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i0.c

/*                                        i0.c
 *
 *    Modified Bessel function of order zero
 *
 *
 *
 * SYNOPSIS:
 *
 * double x, y, i0();
 *
 * y = i0( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns modified Bessel function of order zero of the
 * argument.
 *
 * The function is defined as i0(x) = j0( ix ).
 *
 * The range is partitioned into the two intervals [0,8] and
 * (8, infinity).  Chebyshev polynomial expansions are employed
 * in each interval.
 *
 *
 *
 * ACCURACY:
 *
 *                      Relative error:
 * arithmetic   domain     # trials      peak         rms
 *    DEC       0,30         6000       8.2e-17     1.9e-17
 *    IEEE      0,30        30000       5.8e-16     1.4e-16
 *
 */
/*                                       i0e.c
 *
 *    Modified Bessel function of order zero,
 *    exponentially scaled
 *
 *
 *
 * SYNOPSIS:
 *
 * double x, y, i0e();
 *
 * y = i0e( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns exponentially scaled modified Bessel function
 * of order zero of the argument.
 *
 * The function is defined as i0e(x) = exp(-|x|) j0( ix ).
 *
 *
 *
 * ACCURACY:
 *
 *                      Relative error:
 * arithmetic   domain     # trials      peak         rms
 *    IEEE      0,30        30000       5.4e-16     1.2e-16
 * See i0().
 *
 */

/*                                        i0.c        */


/*
Cephes Math Library Release 2.0:  April, 1987
Copyright 1984, 1987 by Stephen L. Moshier
Direct inquiries to 30 Frost Street, Cambridge, MA 02140
*/

#include <math.h>
#include "mconf.h"
#include "cephes.h"

/* Chebyshev coefficients for exp(-x) I0(x)
 * in the interval [0,8].
 *
 * lim(x->0){ exp(-x) I0(x) } = 1.
 */

#ifdef UNK
static double A[] =
{
-4.41534164647933937950E-18,
 3.33079451882223809783E-17,
-2.43127984654795469359E-16,
 1.71539128555513303061E-15,
-1.16853328779934516808E-14,
 7.67618549860493561688E-14,
-4.85644678311192946090E-13,
 2.95505266312963983461E-12,
-1.72682629144155570723E-11,
 9.67580903537323691224E-11,
-5.18979560163526290666E-10,
 2.65982372468238665035E-9,
-1.30002500998624804212E-8,
 6.04699502254191894932E-8,
-2.67079385394061173391E-7,
 1.11738753912010371815E-6,
-4.41673835845875056359E-6,
 1.64484480707288970893E-5,
-5.75419501008210370398E-5,
 1.88502885095841655729E-4,
-5.76375574538582365885E-4,
 1.63947561694133579842E-3,
-4.32430999505057594430E-3,
 1.05464603945949983183E-2,
-2.37374148058994688156E-2,
 4.93052842396707084878E-2,
-9.49010970480476444210E-2,
 1.71620901522208775349E-1,
-3.04682672343198398683E-1,
 6.76795274409476084995E-1
};
#endif

#ifdef DEC
static unsigned short A[] = {
0121642,0162671,0004646,0103567,
0022431,0115424,0135755,0026104,
0123214,0023533,0110365,0156635,
0023767,0033304,0117662,0172716,
0124522,0100426,0012277,0157531,
0025254,0155062,0054461,0030465,
0126010,0131143,0013560,0153604,
0026517,0170577,0006336,0114437,
0127227,0162253,0152243,0052734,
0027724,0142766,0061641,0160200,
0130416,0123760,0116564,0125262,
0031066,0144035,0021246,0054641,
0131537,0053664,0060131,0102530,
0032201,0155664,0165153,0020652,
0132617,0061434,0074423,0176145,
0033225,0174444,0136147,0122542,
0133624,0031576,0056453,0020470,
0034211,0175305,0172321,0041314,
0134561,0054462,0147040,0165315,
0035105,0124333,0120203,0162532,
0135427,0013750,0174257,0055221,
0035726,0161654,0050220,0100162,
0136215,0131361,0000325,0041110,
0036454,0145417,0117357,0017352,
0136702,0072367,0104415,0133574,
0037111,0172126,0072505,0014544,
0137302,0055601,0120550,0033523,
0037457,0136543,0136544,0043002,
0137633,0177536,0001276,0066150,
0040055,0041164,0100655,0010521
};
#endif

