/* k1.c * * Modified Bessel function, third kind, order one * * * * SYNOPSIS: * * double x, y, k1(); * * y = k1( x ); * * * * DESCRIPTION: * * Computes the modified Bessel function of the third kind * of order one of the argument. * * The range is partitioned into the two intervals [0,2] and * (2, infinity). Chebyshev polynomial expansions are employed * in each interval. * * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * DEC 0, 30 3300 8.9e-17 2.2e-17 * IEEE 0, 30 30000 1.2e-15 1.6e-16 * * ERROR MESSAGES: * * message condition value returned * k1 domain x <= 0 MAXNUM * */ /* k1e.c * * Modified Bessel function, third kind, order one, * exponentially scaled * * * * SYNOPSIS: * * double x, y, k1e(); * * y = k1e( x ); * * * * DESCRIPTION: * * Returns exponentially scaled modified Bessel function * of the third kind of order one of the argument: * * k1e(x) = exp(x) * k1(x). * * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * IEEE 0, 30 30000 7.8e-16 1.2e-16 * See k1(). * */ /* Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 2000 by Stephen L. Moshier */ #include /* Chebyshev coefficients for x(K1(x) - log(x/2) I1(x)) * in the interval [0,2]. * * lim(x->0){ x(K1(x) - log(x/2) I1(x)) } = 1. */ #ifdef UNK static double A[] = { -7.02386347938628759343E-18, -2.42744985051936593393E-15, -6.66690169419932900609E-13, -1.41148839263352776110E-10, -2.21338763073472585583E-8, -2.43340614156596823496E-6, -1.73028895751305206302E-4, -6.97572385963986435018E-3, -1.22611180822657148235E-1, -3.53155960776544875667E-1, 1.52530022733894777053E0 }; #endif #ifdef DEC static unsigned short A[] = { 0122001,0110501,0164746,0151255, 0124056,0165213,0150034,0147377, 0126073,0124026,0167207,0001044, 0130033,0030735,0141061,0033116, 0131676,0020350,0121341,0107175, 0133443,0046631,0062031,0070716, 0135065,0067427,0026435,0164022, 0136344,0112234,0165752,0006222, 0137373,0015622,0017016,0155636, 0137664,0150333,0125730,0067240, 0040303,0036411,0130200,0043120 }; #endif #ifdef IBMPC static unsigned short A[] = { 0xda56,0x3d3c,0x3228,0xbc60, 0x99e0,0x7a03,0xdd51,0xbce5, 0xe045,0xddd0,0x7502,0xbd67, 0x26ca,0xb846,0x663b,0xbde3, 0x31d0,0x145c,0xc41d,0xbe57, 0x2e3a,0x2c83,0x69b3,0xbec4, 0xbd02,0xe5a3,0xade2,0xbf26, 0x4192,0x9d7d,0x9293,0xbf7c, 0xdb74,0x43c1,0x6372,0xbfbf, 0x0dd4,0x757b,0x9a1b,0xbfd6, 0x08ca,0x3610,0x67a1,0x3ff8 }; #endif #ifdef MIEEE static unsigned short A[] = { 0xbc60,0x3228,0x3d3c,0xda56, 0xbce5,0xdd51,0x7a03,0x99e0, 0xbd67,0x7502,0xddd0,0xe045, 0xbde3,0x663b,0xb846,0x26ca, 0xbe57,0xc41d,0x145c,0x31d0, 0xbec4,0x69b3,0x2c83,0x2e3a, 0xbf26,0xade2,0xe5a3,0xbd02, 0xbf7c,0x9293,0x9d7d,0x4192, 0xbfbf,0x6372,0x43c1,0xdb74, 0xbfd6,0x9a1b,0x757b,0x0dd4, 0x3ff8,0x67a1,0x3610,0x08ca }; #endif /* Chebyshev coefficients for exp(x) sqrt(x) K1(x) * in the interval [2,infinity]. * * lim(x->inf){ exp(x) sqrt(x) K1(x) } = sqrt(pi/2). */ #ifdef UNK static double B[] = { -5.75674448366501715755E-18, 1.79405087314755922667E-17, -5.68946255844285935196E-17, 1.83809354436663880070E-16, -6.05704724837331885336E-16, 2.03870316562433424052E-15, -7.01983709041831346144E-15, 2.47715442448130437068E-14, -8.97670518232499435011E-14, 3.34841966607842919884E-13, -1.28917396095102890680E-12, 5.13963967348173025100E-12, -2.12996783842756842877E-11, 9.21831518760500529508E-11, -4.19035475934189648750E-10, 2.01504975519703286596E-9, -1.03457624656780970260E-8, 5.74108412545004946722E-8, -3.50196060308781257119E-7, 