diff options
Diffstat (limited to 'gsl-1.9/specfunc/fermi_dirac.c')
-rw-r--r-- | gsl-1.9/specfunc/fermi_dirac.c | 1633 |
1 files changed, 1633 insertions, 0 deletions
diff --git a/gsl-1.9/specfunc/fermi_dirac.c b/gsl-1.9/specfunc/fermi_dirac.c new file mode 100644 index 0000000..66808af --- /dev/null +++ b/gsl-1.9/specfunc/fermi_dirac.c @@ -0,0 +1,1633 @@ +/* specfunc/fermi_dirac.c + * + * Copyright (C) 1996, 1997, 1998, 1999, 2000 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_exp.h> +#include <gsl/gsl_sf_gamma.h> +#include <gsl/gsl_sf_hyperg.h> +#include <gsl/gsl_sf_pow_int.h> +#include <gsl/gsl_sf_zeta.h> +#include <gsl/gsl_sf_fermi_dirac.h> + +#include "error.h" + +#include "chebyshev.h" +#include "cheb_eval.c" + +#define locEPS (1000.0*GSL_DBL_EPSILON) + + +/* Chebyshev fit for F_{1}(t); -1 < t < 1, -1 < x < 1 + */ +static double fd_1_a_data[22] = { + 1.8949340668482264365, + 0.7237719066890052793, + 0.1250000000000000000, + 0.0101065196435973942, + 0.0, + -0.0000600615242174119, + 0.0, + 6.816528764623e-7, + 0.0, + -9.5895779195e-9, + 0.0, + 1.515104135e-10, + 0.0, + -2.5785616e-12, + 0.0, + 4.62270e-14, + 0.0, + -8.612e-16, + 0.0, + 1.65e-17, + 0.0, + -3.e-19 +}; +static cheb_series fd_1_a_cs = { + fd_1_a_data, + 21, + -1, 1, + 12 +}; + + +/* Chebyshev fit for F_{1}(3/2(t+1) + 1); -1 < t < 1, 1 < x < 4 + */ +static double fd_1_b_data[22] = { + 10.409136795234611872, + 3.899445098225161947, + 0.513510935510521222, + 0.010618736770218426, + -0.001584468020659694, + 0.000146139297161640, + -1.408095734499e-6, + -2.177993899484e-6, + 3.91423660640e-7, + -2.3860262660e-8, + -4.138309573e-9, + 1.283965236e-9, + -1.39695990e-10, + -4.907743e-12, + 4.399878e-12, + -7.17291e-13, + 2.4320e-14, + 1.4230e-14, + -3.446e-15, + 2.93e-16, + 3.7e-17, + -1.6e-17 +}; +static cheb_series fd_1_b_cs = { + fd_1_b_data, + 21, + -1, 1, + 11 +}; + + +/* Chebyshev fit for F_{1}(3(t+1) + 4); -1 < t < 1, 4 < x < 10 + */ +static double fd_1_c_data[23] = { + 56.78099449124299762, + 21.00718468237668011, + 2.24592457063193457, + 0.00173793640425994, + -0.00058716468739423, + 0.00016306958492437, + -0.00003817425583020, + 7.64527252009e-6, + -1.31348500162e-6, + 1.9000646056e-7, + -2.141328223e-8, + 1.23906372e-9, + 2.1848049e-10, + -1.0134282e-10, + 2.484728e-11, + -4.73067e-12, + 7.3555e-13, + -8.740e-14, + 4.85e-15, + 1.23e-15, + -5.6e-16, + 1.4e-16, + -3.e-17 +}; +static cheb_series fd_1_c_cs = { + fd_1_c_data, + 22, + -1, 1, + 13 +}; + + +/* Chebyshev fit for F_{1}(x) / x^2 + * 10 < x < 30 + * -1 < t < 1 + * t = 1/10 (x-10) - 1 = x/10 - 2 + * x = 10(t+2) + */ +static double fd_1_d_data[30] = { + 1.0126626021151374442, + -0.0063312525536433793, + 0.0024837319237084326, + -0.0008764333697726109, + 0.0002913344438921266, + -0.0000931877907705692, + 0.0000290151342040275, + -8.8548707259955e-6, + 2.6603474114517e-6, + -7.891415690452e-7, + 2.315730237195e-7, + -6.73179452963e-8, + 1.94048035606e-8, + -5.5507129189e-9, + 1.5766090896e-9, + -4.449310875e-10, + 1.248292745e-10, + -3.48392894e-11, + 