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Diffstat (limited to 'libm/double/zetac.c')
| -rw-r--r-- | libm/double/zetac.c | 599 |
1 files changed, 599 insertions, 0 deletions
diff --git a/libm/double/zetac.c b/libm/double/zetac.c new file mode 100644 index 000000000..cc28590b3 --- /dev/null +++ b/libm/double/zetac.c @@ -0,0 +1,599 @@ + /* zetac.c + * + * Riemann zeta function + * + * + * + * SYNOPSIS: + * + * double x, y, zetac(); + * + * y = zetac( x ); + * + * + * + * DESCRIPTION: + * + * + * + * inf. + * - -x + * zetac(x) = > k , x > 1, + * - + * k=2 + * + * is related to the Riemann zeta function by + * + * Riemann zeta(x) = zetac(x) + 1. + * + * Extension of the function definition for x < 1 is implemented. + * Zero is returned for x > log2(MAXNUM). + * + * An overflow error may occur for large negative x, due to the + * gamma function in the reflection formula. + * + * ACCURACY: + * + * Tabulated values have full machine accuracy. + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 1,50 10000 9.8e-16 1.3e-16 + * DEC 1,50 2000 1.1e-16 1.9e-17 + * + * + */ + +/* +Cephes Math Library Release 2.8: June, 2000 +Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier +*/ + +#include <math.h> + +extern double MAXNUM, PI; + +/* Riemann zeta(x) - 1 + * for integer arguments between 0 and 30. + */ +#ifdef UNK +static double azetac[] = { +-1.50000000000000000000E0, + 1.70141183460469231730E38, /* infinity. */ + 6.44934066848226436472E-1, + 2.02056903159594285400E-1, + 8.23232337111381915160E-2, + 3.69277551433699263314E-2, + 1.73430619844491397145E-2, + 8.34927738192282683980E-3, + 4.07735619794433937869E-3, + 2.00839282608221441785E-3, + 9.94575127818085337146E-4, + 4.94188604119464558702E-4, + 2.46086553308048298638E-4, + 1.22713347578489146752E-4, + 6.12481350587048292585E-5, + 3.05882363070204935517E-5, + 1.52822594086518717326E-5, + 7.63719763789976227360E-6, + 3.81729326499983985646E-6, + 1.90821271655393892566E-6, + 9.53962033872796113152E-7, + 4.76932986787806463117E-7, + 2.38450502727732990004E-7, + 1.19219925965311073068E-7, + 5.96081890512594796124E-8, + 2.98035035146522801861E-8, + 1.49015548283650412347E-8, + 7.45071178983542949198E-9, + 3.72533402478845705482E-9, + 1.86265972351304900640E-9, + 9.31327432419668182872E-10 +}; +#endif + +#ifdef DEC +static unsigned short azetac[] = { +0140300,0000000,0000000,0000000, +0077777,0177777,0177777,0177777, +0040045,0015146,0022460,0076462, +0037516,0164001,0036001,0104116, +0037250,0114425,0061754,0022033, +0037027,0040616,0145174,0146670, +0036616,0011411,0100444,0104437, +0036410,0145550,0051474,0161067, +0036205,0115527,0141434,0133506, +0036003,0117475,0100553,0053403, +0035602,0056147,0045567,0027703, +0035401,0106157,0111054,0145242, +0035201,0002455,0113151,0101015, +0035000,0126235,0004273,0157260, +0034600,0071127,0112647,0005261, +0034400,0045736,0057610,0157550, +0034200,0031146,0172621,0074172, +0034000,0020603,0115503,0032007, +0033600,0013114,0124672,0023135, +0033400,0007330,0043715,0151117, +0033200,0004742,0145043,0033514, +0033000,0003225,0152624,0004411, +0032600,0002143,0033166,0035746, +0032400,0001354,0074234,0026143, +0032200,0000762,0147776,0170220, +0032000,0000514,0072452,0130631, +0031600,0000335,0114266,0063315, +0031400,0000223,0132710,0041045, +0031200,0000142,0073202,0153426, +0031000,0000101,0121400,0152065, +0030600,0000053,0140525,0072761 +}; +#endif + +#ifdef IBMPC +static unsigned short azetac[] = { +0x0000,0x0000,0x0000,0xbff8, +0xffff,0xffff,0xffff,0x7fef, +0x0fa6,0xc4a6,0xa34c,0x3fe4, +0x310a,0x2780,0xdd00,0x3fc9, +0x8483,0xac7d,0x1322,0x3fb5, +0x99b7,0xd94f,0xe831,0x3fa2, +0x9124,0x3024,0xc261,0x3f91, +0x9c47,0x0a67,0x196d,0x3f81, +0x96e9,0xf863,0xb36a,0x3f70, +0x6ae0,0xb02d,0x73e7,0x3f60, +0xe5f8,0xe96e,0x4b8c,0x3f50, +0x9954,0xf245,0x318d,0x3f40, +0x3042,0xb2cd,0x20a5,0x3f30, +0x7bd6,0xa117,0x1593,0x3f20, +0xe156,0xf2b4,0x0e4a,0x3f10, +0x1bed,0xcbf1,0x097b,0x3f00, +0x2f0f,0xdeb2,0x064c,0x3ef0, +0x6681,0x7368,0x0430,0x3ee0, +0x44cc,0x9537,0x02c9,0x3ed0, +0xba4a,0x08f9,0x01db,0x3ec0, +0x66ea,0x5944,0x013c,0x3eb0, +0x8121,0xbab2,0x00d2,0x3ea0, +0xc77d,0x66ce,0x008c,0x3e90, +0x858c,0x8f13,0x005d,0x3e80, +0xde12,0x59ff,0x003e,0x3e70, +0x5633,0x8ea5,0x0029,0x3e60, +0xccda,0xb316,0x001b,0x3e50, +0x0845,0x76b9,0x0012,0x3e40, +0x5ae3,0x4ed0,0x000c,0x3e30, +0x1a87,0x3460,0x0008,0x3e20, +0xaebe,0x782a,0x0005,0x3e10 +}; +#endif + +#ifdef MIEEE +static unsigned short azetac[] = { +0xbff8,0x0000,0x0000,0x0000, +0x7fef,0xffff,0xffff,0xffff, +0x3fe4,0xa34c,0xc4a6,0x0fa6, +0x3fc9,0xdd00,0x2780,0x310a, +0x3fb5,0x1322,0xac7d,0x8483, +0x3fa2,0xe831,0xd94f,0x99b7, +0x3f91,0xc261,0x3024,0x9124, +0x3f81,0x196d,0x0a67,0x9c47, +0x3f70,0xb36a,0xf863,0x96e9, +0x3f60,0x73e7,0xb02d,0x6ae0, +0x3f50,0x4b8c,0xe96e,0xe5f8, +0x3f40,0x318d,0xf245,0x9954, +0x3f30,0x20a5,0xb2cd,0x3042, +0x3f20,0x1593,0xa117,0x7bd6, +0x3f10,0x0e4a,0xf2b4,0xe156, +0x3f00,0x097b,0xcbf1,0x1bed, +0x3ef0,0x064c,0xdeb2,0x2f0f, +0x3ee0,0x0430,0x7368,0x6681, +0x3ed0,0x02c9,0x9537,0x44cc, +0x3ec0,0x01db,0x08f9,0xba4a, +0x3eb0,0x013c,0x5944,0x66ea, +0x3ea0,0x00d2,0xbab2,0x8121, +0x3e90,0x008c,0x66ce,0xc77d, +0x3e80,0x005d,0x8f13,0x858c, +0x3e70,0x003e,0x59ff,0xde12, +0x3e60,0x0029,0x8ea5,0x5633, +0x3e50,0x001b,0xb316,0xccda, +0x3e40,0x0012,0x76b9,0x0845, +0x3e30,0x000c,0x4ed0,0x5ae3, +0x3e20,0x0008,0x3460,0x1a87, +0x3e10,0x0005,0x782a,0xaebe +}; +#endif + + +/* 2**x (1 - 1/x) (zeta(x) - 1) = P(1/x)/Q(1/x), 1 <= x <= 10 */ +#ifdef UNK +static double P[9] = { + 5.85746514569725319540E11, + 2.57534127756102572888E11, + 4.87781159567948256438E10, + 5.15399538023885770696E9, + 3.41646073514754094281E8, + 1.60837006880656492731E7, + 5.92785467342109522998E5, + 1.51129169964938823117E4, + 2.01822444485997955865E2, +}; +static double Q[8] = { +/* 1.00000000000000000000E0,*/ + 3.90497676373371157516E11, + 5.22858235368272161797E10, + 5.64451517271280543351E9, + 3.39006746015350418834E8, + 