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bessel_i_standalone.c
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bessel_i_standalone.c
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/*
* Mathlib : A C Library of Special Functions
* Copyright (C) 1998-2014 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/ribesl Fortran translated by f2c,...
* ------------------------------=#---- Martin Maechler, ETH Zurich
*/
#include <stdio.h>
#include <math.h>
#include <stdlib.h>
#include <float.h>
#include <stdbool.h>
#define MATHLIB_STANDALONE
#define IEEE_754
#define min0(x, y)(((x) <= (y)) ? (x) : (y))
#define ME_RANGE "range error"
#define ML_WARNING(x, y) printf("%s %s\n", x, y)
#define MATHLIB_ERROR(x, y) printf(x, y);
#define MATHLIB_WARNING4(a, b, c, d, e) fprintf(stderr, a, b, c, d, e)
#define MATHLIB_WARNING2(a, b, c) fprintf(stderr, a, b, c)
#define MATHLIB_WARNING(a, b) fprintf(stderr, a, b)
#define enmten_BESS 8.9e-308
#define enten_BESS 1e308
#define nsig_BESS 16
#define ensig_BESS 1e16
#define rtnsig_BESS 1e-4
#define xlrg_BESS_IJ 1e5
#define exparg_BESS 709.
#define ME_RANGE_NAN 0x7FF8000000000001
double Rf_gamma_cody(double x);
double sinpi(double x);
double bessel_k(double x, double alpha, double expo);
double R_pow_di(double x, int n);
static void I_bessel(double * x, double * alpha, int * nb,
int * ize, double * bi, int * ncalc);
double bessel_i(double x, double alpha, bool expon_scaled) {
int nb, ncalc, ize;
double na, * bi;
const int expo = expon_scaled ? 2 : 1;
// NaNs propagated correctly
if (isnan(x) || isnan(alpha)) {
return x + alpha;
}
if (x < 0) {
MATHLIB_WARNING("%s ME RANGE\n", "bessel_i");
return NAN; //ME_RANGE_NAN;
}
ize = (int) expo;
na = floor(alpha);
if (alpha < 0) {
// Using Abramowitz & Stegun 9.6.2 & 9.6.6
// this may not be quite optimal (CPU and accuracy wise)
return (bessel_i(x, -alpha, expo) +
((alpha == na) ? /* sin(pi * alpha) = 0 */ 0 :
bessel_k(x, -alpha, expo) *
((ize == 1) ? 2. : 2. * exp(-2. * x)) / M_PI * sinpi(-alpha)));
}
nb = 1 + (int) na; /* nb-1 <= alpha < nb */
alpha -= (double)(nb - 1);
bi = (double * ) calloc(nb, sizeof(double));
if (!bi) {
MATHLIB_ERROR("%s", "bessel_i allocation error");
return NAN;
}
I_bessel( & x, & alpha, & nb, & ize, bi, & ncalc);
if (ncalc != nb) {
/* error input */
if (ncalc < 0) {
MATHLIB_WARNING4("bessel_i(%g): ncalc (=%d) != nb (=%d); alpha=%g. Arg. out of range?\n",
x, ncalc, nb, alpha);
} else {
MATHLIB_WARNING2("bessel_i(%g,nu=%g): precision lost in result\n",
x, alpha + (double) nb - 1);
}
}
x = bi[nb - 1];
free(bi);
return x;
}
static void I_bessel(double * x, double * alpha, int * nb,
int * ize, double * bi, int * ncalc) {
/* -------------------------------------------------------------------
This routine calculates Bessel functions I_(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
I's or exponentially scaled I's (I*EXP(-X))
are to be calculated. If I's are to be calculated,
X must be less than exparg_BESS (IZE=1) or xlrg_BESS_IJ (IZE=2),
(see bessel.h).
ALPHA - Fractional part of order for which
I's or exponentially scaled I's (I*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 I's are to be calculated,
= 2 if exponentially scaled I's are to be calculated.
BI - Output vector of length NB. If the routine
terminates normally (NCALC=NB), the vector BI contains the
functions I(ALPHA,X) through I(NB-1+ALPHA,X), or the
corresponding exponentially scaled functions.
NCALC - Output variable indicating possible errors.
Before using the vector BI, 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 I's are
calculated to the desired accuracy.
NCALC < 0: An argument is out of range. For example,
NB <= 0, IZE is not 1 or 2, or IZE=1 and ABS(X) >= EXPARG_BESS.
In this case, the BI-vector is not calculated, and NCALC is
set to MIN0(NB,0)-1 so that NCALC != NB.
NB > NCALC > 0: Not all requested function values could
be calculated accurately. This usually occurs because NB is
much larger than ABS(X). In this case, BI[N] is calculated
to the desired accuracy for N <= NCALC, but precision
is lost for NCALC < N <= NB. If BI[N] does not vanish
for N > NCALC (because it is too small to be represented),
and BI[N]/BI[NCALC] = 10**(-K), then only the first NSIG-K
significant figures of BI[N] can be trusted.
