/* * Mathlib : A C Library of Special Functions * Copyright (C) 1998-2015 Ross Ihaka and the R Core team. * * 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/rybesl 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)) static void Y_bessel(double *x, double *alpha, int *nb, double *by, int *ncalc); // unused now from R double bessel_y(double x, double alpha) { int nb, ncalc; double na, *by; #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_y"); return ML_NAN; } na = floor(alpha); if (alpha < 0) { /* Using Abramowitz & Stegun 9.1.2 * this may not be quite optimal (CPU and accuracy wise) */ return(((alpha - na == 0.5) ? 0 : bessel_y(x, -alpha) * cospi(alpha)) - ((alpha == na ) ? 0 : bessel_j(x, -alpha) * sinpi(alpha))); } else if (alpha > 1e7) { MATHLIB_WARNING(_("besselY(x, nu): nu=%g too large for bessel_y() algorithm"), alpha); return ML_NAN; } nb = 1+ (int)na;/* nb-1 <= alpha < nb */ alpha -= (double)(nb-1); #ifdef MATHLIB_STANDALONE by = (double *) calloc(nb, sizeof(double)); if (!by) MATHLIB_ERROR("%s", _("bessel_y allocation error")); #else vmax = vmaxget(); by = (double *) R_alloc((size_t) nb, sizeof(double)); #endif Y_bessel(&x, &alpha, &nb, by, &ncalc); if(ncalc != nb) {/* error input */ if(ncalc == -1) { #ifdef MATHLIB_STANDALONE free(by); #else vmaxset(vmax); #endif return ML_POSINF; } else if(ncalc < -1) MATHLIB_WARNING4(_("bessel_y(%g): ncalc (=%d) != nb (=%d); alpha=%g. Arg. out of range?\n"), x, ncalc, nb, alpha); else /* ncalc >= 0 */ MATHLIB_WARNING2(_("bessel_y(%g,nu=%g): precision lost in result\n"), x, alpha+(double)nb-1); } x = by[nb-1]; #ifdef MATHLIB_STANDALONE free(by); #else vmaxset(vmax); #endif return x; } /* Called from R: modified version of bessel_y(), accepting a work array * instead of allocating one. */ double bessel_y_ex(double x, double alpha, double *by) { int nb, ncalc; double na; #ifdef IEEE_754 /* NaNs propagated correctly */ if (ISNAN(x) || ISNAN(alpha)) return x + alpha; #endif if (x < 0) { ML_WARNING(ME_RANGE, "bessel_y"); return ML_NAN; } na = floor(alpha); if (alpha < 0) { /* Using Abramowitz & Stegun 9.1.2 * this may not be quite optimal (CPU and accuracy wise) */ return(((alpha - na == 0.5) ? 0 : bessel_y_ex(x, -alpha, by) * cospi(alpha)) - ((alpha == na ) ? 0 : bessel_j_ex(x, -alpha, by) * sinpi(alpha))); } else if (alpha > 1e7) { MATHLIB_WARNING(_("besselY(x, nu): nu=%g too large for bessel_y() algorithm"), alpha); return ML_NAN; } nb = 1+ (int)na;/* nb-1 <= alpha < nb */ alpha -= (double)(nb-1); Y_bessel(&x, &alpha, &nb, by, &ncalc); if(ncalc != nb) {/* error input */ if(ncalc == -1) return ML_POSINF; else if(ncalc < -1) MATHLIB_WARNING4(_("bessel_y(%g): ncalc (=%d) != nb (=%d); alpha=%g. Arg. out of range?\n"), x, ncalc, nb, alpha); else /* ncalc >= 0 */ MATHLIB_WARNING2(_("bessel_y(%g,nu=%g): precision lost in result\n"), x, alpha+(double)nb-1); } x = by[nb-1]; return x; } static void Y_bessel(double *x, double *alpha, int *nb, double *by, int *ncalc) { /* ---------------------------------------------------------------------- This routine calculates Bessel functions Y_(N+ALPHA) (X) v for non-negative argument X, and non-negative order N+ALPHA. Explanation of variables in the calling sequence X - Non-negative argument for which Y's are to be calculated. ALPHA - Fractional part of order for which Y's 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). BY - Output vector of length NB. If the routine terminates normally (NCALC=NB), the vector BY contains the functions Y(ALPHA,X), ... , Y(NB-1+ALPHA,X), If (0 < NCALC < NB), BY(I) contains correct function values for I <= NCALC, and contains the ratios Y(ALPHA+I-1,X)/Y(ALPHA+I-2,X) for the rest of the array. NCALC - Output variable indicating possible errors. Before using the vector BY, 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 Y'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. In this case, BY[0] = 0.0, the remainder of the BY-vector is not calculated, and NCALC is set to MIN0(NB,0)-2 so that NCALC != NB. NCALC = -1: Y(ALPHA,X) >= XINF. The requested function values are set to 0.0. 