1.79410371500126453702E7, + 5.66666825131384797029E5, + 1.60382976810944131506E4, + 1.96436237223387314144E2, +}; +#endif +#ifdef DEC +static unsigned short P[36] = { +0052010,0060466,0101211,0134657, +0051557,0154353,0135060,0064411, +0051065,0133157,0133514,0133633, +0050231,0114735,0035036,0111344, +0047242,0164327,0146036,0033545, +0046165,0065364,0130045,0011005, +0045020,0134427,0075073,0134107, +0043554,0021653,0000440,0177426, +0042111,0151213,0134312,0021402, +}; +static unsigned short Q[32] = { +/*0040200,0000000,0000000,0000000,*/ +0051665,0153363,0054252,0137010, +0051102,0143645,0121415,0036107, +0050250,0034073,0131133,0036465, +0047241,0123250,0150037,0070012, +0046210,0160426,0111463,0116507, +0045012,0054255,0031674,0173612, +0043572,0114460,0151520,0012221, +0042104,0067655,0037037,0137421, +}; +#endif +#ifdef IBMPC +static unsigned short P[36] = { +0x3736,0xd051,0x0c26,0x4261, +0x0d21,0x7746,0xfb1d,0x424d, +0x96f3,0xf6e9,0xb6cd,0x4226, +0xd25c,0xa743,0x333b,0x41f3, +0xc6ed,0xf983,0x5d1a,0x41b4, +0xa241,0x9604,0xad5e,0x416e, +0x7709,0xef47,0x1722,0x4122, +0x1fe3,0x6024,0x8475,0x40cd, +0x4460,0x7719,0x3a51,0x4069, +}; +static unsigned short Q[32] = { +/*0x0000,0x0000,0x0000,0x3ff0,*/ +0x57c1,0x6b15,0xbade,0x4256, +0xa789,0xb461,0x58f4,0x4228, +0x67a7,0x764b,0x0707,0x41f5, +0xee01,0x1a03,0x34d5,0x41b4, +0x73a9,0xd266,0x1c22,0x4171, +0x9ef1,0xa677,0x4b15,0x4121, +0x0292,0x1a6a,0x5326,0x40cf, +0xf7e2,0xa7c3,0x8df5,0x4068, +}; +#endif +#ifdef MIEEE +static unsigned short P[36] = { +0x4261,0x0c26,0xd051,0x3736, +0x424d,0xfb1d,0x7746,0x0d21, +0x4226,0xb6cd,0xf6e9,0x96f3, +0x41f3,0x333b,0xa743,0xd25c, +0x41b4,0x5d1a,0xf983,0xc6ed, +0x416e,0xad5e,0x9604,0xa241, +0x4122,0x1722,0xef47,0x7709, +0x40cd,0x8475,0x6024,0x1fe3, +0x4069,0x3a51,0x7719,0x4460, +}; +static unsigned short Q[32] = { +/*0x3ff0,0x0000,0x0000,0x0000,*/ +0x4256,0xbade,0x6b15,0x57c1, +0x4228,0x58f4,0xb461,0xa789, +0x41f5,0x0707,0x764b,0x67a7, +0x41b4,0x34d5,0x1a03,0xee01, +0x4171,0x1c22,0xd266,0x73a9, +0x4121,0x4b15,0xa677,0x9ef1, +0x40cf,0x5326,0x1a6a,0x0292, +0x4068,0x8df5,0xa7c3,0xf7e2, +}; +#endif + +/* log(zeta(x) - 1 - 2**-x), 10 <= x <= 50 */ +#ifdef UNK +static double A[11] = { + 8.70728567484590192539E6, + 1.76506865670346462757E8, + 2.60889506707483264896E10, + 5.29806374009894791647E11, + 2.26888156119238241487E13, + 3.31884402932705083599E14, + 5.13778997975868230192E15, +-1.98123688133907171455E15, +-9.92763810039983572356E16, + 7.82905376180870586444E16, + 9.26786275768927717187E16, +}; +static double B[10] = { +/* 1.00000000000000000000E0,*/ +-7.92625410563741062861E6, +-1.60529969932920229676E8, +-2.37669260975543221788E10, +-4.80319584350455169857E11, +-2.07820961754173320170E13, +-2.96075404507272223680E14, +-4.86299103694609136686E15, + 5.34589509675789930199E15, + 5.71464111092297631292E16, +-1.79915597658676556828E16, +}; +#endif +#ifdef DEC +static unsigned short