Intrinsic functions required are:
DBLE, EXP, gamma_cody, INT, MAX, MIN, REAL, SQRT
Acknowledgement
This program is based on a program written by David J.
Sookne (2) that computes values of the Bessel functions J or
I of float argument and long order. Modifications include
the restriction of the computation to the I Bessel function
of non-negative float argument, the extension of the computation
to arbitrary positive order, the inclusion of optional
exponential scaling, and the elimination of most underflow.
An earlier version was published in (3).
References: "A Note on Backward Recurrence Algorithms," Olver,
F. W. J., and Sookne, D. J., Math. Comp. 26, 1972,
pp 941-947.
"Bessel Functions of Real Argument and Integer Order,"
Sookne, D. J., NBS Jour. of Res. B. 77B, 1973, pp
125-132.
"ALGORITHM 597, Sequence of Modified Bessel Functions
of the First Kind," Cody, W. J., Trans. Math. Soft.,
1983, pp. 242-245.
Latest modification: May 30, 1989
Modified by: W. J. Cody and L. Stoltz
Applied Mathematics Division
Argonne National Laboratory
Argonne, IL 60439
*/
/*-------------------------------------------------------------------
Mathematical constants
-------------------------------------------------------------------*/
const static double const__ = 1.585;
/* Local variables */
int nend, intx, nbmx, k, l, n, nstart;
double pold, test, p, em, en, empal, emp2al, halfx,
aa, bb, cc, psave, plast, tover, psavel, sum, nu, twonu;
/*Parameter adjustments */
--bi;
nu = * alpha;
twonu = nu + nu;
/*-------------------------------------------------------------------
Check for X, NB, OR IZE out of range.
------------------------------------------------------------------- */
if (
!( *
nb > 0 &&
*
x >= 0. &&
(0. <= nu && nu < 1.) &&
(1 <= * ize && * ize <= 2)
))
{ // argument out of range
* ncalc = min0( * nb, 0) - 1;
return;
}
* ncalc = * nb;
if ( * ize == 1 && * x > exparg_BESS) {
for (k = 1; k <= * nb; k++)
bi[k] = INFINITY; /* the limit *is* = Inf */
return;
}
if ( * ize == 2 && * x > xlrg_BESS_IJ) {
for (k = 1; k <= * nb; k++)
bi[k] = 0.; // The limit exp(-x) * I_nu(x) --> 0 :
return;
}
intx = (int)( * x); // fine, since *x <= xlrg_BESS_IJ <<< LONG_MAX
if ( * x >= rtnsig_BESS) {
// "non-small" x ( >= 1e-4 )
//-------------------------------------------------------------------
// Initialize the forward sweep, the P-sequence of Olver
// -------------------------------------------------------------------
nbmx = * nb - intx;
n = intx + 1;
en = (double)(n + n) + twonu;
plast = 1.;
p = en / * x;
/* ------------------------------------------------
Calculate general significance test
------------------------------------------------ */
test = ensig_BESS + ensig_BESS;
if (intx << 1 > nsig_BESS * 5) {
test = sqrt(test * p);
} else {
test /= R_pow_di(const__, intx);
}
bool L120 = false;
if (nbmx >= 3) {
bool L90 = false;
/* --------------------------------------------------
Calculate P-sequence until N = NB-1
Check for possible overflow.
------------------------------------------------ */
tover = enten_BESS / ensig_BESS;
nstart = intx + 2;
nend = * nb - 1;
for (k = nstart; k <= nend; ++k) {
n = k;
en += 2.;
pold = plast;
plast = p;
p = en * plast / * x + pold;
if (p > tover) {
// ------------------------------------------------
// To avoid overflow, divide P-sequence by TOVER.
// Calculate P-sequence until ABS(P) > 1.
// ----------------------------------------------
tover = enten_BESS;
p /= tover;
plast /= tover;
psave = p;
psavel = plast;
nstart = n + 1;
do {
++n;
en += 2.;
pold = plast;
plast = p;
p = en * plast / * x + pold;
}
while (p <= 1.);
bb = en / * x;
// ------------------------------------------------
// Calculate backward test, and find NCALC,
// the highest N such that the test is passed.
// ------------------------------------------------
test = pold * plast / ensig_BESS;
test *= .5 - .5 / (bb * bb);
p = plast * tover;
--n;
en -= 2.;
nend = min0( * nb, n);
for (l = nstart; l <= nend; ++l) {
* ncalc = l;
pold = psavel;
psavel = psave;
psave = en * psavel / * x + pold;
if (psave * psavel > test) {
L90 = true;
break;
//goto L90;
}
} //for
if (L90 == false) {
* ncalc = nend + 1;
}
//L90:
--( * ncalc);
L120 = true;
break;
//goto L120;
} // if
} // for
if (L120 == false) {
n = nend;
en = (double)(n + n) + twonu;
// ---------------------------------------------------
// Calculate special significance test for NBMX > 2.
// ---------------------------------------------------
test = fmax(test, sqrt(plast * ensig_BESS) * sqrt(p + p));
}
} // if
// --------------------------------------------------------
// Calculate P-sequence until significance test passed.