#ifdef IBMPC
static unsigned short A[] = {
0xd0ef,0x2134,0x5cb7,0xbc54,
0xa589,0x977d,0x3362,0x3c83,
0xbbb4,0x721e,0x84eb,0xbcb1,
0x5eba,0x93f6,0xe6d8,0x3cde,
0xfbeb,0xc297,0x5022,0xbd0a,
0x2627,0x4b26,0x9b46,0x3d35,
0x1af0,0x62ee,0x164c,0xbd61,
0xd324,0xe19b,0xfe2f,0x3d89,
0x6abc,0x7a94,0xfc95,0xbdb2,
0x3c10,0xcc74,0x98be,0x3dda,
0x9556,0x13ae,0xd4fe,0xbe01,
0xcb34,0xa454,0xd903,0x3e26,
0x30ab,0x8c0b,0xeaf6,0xbe4b,
0x6435,0x9d4d,0x3b76,0x3e70,
0x7f8d,0x8f22,0xec63,0xbe91,
0xf4ac,0x978c,0xbf24,0x3eb2,
0x6427,0xcba5,0x866f,0xbed2,
0x2859,0xbe9a,0x3f58,0x3ef1,
0x1d5a,0x59c4,0x2b26,0xbf0e,
0x7cab,0x7410,0xb51b,0x3f28,
0xeb52,0x1f15,0xe2fd,0xbf42,
0x100e,0x8a12,0xdc75,0x3f5a,
0xa849,0x201a,0xb65e,0xbf71,
0xe3dd,0xf3dd,0x9961,0x3f85,
0xb6f0,0xf121,0x4e9e,0xbf98,
0xa32d,0xcea8,0x3e8a,0x3fa9,
0x06ea,0x342d,0x4b70,0xbfb8,
0x88c0,0x77ac,0xf7ac,0x3fc5,
0xcd8d,0xc057,0x7feb,0xbfd3,
0xa22a,0x9035,0xa84e,0x3fe5,
};
#endif

#ifdef MIEEE
static unsigned short A[] = {
0xbc54,0x5cb7,0x2134,0xd0ef,
0x3c83,0x3362,0x977d,0xa589,
0xbcb1,0x84eb,0x721e,0xbbb4,
0x3cde,0xe6d8,0x93f6,0x5eba,
0xbd0a,0x5022,0xc297,0xfbeb,
0x3d35,0x9b46,0x4b26,0x2627,
0xbd61,0x164c,0x62ee,0x1af0,
0x3d89,0xfe2f,0xe19b,0xd324,
0xbdb2,0xfc95,0x7a94,0x6abc,
0x3dda,0x98be,0xcc74,0x3c10,
0xbe01,0xd4fe,0x13ae,0x9556,
0x3e26,0xd903,0xa454,0xcb34,
0xbe4b,0xeaf6,0x8c0b,0x30ab,
0x3e70,0x3b76,0x9d4d,0x6435,
0xbe91,0xec63,0x8f22,0x7f8d,
0x3eb2,0xbf24,0x978c,0xf4ac,
0xbed2,0x866f,0xcba5,0x6427,
0x3ef1,0x3f58,0xbe9a,0x2859,
0xbf0e,0x2b26,0x59c4,0x1d5a,
0x3f28,0xb51b,0x7410,0x7cab,
0xbf42,0xe2fd,0x1f15,0xeb52,
0x3f5a,0xdc75,0x8a12,0x100e,
0xbf71,0xb65e,0x201a,0xa849,
0x3f85,0x9961,0xf3dd,0xe3dd,
0xbf98,0x4e9e,0xf121,0xb6f0,
0x3fa9,0x3e8a,0xcea8,0xa32d,
0xbfb8,0x4b70,0x342d,0x06ea,
0x3fc5,0xf7ac,0x77ac,0x88c0,
0xbfd3,0x7feb,0xc057,0xcd8d,
0x3fe5,0xa84e,0x9035,0xa22a
};
#endif


/* Chebyshev coefficients for exp(-x) sqrt(x) I0(x)
 * in the inverted interval [8,infinity].
 *
 * lim(x->inf){ exp(-x) sqrt(x) I0(x) } = 1/sqrt(2pi).
 */

#ifdef UNK
static double B[] =
{
-7.23318048787475395456E-18,
-4.83050448594418207126E-18,
 4.46562142029675999901E-17,
 3.46122286769746109310E-17,
-2.82762398051658348494E-16,
-3.42548561967721913462E-16,
 1.77256013305652638360E-15,
 3.81168066935262242075E-15,
-9.55484669882830764870E-15,
-4.15056934728722208663E-14,
 1.54008621752140982691E-14,
 3.85277838274214270114E-13,
 7.18012445138366623367E-13,
-1.79417853150680611778E-12,
-1.32158118404477131188E-11,
-3.14991652796324136454E-11,
 1.18891471078464383424E-11,
 4.94060238822496958910E-10,
 3.39623202570838634515E-9,
 2.26666899049817806459E-8,
 2.04891858946906374183E-7,
 2.89137052083475648297E-6,
 6.88975834691682398426E-5,
 3.36911647825569408990E-3,
 8.04490411014108831608E-1
};
#endif