2.40648494783721712015E-6, -1.93619797416608296024E-5, 1.95215518471351631108E-4, -2.85781685962277938680E-3, 1.03923736576817238437E-1, 2.72062619048444266945E0 }; #endif #ifdef DEC static unsigned short B[] = { 0121724,0061352,0013041,0150076, 0022245,0074324,0016172,0173232, 0122603,0030250,0135670,0165221, 0023123,0165362,0023561,0060124, 0123456,0112436,0141654,0073623, 0024022,0163557,0077564,0006753, 0124374,0165221,0131014,0026524, 0024737,0017512,0144250,0175451, 0125312,0021456,0123136,0076633, 0025674,0077720,0020125,0102607, 0126265,0067543,0007744,0043701, 0026664,0152702,0033002,0074202, 0127273,0055234,0120016,0071733, 0027712,0133200,0042441,0075515, 0130346,0057000,0015456,0074470, 0031012,0074441,0051636,0111155, 0131461,0136444,0177417,0002101, 0032166,0111743,0032176,0021410, 0132674,0001224,0076555,0027060, 0033441,0077430,0135226,0106663, 0134242,0065610,0167155,0113447, 0035114,0131304,0043664,0102163, 0136073,0045065,0171465,0122123, 0037324,0152767,0147401,0017732, 0040456,0017275,0050061,0062120, }; #endif #ifdef IBMPC static unsigned short B[] = { 0x3a08,0x42c4,0x8c5d,0xbc5a, 0x5ed3,0x838f,0xaf1a,0x3c74, 0x1d52,0x1777,0x6615,0xbc90, 0x2c0b,0x44ee,0x7d5e,0x3caa, 0x8ef2,0xd875,0xd2a3,0xbcc5, 0x81bd,0xefee,0x5ced,0x3ce2, 0x85ab,0x3641,0x9d52,0xbcff, 0x1f65,0x5915,0xe3e9,0x3d1b, 0xcfb3,0xd4cb,0x4465,0xbd39, 0xb0b1,0x040a,0x8ffa,0x3d57, 0x88f8,0x61fc,0xadec,0xbd76, 0x4f10,0x46c0,0x9ab8,0x3d96, 0xce7b,0x9401,0x6b53,0xbdb7, 0x2f6a,0x08a4,0x56d0,0x3dd9, 0xcf27,0x0365,0xcbc0,0xbdfc, 0xd24e,0x2a73,0x4f24,0x3e21, 0xe088,0x9fe1,0x37a4,0xbe46, 0xc461,0x668f,0xd27c,0x3e6e, 0xa5c6,0x8fad,0x8052,0xbe97, 0xd1b6,0x1752,0x2fe3,0x3ec4, 0xb2e5,0x1dcd,0x4d71,0xbef4, 0x908e,0x88f6,0x9658,0x3f29, 0xb48a,0xbe66,0x6946,0xbf67, 0x23fb,0xf9e0,0x9abe,0x3fba, 0x2c8a,0xaa06,0xc3d7,0x4005 }; #endif #ifdef MIEEE static unsigned short B[] = { 0xbc5a,0x8c5d,0x42c4,0x3a08, 0x3c74,0xaf1a,0x838f,0x5ed3, 0xbc90,0x6615,0x1777,0x1d52, 0x3caa,0x7d5e,0x44ee,0x2c0b, 0xbcc5,0xd2a3,0xd875,0x8ef2, 0x3ce2,0x5ced,0xefee,0x81bd, 0xbcff,0x9d52,0x3641,0x85ab, 0x3d1b,0xe3e9,0x5915,0x1f65, 0xbd39,0x4465,0xd4cb,0xcfb3, 0x3d57,0x8ffa,0x040a,0xb0b1, 0xbd76,0xadec,0x61fc,0x88f8, 0x3d96,0x9ab8,0x46c0,0x4f10, 0xbdb7,0x6b53,0x9401,0xce7b, 0x3dd9,0x56d0,0x08a4,0x2f6a, 0xbdfc,0xcbc0,0x0365,0xcf27, 0x3e21,0x4f24,0x2a73,0xd24e, 0xbe46,0x37a4,0x9fe1,0xe088, 0x3e6e,0xd27c,0x668f,0xc461, 0xbe97,0x8052,0x8fad,0xa5c6, 0x3ec4,0x2fe3,0x1752,0xd1b6, 0xbef4,0x4d71,0x1dcd,0xb2e5, 0x3f29,0x9658,0x88f6,0x908e, 0xbf67,0x6946,0xbe66,0xb48a, 0x3fba,0x9abe,0xf9e0,0x23fb, 0x4005,0xc3d7,0xaa06,0x2c8a }; #endif #ifdef ANSIPROT extern double chbevl ( double, void *, int ); extern double exp ( double ); extern double i1 ( double ); extern double log ( double ); extern double sqrt ( double ); #else double chbevl(), exp(), i1(), log(), sqrt(); #endif extern double PI; extern double MINLOG, MAXNUM; double k1(x) double x; { double y, z; z = 0.5 * x; if( z <= 0.0 ) { mtherr( "k1", DOMAIN ); return( MAXNUM ); } if( x <= 2.0 ) { y = x * x - 2.0; y = log(z) * i1(x) + chbevl( y, A, 11 ) / x; return( y ); } return( exp(-x) * chbevl( 8.0/x - 2.0, B, 25 ) / sqrt(x) ); } double k1e( x ) double x; { double y; if( x <= 0.0 ) { mtherr( "k1e", DOMAIN ); return( MAXNUM ); } if( x <= 2.0 ) { y = x * x - 2.0; y = log( 0.5 * x ) * i1(x) + chbevl( y, A, 11 ) / x; return( y * exp(x) ); } return( chbevl( 8.0/x - 2.0, B, 25 ) / sqrt(x) ); }