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Diffstat (limited to 'gsl-1.9/specfunc/dilog.c')
-rw-r--r-- | gsl-1.9/specfunc/dilog.c | 662 |
1 files changed, 662 insertions, 0 deletions
diff --git a/gsl-1.9/specfunc/dilog.c b/gsl-1.9/specfunc/dilog.c new file mode 100644 index 0000000..1d8a7b1 --- /dev/null +++ b/gsl-1.9/specfunc/dilog.c @@ -0,0 +1,662 @@ +/* specfunc/dilog.c + * + * Copyright (C) 1996, 1997, 1998, 1999, 2000, 2004 Gerard Jungman + * + * This program is free software; you can redistribute it and/or modify + * it under the terms of the GNU General Public License as published by + * the Free Software Foundation; either version 2 of the License, or (at + * your option) any later version. + * + * This program is distributed in the hope that it will be useful, but + * WITHOUT ANY WARRANTY; without even the implied warranty of + * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + * General Public License for more details. + * + * You should have received a copy of the GNU General Public License + * along with this program; if not, write to the Free Software + * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA. + */ + +/* Author: G. Jungman */ + +#include <config.h> +#include <gsl/gsl_math.h> +#include <gsl/gsl_errno.h> +#include <gsl/gsl_sf_clausen.h> +#include <gsl/gsl_sf_trig.h> +#include <gsl/gsl_sf_log.h> +#include <gsl/gsl_sf_dilog.h> + + +/* Evaluate series for real dilog(x) + * Sum[ x^k / k^2, {k,1,Infinity}] + * + * Converges rapidly for |x| < 1/2. + */ +static +int +dilog_series_1(const double x, gsl_sf_result * result) +{ + const int kmax = 1000; + double sum = x; + double term = x; + int k; + for(k=2; k<kmax; k++) { + const double rk = (k-1.0)/k; + term *= x; + term *= rk*rk; + sum += term; + if(fabs(term/sum) < GSL_DBL_EPSILON) break; + } + + result->val = sum; + result->err = 2.0 * fabs(term); + result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val); + + if(k == kmax) + GSL_ERROR ("error", GSL_EMAXITER); + else + return GSL_SUCCESS; +} + + +/* Compute the associated series + * + * sum_{k=1}{infty} r^k / (k^2 (k+1)) + * + * This is a series which appears in the one-step accelerated + * method, which splits out one elementary function from the + * full definition of Li_2(x). See below. + */ +static int +series_2(double r, gsl_sf_result * result) +{ + static const int kmax = 100; + double rk = r; + double sum = 0.5 * r; + int k; + for(k=2; k<10; k++) + { + double ds; + rk *= r; + ds = rk/(k*k*(k+1.0)); + sum += ds; + } + for(; k<kmax; k++) + { + double ds; + rk *= r; + ds = rk/(k*k*(k+1.0)); + sum += ds; + if(fabs(ds/sum) < 0.5*GSL_DBL_EPSILON) break; + } + + result->val = sum; + result->err = 2.0 * kmax * GSL_DBL_EPSILON * fabs(sum); + + return GSL_SUCCESS; +} + + +/* Compute Li_2(x) using the accelerated series representation. + * + * Li_2(x) = 1 + (1-x)ln(1-x)/x + series_2(x) + * + * assumes: -1 < x < 1 + */ +static int +dilog_series_2(double x, gsl_sf_result * result) +{ + const int stat_s3 = series_2(x, result); + double t; + if(x > 0.01) + t = (1.0 - x) * log(1.0-x) / x; + else + { + static const double c3 = 1.0/3.0; + static const double c4 = 1.0/4.0; + static const double c5 = 1.0/5.0; + static const double c6 = 1.0/6.0; + static const double c7 = 1.0/7.0; + static const double c8 = 1.0/8.0; + const double t68 = c6 + x*(c7 + x*c8); + const double t38 = c3 + x *(c4 + x *(c5 + x * t68)); + t = (x - 1.0) * (1.0 + x*(0.5 + x*t38)); + } + result->val += 1.0 + t; + result->err += 2.0 * GSL_DBL_EPSILON * fabs(t); + return stat_s3; +} + + +/* Calculates Li_2(x) for real x. Assumes x >= 0.0. + */ +static +int +dilog_xge0(const double x, gsl_sf_result * result) +{ + if(x > 2.0) { + gsl_sf_result ser; + const int stat_ser = dilog_series_2(1.0/x, &ser); + const double log_x = log(x); + const double t1 = M_PI*M_PI/3.0; + const double t2 = ser.val; + const double t3 = 0.5*log_x*log_x; + result->val = t1 - t2 - t3; + result->err = GSL_DBL_EPSILON * fabs(log_x) + ser.err; + result->err += GSL_DBL_EPSILON * (fabs(t1) + fabs(t2) + fabs(t3)); + result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val); + return stat_ser; + } + else if(x > 1.01) { + gsl_sf_result ser; + const int stat_ser = dilog_series_2(1.0 - 1.0/x, &ser); + const double log_x = log(x); + const double log_term = log_x * (log(1.0-1.0/x) + 0.5*log_x); + const double t1 = M_PI*M_PI/6.0; + const double t2 = ser.val; + const double t3 = log_term; + result->val = t1 + t2 - t3; + result->err = GSL_DBL_EPSILON * fabs(log_x) + ser.err; + result->err += GSL_DBL_EPSILON * (fabs(t1) + fabs(t2) + fabs(t3)); + result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val); + return stat_ser; + } + else if(x > 1.0) { + /* series around x = 1.0 */ + const double eps = x - 1.0; + const double lne = log(eps); + const double c0 = M_PI*M_PI/6.0; + const double c1 = 1.0 - lne; + const double c2 = -(1.0 - 2.0*lne)/4.0; + const double c3 = (1.0 - 3.0*lne)/9.0; + const double c4 = -(1.0 - 4.0*lne)/16.0; + const double c5 = (1.0 - 5.0*lne)/25.0; + const double c6 = -(1.0 - 6.0*lne)/36.0; + const double c7 = (1.0 - 7.0*lne)/49.0; + const double c8 = -(1.0 - 8.0*lne)/64.0; + result->val = c0+eps*(c1+eps*(c2+eps*(c3+eps*(c4+eps*(c5+eps*(c6+eps*(c7+eps*c8))))))); + result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val); + return GSL_SUCCESS; + } + else if(x == 1.0) { + result->val = M_PI*M_PI/6.0; + result->err = 2.0 * GSL_DBL_EPSILON * M_PI*M_PI/6.0; + return GSL_SUCCESS; + } + else if(x > 0.5) { + gsl_sf_result ser; + const int stat_ser = dilog_series_2(1.0-x, &ser); + const double log_x = log(x); + const double t1 = M_PI*M_PI/6.0; + const double t2 = ser.val; + const double t3 = log_x*log(1.0-x); + result->val = t1 - t2 - t3; + result->err = GSL_DBL_EPSILON * fabs(log_x) + ser.err; + result->err += GSL_DBL_EPSILON * (fabs(t1) + fabs(t2) + fabs(t3)); + result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val); + return stat_ser; + } + else if(x > 0.25) { + return dilog_series_2(x, result); + } + else if(x > 