1 < NCALC < NB: Not all requested function values could be calculated accurately. BY(I) contains correct function values for I <= NCALC, and and the remaining NB-NCALC array elements contain 0.0. Intrinsic functions required are: DBLE, EXP, INT, MAX, MIN, REAL, SQRT Acknowledgement This program draws heavily on Temme's Algol program for Y(a,x) and Y(a+1,x) and on Campbell's programs for Y_nu(x). Temme's scheme is used for x < THRESH, and Campbell's scheme is used in the asymptotic region. Segments of code from both sources have been translated into Fortran 77, merged, and heavily modified. Modifications include parameterization of machine dependencies, use of a new approximation for ln(gamma(x)), and built-in protection against over/underflow. References: "Bessel functions J_nu(x) and Y_nu(x) of float order and float argument," Campbell, J. B., Comp. Phy. Comm. 18, 1979, pp. 133-142. "On the numerical evaluation of the ordinary Bessel function of the second kind," Temme, N. M., J. Comput. Phys. 21, 1976, pp. 343-350. Latest modification: March 19, 1990 Modified by: W. J. Cody Applied Mathematics Division Argonne National Laboratory Argonne, IL 60439 ----------------------------------------------------------------------*/ /* ---------------------------------------------------------------------- Mathematical constants FIVPI = 5*PI PIM5 = 5*PI - 15 ----------------------------------------------------------------------*/ const static double fivpi = 15.707963267948966192; const static double pim5 = .70796326794896619231; /*---------------------------------------------------------------------- Coefficients for Chebyshev polynomial expansion of 1/gamma(1-x), abs(x) <= .5 ----------------------------------------------------------------------*/ const static double ch[21] = { -6.7735241822398840964e-24, -6.1455180116049879894e-23,2.9017595056104745456e-21, 1.3639417919073099464e-19,2.3826220476859635824e-18, -9.0642907957550702534e-18,-1.4943667065169001769e-15, -3.3919078305362211264e-14,-1.7023776642512729175e-13, 9.1609750938768647911e-12,2.4230957900482704055e-10, 1.7451364971382984243e-9,-3.3126119768180852711e-8, -8.6592079961391259661e-7,-4.9717367041957398581e-6, 7.6309597585908126618e-5,.0012719271366545622927, .0017063050710955562222,-.07685284084478667369, -.28387654227602353814,.92187029365045265648 }; /* Local variables */ int i, k, na; double alfa, div, ddiv, even, gamma, term, cosmu, sinmu, b, c, d, e, f, g, h, p, q, r, s, d1, d2, q0, pa,pa1, qa,qa1, en, en1, nu, ex, ya,ya1, twobyx, den, odd, aye, dmu, x2, xna; en1 = ya = ya1 = 0; /* -Wall */ ex = *x; nu = *alpha; if (*nb > 0 && 0. <= nu && nu < 1.) { if(ex < DBL_MIN || ex > xlrg_BESS_Y) { /* Warning is not really appropriate, give * proper limit: * ML_WARNING(ME_RANGE, "Y_bessel"); */ *ncalc = *nb; if(ex > xlrg_BESS_Y) by[0]= 0.; /*was ML_POSINF */ else if(ex < DBL_MIN) by[0]=ML_NEGINF; for(i=0; i < *nb; i++) by[i] = by[0]; return; } xna = trunc(nu + .5); na = (int) xna; if (na == 1) {/* <==> .5 <= *alpha < 1 <==> -5. <= nu < 0 */ nu -= xna; } if (nu == -.5) { p = M_SQRT_2dPI / sqrt(ex); ya = p * sin(ex); ya1 = -p * cos(ex); } else if (ex < 3.) { /* ------------------------------------------------------------- Use Temme's scheme for small X ------------------------------------------------------------- */ b = ex * .5; d = -log(b); f = nu * d; e = pow(b, -nu); if (fabs(nu) < M_eps_sinc) c = M_1_PI; else c = nu / sinpi(nu); /* ------------------------------------------------------------ Computation of sinh(f)/f ------------------------------------------------------------ */ if (fabs(f) < 1.) { x2 = f * f; en = 19.; s = 1.; for (i = 1; i <= 9; ++i) { s = s * x2 / en / (en - 1.) + 1.; en -= 2.; } } else { s = (e - 