A[44] = { +0046004,0156325,0126302,0131567, +0047050,0052177,0015271,0136466, +0050702,0060271,0070727,0171112, +0051766,0132727,0064363,0145042, +0053245,0012466,0056000,0117230, +0054226,0166155,0174275,0170213, +0055222,0003127,0112544,0101322, +0154741,0036625,0010346,0053767, +0156260,0054653,0154052,0031113, +0056213,0011152,0021000,0007111, +0056244,0120534,0040576,0163262, +}; +static unsigned short B[40] = { +/*0040200,0000000,0000000,0000000,*/ +0145761,0161734,0033026,0015520, +0147031,0013743,0017355,0036703, +0150661,0011720,0061061,0136402, +0151737,0125216,0070274,0164414, +0153227,0032653,0127211,0145250, +0154206,0121666,0123774,0042035, +0155212,0033352,0125154,0132533, +0055227,0170201,0110775,0072132, +0056113,0003133,0127132,0122303, +0155577,0126351,0141462,0171037, +}; +#endif +#ifdef IBMPC +static unsigned short A[44] = { +0x566f,0xb598,0x9b9a,0x4160, +0x37a7,0xe357,0x0a8f,0x41a5, +0xfe49,0x2e3a,0x4c17,0x4218, +0x7944,0xed1e,0xd6ba,0x425e, +0x13d3,0xcb80,0xa2a6,0x42b4, +0xbe11,0xbf17,0xdd8d,0x42f2, +0x905a,0xf2ac,0x40ca,0x4332, +0xcaff,0xa21c,0x27b2,0xc31c, +0x4649,0x7b05,0x0b35,0xc376, +0x01c9,0x4440,0x624d,0x4371, +0xdcd6,0x882f,0x942b,0x4374, +}; +static unsigned short B[40] = { +/*0x0000,0x0000,0x0000,0x3ff0,*/ +0xc36a,0x86c2,0x3c7b,0xc15e, +0xa7b8,0x63dd,0x22fc,0xc1a3, +0x37a0,0x0c46,0x227a,0xc216, +0x9d22,0xce17,0xf551,0xc25b, +0x3955,0x75d1,0xe6b5,0xc2b2, +0x8884,0xd4ff,0xd476,0xc2f0, +0x96ab,0x554d,0x46dd,0xc331, +0xae8b,0x323f,0xfe10,0x4332, +0x5498,0x75cb,0x60cb,0x4369, +0x5e44,0x3866,0xf59d,0xc34f, +}; +#endif +#ifdef MIEEE +static unsigned short A[44] = { +0x4160,0x9b9a,0xb598,0x566f, +0x41a5,0x0a8f,0xe357,0x37a7, +0x4218,0x4c17,0x2e3a,0xfe49, +0x425e,0xd6ba,0xed1e,0x7944, +0x42b4,0xa2a6,0xcb80,0x13d3, +0x42f2,0xdd8d,0xbf17,0xbe11, +0x4332,0x40ca,0xf2ac,0x905a, +0xc31c,0x27b2,0xa21c,0xcaff, +0xc376,0x0b35,0x7b05,0x4649, +0x4371,0x624d,0x4440,0x01c9, +0x4374,0x942b,0x882f,0xdcd6, +}; +static unsigned short B[40] = { +/*0x3ff0,0x0000,0x0000,0x0000,*/ +0xc15e,0x3c7b,0x86c2,0xc36a, +0xc1a3,0x22fc,0x63dd,0xa7b8, +0xc216,0x227a,0x0c46,0x37a0, +0xc25b,0xf551,0xce17,0x9d22, +0xc2b2,0xe6b5,0x75d1,0x3955, +0xc2f0,0xd476,0xd4ff,0x8884, +0xc331,0x46dd,0x554d,0x96ab, +0x4332,0xfe10,0x323f,0xae8b, +0x4369,0x60cb,0x75cb,0x5498, +0xc34f,0xf59d,0x3866,0x5e44, +}; +#endif + +/* (1-x) (zeta(x) - 1), 0 <= x <= 1 */ + +#ifdef UNK +static double R[6] = { +-3.28717474506562731748E-1, + 1.55162528742623950834E1, +-2.48762831680821954401E2, + 1.01050368053237678329E3, + 1.26726061410235149405E4, +-1.11578094770515181334E5, +}; +static double S[5] = { +/* 1.00000000000000000000E0,*/ + 1.95107674914060531512E1, + 3.17710311750646984099E2, + 3.03835500874445748734E3, + 2.03665876435770579345E4, + 7.43853965136767874343E4, +}; +#endif +#ifdef DEC +static unsigned short R[24] = { +0137650,0046650,0022502,0040316, +0041170,0041222,0057666,0142216, +0142170,0141510,0167741,0075646, +0042574,0120074,0046505,0106053, +0043506,0001154,0130073,0101413, +0144331,0166414,0020560,0131652, +}; +static