/* * Mathlib : A C Library of Special Functions * Copyright (C) 1998-2014 Ross Ihaka and the R Core team. * Copyright (C) 2002-3 The R Foundation * * 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; 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. * * You should have received a copy of the GNU General Public License * along with this program; if not, a copy is available at * https://www.R-project.org/Licenses/ */ /* DESCRIPTION --> see below */ /* From http://www.netlib.org/specfun/rkbesl Fortran translated by f2c,... * ------------------------------=#---- Martin Maechler, ETH Zurich */ #include "nmath.h" #include "bessel.h" #ifndef MATHLIB_STANDALONE #include #endif #define min0(x, y) (((x) <= (y)) ? (x) : (y)) #define max0(x, y) (((x) <= (y)) ? (y) : (x)) static void K_bessel(double *x, double *alpha, int *nb, int *ize, double *bk, int *ncalc); double bessel_k(double x, double alpha, double expo) { int nb, ncalc, ize; double *bk; #ifndef MATHLIB_STANDALONE const void *vmax; #endif #ifdef IEEE_754 /* NaNs propagated correctly */ if (ISNAN(x) || ISNAN(alpha)) return x + alpha; #endif if (x < 0) { ML_WARNING(ME_RANGE, "bessel_k"); return ML_NAN; } ize = (int)expo; if(alpha < 0) alpha = -alpha; nb = 1+ (int)floor(alpha);/* nb-1 <= |alpha| < nb */ alpha -= (double)(nb-1); #ifdef MATHLIB_STANDALONE bk = (double *) calloc(nb, sizeof(double)); if (!bk) MATHLIB_ERROR("%s", _("bessel_k allocation error")); #else vmax = vmaxget(); bk = (double *) R_alloc((size_t) nb, sizeof(double)); #endif K_bessel(&x, &alpha, &nb, &ize, bk, &ncalc); if(ncalc != nb) {/* error input */ if(ncalc < 0) MATHLIB_WARNING4(_("bessel_k(%g): ncalc (=%d) != nb (=%d); alpha=%g. Arg. out of range?\n"), x, ncalc, nb, alpha); else MATHLIB_WARNING2(_("bessel_k(%g,nu=%g): precision lost in result\n"), x, alpha+(double)nb-1); } x = bk[nb-1]; #ifdef MATHLIB_STANDALONE free(bk); #else vmaxset(vmax); #endif return x; } /* modified version of bessel_k that accepts a work array instead of allocating one. */ double bessel_k_ex(double x, double alpha, double expo, double *bk) { int nb, ncalc, ize; #ifdef IEEE_754 /* NaNs propagated correctly */ if (ISNAN(x) || ISNAN(alpha)) return x + alpha; #endif if (x < 0) { ML_WARNING(ME_RANGE, "bessel_k"); return ML_NAN; } ize = (int)expo; if(alpha < 0) alpha = -alpha; nb = 1+ (int)floor(alpha);/* nb-1 <= |alpha| < nb */ alpha -= (double)(nb-1); K_bessel(&x, &alpha, &nb, &ize, bk, &ncalc); if(ncalc != nb) {/* error input */ if(ncalc < 0) MATHLIB_WARNING4(_("bessel_k(%g): ncalc (=%d) != nb (=%d); alpha=%g. Arg. out of range?\n"), x, ncalc, nb, alpha); else MATHLIB_WARNING2(_("bessel_k(%g,nu=%g): precision lost in result\n"), x, alpha+(double)nb-1); } x = bk[nb-1]; return x; } static void K_bessel(double *x, double *alpha, int *nb, int *ize, double *bk, int *ncalc) { /*------------------------------------------------------------------- This routine calculates modified Bessel functions of the third kind, K_(N+ALPHA) (X), for non-negative argument X, and non-negative order N+ALPHA, with or without exponential scaling. Explanation of variables in the calling sequence X - Non-negative argument for which K's or exponentially scaled K's (K*EXP(X)) are to be calculated. If K's are to be calculated, X must not be greater than XMAX_BESS_K. ALPHA - Fractional part of order for which K's or exponentially scaled K's (K*EXP(X)) are to be calculated. 