9.6791550e-12, + -2.6786240e-12, + 7.388852e-13, + -2.032828e-13, + 5.58115e-14, + -1.52987e-14, + 4.1886e-15, + -1.1458e-15, + 3.132e-16, + -8.56e-17, + 2.33e-17, + -5.9e-18 +}; +static cheb_series fd_1_d_cs = { + fd_1_d_data, + 29, + -1, 1, + 14 +}; + + +/* Chebyshev fit for F_{1}(x) / x^2 + * 30 < x < Inf + * -1 < t < 1 + * t = 60/x - 1 + * x = 60/(t+1) + */ +static double fd_1_e_data[10] = { + 1.0013707783890401683, + 0.0009138522593601060, + 0.0002284630648400133, + -1.57e-17, + -1.27e-17, + -9.7e-18, + -6.9e-18, + -4.6e-18, + -2.9e-18, + -1.7e-18 +}; +static cheb_series fd_1_e_cs = { + fd_1_e_data, + 9, + -1, 1, + 4 +}; + + +/* Chebyshev fit for F_{2}(t); -1 < t < 1, -1 < x < 1 + */ +static double fd_2_a_data[21] = { + 2.1573661917148458336, + 0.8849670334241132182, + 0.1784163467613519713, + 0.0208333333333333333, + 0.0012708226459768508, + 0.0, + -5.0619314244895e-6, + 0.0, + 4.32026533989e-8, + 0.0, + -4.870544166e-10, + 0.0, + 6.4203740e-12, + 0.0, + -9.37424e-14, + 0.0, + 1.4715e-15, + 0.0, + -2.44e-17, + 0.0, + 4.e-19 +}; +static cheb_series fd_2_a_cs = { + fd_2_a_data, + 20, + -1, 1, + 12 +}; + + +/* Chebyshev fit for F_{2}(3/2(t+1) + 1); -1 < t < 1, 1 < x < 4 + */ +static double fd_2_b_data[22] = { + 16.508258811798623599, + 7.421719394793067988, + 1.458309885545603821, + 0.128773850882795229, + 0.001963612026198147, + -0.000237458988738779, + 0.000018539661382641, + -1.92805649479e-7, + -2.01950028452e-7, + 3.2963497518e-8, + -1.885817092e-9, + -2.72632744e-10, + 8.0554561e-11, + -8.313223e-12, + -2.24489e-13, + 2.18778e-13, + -3.4290e-14, + 1.225e-15, + 5.81e-16, + -1.37e-16, + 1.2e-17, + 1.e-18 +}; +static cheb_series fd_2_b_cs = { + fd_2_b_data, + 21, + -1, 1, + 12 +}; + + +/* Chebyshev fit for F_{1}(3(t+1) + 4); -1 < t < 1, 4 < x < 10 + */ +static double fd_2_c_data[20] = { + 168.87129776686440711, + 81.80260488091659458, + 15.75408505947931513, + 1.12325586765966440, + 0.00059057505725084, + -0.00016469712946921, + 0.00003885607810107, + -7.89873660613e-6, + 1.39786238616e-6, + -2.1534528656e-7, + 2.831510953e-8, + -2.94978583e-9, + 1.6755082e-10, + 2.234229e-11, + -1.035130e-11, + 2.41117e-12, + -4.3531e-13, + 6.447e-14, + -7.39e-15, + 4.3e-16 +}; +static cheb_series fd_2_c_cs = { + fd_2_c_data, + 19, + -1, 1, + 12 +}; + + +/* Chebyshev fit for F_{1}(x) / x^3 + * 10 < x < 30 + * -1 < t < 1 + * t = 1/10 (x-10) - 1 = x/10 - 2 + * x = 10(t+2) + */ +static double fd_2_d_data[30] = { + 0.3459960518965277589, + -0.00633136397691958024, + 0.00248382959047594408, + -0.00087651191884005114, + 0.00029139255351719932, + -0.00009322746111846199, + 0.00002904021914564786, + -8.86962264810663e-6, + 2.66844972574613e-6, + -7.9331564996004e-7, + 2.3359868615516e-7, + -6.824790880436e-8, + 1.981036528154e-8, + -5.71940426300e-9, + 1.64379426579e-9, + -4.7064937566e-10, + 1.3432614122e-10, + -3.823400534e-11, + 1.085771994e-11, + -3.07727465e-12, + 8.7064848e-13, + -2.4595431e-13, + 6.938531e-14, + -1.954939e-14, + 5.50162e-15, + -1.54657e-15, + 4.3429e-16, + -1.2178e-16, + 3.394e-17, + -8.81e-18 +}; +static cheb_series fd_2_d_cs = { + fd_2_d_data, + 29, + -1, 1, + 14 +}; + + +/* Chebyshev fit for F_{2}(x) / x^3 + * 30 < x < Inf + * -1 < t < 1 + * t = 60/x - 1 + * x = 60/(t+1) + */ +static double fd_2_e_data[4] = { + 0.3347041117223735227, + 0.00091385225936012645, + 0.00022846306484003205, + 5.2e-19 +}; +static cheb_series fd_2_e_cs = { + fd_2_e_data, + 3, + -1, 1, + 3 +}; + + +/* Chebyshev fit for F_{-1/2}(t); -1 < t < 1, -1 < x < 1 + */ +static double fd_mhalf_a_data[20] = { + 1.2663290042859741974, + 0.3697876251911153071, + 0.0278131011214405055, + -0.0033332848565672007, + -0.0004438108265412038, + 0.0000616495177243839, + 8.7589611449897e-6, + -1.2622936986172e-6, + -1.837464037221e-7, + 2.69495091400e-8, + 3.9760866257e-9, + -5.894468795e-10, + -8.77321638e-11, + 1.31016571e-11, + 1.9621619e-12, + -2.945887e-13, + -4.43234e-14, + 6.6816e-15, + 1.0084e-15, + -1.561e-16 +}; +static cheb_series fd_mhalf_a_cs = { + fd_mhalf_a_data, + 19, + -1, 1, + 12 +}; + + +/* Chebyshev fit for F_{-1/2}(3/2(t+1) + 1); -1 < t < 1, 1 < x < 4 + */ +static double fd_mhalf_b_data[20] = { + 3.270796131942071484, + 0.5809004935853417887, + -0.0299313438794694987, + -0.0013287935412612198, + 0.0009910221228704198, + -0.0001690954939688554, + 6.5955849946915e-6, + 3.5953966033618e-6, + -9.430672023181e-7, + 8.75773958291e-8, + 1.06247652607e-8, + -4.9587006215e-9, + 7.160432795e-10, + 4.5072219e-12, + -2.3695425e-11, + 4.9122208e-12, + -2.905277e-13, + -9.59291e-14, + 3.00028e-14, + -3.4970e-15 +}; +static cheb_series fd_mhalf_b_cs = { + fd_mhalf_b_data, + 19, + -1, 1, + 12 +}; + + +/* Chebyshev fit for F_{-1/2}(3(t+1) + 4); -1 < t < 1, 4 < x < 10 + */ +static double fd_mhalf_c_data[25] = { + 5.828283273430595507, + 0.677521118293264655, + -0.043946248736481554, + 0.005825595781828244, + -0.000864858907380668, + 0.000110017890076539, + -6.973305225404e-6, + -1.716267414672e-6, + 8.59811582041e-7, + -2.33066786976e-7, + 4.8503191159e-8, + -8.130620247e-9, + 1.021068250e-9, + -5.3188423e-11, + -1.9430559e-11, + 8.750506e-12, + -2.324897e-12, + 4.83102e-13, + -8.1207e-14, + 1.0132e-14, + -4.64e-16, + -2.24e-16, + 9.7e-17, + -2.6e-17, + 5.e-18 +}; +static cheb_series fd_mhalf_c_cs = { + fd_mhalf_c_data, + 24, + -1, 1, + 13 +}; + + +/* Chebyshev fit for F_{-1/2}(x) / x^(1/2) + * 10 < x < 30 + * -1 < t < 1 + * t = 1/10 (x-10) - 1 = x/10 - 2 + */ +static double fd_mhalf_d_data[30] = { + 2.2530744202862438709, + 0.0018745152720114692, + -0.0007550198497498903, + 0.0002759818676644382, + -0.0000959406283465913, + 0.0000324056855537065, + -0.0000107462396145761, + 3.5126865219224e-6, + -1.1313072730092e-6, + 3.577454162766e-7, + -1.104926666238e-7, + 3.31304165692e-8, + -9.5837381008e-9, + 2.6575790141e-9, + -7.015201447e-10, + 1.747111336e-10, + -4.04909605e-11, + 8.5104999e-12, + -1.5261885e-12, + 1.876851e-13, + 1.00574e-14, + -1.82002e-14, + 8.6634e-15, + -3.2058e-15, + 1.0572e-15, + -3.259e-16, + 9.60e-17, + -2.74e-17, + 7.6e-18, + -1.9e-18 +}; +static cheb_series fd_mhalf_d_cs = { + fd_mhalf_d_data, + 