bessel_k.c from R math library.
/*
* 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 <R_ext/Memory.h>
#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_ERROR(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_ERROR(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_ERROR(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)++;
}
}
}