unsigned short S[20] = { +/*0040200,0000000,0000000,0000000,*/ +0041234,0013015,0042073,0113570, +0042236,0155353,0077325,0077445, +0043075,0162656,0016646,0031723, +0043637,0016454,0157636,0071126, +0044221,0044262,0140365,0146434, +}; +#endif +#ifdef IBMPC +static unsigned short R[24] = { +0x481a,0x04a8,0x09b5,0xbfd5, +0xd892,0x4bf6,0x0852,0x402f, +0x2f75,0x1dfc,0x1869,0xc06f, +0xb185,0x89a8,0x9407,0x408f, +0x7061,0x9607,0xc04d,0x40c8, +0x1675,0x842e,0x3da1,0xc0fb, +}; +static unsigned short S[20] = { +/*0x0000,0x0000,0x0000,0x3ff0,*/ +0x72ef,0xa887,0x82c1,0x4033, +0xafe5,0x6fda,0xdb5d,0x4073, +0xc67a,0xc3b4,0xbcb5,0x40a7, +0xce4b,0x9bf3,0xe3a5,0x40d3, +0xb9a3,0x581e,0x2916,0x40f2, +}; +#endif +#ifdef MIEEE +static unsigned short R[24] = { +0xbfd5,0x09b5,0x04a8,0x481a, +0x402f,0x0852,0x4bf6,0xd892, +0xc06f,0x1869,0x1dfc,0x2f75, +0x408f,0x9407,0x89a8,0xb185, +0x40c8,0xc04d,0x9607,0x7061, +0xc0fb,0x3da1,0x842e,0x1675, +}; +static unsigned short S[20] = { +/*0x3ff0,0x0000,0x0000,0x0000,*/ +0x4033,0x82c1,0xa887,0x72ef, +0x4073,0xdb5d,0x6fda,0xafe5, +0x40a7,0xbcb5,0xc3b4,0xc67a, +0x40d3,0xe3a5,0x9bf3,0xce4b, +0x40f2,0x2916,0x581e,0xb9a3, +}; +#endif + +#define MAXL2 127 + +/* + * Riemann zeta function, minus one + */ +#ifdef ANSIPROT +extern double sin ( double ); +extern double floor ( double ); +extern double gamma ( double ); +extern double pow ( double, double ); +extern double exp ( double ); +extern double polevl ( double, void *, int ); +extern double p1evl ( double, void *, int ); +double zetac ( double ); +#else +double sin(), floor(), gamma(), pow(), exp(); +double polevl(), p1evl(), zetac(); +#endif +extern double MACHEP; + +double zetac(x) +double x; +{ +int i; +double a, b, s, w; + +if( x < 0.0 ) + { +#ifdef DEC + if( x < -30.8148 ) +#else + if( x < -170.6243 ) +#endif + { + mtherr( "zetac", OVERFLOW ); + return(0.0); + } + s = 1.0 - x; + w = zetac( s ); + b = sin(0.5*PI*x) * pow(2.0*PI, x) * gamma(s) * (1.0 + w) / PI; + return(b - 1.0); + } + +if( x >= MAXL2 ) + return(0.0); /* because first term is 2**-x */ + +/* Tabulated values for integer argument */ +w = floor(x); +if( w == x ) + { + i = x; + if( i < 31 ) + { +#ifdef UNK + return( azetac[i] ); +#else + return( *(double *)&azetac[4*i] ); +#endif + } + } + + +if( x < 1.0 ) + { + w = 1.0 - x; + a = polevl( x, R, 5 ) / ( w * p1evl( x, S, 5 )); + return( a ); + } + +if( x == 1.0 ) + { + mtherr( "zetac", SING ); + return( MAXNUM ); + } + +if( x <= 10.0 ) + { + b = pow( 2.0, x ) * (x - 1.0); + w = 1.0/x; + s = (x * polevl( w, P, 8 )) / (b * p1evl( w, Q, 8 )); + return( s ); + } + +if( x <= 50.0 ) + { + b = pow( 2.0, -x ); + w = polevl( x, A, 10 ) / p1evl( x, B, 10 ); + w = exp(w) + b; + return(w); + } + + +/* Basic sum of inverse powers */ + + +s = 0.0; +a = 1.0; +do + { + a += 2.0; + b = pow( a, -x ); + s += b; + } +while( b/s > MACHEP ); + +b = pow( 2.0, -x ); +s = (s + b)/(1.0-b); +return(s); +} |