0 <= ALPHA < 1.0. NB - Number of functions to be calculated, NB > 0. The first function calculated is of order ALPHA, and the last is of order (NB - 1 + ALPHA). IZE - Type. IZE = 1 if unscaled K's are to be calculated, = 2 if exponentially scaled K's are to be calculated. BK - Output vector of length NB. If the routine terminates normally (NCALC=NB), the vector BK contains the functions K(ALPHA,X), ... , K(NB-1+ALPHA,X), or the corresponding exponentially scaled functions. If (0 < NCALC < NB), BK(I) contains correct function values for I <= NCALC, and contains the ratios K(ALPHA+I-1,X)/K(ALPHA+I-2,X) for the rest of the array. NCALC - Output variable indicating possible errors. Before using the vector BK, the user should check that NCALC=NB, i.e., all orders have been calculated to the desired accuracy. See error returns below. ******************************************************************* Error returns In case of an error, NCALC != NB, and not all K's are calculated to the desired accuracy. NCALC < -1: An argument is out of range. For example, NB <= 0, IZE is not 1 or 2, or IZE=1 and ABS(X) >= XMAX_BESS_K. In this case, the B-vector is not calculated, and NCALC is set to MIN0(NB,0)-2 so that NCALC != NB. NCALC = -1: Either K(ALPHA,X) >= XINF or K(ALPHA+NB-1,X)/K(ALPHA+NB-2,X) >= XINF. In this case, the B-vector is not calculated. Note that again NCALC != NB. 0 < NCALC < NB: Not all requested function values could be calculated accurately. BK(I) contains correct function values for I <= NCALC, and contains the ratios K(ALPHA+I-1,X)/K(ALPHA+I-2,X) for the rest of the array. Intrinsic functions required are: ABS, AINT, EXP, INT, LOG, MAX, MIN, SINH, SQRT Acknowledgement This program is based on a program written by J. B. Campbell (2) that computes values of the Bessel functions K of float argument and float order. Modifications include the addition of non-scaled functions, parameterization of machine dependencies, and the use of more accurate approximations for SINH and SIN. References: "On Temme's Algorithm for the Modified Bessel Functions of the Third Kind," Campbell, J. B., TOMS 6(4), Dec. 1980, pp. 581-586. "A FORTRAN IV Subroutine for the Modified Bessel Functions of the Third Kind of Real Order and Real Argument," Campbell, J. B., Report NRC/ERB-925, National Research Council, Canada. Latest modification: May 30, 1989 Modified by: W. J. Cody and L. Stoltz Applied Mathematics Division Argonne National Laboratory Argonne, IL 60439 ------------------------------------------------------------------- */ /*--------------------------------------------------------------------- * Mathematical constants * A = LOG(2) - Euler's constant * D = SQRT(2/PI) ---------------------------------------------------------------------*/ const static double a = .11593151565841244881; /*--------------------------------------------------------------------- P, Q - Approximation for LOG(GAMMA(1+ALPHA))/ALPHA + Euler's constant Coefficients converted from hex to decimal and modified by W. J. Cody, 2/26/82 */ const static double p[8] = { .805629875690432845,20.4045500205365151, 157.705605106676174,536.671116469207504,900.382759291288778, 730.923886650660393,229.299301509425145,.822467033424113231 }; const static double q[7] = { 29.4601986247850434,277.577868510221208, 1206.70325591027438,2762.91444159791519,3443.74050506564618, 2210.63190113378647,572.267338359892221 }; /* R, S - Approximation for (1-ALPHA*PI/SIN(ALPHA*PI))/(2.D0*ALPHA) */ const static double r[5] = { -.48672575865218401848,13.079485869097804016, -101.96490580880537526,347.65409106507813131, 3.495898124521934782e-4 }; const static double s[4] = { -25.579105509976461286,212.57260432226544008, -610.69018684944109624,422.69668805777760407 }; /* T - Approximation for SINH(Y)/Y */ const static double t[6] = { 1.6125990452916363814e-10, 2.5051878502858255354e-8,2.7557319615147964774e-6, 1.9841269840928373686e-4,.0083333333333334751799, .16666666666666666446 }; /*---------------------------------------------------------------------*/ const static double estm[6] = { 52.0583,5.7607,2.7782,14.4303,185.3004, 9.3715 }; const static double estf[7] = { 41.8341,7.1075,6.4306,42.511,1.35633,84.5096,20.