xdefault); /************************************************************************* 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 *************************************************************************/ double normalcdf(const double x, const xparams _xparams = alglib::xdefault); /************************************************************************* Inverse of the error function Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier *************************************************************************/ double inverf(const double e, const xparams _xparams = alglib::xdefault); /************************************************************************* Same as invnormalcdf(), deprecated name *************************************************************************/ double invnormaldistribution(const double y0, const xparams _xparams = alglib::xdefault); /************************************************************************* 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 *************************************************************************/ double invnormalcdf(const double y0, const xparams _xparams = alglib::xdefault); /************************************************************************* 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 *************************************************************************/ double hermitecalculate(const ae_int_t n, const double x, const xparams _xparams = alglib::xdefault); /************************************************************************* 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 *************************************************************************/ double hermitesum(const real_1d_array &c, const ae_int_t n, const double x, const xparams _xparams = alglib::xdefault); /************************************************************************* Representation of Hn as C[0] + C[1]*X + ... + C[N]*X^N Input parameters: N - polynomial degree, n>=0 Output parameters: C - coefficients *************************************************************************/ void hermitecoefficients(const ae_int_t n, real_1d_array &c, const xparams _xparams = alglib::xdefault); #endif #if defined(AE_COMPILE_DAWSON) || !defined(AE_PARTIAL_BUILD) /************************************************************************* 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 *************************************************************************/ double dawsonintegral(const double x, const xparams _xparams = alglib::xdefault); #endif #if defined(AE_COMPILE_TRIGINTEGRALS) || !defined(AE_PARTIAL_BUILD) /************************************************************************* 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 *************************************************************************/ void sinecosineintegrals(const double x, double &si, double &ci, const xparams _xparams = alglib::xdefault); /************************************************************************* 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 *************************************************************************/ void hyperbolicsinecosineintegrals(const double x, double &shi, double &chi, const xparams _xparams = alglib::xdefault); #endif #if defined(AE_COMPILE_POISSONDISTR) || !defined(AE_PARTIAL_BUILD) /************************************************************************* 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 *************************************************************************/ double poissondistribution(const ae_int_t k, const double m, const xparams _xparams = alglib::xdefault); /************************************************************************* 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 *************************************************************************/ double poissoncdistribution(const ae_int_t k, const double m, const xparams _xparams = alglib::xdefault); /************************************************************************* 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 *************************************************************************/ double invpoissondistribution(const ae_int_t k, const double y, const xparams _xparams = alglib::xdefault); #endif #if defined(AE_COMPILE_BESSEL) || !defined(AE_PARTIAL_BUILD) /************************************************************************* 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 *************************************************************************/ double besselj0(const double x, const xparams _xparams = alglib::xdefault); /************************************************************************* 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 *************************************************************************/ double besselj1(const double x, const xparams _xparams = alglib::xdefault); /************************************************************************* 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 *************************************************************************/ double besseljn(const ae_int_t n, const double x, const xparams _xparams = alglib::xdefault); /************************************************************************* 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 *************************************************************************/ double bessely0(const double x, const xparams _xparams = alglib::xdefault); /************************************************************************* 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 *************************************************************************/ double bessely1(const double x, const xparams _xparams = alglib::xdefault); /************************************************************************* 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 *************************************************************************/ double besselyn(const ae_int_t n, const double x, const xparams _xparams = alglib::xdefault); /************************************************************************* 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 *************************************************************************/ double besseli0(const double x, const xparams _xparams = alglib::xdefault); /************************************************************************* 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 *************************************************************************/ double besseli1(const double x, const xparams _xparams = alglib::xdefault); /************************************************************************* 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 *************************************************************************/ double besselk0(const double x, const xparams _xparams = alglib::xdefault); /************************************************************************* 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 *************************************************************************/ double besselk1(const double x, const xparams _xparams = alglib::xdefault); /************************************************************************* 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 *************************************************************************/ double besselkn(const ae_int_t nn, const double x, const xparams _xparams = alglib::xdefault); #endif #if defined(AE_COMPILE_IBETAF) || !defined(AE_PARTIAL_BUILD) /************************************************************************* 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 *************************************************************************/ double incompletebeta(const double a, const double b, const double x, const xparams _xparams = alglib::xdefault); /************************************************************************* 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 *************************************************************************/ double invincompletebeta(const double a, const double b, const double y, const xparams _xparams = alglib::xdefault); #endif #if defined(AE_COMPILE_FDISTR) || !defined(AE_PARTIAL_BUILD) /************************************************************************* 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 *************************************************************************/ double fdistribution(const ae_int_t a, const ae_int_t b, const double x, const xparams _xparams = alglib::xdefault); /************************************************************************* 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 *************************************************************************/ double fcdistribution(const ae_int_t a, const ae_int_t b, const double x, const xparams _xparams = alglib::xdefault); /************************************************************************* 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 *************************************************************************/ double invfdistribution(const ae_int_t a, const ae_int_t b, const double y, const xparams _xparams = alglib::xdefault); #endif #if defined(AE_COMPILE_FRESNEL) || !defined(AE_PARTIAL_BUILD) /************************************************************************* 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 *************************************************************************/ void fresnelintegral(const double x, double &c, double &s, const xparams _xparams = alglib::xdefault); #endif #if defined(AE_COMPILE_JACOBIANELLIPTIC) || !defined(AE_PARTIAL_BUILD) /************************************************************************* 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 *************************************************************************/ void jacobianellipticfunctions(const double u, const double m, double &sn, double &cn, double &dn, double &ph, const xparams _xparams = alglib::xdefault); #endif #if defined(AE_COMPILE_PSIF) || !defined(AE_PARTIAL_BUILD) /************************************************************************* 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 *************************************************************************/ double psi(const double x, const xparams _xparams = alglib::xdefault); #endif #if defined(AE_COMPILE_EXPINTEGRALS) || !defined(AE_PARTIAL_BUILD) /************************************************************************* 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 *************************************************************************/ double exponentialintegralei(const double x, const xparams _xparams = alglib::xdefault); /************************************************************************* 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 *************************************************************************/ double exponentialintegralen(const double x, const ae_int_t n, const xparams _xparams = alglib::xdefault); #endif #if defined(AE_COMPILE_LAGUERRE) || !defined(AE_PARTIAL_BUILD) /************************************************************************* 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 *************************************************************************/ double laguerrecalculate(const ae_int_t n, const double x, const xparams _xparams = alglib::xdefault); /************************************************************************* 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 *************************************************************************/ double laguerresum(const real_1d_array &c, const ae_int_t n, const double x, const xparams _xparams = alglib::xdefault); /************************************************************************* Representation of Ln as C[0] + C[1]*X + ... + C[N]*X^N Input parameters: N - polynomial degree, n>=0 Output parameters: C - coefficients *************************************************************************/ void laguerrecoefficients(const ae_int_t n, real_1d_array &c, const xparams _xparams = alglib::xdefault); #endif #if defined(AE_COMPILE_CHISQUAREDISTR) || !defined(AE_PARTIAL_BUILD) /************************************************************************* 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 *************************************************************************/ double chisquaredistribution(const double v, const double x, const xparams _xparams = alglib::xdefault); /************************************************************************* 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 *************************************************************************/ double chisquarecdistribution(const double v, const double x, const xparams _xparams = alglib::xdefault); /************************************************************************* 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 *************************************************************************/ double invchisquaredistribution(const double v, const double y, const xparams _xparams = alglib::xdefault); #endif #if defined(AE_COMPILE_LEGENDRE) || !defined(AE_PARTIAL_BUILD) /************************************************************************* 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 *************************************************************************/ double legendrecalculate(const ae_int_t n, const double x, const xparams _xparams = alglib::xdefault); /************************************************************************* 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 *************************************************************************/ double legendresum(const real_1d_array &c, const ae_int_t n, const double x, const xparams _xparams = alglib::xdefault); /************************************************************************* Representation of Pn as C[0] + C[1]*X + ... + C[N]*X^N Input parameters: N - polynomial degree, n>=0 Output parameters: C - coefficients *************************************************************************/ void legendrecoefficients(const ae_int_t n, real_1d_array &c, const xparams _xparams = alglib::xdefault); #endif #if defined(AE_COMPILE_BETAF) || !defined(AE_PARTIAL_BUILD) /************************************************************************* 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 *************************************************************************/ double beta(const double a, const double b, const xparams _xparams = alglib::xdefault); #endif #if defined(AE_COMPILE_CHEBYSHEV) || !defined(AE_PARTIAL_BUILD) /************************************************************************* 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 *************************************************************************/ double chebyshevcalculate(const ae_int_t r, const ae_int_t n, const double x, const xparams _xparams = alglib::xdefault); /************************************************************************* 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 *************************************************************************/ double chebyshevsum(const real_1d_array &c, const ae_int_t r, const ae_int_t n, const double x, const xparams _xparams = alglib::xdefault); /************************************************************************* Representation of Tn as C[0] + C[1]*X + ... + C[N]*X^N Input parameters: N - polynomial degree, n>=0 Output parameters: C - coefficients *************************************************************************/ void chebyshevcoefficients(const ae_int_t n, real_1d_array &c, const xparams _xparams = alglib::xdefault); /************************************************************************* 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 *************************************************************************/ void fromchebyshev(const real_1d_array &a, const ae_int_t n, real_1d_array &b, const xparams _xparams = alglib::xdefault); #endif #if defined(AE_COMPILE_STUDENTTDISTR) || !defined(AE_PARTIAL_BUILD) /************************************************************************* 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 *************************************************************************/ double studenttdistribution(const ae_int_t k, const double t, const xparams _xparams = alglib::xdefault); /************************************************************************* 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 *************************************************************************/ double invstudenttdistribution(const ae_int_t k, const double p, const xparams _xparams = alglib::xdefault); #endif #if defined(AE_COMPILE_BINOMIALDISTR) || !defined(AE_PARTIAL_BUILD) /************************************************************************* 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 *************************************************************************/ double binomialdistribution(const ae_int_t k, const ae_int_t n, const double p, const xparams _xparams = alglib::xdefault); /************************************************************************* 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 *************************************************************************/ double binomialcdistribution(const ae_int_t k, const ae_int_t n, const double p, const xparams _xparams = alglib::xdefault); /************************************************************************* 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 *************************************************************************/ double invbinomialdistribution(const ae_int_t k, const ae_int_t n, const double y, const xparams _xparams = alglib::xdefault); #endif #if defined(AE_COMPILE_AIRYF) || !defined(AE_PARTIAL_BUILD) /************************************************************************* 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 *************************************************************************/ void airy(const double x, double &ai, double &aip, double &bi, double &bip, const xparams _xparams = alglib::xdefault); #endif } ///////////////////////////////////////////////////////////////////////// // // THIS SECTION CONTAINS COMPUTATIONAL CORE DECLARATIONS (FUNCTIONS) // ///////////////////////////////////////////////////////////////////////// namespace alglib_impl { #if defined(AE_COMPILE_GAMMAFUNC) || !defined(AE_PARTIAL_BUILD) double gammafunction(double x, ae_state *_state); double lngamma(double x, double* sgngam, ae_state *_state); #endif #if defined(AE_COMPILE_NORMALDISTR) || !defined(AE_PARTIAL_BUILD) double errorfunction(double x, ae_state *_state); double errorfunctionc(double x, ae_state *_state); double normaldistribution(double x, ae_state *_state); double normalpdf(double x, ae_state *_state); double normalcdf(double x, ae_state *_state); double inverf(double e, ae_state *_state); double invnormaldistribution(double y0, ae_state *_state); double invnormalcdf(double y0, ae_state *_state); double bivariatenormalpdf(double x, double y, double rho, ae_state *_state); double bivariatenormalcdf(double x, double y, double rho, ae_state *_state); #endif #if defined(AE_COMPILE_IGAMMAF) || !defined(AE_PARTIAL_BUILD) double incompletegamma(double a, double x, ae_state *_state); double incompletegammac(double a, double x, ae_state *_state); double invincompletegammac(double a, double y0, ae_state *_state); #endif #if defined(AE_COMPILE_ELLIPTIC) || !defined(AE_PARTIAL_BUILD) double ellipticintegralk(double m, ae_state *_state); double ellipticintegralkhighprecision(double m1, ae_state *_state); double incompleteellipticintegralk(double phi, double m, ae_state *_state); double ellipticintegrale(double m, ae_state *_state); double incompleteellipticintegrale(double phi, double m, ae_state *_state); #endif #if defined(AE_COMPILE_HERMITE) || !defined(AE_PARTIAL_BUILD) double hermitecalculate(ae_int_t n, double x, ae_state *_state); double hermitesum(/* Real */ ae_vector* c, ae_int_t n, double x, ae_state *_state); void hermitecoefficients(ae_int_t n, /* Real */ ae_vector* c, ae_state *_state); #endif #if defined(AE_COMPILE_DAWSON) || !defined(AE_PARTIAL_BUILD) double dawsonintegral(double x, ae_state *_state); #endif #if defined(AE_COMPILE_TRIGINTEGRALS) || !defined(AE_PARTIAL_BUILD) void sinecosineintegrals(double x, double* si, double* ci, ae_state *_state); void hyperbolicsinecosineintegrals(double x, double* shi, double* chi, ae_state *_state); #endif #if defined(AE_COMPILE_POISSONDISTR) || !defined(AE_PARTIAL_BUILD) double poissondistribution(ae_int_t k, double m, ae_state *_state); double poissoncdistribution(ae_int_t k, double m, ae_state *_state); double invpoissondistribution(ae_int_t k, double y, ae_state *_state); #endif #if defined(AE_COMPILE_BESSEL) || !defined(AE_PARTIAL_BUILD) double besselj0(double x, ae_state *_state); double besselj1(double x, ae_state *_state); double besseljn(ae_int_t n, double x, ae_state *_state); double bessely0(double x, ae_state *_state); double bessely1(double x, ae_state *_state); double besselyn(ae_int_t n, double x, ae_state *_state); double besseli0(double x, ae_state *_state); double besseli1(double x, ae_state *_state); double besselk0(double x, ae_state *_state); double besselk1(double x, ae_state *_state); double besselkn(ae_int_t nn, double x, ae_state *_state); #endif #if defined(AE_COMPILE_IBETAF) || !defined(AE_PARTIAL_BUILD) double incompletebeta(double a, double b, double x, ae_state *_state); double invincompletebeta(double a, double b, double y, ae_state *_state); #endif #if defined(AE_COMPILE_FDISTR) || !defined(AE_PARTIAL_BUILD) double fdistribution(ae_int_t a, ae_int_t b, double x, ae_state *_state); double fcdistribution(ae_int_t a, ae_int_t b, double x, ae_state *_state); double invfdistribution(ae_int_t a, ae_int_t b, double y, ae_state *_state); #endif #if defined(AE_COMPILE_FRESNEL) || !defined(AE_PARTIAL_BUILD) void fresnelintegral(double x, double* c, double* s, ae_state *_state); #endif #if defined(AE_COMPILE_JACOBIANELLIPTIC) || !defined(AE_PARTIAL_BUILD) void jacobianellipticfunctions(double u, double m, double* sn, double* cn, double* dn, double* ph, ae_state *_state); #endif #if defined(AE_COMPILE_PSIF) || !defined(AE_PARTIAL_BUILD) double psi(double x, ae_state *_state); #endif #if defined(AE_COMPILE_EXPINTEGRALS) || !defined(AE_PARTIAL_BUILD) double exponentialintegralei(double x, ae_state *_state); double exponentialintegralen(double x, ae_int_t n, ae_state *_state); #endif #if defined(AE_COMPILE_LAGUERRE) || !defined(AE_PARTIAL_BUILD) double laguerrecalculate(ae_int_t n, double x, ae_state *_state); double laguerresum(/* Real */ ae_vector* c, ae_int_t n, double x, ae_state *_state); void laguerrecoefficients(ae_int_t n, /* Real */ ae_vector* c, ae_state *_state); #endif #if defined(AE_COMPILE_CHISQUAREDISTR) || !defined(AE_PARTIAL_BUILD) double chisquaredistribution(double v, double x, ae_state *_state); double chisquarecdistribution(double v, double x, ae_state *_state); double invchisquaredistribution(double v, double y, ae_state *_state); #endif #if defined(AE_COMPILE_LEGENDRE) || !defined(AE_PARTIAL_BUILD) double legendrecalculate(ae_int_t n, double x, ae_state *_state); double legendresum(/* Real */ ae_vector* c, ae_int_t n, double x, ae_state *_state); void legendrecoefficients(ae_int_t n, /* Real */ ae_vector* c, ae_state *_state); #endif #if defined(AE_COMPILE_BETAF) || !defined(AE_PARTIAL_BUILD) double beta(double a, double b, ae_state *_state); #endif #if defined(AE_COMPILE_CHEBYSHEV) || !defined(AE_PARTIAL_BUILD) double chebyshevcalculate(ae_int_t r, ae_int_t n, double x, ae_state *_state); double chebyshevsum(/* Real */ ae_vector* c, ae_int_t r, ae_int_t n, double x, ae_state *_state); void chebyshevcoefficients(ae_int_t n, /* Real */ ae_vector* c, ae_state *_state); void fromchebyshev(/* Real */ ae_vector* a, ae_int_t n, /* Real */ ae_vector* b, ae_state *_state); #endif #if defined(AE_COMPILE_STUDENTTDISTR) || !defined(AE_PARTIAL_BUILD) double studenttdistribution(ae_int_t k, double t, ae_state *_state); double invstudenttdistribution(ae_int_t k, double p, ae_state *_state); #endif #if defined(AE_COMPILE_BINOMIALDISTR) || !defined(AE_PARTIAL_BUILD) double binomialdistribution(ae_int_t k, ae_int_t n, double p, ae_state *_state); double binomialcdistribution(ae_int_t k, ae_int_t n, double p, ae_state *_state); double invbinomialdistribution(ae_int_t k, ae_int_t n, double y, ae_state *_state); #endif #if defined(AE_COMPILE_AIRYF) || !defined(AE_PARTIAL_BUILD) void airy(double x, double* ai, double* aip, double* bi, double* bip, ae_state *_state); #endif } #endif