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Diffstat (limited to 'libm/double/polylog.c')
| -rw-r--r-- | libm/double/polylog.c | 467 |
1 files changed, 0 insertions, 467 deletions
diff --git a/libm/double/polylog.c b/libm/double/polylog.c deleted file mode 100644 index c21e04449..000000000 --- a/libm/double/polylog.c +++ /dev/null @@ -1,467 +0,0 @@ -/* polylog.c - * - * Polylogarithms - * - * - * - * SYNOPSIS: - * - * double x, y, polylog(); - * int n; - * - * y = polylog( n, x ); - * - * - * The polylogarithm of order n is defined by the series - * - * - * inf k - * - x - * Li (x) = > --- . - * n - n - * k=1 k - * - * - * For x = 1, - * - * inf - * - 1 - * Li (1) = > --- = Riemann zeta function (n) . - * n - n - * k=1 k - * - * - * When n = 2, the function is the dilogarithm, related to Spence's integral: - * - * x 1-x - * - - - * | | -ln(1-t) | | ln t - * Li (x) = | -------- dt = | ------ dt = spence(1-x) . - * 2 | | t | | 1 - t - * - - - * 0 1 - * - * - * See also the program cpolylog.c for the complex polylogarithm, - * whose definition is extended to x > 1. - * - * References: - * - * Lewin, L., _Polylogarithms and Associated Functions_, - * North Holland, 1981. - * - * Lewin, L., ed., _Structural Properties of Polylogarithms_, - * American Mathematical Society, 1991. - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain n # trials peak rms - * IEEE 0, 1 2 50000 6.2e-16 8.0e-17 - * IEEE 0, 1 3 100000 2.5e-16 6.6e-17 - * IEEE 0, 1 4 30000 1.7e-16 4.9e-17 - * IEEE 0, 1 5 30000 5.1e-16 7.8e-17 - * - */ - -/* -Cephes Math Library Release 2.8: July, 1999 -Copyright 1999 by Stephen L. Moshier -*/ - -#include <math.h> -extern double PI; - -/* polylog(4, 1-x) = zeta(4) - x zeta(3) + x^2 A4(x)/B4(x) - 0 <= x <= 0.125 - Theoretical peak absolute error 4.5e-18 */ -#if UNK -static double A4[13] = { - 3.056144922089490701751E-2, - 3.243086484162581557457E-1, - 2.877847281461875922565E-1, - 7.091267785886180663385E-2, - 6.466460072456621248630E-3, - 2.450233019296542883275E-4, - 4.031655364627704957049E-6, - 2.884169163909467997099E-8, - 8.680067002466594858347E-11, - 1.025983405866370985438E-13, - 4.233468313538272640380E-17, - 4.959422035066206902317E-21, - 1.059365867585275714599E-25, -}; -static double B4[12] = { - /* 1.000000000000000000000E0, */ - 2.821262403600310974875E0, - 1.780221124881327022033E0, - 3.778888211867875721773E-1, - 3.193887040074337940323E-2, - 1.161252418498096498304E-3, - 1.867362374829870620091E-5, - 1.319022779715294371091E-7, - 3.942755256555603046095E-10, - 4.644326968986396928092E-13, - 1.913336021014307074861E-16, - 2.240041814626069927477E-20, - 4.784036597230791011855E-25, -}; -#endif -#if DEC -static short A4[52] = { -0036772,0056001,0016601,0164507, -0037646,0005710,0076603,0176456, -0037623,0054205,0013532,0026476, -0037221,0035252,0101064,0065407, -0036323,0162231,0042033,0107244, -0035200,0073170,0106141,0136543, -0033607,0043647,0163672,0055340, -0031767,0137614,0173376,0072313, -0027676,0160156,0161276,0034203, -0025347,0003752,0123106,0064266, -0022503,0035770,0160173,0177501, -0017273,0056226,0033704,0132530, -0013403,0022244,0175205,0052161, -}; -static