}; /* Local variables */ int iend, i, j, k, m, ii, mplus1; double x2by4, twox, c, blpha, ratio, wminf; double d1, d2, d3, f0, f1, f2, p0, q0, t1, t2, twonu; double dm, ex, bk1, bk2, nu; ii = 0; /* -Wall */ ex = *x; nu = *alpha; *ncalc = min0(*nb,0) - 2; if (*nb > 0 && (0. <= nu && nu < 1.) && (1 <= *ize && *ize <= 2)) { if(ex <= 0 || (*ize == 1 && ex > xmax_BESS_K)) { if(ex <= 0) { if(ex < 0) ML_WARNING(ME_RANGE, "K_bessel"); for(i=0; i < *nb; i++) bk[i] = ML_POSINF; } else /* would only have underflow */ for(i=0; i < *nb; i++) bk[i] = 0.; *ncalc = *nb; return; } k = 0; if (nu < sqxmin_BESS_K) { nu = 0.; } else if (nu > .5) { k = 1; nu -= 1.; } twonu = nu + nu; iend = *nb + k - 1; c = nu * nu; d3 = -c; if (ex <= 1.) { /* ------------------------------------------------------------ Calculation of P0 = GAMMA(1+ALPHA) * (2/X)**ALPHA Q0 = GAMMA(1-ALPHA) * (X/2)**ALPHA ------------------------------------------------------------ */ d1 = 0.; d2 = p[0]; t1 = 1.; t2 = q[0]; for (i = 2; i <= 7; i += 2) { d1 = c * d1 + p[i - 1]; d2 = c * d2 + p[i]; t1 = c * t1 + q[i - 1]; t2 = c * t2 + q[i]; } d1 = nu * d1; t1 = nu * t1; f1 = log(ex); f0 = a + nu * (p[7] - nu * (d1 + d2) / (t1 + t2)) - f1; q0 = exp(-nu * (a - nu * (p[7] + nu * (d1-d2) / (t1-t2)) - f1)); f1 = nu * f0; p0 = exp(f1); /* ----------------------------------------------------------- Calculation of F0 = ----------------------------------------------------------- */ d1 = r[4]; t1 = 1.; for (i = 0; i < 4; ++i) { d1 = c * d1 + r[i]; t1 = c * t1 + s[i]; } /* d2 := sinh(f1)/ nu = sinh(f1)/(f1/f0) * = f0 * sinh(f1)/f1 */ if (fabs(f1) <= .5) { f1 *= f1; d2 = 0.; for (i = 0; i < 6; ++i) { d2 = f1 * d2 + t[i]; } d2 = f0 + f0 * f1 * d2; } else { d2 = sinh(f1) / nu; } f0 = d2 - nu * d1 / (t1 * p0); if (ex <= 1e-10) { /* --------------------------------------------------------- X <= 1.0E-10 Calculation of K(ALPHA,X) and X*K(ALPHA+1,X)/K(ALPHA,X) --------------------------------------------------------- */ bk[0] = f0 + ex * f0; if (*ize == 1) { bk[0] -= ex * bk[0]; } ratio = p0 / f0; c = ex * DBL_MAX; if (k != 0) { /* --------------------------------------------------- Calculation of K(ALPHA,X) and X*K(ALPHA+1,X)/K(ALPHA,X), ALPHA >= 1/2 --------------------------------------------------- */ *ncalc = -1; if (bk[0] >= c / ratio) { return; } bk[0] = ratio * bk[0] / ex; twonu += 2.; ratio = twonu; } *ncalc = 1; if (*nb == 1) return; /* ----------------------------------------------------- Calculate K(ALPHA+L,X)/K(ALPHA+L-1,X), L = 1, 2, ... , NB-1 ----------------------------------------------------- */ *ncalc = -1; for (i = 1; i < *nb; ++i) { if (ratio >= c) return; bk[i] = ratio / ex; twonu += 2.; ratio = twonu; } *ncalc = 1; goto L420; } else { /* ------------------------------------------------------ 10^-10 < X <= 1.0 ------------------------------------------------------ */ c = 1.; x2by4 = ex * ex / 4.; p0 = .5 * p0; q0 = .5 * q0; d1 = -1.; d2 = 0.; bk1 = 0.; bk2 = 0.; f1 = f0; f2 = p0; do { d1 += 2.; d2 += 1.; d3 = d1 + d3; c = x2by4 * c / d2; f0 = (d2 * f0 + p0 + q0) / d3; p0 /= d2 - nu; q0 /= d2 + nu; t1 = c * f0; t2 = c * (p0 - d2 * f0); bk1 += t1; bk2 += t2; } while (fabs(t1 / (f1 + bk1)) > DBL_EPSILON || fabs(t2 / (f2 + bk2)) > DBL_EPSILON); bk1 = f1 + bk1; bk2 = 2. * (f2 + bk2) / ex; if (*ize == 2) { d1 = exp(ex); bk1 *= d1; bk2 *= d1; } wminf = estf[0] * ex + estf[1]; } } else