0.0) { + return dilog_series_1(x, result); + } + else { + /* x == 0.0 */ + result->val = 0.0; + result->err = 0.0; + return GSL_SUCCESS; + } +} + + +/* Evaluate the series representation for Li2(z): + * + * Li2(z) = Sum[ |z|^k / k^2 Exp[i k arg(z)], {k,1,Infinity}] + * |z| = r + * arg(z) = theta + * + * Assumes 0 < r < 1. + * It is used only for small r. + */ +static +int +dilogc_series_1( + const double r, + const double x, + const double y, + gsl_sf_result * real_result, + gsl_sf_result * imag_result + ) +{ + const double cos_theta = x/r; + const double sin_theta = y/r; + const double alpha = 1.0 - cos_theta; + const double beta = sin_theta; + double ck = cos_theta; + double sk = sin_theta; + double rk = r; + double real_sum = r*ck; + double imag_sum = r*sk; + const int kmax = 50 + (int)(22.0/(-log(r))); /* tuned for double-precision */ + int k; + for(k=2; k<kmax; k++) { + double dr, di; + double ck_tmp = ck; + ck = ck - (alpha*ck + beta*sk); + sk = sk - (alpha*sk - beta*ck_tmp); + rk *= r; + dr = rk/((double)k*k) * ck; + di = rk/((double)k*k) * sk; + real_sum += dr; + imag_sum += di; + if(fabs((dr*dr + di*di)/(real_sum*real_sum + imag_sum*imag_sum)) < GSL_DBL_EPSILON*GSL_DBL_EPSILON) break; + } + + real_result->val = real_sum; + real_result->err = 2.0 * kmax * GSL_DBL_EPSILON * fabs(real_sum); + imag_result->val = imag_sum; + imag_result->err = 2.0 * kmax * GSL_DBL_EPSILON * fabs(imag_sum); + + return GSL_SUCCESS; +} + + +/* Compute + * + * sum_{k=1}{infty} z^k / (k^2 (k+1)) + * + * This is a series which appears in the one-step accelerated + * method, which splits out one elementary function from the + * full definition of Li_2. + */ +static int +series_2_c( + double r, + double x, + double y, + gsl_sf_result * sum_re, + gsl_sf_result * sum_im + ) +{ + const double cos_theta = x/r; + const double sin_theta = y/r; + const double alpha = 1.0 - cos_theta; + const double beta = sin_theta; + double ck = cos_theta; + double sk = sin_theta; + double rk = r; + double real_sum = 0.5 * r*ck; + double imag_sum = 0.5 * r*sk; + const int kmax = 30 + (int)(18.0/(-log(r))); /* tuned for double-precision */ + int k; + for(k=2; k<kmax; k++) + { + double dr, di; + const double ck_tmp = ck; + ck = ck - (alpha*ck + beta*sk); + sk = sk - (alpha*sk - beta*ck_tmp); + rk *= r; + dr = rk/((double)k*k*(k+1.0)) * ck; + di = rk/((double)k*k*(k+1.0)) * sk; + real_sum += dr; + imag_sum += di; + if(fabs((dr*dr + di*di)/(real_sum*real_sum + imag_sum*imag_sum)) < GSL_DBL_EPSILON*GSL_DBL_EPSILON) break; + } + + sum_re->val = real_sum; + sum_re->err = 2.0 * kmax * GSL_DBL_EPSILON * fabs(real_sum); + sum_im->val = imag_sum; + sum_im->err = 2.0 * kmax * GSL_DBL_EPSILON * fabs(imag_sum); + + return GSL_SUCCESS; +} + + +/* Compute Li_2(z) using the one-step accelerated series. + * + * Li_2(z) = 1 + (1-z)ln(1-z)/z + series_2_c(z) + * + * z = r exp(i theta) + * assumes: r < 1 + * assumes: r > epsilon, so that we take no special care with log(1-z) + */ +static +int +dilogc_series_2( + const double