short B4[48] = { - /*0040200,0000000,0000000,0000000, */ -0040464,0107620,0027471,0071672, -0040343,0157111,0025601,0137255, -0037701,0075244,0140412,0160220, -0037002,0151125,0036572,0057163, -0035630,0032452,0050727,0161653, -0034234,0122515,0034323,0172615, -0032415,0120405,0123660,0003160, -0030330,0140530,0161045,0150177, -0026002,0134747,0014542,0002510, -0023134,0113666,0035730,0035732, -0017723,0110343,0041217,0007764, -0014024,0007412,0175575,0160230, -}; -#endif -#if IBMPC -static short A4[52] = { -0x3d29,0x23b0,0x4b80,0x3f9f, -0x7fa6,0x0fb0,0xc179,0x3fd4, -0x45a8,0xa2eb,0x6b10,0x3fd2, -0x8d61,0x5046,0x2755,0x3fb2, -0x71d4,0x2883,0x7c93,0x3f7a, -0x37ac,0x118c,0x0ecf,0x3f30, -0x4b5c,0xfcf7,0xe8f4,0x3ed0, -0xce99,0x9edf,0xf7f1,0x3e5e, -0xc710,0xdc57,0xdc0d,0x3dd7, -0xcd17,0x54c8,0xe0fd,0x3d3c, -0x7fe8,0x1c0f,0x677f,0x3c88, -0x96ab,0xc6f8,0x6b92,0x3bb7, -0xaa8e,0x9f50,0x6494,0x3ac0, -}; -static short B4[48] = { - /*0x0000,0x0000,0x0000,0x3ff0,*/ -0x2e77,0x05e7,0x91f2,0x4006, -0x37d6,0x2570,0x7bc9,0x3ffc, -0x5c12,0x9821,0x2f54,0x3fd8, -0x4bce,0xa7af,0x5a4a,0x3fa0, -0xfc75,0x4a3a,0x06a5,0x3f53, -0x7eb2,0xa71a,0x94a9,0x3ef3, -0x00ce,0xb4f6,0xb420,0x3e81, -0xba10,0x1c44,0x182b,0x3dfb, -0x40a9,0xe32c,0x573c,0x3d60, -0x077b,0xc77b,0x92f6,0x3cab, -0xe1fe,0x6851,0x721c,0x3bda, -0xbc13,0x5f6f,0x81e1,0x3ae2, -}; -#endif -#if MIEEE -static short A4[52] = { -0x3f9f,0x4b80,0x23b0,0x3d29, -0x3fd4,0xc179,0x0fb0,0x7fa6, -0x3fd2,0x6b10,0xa2eb,0x45a8, -0x3fb2,0x2755,0x5046,0x8d61, -0x3f7a,0x7c93,0x2883,0x71d4, -0x3f30,0x0ecf,0x118c,0x37ac, -0x3ed0,0xe8f4,0xfcf7,0x4b5c, -0x3e5e,0xf7f1,0x9edf,0xce99, -0x3dd7,0xdc0d,0xdc57,0xc710, -0x3d3c,0xe0fd,0x54c8,0xcd17, -0x3c88,0x677f,0x1c0f,0x7fe8, -0x3bb7,0x6b92,0xc6f8,0x96ab, -0x3ac0,0x6494,0x9f50,0xaa8e, -}; -static short B4[48] = { - /*0x3ff0,0x0000,0x0000,0x0000,*/ -0x4006,0x91f2,0x05e7,0x2e77, -0x3ffc,0x7bc9,0x2570,0x37d6, -0x3fd8,0x2f54,0x9821,0x5c12, -0x3fa0,0x5a4a,0xa7af,0x4bce, -0x3f53,0x06a5,0x4a3a,0xfc75, -0x3ef3,0x94a9,0xa71a,0x7eb2, -0x3e81,0xb420,0xb4f6,0x00ce, -0x3dfb,0x182b,0x1c44,0xba10, -0x3d60,0x573c,0xe32c,0x40a9, -0x3cab,0x92f6,0xc77b,0x077b, -0x3bda,0x721c,0x6851,0xe1fe, -0x3ae2,0x81e1,0x5f6f,0xbc13, -}; -#endif - -#ifdef ANSIPROT -extern double spence ( double ); -extern double polevl ( double, void *, int ); -extern double p1evl ( double, void *, int ); -extern double zetac ( double ); -extern double pow ( double, double ); -extern double powi ( double, int ); -extern double log ( double ); -extern double fac ( int i ); -extern double fabs (double); -double polylog (int, double); -#else -extern double spence(), polevl(), p1evl(), zetac(); -extern double pow(), powi(), log(); -extern double fac(); /* factorial */ -extern double fabs(); -double polylog(); -#endif -extern double MACHEP; - -double -polylog (n, x) - int n; - double x; -{ - double h, k, p, s, t, u, xc, z; - int i, j; - -/* This recurrence provides formulas for n < 2. - - d 1 - -- Li (x) = --- Li (x) . - dx n x n-1 - -*/ - - if (n == -1) - { - p = 1.0 - x; - u = x / p; - s = u * u + u; - return s; - } - - if (n == 0) - { - s = x / (1.0 - x); - return s; - } - - /* Not implemented