/* 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 "mconf.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) { /* Thanks to Oscar van Vlijmen for detecting an error here. */ u = log(x); s = u * u * u / 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; }