1. / e) * .5 / f; } /* -------------------------------------------------------- Computation of 1/gamma(1-a) using Chebyshev polynomials */ x2 = nu * nu * 8.; aye = ch[0]; even = 0.; alfa = ch[1]; odd = 0.; for (i = 3; i <= 19; i += 2) { even = -(aye + aye + even); aye = -even * x2 - aye + ch[i - 1]; odd = -(alfa + alfa + odd); alfa = -odd * x2 - alfa + ch[i]; } even = (even * .5 + aye) * x2 - aye + ch[20]; odd = (odd + alfa) * 2.; gamma = odd * nu + even; /* End of computation of 1/gamma(1-a) ----------------------------------------------------------- */ g = e * gamma; e = (e + 1. / e) * .5; f = 2. * c * (odd * e + even * s * d); e = nu * nu; p = g * c; q = M_1_PI / g; c = nu * M_PI_2; if (fabs(c) < M_eps_sinc) r = 1.; else r = sinpi(nu/2) / c; r = M_PI * c * r * r; c = 1.; d = -b * b; h = 0.; ya = f + r * q; ya1 = p; en = 1.; while (fabs(g / (1. + fabs(ya))) + fabs(h / (1. + fabs(ya1))) > DBL_EPSILON) { f = (f * en + p + q) / (en * en - e); c *= (d / en); p /= en - nu; q /= en + nu; g = c * (f + r * q); h = c * p - en * g; ya += g; ya1+= h; en += 1.; } ya = -ya; ya1 = -ya1 / b; } else if (ex < thresh_BESS_Y) { /* -------------------------------------------------------------- Use Temme's scheme for moderate X : 3 <= x < 16 -------------------------------------------------------------- */ c = (.5 - nu) * (.5 + nu); b = ex + ex; e = ex * M_1_PI * cospi(nu) / DBL_EPSILON; e *= e; p = 1.; q = -ex; r = 1. + ex * ex; s = r; en = 2.; while (r * en * en < e) { en1 = en + 1.; d = (en - 1. + c / en) / s; p = (en + en - p * d) / en1; q = (-b + q * d) / en1; s = p * p + q * q; r *= s; en = en1; } f = p / s; p = f; g = -q / s; q = g; L220: en -= 1.; if (en > 0.) { r = en1 * (2. - p) - 2.; s = b + en1 * q; d = (en - 1. + c / en) / (r * r + s * s); p = d * r; q = d * s; e = f + 1.; f = p * e - g * q; g = q * e + p * g; en1 = en; goto L220; } f = 1. + f; d = f * f + g * g; pa = f / d; qa = -g / d; d = nu + .5 - p; q += ex; pa1 = (pa * q - qa * d) / ex; qa1 = (qa * q + pa * d) / ex; b = ex - M_PI_2 * (nu + .5); c = cos(b); s = sin(b); d = M_SQRT_2dPI / sqrt(ex); ya = d * (pa * s + qa * c); ya1 = d * (qa1 * s - pa1 * c); } else { /* x > thresh_BESS_Y */ /* ---------------------------------------------------------- Use Campbell's asymptotic scheme. ---------------------------------------------------------- */ na = 0; d1 = trunc(ex / fivpi); i = (int) d1; dmu = ex - 15. * d1 - d1 * pim5 - (*alpha + .5) * M_PI_2; if (i - (i / 2 << 1) == 0) { cosmu = cos(dmu); sinmu = sin(dmu); } else { cosmu = -cos(dmu); sinmu = -sin(dmu); } ddiv = 8. * ex; dmu = *alpha; den = sqrt(ex); for (k = 1; k <= 2; ++k) { p = cosmu; cosmu = sinmu; sinmu = -p; d1 = (2. * dmu - 1.) * (2. * dmu + 1.); d2 = 0.; div = ddiv; p = 0.; q = 0.; q0 = d1 / div; term = q0; for (i = 2; i <= 20; ++i) { d2 += 8.; d1 -= d2; div += ddiv; term = -term * d1 / div; p += term; d2 += 8.; d1 -= d2; div += ddiv; term *= (d1 / div); q += term; if (fabs(term) <= DBL_EPSILON) { break; } } p += 1.; q += q0; if (k == 1) ya = M_SQRT_2dPI * (p * cosmu - q * sinmu) / den; else ya1 = M_SQRT_2dPI * (p * cosmu - q * sinmu) / den; dmu += 1.; } } if (na == 1) { h = 2. * (nu + 1.) / ex; if (h > 1.) { if (fabs(ya1) > DBL_MAX / h) { h = 0.; ya = 0.; } } h = h * ya1 - ya; ya = ya1; ya1 = h; } /* --------------------------------------------------------------- Now have first one or two Y's --------------------------------------------------------------- */ by[0] = ya; *ncalc = 1; if(*nb > 1) { by[1] = ya1; if (ya1 != 0.) { aye = 1. + *alpha; twobyx = 2. / ex; *ncalc = 2; for (i = 2; i < *nb; ++i) { if (twobyx < 1.) { if (fabs(by[i - 1]) * twobyx >= DBL_MAX / aye) goto L450; } else { if (fabs(by[i - 1]) >= DBL_MAX / aye / twobyx) goto L450; } by[i] = twobyx * aye * by[i - 1] - by[i - 2]; aye += 1.; ++(*ncalc); } } } L450: for (i = *ncalc; i < *nb; ++i) by[i] = ML_NEGINF;/* was 0 */ } else { by[0] = 0.; *ncalc = min0(*nb,0) - 1; } }