for n < -1. - Not defined for x > 1. Use cpolylog if you need that. */ - if (x > 1.0 || n < -1) - { - mtherr("polylog", DOMAIN); - return 0.0; - } - - if (n == 1) - { - s = -log (1.0 - x); - return s; - } - - /* Argument +1 */ - if (x == 1.0 && n > 1) - { - s = zetac ((double) n) + 1.0; - return s; - } - - /* Argument -1. - 1-n - Li (-z) = - (1 - 2 ) Li (z) - n n - */ - if (x == -1.0 && n > 1) - { - /* Li_n(1) = zeta(n) */ - s = zetac ((double) n) + 1.0; - s = s * (powi (2.0, 1 - n) - 1.0); - return s; - } - -/* Inversion formula: - * [n/2] n-2r - * n 1 n - log (z) - * Li (-z) + (-1) Li (-1/z) = - --- log (z) + 2 > ----------- Li (-1) - * n n n! - (n - 2r)! 2r - * r=1 - */ - if (x < -1.0 && n > 1) - { - double q, w; - int r; - - w = log (-x); - s = 0.0; - for (r = 1; r <= n / 2; r++) - { - j = 2 * r; - p = polylog (j, -1.0); - j = n - j; - if (j == 0) - { - s = s + p; - break; - } - q = (double) j; - q = pow (w, q) * p / fac (j); - s = s + q; - } - s = 2.0 * s; - q = polylog (n, 1.0 / x); - if (n & 1) - q = -q; - s = s - q; - s = s - pow (w, (double) n) / fac (n); - return s; - } - - if (n == 2) - { - if (x < 0.0 || x > 1.0) - return (spence (1.0 - x)); - } - - - - /* The power series converges slowly when x is near 1. For n = 3, this - identity helps: - - Li (-x/(1-x)) + Li (1-x) + Li (x) - 3 3 3 - 2 2 3 - = Li (1) + (pi /6) log(1-x) - (1/2) log(x) log (1-x) + (1/6) log (1-x) - 3 - */ - - if (n == 3) - { - p = x * x * x; - if (x > 0.8) - { - u = log(x); - s = p / 6.0; - xc = 1.0 - x; - s = s - 0.5 * u * u * log(xc); - s = s + PI * PI * u / 6.0; - s = s - polylog (3, -xc/x); - s = s - polylog (3, xc); - s = s + zetac(3.0); - s = s + 1.0; - return s; - } - /* Power series */ - t = p / 27.0; - t = t + .125 * x * x; - t = t + x; - - s = 0.0; - k = 4.0; - do - { - p = p * x; - h = p / (k * k * k); - s = s + h; - k += 1.0; - } - while (fabs(h/s) > 1.1e-16); - return (s + t); - } - -if (n == 4) - { - if (x >= 0.875) - { - u = 1.0 - x; - s = polevl(u, A4, 12) / p1evl(u, B4, 12); - s = s * u * u - 1.202056903159594285400 * u; - s += 1.0823232337111381915160; - return s; - } - goto pseries; - } - - - if (x < 0.75) - goto pseries; - - -/* This expansion in powers of log(x) is especially useful when - x is near 1. - - See also the pari gp calculator. - - inf j - - z(n-j) (log(x)) - polylog(n,x) = > ----------------- - - j! - j=0 - - where - - z(j) = Riemann zeta function (j), j != 1 - - n-1 - - - z(1) = -log(-log(x)) + > 1/k - - - k=1 - */ - - z = log(x); - h = -log(-z); - for (i = 1; i < n; i++) - h = h + 1.0/i; - p = 1.0; - s = zetac((double)n) + 1.0; - for (j=1; j<=n+1; j++) - { - p = p * z / j; - if (j == n-1) - s = s + h * p; - else - s = s + (zetac((double)(n-j)) + 1.0) * p; - } - j = n + 3; - z = z * z; - for(;;) - { - p = p * z / ((j-1)*j); - h = (zetac((double)(n-j)) + 1.0); - h = h * p; - s = s + h; - if (fabs(h/s) < MACHEP) - break; - j += 2; - } - return s; - - -pseries: - - p = x * x * x; - k = 3.0; - s = 0.0; - do - { - p = p * x; - k += 1.0; - h = p / powi(k, n); - s = s + h; - } - while (fabs(h/s) > MACHEP); - s += x * x * x / powi(3.0,n); - s += x * x / powi(2.0,n); - s += x; - return s; -} |