/* specfunc/bessel_K1.c

 * 

 * Copyright (C) 1996, 1997, 1998, 1999, 2000 Gerard Jungman

 * Copyright (C) 2016 Pavel Holoborodko, Patrick Alken

 * 

 * 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 3 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, write to the Free Software

 * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.

 */

/* Author:  G. Jungman */

#include <config.h>

#include <gsl/gsl_math.h>

#include <gsl/gsl_errno.h>

#include <gsl/gsl_poly.h>

#include <gsl/gsl_sf_exp.h>

#include <gsl/gsl_sf_bessel.h>

#include "error.h"

#include "chebyshev.h"

#include "cheb_eval.c"

/*-*-*-*-*-*-*-*-*-*-*-* Private Section *-*-*-*-*-*-*-*-*-*-*-*/

/*

 Minimax rational approximation for [0,1), peak relative error = 1.83*GSL_DBL_EPSILON.

 Source: http://www.advanpix.com/?p=3987

*/

static double k1_poly[9] = {

  -3.0796575782920622440538935e-01,

  -8.5370719728650778045782736e-02,

  -4.6421827664715603298154971e-03,

  -1.1253607036630425931072996e-04,

  -1.5592887702110907110292728e-06,

  -1.4030163679125934402498239e-08,

  -8.8718998640336832196558868e-11,

  -4.1614323580221539328960335e-13,

  -1.5261293392975541707230366e-15

};

static double i1_poly[7] = {

  8.3333333333333325191635191e-02,

  6.9444444444467956461838830e-03,

  3.4722222211230452695165215e-04,

  1.1574075952009842696580084e-05,

  2.7555870002088181016676934e-07,

  4.9724386164128529514040614e-09

};

/*

 Chebyshev expansion for [1,8], peak relative error = 1.28*GSL_DBL_EPSILON. 

 Source: Pavel Holoborodko.

*/

static double ak1_data[25] = {

  +2.07996868001418246e-01,

  +1.62581565017881476e-01,

  -5.87070423518863640e-03,

  +4.95021520115789501e-04,

  -5.78958347598556986e-05,

  +8.18614610209334726e-06,

  -1.31604832009487277e-06,

  +2.32546031520101213e-07,

  -4.42206518311557987e-08,

  +8.92163994883100361e-09,

  -1.89046270526983427e-09,

  +4.17568808108504702e-10,

  -9.55912361791375794e-11,

  +2.25769353153867758e-11,

  -5.48128000211158482e-12,

  +1.36386122546441926e-12,

  -3.46936690565986409e-13,

  +9.00354564415705942e-14,

  -2.37950577776254432e-14,

  +6.39447503964025336e-15,

  -1.74498363492322044e-15,

  +4.82994547989290473e-16,

  -1.35460927805445606e-16,

  +3.84604274446777234e-17,

  -1.10456856122581316e-17

};

static cheb_series ak1_cs = {

  ak1_data,

  24,

  -1, 1,

  9

};

/* 

 Chebyshev expansion for [8,inf), peak relative error = 1.25*GSL_DBL_EPSILON.

 Source: SLATEC/dbsk1e.f

*/

static double ak12_data[14] = {

  +.637930834373900104E-1,

  +.283288781304972094E-1,

  -.247537067390525035E-3,

  +.577197245160724882E-5,

  -.206893921953654830E-6,

  +.973998344138180418E-8,

  -.558533614038062498E-9,

  +.373299663404618524E-10,

  -.282505196102322545E-11,

  +.237201900248414417E-12,

  -.217667738799175398E-13,

  +.215791416161603245E-14,

  -.229019693071826928E-15,

  +.258288572982327496E-16

};

static cheb_series ak12_cs = {

  ak12_data,

  13,

  -1, 1,

  7

};

/*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/

int gsl_sf_bessel_K1_scaled_e(const double x, gsl_sf_result * result)

{

  /* CHECK_POINTER(result) */

  if(x <= 0.0) {

    DOMAIN_ERROR(result);

  }

  else if(x < 2.0*GSL_DBL_MIN) {

    OVERFLOW_ERROR(result);

  }

  else if(x < 1.0) {

    const double lx = log(x);

    const double ex = exp(x);

    const double x2 = x*x;

    const double t  = 0.25*x2;    

    const double i1 = 0.5 * x * (1.0 + t * (0.5 + t * gsl_poly_eval(i1_poly,6,t)));

    result->val  = ex * (x2 * gsl_poly_eval(k1_poly,9,x2) + x * lx * i1 + 1) / x;

    result->err  = ex * (1.6+fabs(lx)*0.6) * GSL_DBL_EPSILON;

    result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);

    return GSL_SUCCESS;

  }

  else if(x <= 8.0) {

    const double sx = sqrt(x);

    gsl_sf_result c;

    cheb_eval_e(&ak1_cs, (16.0/x-9.0)/7.0, &c);

    result->val  = (1.375 + c.val) / sx; /* 1.375 = 11/8 */

    result->err  = c.err / sx;

    result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);

    return GSL_SUCCESS;

  }

  else {

    const double sx = sqrt(x);

    gsl_sf_result c;

    cheb_eval_e(&ak12_cs, 16.0/x-1.0, &c);

    result->val  = (1.25 + c.val) / sx;

    result->err  = c.err / sx;

    result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);

    return GSL_SUCCESS;

  }

}

int gsl_sf_bessel_K1_e(const double x, gsl_sf_result * result)

{

  /* CHECK_POINTER(result) */

  if(x <= 0.0) {

    DOMAIN_ERROR(result);

  }

  else if(x < 2.0*GSL_DBL_MIN) {

    OVERFLOW_ERROR(result);

  }

  else if(x < 1.0) {

    const double lx = log(x);

    const double x2 = x*x;

    const double t  = 0.25*x2;    

    const double i1 = 0.5 * x * (1.0 + t * (0.5 + t * gsl_poly_eval(i1_poly,6,t)));

    result->val  = (x2 * gsl_poly_eval(k1_poly,9,x2) + x * lx * i1 + 1) / x;

    result->err  = (1.6+fabs(lx)*0.6) * GSL_DBL_EPSILON;

    result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);

    return GSL_SUCCESS;      

  }

  else {

    gsl_sf_result K1_scaled;

    int stat_K1 = gsl_sf_bessel_K1_scaled_e(x, &K1_scaled);

    int stat_e  = gsl_sf_exp_mult_err_e(-x, 0.0,

                                           K1_scaled.val, K1_scaled.err,

                                           result);

    result->err = fabs(result->val) * (GSL_DBL_EPSILON*fabs(x) + K1_scaled.err/K1_scaled.val);

    return GSL_ERROR_SELECT_2(stat_e, stat_K1);

  }

}

/*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/

#include "eval.h"

double gsl_sf_bessel_K1_scaled(const double x)

{

  EVAL_RESULT(gsl_sf_bessel_K1_scaled_e(x, &result));

}

double gsl_sf_bessel_K1(const double x)

{

  EVAL_RESULT(gsl_sf_bessel_K1_e(x, &result));

}