29, + -1, 1, + 15 +}; + + +/* Chebyshev fit for F_{1/2}(t); -1 < t < 1, -1 < x < 1 + */ +static double fd_half_a_data[23] = { + 1.7177138871306189157, + 0.6192579515822668460, + 0.0932802275119206269, + 0.0047094853246636182, + -0.0004243667967864481, + -0.0000452569787686193, + 5.2426509519168e-6, + 6.387648249080e-7, + -8.05777004848e-8, + -1.04290272415e-8, + 1.3769478010e-9, + 1.847190359e-10, + -2.51061890e-11, + -3.4497818e-12, + 4.784373e-13, + 6.68828e-14, + -9.4147e-15, + -1.3333e-15, + 1.898e-16, + 2.72e-17, + -3.9e-18, + -6.e-19, + 1.e-19 +}; +static cheb_series fd_half_a_cs = { + fd_half_a_data, + 22, + -1, 1, + 11 +}; + + +/* Chebyshev fit for F_{1/2}(3/2(t+1) + 1); -1 < t < 1, 1 < x < 4 + */ +static double fd_half_b_data[20] = { + 7.651013792074984027, + 2.475545606866155737, + 0.218335982672476128, + -0.007730591500584980, + -0.000217443383867318, + 0.000147663980681359, + -0.000021586361321527, + 8.07712735394e-7, + 3.28858050706e-7, + -7.9474330632e-8, + 6.940207234e-9, + 6.75594681e-10, + -3.10200490e-10, + 4.2677233e-11, + -2.1696e-14, + -1.170245e-12, + 2.34757e-13, + -1.4139e-14, + -3.864e-15, + 1.202e-15 +}; +static cheb_series fd_half_b_cs = { + fd_half_b_data, + 19, + -1, 1, + 12 +}; + + +/* Chebyshev fit for F_{1/2}(3(t+1) + 4); -1 < t < 1, 4 < x < 10 + */ +static double fd_half_c_data[23] = { + 29.584339348839816528, + 8.808344283250615592, + 0.503771641883577308, + -0.021540694914550443, + 0.002143341709406890, + -0.000257365680646579, + 0.000027933539372803, + -1.678525030167e-6, + -2.78100117693e-7, + 1.35218065147e-7, + -3.3740425009e-8, + 6.474834942e-9, + -1.009678978e-9, + 1.20057555e-10, + -6.636314e-12, + -1.710566e-12, + 7.75069e-13, + -1.97973e-13, + 3.9414e-14, + -6.374e-15, + 7.77e-16, + -4.0e-17, + -1.4e-17 +}; +static cheb_series fd_half_c_cs = { + fd_half_c_data, + 22, + -1, 1, + 13 +}; + + +/* Chebyshev fit for F_{1/2}(x) / x^(3/2) + * 10 < x < 30 + * -1 < t < 1 + * t = 1/10 (x-10) - 1 = x/10 - 2 + */ +static double fd_half_d_data[30] = { + 1.5116909434145508537, + -0.0036043405371630468, + 0.0014207743256393359, + -0.0005045399052400260, + 0.0001690758006957347, + -0.0000546305872688307, + 0.0000172223228484571, + -5.3352603788706e-6, + 1.6315287543662e-6, + -4.939021084898e-7, + 1.482515450316e-7, + -4.41552276226e-8, + 1.30503160961e-8, + -3.8262599802e-9, + 1.1123226976e-9, + -3.204765534e-10, + 9.14870489e-11, + -2.58778946e-11, + 7.2550731e-12, + -2.0172226e-12, + 5.566891e-13, + -1.526247e-13, + 4.16121e-14, + -1.12933e-14, + 3.0537e-15, + -8.234e-16, + 2.215e-16, + -5.95e-17, + 1.59e-17, + -4.0e-18 +}; +static cheb_series fd_half_d_cs = { + fd_half_d_data, + 29, + -1, 1, + 15 +}; + + + +/* Chebyshev fit for F_{3/2}(t); -1 < t < 1, -1 < x < 1 + */ +static double fd_3half_a_data[20] = { + 2.0404775940601704976, + 0.8122168298093491444, + 0.1536371165644008069, + 0.0156174323847845125, + 0.0005943427879290297, + -0.0000429609447738365, + -3.8246452994606e-6, + 3.802306180287e-7, + 4.05746157593e-8, + -4.5530360159e-9, + -5.306873139e-10, + 