/* k0.c * * Modified Bessel function, third kind, order zero * * * * SYNOPSIS: * * double x, y, k0(); * * y = k0( x ); * * * * DESCRIPTION: * * Returns modified Bessel function of the third kind * of order zero of the argument. * * The range is partitioned into the two intervals [0,8] and * (8, infinity). Chebyshev polynomial expansions are employed * in each interval. * * * * ACCURACY: * * Tested at 2000 random points between 0 and 8. Peak absolute * error (relative when K0 > 1) was 1.46e-14; rms, 4.26e-15. * Relative error: * arithmetic domain # trials peak rms * DEC 0, 30 3100 1.3e-16 2.1e-17 * IEEE 0, 30 30000 1.2e-15 1.6e-16 * * ERROR MESSAGES: * * message condition value returned * K0 domain x <= 0 MAXNUM * */ /* k0e() * * Modified Bessel function, third kind, order zero, * exponentially scaled * * * * SYNOPSIS: * * double x, y, k0e(); * * y = k0e( x ); * * * * DESCRIPTION: * * Returns exponentially scaled modified Bessel function * of the third kind of order zero of the argument. * * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * IEEE 0, 30 30000 1.4e-15 1.4e-16 * See k0(). * */ /* Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 2000 by Stephen L. Moshier */ #include "mconf.h" /* Chebyshev coefficients for K0(x) + log(x/2) I0(x) * in the interval [0,2]. The odd order coefficients are all * zero; only the even order coefficients are listed. * * lim(x->0){ K0(x) + log(x/2) I0(x) } = -EUL. */ #ifdef UNK static double A[] = { 1.37446543561352307156E-16, 4.25981614279661018399E-14, 1.03496952576338420167E-11, 1.90451637722020886025E-9, 2.53479107902614945675E-7, 2.28621210311945178607E-5, 1.26461541144692592338E-3, 3.59799365153615016266E-2, 3.44289899924628486886E-1, -5.35327393233902768720E-1 }; #endif #ifdef DEC static unsigned short A[] = { 0023036,0073417,0032477,0165673, 0025077,0154126,0016046,0012517, 0027066,0011342,0035211,0005041, 0031002,0160233,0037454,0050224, 0032610,0012747,0037712,0173741, 0034277,0144007,0172147,0162375, 0035645,0140563,0125431,0165626, 0037023,0057662,0125124,0102051, 0037660,0043304,0004411,0166707, 0140011,0005467,0047227,0130370 }; #endif #ifdef IBMPC static unsigned short A[] = { 0xfd77,0xe6a7,0xcee1,0x3ca3, 0xc2aa,0xc384,0xfb0a,0x3d27, 0x2144,0x4751,0xc25c,0x3da6, 0x8a13,0x67e5,0x5c13,0x3e20, 0x5efc,0xe7f9,0x02bc,0x3e91, 0xfca0,0xfe8c,0xf900,0x3ef7, 0x3d73,0x7563,0xb82e,0x3f54, 0x9085,0x554a,0x6bf6,0x3fa2, 0x3db9,0x8121,0x08d8,0x3fd6, 0xf61f,0xe9d2,0x2166,0xbfe1 }; #endif #ifdef MIEEE static unsigned short A[] = { 0x3ca3,0xcee1,0xe6a7,0xfd77, 0x3d27,0xfb0a,0xc384,0xc2aa, 0x3da6,0xc25c,0x4751,0x2144, 0x3e20,0x5c13,0x67e5,0x8a13, 0x3e91,0x02bc,0xe7f9,0x5efc, 0x3ef7,0xf900,0xfe8c,0xfca0, 0x3f54,0xb82e,0x7563,0x3d73, 0x3fa2,0x6bf6,0x554a,0x9085, 0x3fd6,0x08d8,0x8121,0x3db9, 0xbfe1,0x2166,0xe9d2,0xf61f }; #endif /* Chebyshev coefficients for exp(x) sqrt(x) K0(x) * in the inverted interval [2,infinity]. * * lim(x->inf){ exp(x) sqrt(x) K0(x) } = sqrt(pi/2). */ #ifdef UNK static double B[] = { 5.30043377268626276149E-18, -1.64758043015242134646E-17, 5.21039150503902756861E-17, -1.67823109680541210385E-16, 5.51205597852431940784E-16, -1.84859337734377901440E-15, 6.34007647740507060557E-15, -2.22751332699166985548E-14, 8.03289077536357521100E-14, -2.98009692317273043925E-13, 1.14034058820847496303E-12, -4.51459788337394416547E-12, 1.85594911495471785253E-11, -7.95748924447710747776E-11, 3.57739728140030116597E-10, -1.69753450938905987466E-9, 8.57403401741422608519E-9, -4.66048989768794782956E-8, 