if (DBL_EPSILON * ex > 1.) { /* ------------------------------------------------- X > 1./EPS ------------------------------------------------- */ *ncalc = *nb; bk1 = 1. / (M_SQRT_2dPI * sqrt(ex)); for (i = 0; i < *nb; ++i) bk[i] = bk1; return; } else { /* ------------------------------------------------------- X > 1.0 ------------------------------------------------------- */ twox = ex + ex; blpha = 0.; ratio = 0.; if (ex <= 4.) { /* ---------------------------------------------------------- Calculation of K(ALPHA+1,X)/K(ALPHA,X), 1.0 <= X <= 4.0 ----------------------------------------------------------*/ d2 = trunc(estm[0] / ex + estm[1]); m = (int) d2; d1 = d2 + d2; d2 -= .5; d2 *= d2; for (i = 2; i <= m; ++i) { d1 -= 2.; d2 -= d1; ratio = (d3 + d2) / (twox + d1 - ratio); } /* ----------------------------------------------------------- Calculation of I(|ALPHA|,X) and I(|ALPHA|+1,X) by backward recurrence and K(ALPHA,X) from the wronskian -----------------------------------------------------------*/ d2 = trunc(estm[2] * ex + estm[3]); m = (int) d2; c = fabs(nu); d3 = c + c; d1 = d3 - 1.; f1 = DBL_MIN; f0 = (2. * (c + d2) / ex + .5 * ex / (c + d2 + 1.)) * DBL_MIN; for (i = 3; i <= m; ++i) { d2 -= 1.; f2 = (d3 + d2 + d2) * f0; blpha = (1. + d1 / d2) * (f2 + blpha); f2 = f2 / ex + f1; f1 = f0; f0 = f2; } f1 = (d3 + 2.) * f0 / ex + f1; d1 = 0.; t1 = 1.; for (i = 1; i <= 7; ++i) { d1 = c * d1 + p[i - 1]; t1 = c * t1 + q[i - 1]; } p0 = exp(c * (a + c * (p[7] - c * d1 / t1) - log(ex))) / ex; f2 = (c + .5 - ratio) * f1 / ex; bk1 = p0 + (d3 * f0 - f2 + f0 + blpha) / (f2 + f1 + f0) * p0; if (*ize == 1) { bk1 *= exp(-ex); } wminf = estf[2] * ex + estf[3]; } else { /* --------------------------------------------------------- Calculation of K(ALPHA,X) and K(ALPHA+1,X)/K(ALPHA,X), by backward recurrence, for X > 4.0 ----------------------------------------------------------*/ dm = trunc(estm[4] / ex + estm[5]); m = (int) dm; d2 = dm - .5; d2 *= d2; d1 = dm + dm; for (i = 2; i <= m; ++i) { dm -= 1.; d1 -= 2.; d2 -= d1; ratio = (d3 + d2) / (twox + d1 - ratio); blpha = (ratio + ratio * blpha) / dm; } bk1 = 1. / ((M_SQRT_2dPI + M_SQRT_2dPI * blpha) * sqrt(ex)); if (*ize == 1) bk1 *= exp(-ex); wminf = estf[4] * (ex - fabs(ex - estf[6])) + estf[5]; } /* --------------------------------------------------------- Calculation of K(ALPHA+1,X) from K(ALPHA,X) and K(ALPHA+1,X)/K(ALPHA,X) --------------------------------------------------------- */ bk2 = bk1 + bk1 * (nu + .5 - ratio) / ex; } /*-------------------------------------------------------------------- Calculation of 'NCALC', K(ALPHA+I,X), I = 0, 1, ... , NCALC-1, & K(ALPHA+I,X)/K(ALPHA+I-1,X), I = NCALC, NCALC+1, ... , NB-1 -------------------------------------------------------------------*/ *ncalc = *nb; bk[0] = bk1; if (iend == 0) return; j = 1 - k; if (j >= 0) bk[j] = bk2; if (iend == 1) return; m = min0((int) (wminf - nu),iend); for (i = 2; i <= m; ++i) { t1 = bk1; bk1 = bk2; twonu += 2.; if (ex < 1.) { if (bk1 >= DBL_MAX / twonu * ex) break; } else { if (bk1 / ex >= DBL_MAX / twonu) break; } bk2 = twonu / ex * bk1 + t1; ii = i; ++j; if (j >= 0) { bk[j] = bk2; } } m = ii; if (m == iend) { return; } ratio = bk2 / bk1; mplus1 = m + 1; *ncalc = -1; for (i = mplus1; i <= iend; ++i) { twonu += 2.; ratio = twonu / ex + 1./ratio; ++j; if (j >= 1) { bk[j] = ratio; } else { if (bk2 >= DBL_MAX / ratio) return; bk2 *= ratio; } } *ncalc = max0(1, mplus1 - k); if (*ncalc == 1) bk[0] = bk2; if (*nb == 1) return; L420: for (i = *ncalc; i < *nb; ++i) { /* i == *ncalc */ #ifndef IEEE_754 if (bk[i-1] >= DBL_MAX / bk[i]) return; #endif bk[i] *= bk[i-1]; (*ncalc)++; } } }