r, + const double x, + const double y, + gsl_sf_result * real_dl, + gsl_sf_result * imag_dl + ) +{ + if(r == 0.0) + { + real_dl->val = 0.0; + imag_dl->val = 0.0; + real_dl->err = 0.0; + imag_dl->err = 0.0; + return GSL_SUCCESS; + } + else + { + gsl_sf_result sum_re; + gsl_sf_result sum_im; + const int stat_s3 = series_2_c(r, x, y, &sum_re, &sum_im); + + /* t = ln(1-z)/z */ + gsl_sf_result ln_omz_r; + gsl_sf_result ln_omz_theta; + const int stat_log = gsl_sf_complex_log_e(1.0-x, -y, &ln_omz_r, &ln_omz_theta); + const double t_x = ( ln_omz_r.val * x + ln_omz_theta.val * y)/(r*r); + const double t_y = (-ln_omz_r.val * y + ln_omz_theta.val * x)/(r*r); + + /* r = (1-z) ln(1-z)/z */ + const double r_x = (1.0 - x) * t_x + y * t_y; + const double r_y = (1.0 - x) * t_y - y * t_x; + + real_dl->val = sum_re.val + r_x + 1.0; + imag_dl->val = sum_im.val + r_y; + real_dl->err = sum_re.err + 2.0*GSL_DBL_EPSILON*(fabs(real_dl->val) + fabs(r_x)); + imag_dl->err = sum_im.err + 2.0*GSL_DBL_EPSILON*(fabs(imag_dl->val) + fabs(r_y)); + return GSL_ERROR_SELECT_2(stat_s3, stat_log); + } +} + + +/* Evaluate a series for Li_2(z) when |z| is near 1. + * This is uniformly good away from z=1. + * + * Li_2(z) = Sum[ a^n/n! H_n(theta), {n, 0, Infinity}] + * + * where + * H_n(theta) = Sum[ e^(i m theta) m^n / m^2, {m, 1, Infinity}] + * a = ln(r) + * + * H_0(t) = Gl_2(t) + i Cl_2(t) + * H_1(t) = 1/2 ln(2(1-c)) + I atan2(-s, 1-c) + * H_2(t) = -1/2 + I/2 s/(1-c) + * H_3(t) = -1/2 /(1-c) + * H_4(t) = -I/2 s/(1-c)^2 + * H_5(t) = 1/2 (2 + c)/(1-c)^2 + * H_6(t) = I/2 s/(1-c)^5 (8(1-c) - s^2 (3 + c)) + */ +static +int +dilogc_series_3( + const double r, + const double x, + const double y, + gsl_sf_result * real_result, + gsl_sf_result * imag_result + ) +{ + const double theta = atan2(y, x); + const double cos_theta = x/r; + const double sin_theta = y/r; + const double a = log(r); + const double omc = 1.0 - cos_theta; + const double omc2 = omc*omc; + double H_re[7]; + double H_im[7]; + double an, nfact; + double sum_re, sum_im; + gsl_sf_result Him0; + int n; + + H_re[0] = M_PI*M_PI/6.0 + 0.25*(theta*theta - 2.0*M_PI*fabs(theta)); + gsl_sf_clausen_e(theta, &Him0); + H_im[0] = Him0.val; + + H_re[1] = -0.5*log(2.0*omc); + H_im[1] = -atan2(-sin_theta, omc); + + H_re[2] = -0.5; + H_im[2] = 0.5 * sin_theta/omc; + + H_re[3] = -0.5/omc; + H_im[3] = 0.0; + + H_re[4] = 0.0; + H_im[4] = -0.5*sin_theta/omc2; + + H_re[5] = 0.5 * (2.0 + cos_theta)/omc2; + H_im[5] = 0.0; + + H_re[6] = 0.0; + H_im[6] = 0.5 * sin_theta/(omc2*omc2*omc) * (8.0*omc - sin_theta*sin_theta*(3.0 + cos_theta)); + + sum_re = H_re[0]; + sum_im = H_im[0]; + an = 1.0; + nfact = 1.0; + for(n=1; n<=6; n++) { + double t; + an *= a; + nfact *= n; + t = an/nfact; + sum_re += t * H_re[n]; + sum_im += t * H_im[n]; + } + + real_result->val = sum_re; + real_result->err = 2.0 * 6.0 * GSL_DBL_EPSILON * fabs(sum_re) + fabs(an/nfact); + imag_result->val = sum_im; + imag_result->err = 2.0 * 6.0 * GSL_DBL_EPSILON * fabs(sum_im) + Him0.err + fabs(an/nfact); + + return GSL_SUCCESS; +} + + +/* Calculate complex dilogarithm Li_2(z) in the fundamental region, + * which we take to be the intersection of the unit disk with the + * half-space x < MAGIC_SPLIT_VALUE. It turns out that 0.732 is a + * nice choice for MAGIC_SPLIT_VALUE since then points mapped out + * of the x > MAGIC_SPLIT_VALUE region and into another part of the + * unit disk are bounded in radius by MAGIC_SPLIT_VALUE itself. + * + * If |z| < 0.98 we use a direct series summation. Otherwise z is very + * near the unit circle, and the series_2 expansion is used; see above. + * Because the fundamental region is bounded away from z = 1, this + * works well. + */ +static +int +dilogc_fundamental(double r, double x, double y, gsl_sf_result * real_dl, gsl_sf_result * imag_dl) +{ + if(r > 0.98) + return dilogc_series_3(r, x, y, real_dl, imag_dl); + else if(r > 0.25) + return dilogc_series_2(r, x, y, real_dl, imag_dl); + else + return dilogc_series_1(r, x, y, real_dl, imag_dl); +} + + +/* Compute Li_2(z) for z in the unit disk, |z| < 1. If z is outside + * the fundamental region, which means that it is too close to z = 1, + * then it is reflected into the fundamental region using the identity + * + * Li2(z) = -Li2(1-z) + zeta(2) - ln(z) ln(1-z). + */ +static +int +dilogc_unitdisk(double x, double y, gsl_sf_result * real_dl, gsl_sf_result * imag_dl) +{ + static const double MAGIC_SPLIT_VALUE = 0.732; + static const double zeta2 = M_PI*M_PI/6.0; + const double r = hypot(x, y); + + if(x > MAGIC_SPLIT_VALUE) + { + /* Reflect away from z = 1 if we are too close. The magic value + * insures that the reflected value of the radius satisfies the + * related inequality r_tmp < MAGIC_SPLIT_VALUE. + */ + const double x_tmp = 1.0 - x; + const double y_tmp = - y; + const double r_tmp = hypot(x_tmp, y_tmp); + /* const double cos_theta_tmp = x_tmp/r_tmp; */ + /* const double sin_theta_tmp = y_tmp/r_tmp; */ + + gsl_sf_result result_re_tmp; + gsl_sf_result result_im_tmp; + + const int stat_dilog = dilogc_fundamental(r_tmp, x_tmp, y_tmp, &result_re_tmp, &result_im_tmp); + + const double lnz = log(r); /* log(|z|) */ + const double lnomz = log(r_tmp); /* log(|1-z|) */ + const double argz = atan2(y, x); /* arg(z) assuming principal branch */ + const double argomz = atan2(y_tmp, x_tmp); /* arg(1-z) */ + real_dl->val = -result_re_tmp.val + zeta2 - lnz*lnomz + argz*argomz; + real_dl->err = result_re_tmp.err; + real_dl->err += 2.0 * GSL_DBL_EPSILON * (zeta2 + fabs(lnz*lnomz) + fabs(argz*argomz)); + imag_dl->val = -result_im_tmp.val - argz*lnomz - argomz*lnz; + imag_dl->err = result_im_tmp.err; + imag_dl->err += 2.0 * GSL_DBL_EPSILON * (fabs(argz*lnomz) + fabs(argomz*lnz)); + + return stat_dilog; + } + else + { + return dilogc_fundamental(r, x, y, real_dl, imag_dl); + } +} + + + +/*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/ + + +int +gsl_sf_dilog_e(const double x, gsl_sf_result * result) +{ + if(x >= 0.0) { + return dilog_xge0(x, result); + } + else { + gsl_sf_result