2.76681363944501510342E-7, -1.83175552271911948767E-6, 1.39498137188764993662E-5, -1.28495495816278026384E-4, 1.56988388573005337491E-3, -3.14481013119645005427E-2, 2.44030308206595545468E0 }; #endif #ifdef DEC static unsigned short B[] = { 0021703,0106456,0076144,0173406, 0122227,0173144,0116011,0030033, 0022560,0044562,0006506,0067642, 0123101,0076243,0123273,0131013, 0023436,0157713,0056243,0141331, 0124005,0032207,0063726,0164664, 0024344,0066342,0051756,0162300, 0124710,0121365,0154053,0077022, 0025264,0161166,0066246,0077420, 0125647,0141671,0006443,0103212, 0026240,0076431,0077147,0160445, 0126636,0153741,0174002,0105031, 0027243,0040102,0035375,0163073, 0127656,0176256,0113476,0044653, 0030304,0125544,0006377,0130104, 0130751,0047257,0110537,0127324, 0031423,0046400,0014772,0012164, 0132110,0025240,0155247,0112570, 0032624,0105314,0007437,0021574, 0133365,0155243,0174306,0116506, 0034152,0004776,0061643,0102504, 0135006,0136277,0036104,0175023, 0035715,0142217,0162474,0115022, 0137000,0147671,0065177,0134356, 0040434,0026754,0175163,0044070 }; #endif #ifdef IBMPC static unsigned short B[] = { 0x9ee1,0xcf8c,0x71a5,0x3c58, 0x2603,0x9381,0xfecc,0xbc72, 0xcdf4,0x41a8,0x092e,0x3c8e, 0x7641,0x74d7,0x2f94,0xbca8, 0x785b,0x6b94,0xdbf9,0x3cc3, 0xdd36,0xecfa,0xa690,0xbce0, 0xdc98,0x4a7d,0x8d9c,0x3cfc, 0x6fc2,0xbb05,0x145e,0xbd19, 0xcfe2,0xcd94,0x9c4e,0x3d36, 0x70d1,0x21a4,0xf877,0xbd54, 0xfc25,0x2fcc,0x0fa3,0x3d74, 0x5143,0x3f00,0xdafc,0xbd93, 0xbcc7,0x475f,0x6808,0x3db4, 0xc935,0xd2e7,0xdf95,0xbdd5, 0xf608,0x819f,0x956c,0x3df8, 0xf5db,0xf22b,0x29d5,0xbe1d, 0x428e,0x033f,0x69a0,0x3e42, 0xf2af,0x1b54,0x0554,0xbe69, 0xe46f,0x81e3,0x9159,0x3e92, 0xd3a9,0x7f18,0xbb54,0xbebe, 0x70a9,0xcc74,0x413f,0x3eed, 0x9f42,0xe788,0xd797,0xbf20, 0x9342,0xfca7,0xb891,0x3f59, 0xf71e,0x2d4f,0x19f7,0xbfa0, 0x6907,0x9f4e,0x85bd,0x4003 }; #endif #ifdef MIEEE static unsigned short B[] = { 0x3c58,0x71a5,0xcf8c,0x9ee1, 0xbc72,0xfecc,0x9381,0x2603, 0x3c8e,0x092e,0x41a8,0xcdf4, 0xbca8,0x2f94,0x74d7,0x7641, 0x3cc3,0xdbf9,0x6b94,0x785b, 0xbce0,0xa690,0xecfa,0xdd36, 0x3cfc,0x8d9c,0x4a7d,0xdc98, 0xbd19,0x145e,0xbb05,0x6fc2, 0x3d36,0x9c4e,0xcd94,0xcfe2, 0xbd54,0xf877,0x21a4,0x70d1, 0x3d74,0x0fa3,0x2fcc,0xfc25, 0xbd93,0xdafc,0x3f00,0x5143, 0x3db4,0x6808,0x475f,0xbcc7, 0xbdd5,0xdf95,0xd2e7,0xc935, 0x3df8,0x956c,0x819f,0xf608, 0xbe1d,0x29d5,0xf22b,0xf5db, 0x3e42,0x69a0,0x033f,0x428e, 0xbe69,0x0554,0x1b54,0xf2af, 0x3e92,0x9159,0x81e3,0xe46f, 0xbebe,0xbb54,0x7f18,0xd3a9, 0x3eed,0x413f,0xcc74,0x70a9, 0xbf20,0xd797,0xe788,0x9f42, 0x3f59,0xb891,0xfca7,0x9342, 0xbfa0,0x19f7,0x2d4f,0xf71e, 0x4003,0x85bd,0x9f4e,0x6907 }; #endif /* k0.c */ #ifdef ANSIPROT extern double chbevl ( double, void *, int ); extern double exp ( double ); extern double i0 ( double ); extern double log ( double ); extern double sqrt ( double ); #else double chbevl(), exp(), i0(), log(), sqrt(); #endif extern double PI; extern double MAXNUM; double k0(x) double x; { double y, z; if( x <= 0.0 ) { mtherr( "k0", DOMAIN ); return( MAXNUM ); } if( x <= 2.0 ) { y = x * x - 2.0; y = chbevl( y, A, 10 ) - log( 0.5 * x ) * i0(x); return( y ); } z = 8.0/x - 2.0; y = exp(-x) * chbevl( z, B, 25 ) / sqrt(x); return(y); } double k0e( x ) double x; { double y; if( x <= 0.0 ) { mtherr( "k0e", DOMAIN ); return( MAXNUM ); } if( x <= 2.0 ) { y = x * x - 2.0; y = chbevl( y, A, 10 ) - log( 0.5 * x ) * i0(x); return( y * exp(x) ); } y = chbevl( 8.0/x - 2.0, B, 25 ) / sqrt(x); return(y); }