bessel_k.c

/*
 *  Mathlib : A C Library of Special Functions
 *  Copyright (C) 1998-2012 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
 *  http://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, long *nb,
		     long *ize, double *bk, long *ncalc);

double bessel_k(double x, double alpha, double expo)
{
    long 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 = (long)expo;
    if(alpha < 0)
	alpha = -alpha;
    nb = 1+ (long)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 (=%ld) != nb (=%ld); 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)
{
    long 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 = (long)expo;
    if(alpha < 0)
	alpha = -alpha;
    nb = 1+ (long)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 (=%ld) != nb (=%ld); 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, long *nb,
		     long *ize, double *bk, long *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 */
    long 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 = ftrunc(estm[0] / ex + estm[1]);
		m = (long) 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 = ftrunc(estm[2] * ex + estm[3]);
		m = (long) 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 = ftrunc(estm[4] / ex + estm[5]);
		m = (long) 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((long) (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)++;
	}
    }
}