d1, d2; + int stat_d1 = dilog_xge0( -x, &d1); + int stat_d2 = dilog_xge0(x*x, &d2); + result->val = -d1.val + 0.5 * d2.val; + result->err = d1.err + 0.5 * d2.err; + result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val); + return GSL_ERROR_SELECT_2(stat_d1, stat_d2); + } +} + + +int +gsl_sf_complex_dilog_xy_e( + const double x, + const double y, + gsl_sf_result * real_dl, + gsl_sf_result * imag_dl + ) +{ + const double zeta2 = M_PI*M_PI/6.0; + const double r2 = x*x + y*y; + + if(y == 0.0) + { + if(x >= 1.0) + { + imag_dl->val = -M_PI * log(x); + imag_dl->err = 2.0 * GSL_DBL_EPSILON * fabs(imag_dl->val); + } + else + { + imag_dl->val = 0.0; + imag_dl->err = 0.0; + } + return gsl_sf_dilog_e(x, real_dl); + } + else if(fabs(r2 - 1.0) < GSL_DBL_EPSILON) + { + /* Lewin A.2.4.1 and A.2.4.2 */ + + const double theta = atan2(y, x); + const double term1 = theta*theta/4.0; + const double term2 = M_PI*fabs(theta)/2.0; + real_dl->val = zeta2 + term1 - term2; + real_dl->err = 2.0 * GSL_DBL_EPSILON * (zeta2 + term1 + term2); + return gsl_sf_clausen_e(theta, imag_dl); + } + else if(r2 < 1.0) + { + return dilogc_unitdisk(x, y, real_dl, imag_dl); + } + else + { + /* Reduce argument to unit disk. */ + const double r = sqrt(r2); + const double x_tmp = x/r2; + const double y_tmp = -y/r2; + /* const double r_tmp = 1.0/r; */ + gsl_sf_result result_re_tmp, result_im_tmp; + + const int stat_dilog = + dilogc_unitdisk(x_tmp, y_tmp, &result_re_tmp, &result_im_tmp); + + /* Unwind the inversion. + * + * Li_2(z) + Li_2(1/z) = -zeta(2) - 1/2 ln(-z)^2 + */ + const double theta = atan2(y, x); + const double theta_abs = fabs(theta); + const double theta_sgn = ( theta < 0.0 ? -1.0 : 1.0 ); + const double ln_minusz_re = log(r); + const double ln_minusz_im = theta_sgn * (theta_abs - M_PI); + const double lmz2_re = ln_minusz_re*ln_minusz_re - ln_minusz_im*ln_minusz_im; + const double lmz2_im = 2.0*ln_minusz_re*ln_minusz_im; + real_dl->val = -result_re_tmp.val - 0.5 * lmz2_re - zeta2; + real_dl->err = result_re_tmp.err + 2.0*GSL_DBL_EPSILON*(0.5 * fabs(lmz2_re) + zeta2); + imag_dl->val = -result_im_tmp.val - 0.5 * lmz2_im; + imag_dl->err = result_im_tmp.err + 2.0*GSL_DBL_EPSILON*fabs(lmz2_im); + return stat_dilog; + } +} + + +int +gsl_sf_complex_dilog_e( + const double r, + const double theta, + gsl_sf_result * real_dl, + gsl_sf_result * imag_dl + ) +{ + const double cos_theta = cos(theta); + const double sin_theta = sin(theta); + const double x = r * cos_theta; + const double y = r * sin_theta; + return gsl_sf_complex_dilog_xy_e(x, y, real_dl, imag_dl); +} + + +int +gsl_sf_complex_spence_xy_e( + const double x, + const double y, + gsl_sf_result * real_sp, + gsl_sf_result * imag_sp + ) +{ + const double oms_x = 1.0 - x; + const double oms_y = - y; + return gsl_sf_complex_dilog_xy_e(oms_x, oms_y, real_sp, imag_sp); +} + + + +/*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/ + +#include "eval.h" + +double gsl_sf_dilog(const double x) +{ + EVAL_RESULT(gsl_sf_dilog_e(x, &result)); +} |