#ifdef DEC
static unsigned short B[] = {
0122005,0066672,0123124,0054311,
0121662,0033323,0030214,0104602,
0022515,0170300,0113314,0020413,
0022437,0117350,0035402,0007146,
0123243,0000135,0057220,0177435,
0123305,0073476,0144106,0170702,
0023777,0071755,0017527,0154373,
0024211,0052214,0102247,0033270,
0124454,0017763,0171453,0012322,
0125072,0166316,0075505,0154616,
0024612,0133770,0065376,0025045,
0025730,0162143,0056036,0001632,
0026112,0015077,0150464,0063542,
0126374,0101030,0014274,0065457,
0127150,0077271,0125763,0157617,
0127412,0104350,0040713,0120445,
0027121,0023765,0057500,0001165,
0030407,0147146,0003643,0075644,
0031151,0061445,0044422,0156065,
0031702,0132224,0003266,0125551,
0032534,0000076,0147153,0005555,
0033502,0004536,0004016,0026055,
0034620,0076433,0142314,0171215,
0036134,0146145,0013454,0101104,
0040115,0171425,0062500,0047133
};
#endif

#ifdef IBMPC
static unsigned short B[] = {
0x8b19,0x54ca,0xadb7,0xbc60,
0x9130,0x6611,0x46da,0xbc56,
0x8421,0x12d9,0xbe18,0x3c89,
0x41cd,0x0760,0xf3dd,0x3c83,
0x1fe4,0xabd2,0x600b,0xbcb4,
0xde38,0xd908,0xaee7,0xbcb8,
0xfb1f,0xa3ea,0xee7d,0x3cdf,
0xe6d7,0x9094,0x2a91,0x3cf1,
0x629a,0x7e65,0x83fe,0xbd05,
0xbb32,0xcf68,0x5d99,0xbd27,
0xc545,0x0d5f,0x56ff,0x3d11,
0xc073,0x6b83,0x1c8c,0x3d5b,
0x8cec,0xfa26,0x4347,0x3d69,
0x8d66,0x0317,0x9043,0xbd7f,
0x7bf2,0x357e,0x0fd7,0xbdad,
0x7425,0x0839,0x511d,0xbdc1,
0x004f,0xabe8,0x24fe,0x3daa,
0x6f75,0xc0f4,0xf9cc,0x3e00,
0x5b87,0xa922,0x2c64,0x3e2d,
0xd56d,0x80d6,0x5692,0x3e58,
0x616e,0xd9cd,0x8007,0x3e8b,
0xc586,0xc101,0x412b,0x3ec8,
0x9e52,0x7899,0x0fa3,0x3f12,
0x9049,0xa2e5,0x998c,0x3f6b,
0x09cb,0xaca8,0xbe62,0x3fe9
};
#endif

#ifdef MIEEE
static unsigned short B[] = {
0xbc60,0xadb7,0x54ca,0x8b19,
0xbc56,0x46da,0x6611,0x9130,
0x3c89,0xbe18,0x12d9,0x8421,
0x3c83,0xf3dd,0x0760,0x41cd,
0xbcb4,0x600b,0xabd2,0x1fe4,
0xbcb8,0xaee7,0xd908,0xde38,
0x3cdf,0xee7d,0xa3ea,0xfb1f,
0x3cf1,0x2a91,0x9094,0xe6d7,
0xbd05,0x83fe,0x7e65,0x629a,
0xbd27,0x5d99,0xcf68,0xbb32,
0x3d11,0x56ff,0x0d5f,0xc545,
0x3d5b,0x1c8c,0x6b83,0xc073,
0x3d69,0x4347,0xfa26,0x8cec,
0xbd7f,0x9043,0x0317,0x8d66,
0xbdad,0x0fd7,0x357e,0x7bf2,
0xbdc1,0x511d,0x0839,0x7425,
0x3daa,0x24fe,0xabe8,0x004f,
0x3e00,0xf9cc,0xc0f4,0x6f75,
0x3e2d,0x2c64,0xa922,0x5b87,
0x3e58,0x5692,0x80d6,0xd56d,
0x3e8b,0x8007,0xd9cd,0x616e,
0x3ec8,0x412b,0xc101,0xc586,
0x3f12,0x0fa3,0x7899,0x9e52,
0x3f6b,0x998c,0xa2e5,0x9049,
0x3fe9,0xbe62,0xaca8,0x09cb
};
#endif

double i0(double x)
{
double y;

if( x < 0 )
      x = -x;
if( x <= 8.0 )
      {
      y = (x/2.0) - 2.0;
      return( exp(x) * chbevl( y, A, 30 ) );
      }

return(  exp(x) * chbevl( 32.0/x - 2.0, B, 25 ) / sqrt(x) );

}




double i0e(double x )
{
double y;

if( x < 0 )
      x = -x;
if( x <= 8.0 )
      {
      y = (x/2.0) - 2.0;
      return( chbevl( y, A, 30 ) );
      }

return(  chbevl( 32.0/x - 2.0, B, 25 ) / sqrt(x) );

}

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