// --------------------------------------------------------
if (L120 == false) {
do {
++n;
en += 2.;
pold = plast;
plast = p;
p = en * plast / * x + pold;
} while (p < test);
}
//L120:
/* -------------------------------------------------------------------
Initialize the backward recursion and the normalization sum.
------------------------------------------------------------------- */
++n;
en += 2.;
bb = 0.;
aa = 1. / p;
em = (double) n - 1.;
empal = em + nu;
emp2al = em - 1. + twonu;
sum = aa * empal * emp2al / em;
nend = n - * nb;
if (nend < 0) {
/* -----------------------------------------------------
N < NB, so store BI[N] and set higher orders to 0..
----------------------------------------------------- */
bi[n] = aa;
nend = -nend;
for (l = 1; l <= nend; ++l) {
bi[n + l] = 0.;
}
} else {
if (nend > 0) {
// -----------------------------------------------------
// Recur backward via difference equation,
// calculating (but not storing) BI[N], until N = NB.
// ---------------------------------------------------
for (l = 1; l <= nend; ++l) {
--n;
en -= 2.;
cc = bb;
bb = aa;
// for x ~= 1500, sum would overflow to 'inf' here,
// and the final bi[] /= sum would give 0 wrongly;
// RE-normalize (aa, sum) here -- no need to undo
if (nend > 100 && aa > 1e200) {
/* multiply by 2^-900 = 1.18e-271 */
cc = ldexp(cc, -900);
bb = ldexp(bb, -900);
sum = ldexp(sum, -900);
}
aa = en * bb / * x + cc;
em -= 1.;
emp2al -= 1.;
if (n == 1) {
break;
}
if (n == 2) {
emp2al = 1.;
}
empal -= 1.;
sum = (sum + aa * empal) * emp2al / em;
}
}
// ---------------------------------------------------
// Store BI[NB]
// ---------------------------------------------------
bi[n] = aa;
if ( * nb <= 1) {
sum = sum + sum + aa;
goto L230;
}
// -------------------------------------------------
// Calculate and Store BI[NB-1]
// -------------------------------------------------
--n;
en -= 2.;
bi[n] = en * aa / * x + bb;
if (n == 1) {
goto L220;
}
em -= 1.;
if (n == 2)
emp2al = 1.;
else
emp2al -= 1.;
empal -= 1.;
sum = (sum + bi[n] * empal) * emp2al / em;
}
nend = n - 2;
if (nend > 0) {
// --------------------------------------------
// Calculate via difference equation
// and store BI[N], until N = 2.
// ------------------------------------------
for (l = 1; l <= nend; ++l) {
--n;
en -= 2.;
bi[n] = en * bi[n + 1] / * x + bi[n + 2];
em -= 1.;
if (n == 2)
emp2al = 1.;
else
emp2al -= 1.;
empal -= 1.;
sum = (sum + bi[n] * empal) * emp2al / em;
}
}
// ----------------------------------------------
// Calculate BI[1]
// --------------------------------------------
bi[1] = 2. * empal * bi[2] / * x + bi[3];
L220:
sum = sum + sum + bi[1];
L230:
// ---------------------------------------------------------
// Normalize. Divide all BI[N] by sum.
// ---------------------------------------------------------
if (nu != 0.) {
sum *= (Rf_gamma_cody(1. + nu) * pow( * x * .5, -nu));
}
if ( * ize == 1) {
sum *= exp(-( * x));
}
aa = enmten_BESS;
if (sum > 1.) {
aa *= sum;
}
for (n = 1; n <= * nb; ++n) {
if (bi[n] < aa) {
bi[n] = 0.;
} else {
bi[n] /= sum;
}
}
return;
} else { // small x < 1e-4
// -----------------------------------------------------------
// Two-term ascending series for small X.
// -----------------------------------------------------------
aa = 1.;
empal = 1. + nu;
// No need to check for underflow
halfx = .5 * * x;
if (nu != 0.) {
aa = pow(halfx, nu) / Rf_gamma_cody(empal);
}
if ( * ize == 2) {
aa *= exp(-( * x));
}
bb = halfx * halfx;
bi[1] = aa + aa * bb / empal;
if ( * x != 0. && bi[1] == 0.) {
* ncalc = 0;
}
if ( * nb > 1) {
if ( * x == 0.) {
for (n = 2; n <= * nb; ++n)
bi[n] = 0.;
} else {
// -------------------------------------------------
// Calculate higher-order functions.
// -------------------------------------------------
cc = halfx;
tover = (enmten_BESS + enmten_BESS) / * x;
if (bb != 0.) {
tover = enmten_BESS / bb;
}
for (n = 2; n <= * nb; ++n) {
aa /= empal;
empal += 1.;
aa *= cc;
if (aa <= tover * empal)
bi[n] = aa = 0.;
else
bi[n] = aa + aa * bb / empal;
if (bi[n] == 0. && * ncalc > n)
*
ncalc = n - 1;
}
}
}
}
}