6.37297268e-11, + 7.8403674e-12, + -9.840241e-13, + -1.255952e-13, + 1.62617e-14, + 2.1318e-15, + -2.825e-16, + -3.78e-17, + 5.1e-18 +}; +static cheb_series fd_3half_a_cs = { + fd_3half_a_data, + 19, + -1, 1, + 11 +}; + + +/* Chebyshev fit for F_{3/2}(3/2(t+1) + 1); -1 < t < 1, 1 < x < 4 + */ +static double fd_3half_b_data[22] = { + 13.403206654624176674, + 5.574508357051880924, + 0.931228574387527769, + 0.054638356514085862, + -0.001477172902737439, + -0.000029378553381869, + 0.000018357033493246, + -2.348059218454e-6, + 8.3173787440e-8, + 2.6826486956e-8, + -6.011244398e-9, + 4.94345981e-10, + 3.9557340e-11, + -1.7894930e-11, + 2.348972e-12, + -1.2823e-14, + -5.4192e-14, + 1.0527e-14, + -6.39e-16, + -1.47e-16, + 4.5e-17, + -5.e-18 +}; +static cheb_series fd_3half_b_cs = { + fd_3half_b_data, + 21, + -1, 1, + 12 +}; + + +/* Chebyshev fit for F_{3/2}(3(t+1) + 4); -1 < t < 1, 4 < x < 10 + */ +static double fd_3half_c_data[21] = { + 101.03685253378877642, + 43.62085156043435883, + 6.62241373362387453, + 0.25081415008708521, + -0.00798124846271395, + 0.00063462245101023, + -0.00006392178890410, + 6.04535131939e-6, + -3.4007683037e-7, + -4.072661545e-8, + 1.931148453e-8, + -4.46328355e-9, + 7.9434717e-10, + -1.1573569e-10, + 1.304658e-11, + -7.4114e-13, + -1.4181e-13, + 6.491e-14, + -1.597e-14, + 3.05e-15, + -4.8e-16 +}; +static cheb_series fd_3half_c_cs = { + fd_3half_c_data, + 20, + -1, 1, + 12 +}; + + +/* Chebyshev fit for F_{3/2}(x) / x^(5/2) + * 10 < x < 30 + * -1 < t < 1 + * t = 1/10 (x-10) - 1 = x/10 - 2 + */ +static double fd_3half_d_data[25] = { + 0.6160645215171852381, + -0.0071239478492671463, + 0.0027906866139659846, + -0.0009829521424317718, + 0.0003260229808519545, + -0.0001040160912910890, + 0.0000322931223232439, + -9.8243506588102e-6, + 2.9420132351277e-6, + -8.699154670418e-7, + 2.545460071999e-7, + -7.38305056331e-8, + 2.12545670310e-8, + -6.0796532462e-9, + 1.7294556741e-9, + -4.896540687e-10, + 1.380786037e-10, + -3.88057305e-11, + 1.08753212e-11, + -3.0407308e-12, + 8.485626e-13, + -2.364275e-13, + 6.57636e-14, + -1.81807e-14, + 4.6884e-15 +}; +static cheb_series fd_3half_d_cs = { + fd_3half_d_data, + 24, + -1, 1, + 16 +}; + + +/* Goano's modification of the Levin-u implementation. + * This is a simplification of the original WHIZ algorithm. + * See [Fessler et al., ACM Toms 9, 346 (1983)]. + */ +static +int +fd_whiz(const double term, const int iterm, + double * qnum, double * qden, + double * result, double * s) +{ + if(iterm == 0) *s = 0.0; + + *s += term; + + qden[iterm] = 1.0/(term*(iterm+1.0)*(iterm+1.0)); + qnum[iterm] = *s * qden[iterm]; + + if(iterm > 0) { + double factor = 1.0; + double ratio = iterm/(iterm+1.0); + int j; + for(j=iterm-1; j>=0; j--) { + double c = factor * (j+1.0) / (iterm+1.0); + factor *= ratio; + qden[j] = qden[j+1] - c * qden[j]; + qnum[j] = qnum[j+1] - c * qnum[j]; + } + } + + *result = qnum[0] / qden[0]; + return GSL_SUCCESS; +} + + +/* Handle case of integer j <= -2. + */ +static +int +fd_nint(const int j, const double x, gsl_sf_result * result) +{ +/* const int nsize = 100 + 1; */ + enum { + nsize = 100+1 + }; + double qcoeff[nsize]; + + if(j >= -1) { + result->val = 0.0; + result->err = 0.0; + GSL_ERROR ("error", GSL_ESANITY); + } + else if(j < -(nsize)) { + result->val = 0.0; + result->err = 0.0; + GSL_ERROR ("error", GSL_EUNIMPL); + } + else { + double a, p, f; + int i, k; + int n = -(j+1); + + qcoeff[1] = 1.0; + + for(k=2; k<=n; k++) { + qcoeff[k] = -qcoeff[k-1]; + for(i=k-1; i>=2; i--) { + qcoeff[i] = i*qcoeff[i] - (k-(i-1))*qcoeff[i-1]; + } + } + + if(x >= 0.0) { + a = exp(-x); + f = qcoeff[1]; + for(i=2; i<=n; i++) { + f = f*a + qcoeff[i]; + } + } + else { + a = exp(x); + f = qcoeff[n]; + for(i=n-1; i>=1; i--) { + f = f*a + qcoeff[i]; + } + } + + p = gsl_sf_pow_int(1.0+a, j); + result->val = f*a*p; + result->err = 3.0 * GSL_DBL_EPSILON * fabs(f*a*p); + return GSL_SUCCESS; + } +} + + +/* x < 0 + */ +static +int +fd_neg(const double j, const double x, gsl_sf_result * result) +{ + enum { + itmax = 100, + qsize = 100+1 + }; +/* const int itmax = 100; */ +/* const int qsize = 100 + 1; */ + double qnum[qsize], qden[qsize]; + + if(x < GSL_LOG_DBL_MIN) { + result->val = 0.0; + result->err = 0.0; + return GSL_SUCCESS; + } + else if(x < -1.0 && x < -fabs(j+1.0)) { + /* Simple series implementation. Avoid the + * complexity and extra work of the series + * acceleration method below. + */ + double ex = exp(x); + double term = ex; + double sum = term; + int n; + for(n=2; n<100; n++) { + double rat = (n-1.0)/n; + double p = pow(rat, j+1.0); + term *= -ex * p; + sum += term; + if(fabs(term/sum) < GSL_DBL_EPSILON) break; + } + result->val = sum; + result->err = 2.0 * GSL_DBL_EPSILON * fabs(sum); + return GSL_SUCCESS; + } + else { + double s = 0.0; + double xn = x; + double ex = -exp(x); + double enx = -ex; + double f = 0.0; + double f_previous; + int jterm; + for(jterm=0; jterm<=itmax; jterm++) { + double p = pow(jterm+1.0, j+1.0); + double term = enx/p; + f_previous = f; + fd_whiz(term, jterm, qnum, qden, &f, &s); + xn += x; + if(fabs(f-f_previous) < fabs(f)*2.0*GSL_DBL_EPSILON || xn < GSL_LOG_DBL_MIN) break; + enx *= ex; + } + + result->val = f; + result->err = fabs(f-f_previous); + result->err += 2.0 * GSL_DBL_EPSILON * fabs(f); + + if(jterm == itmax) + GSL_ERROR ("error", GSL_EMAXITER); + else + return GSL_SUCCESS; + } +} + + +/* asymptotic expansion + * j + 2.0 > 0.0 + */ +static +int +fd_asymp(const double j, const double x, gsl_sf_result * result) +{ + const int j_integer = ( fabs(j - floor(j+0.5)) < 100.0*GSL_DBL_EPSILON ); + const int itmax = 200; + gsl_sf_result lg; + int stat_lg = gsl_sf_lngamma_e(j + 2.0, &lg); + double seqn_val = 0.5; + double seqn_err = 0.0; + double xm2 = (1.0/x)/x; + double xgam = 1.0; + double add = GSL_DBL_MAX; + double cos_term; + double ln_x; + double ex_term_1; + double ex_term_2; + gsl_sf_result fneg; + gsl_sf_result ex_arg; + gsl_sf_result ex; + int stat_fneg; + int stat_e; + int n; + for(n=1; n<=itmax; n++) { + double add_previous = add; + gsl_sf_result eta; + gsl_sf_eta_int_e(2*n, &eta); + xgam = xgam * xm2 * (j + 1.0 - (2*n-2)) * (j + 1.0 - (2*n-1)); + add = eta.val * xgam; + if(!j_integer && fabs(add) > fabs(add_previous)) break; + if(fabs(add/seqn_val) < GSL_DBL_EPSILON) break; + seqn_val += add; + seqn_err += 2.0 * GSL_DBL_EPSILON * fabs(add); + } + seqn_err += fabs(add); + + stat_fneg = fd_neg(j, -x, &fneg); + ln_x = log(x); + ex_term_1 = (j+1.0)*ln_x; + ex_term_2 = lg.val; + ex_arg.val = ex_term_1 - ex_term_2; /*(j+1.0)*ln_x - lg.val; */ + ex_arg.err = GSL_DBL_EPSILON*(fabs(ex_term_1) + fabs(ex_term_2)) + lg.err; + stat_e = gsl_sf_exp_err_e(ex_arg.val, ex_arg.err, &ex); + cos_term = cos(j*M_PI); + result->val = cos_term * fneg.val + 2.0 * seqn_val * ex.val; + result->err = fabs(2.0 * ex.err * seqn_val); + result->err += fabs(2.0 * ex.val * seqn_err); + result->err += fabs(cos_term) * fneg.err; + result->err += 4.0 * GSL_DBL_EPSILON * fabs(result->val); + return GSL_ERROR_SELECT_3(stat_e, stat_fneg, stat_lg); +} + + +/* Series evaluation for small x, generic j. + * [Goano (8)] + */ +#if 0 +static +int +fd_series(const double j, const double x, double * result) +{ + const int nmax = 1000; + int n; + double sum = 0.0; + double prev; + double pow_factor = 1.0; + double eta_factor; + gsl_sf_eta_e(j + 1.0, &eta_factor); + prev = pow_factor * eta_factor; + sum += prev; + for(n=1; n<nmax; n++) { + double term; + gsl_sf_eta_e(j+1.0-n, &eta_factor); + pow_factor *= x/n; + term = pow_factor * eta_factor; + sum += term; + if(fabs(term/sum) < GSL_DBL_EPSILON && fabs(prev/sum) < GSL_DBL_EPSILON) break; + prev = term; + } + + *result = sum; + return GSL_SUCCESS; +} +#endif /* 0 */ + + +/* Series evaluation for small x > 0, integer j > 0; x < Pi. + * [Goano (8)] + */ +static +int +fd_series_int(const int j, const double x, gsl_sf_result * result) +{ + int n; + double sum = 0.0; + double del; + double pow_factor = 1.0; + gsl_sf_result eta_factor; + gsl_sf_eta_int_e(j + 1, &eta_factor); + del = pow_factor * eta_factor.val; + sum += del; + + /* Sum terms where the argument + * of eta() is positive. + */ + for(n=1; n<=j+2; n++) { + gsl_sf_eta_int_e(j+1-n, &eta_factor); + pow_factor *= x/n; + del = pow_factor * eta_factor.val; + sum += del; + if(fabs(del/sum) < GSL_DBL_EPSILON) break; + } + + /* Now sum the terms where eta() is negative. + * The argument of eta() must be odd as well, + * so it is convenient to transform the series + * as follows: + * + * Sum[ eta(j+1-n) x^n / n!, {n,j+4,Infinity}] + * = x^j / j! Sum[ eta(1-2m) x^(2m) j! / (2m+j)! , {m,2,Infinity}] + * + * We do not need to do this sum if j is large enough. + */ + if(j < 32) { + int m; + gsl_sf_result jfact; + double sum2; + double pre2; + + gsl_sf_fact_e((unsigned int)j, &jfact); + pre2 = gsl_sf_pow_int(x, j) / jfact.val; + + gsl_sf_eta_int_e(-3, &eta_factor); + pow_factor = x*x*x*x / ((j+4)*(j+3)*(j+2)*(j+1)); + sum2 = eta_factor.val * pow_factor; + + for(m=3; m<24; m++) { + gsl_sf_eta_int_e(1-2*m, &eta_factor); + pow_factor *= x*x / ((j+2*m)*(j+2*m-1)); + sum2 += eta_factor.val * pow_factor; + } + + sum += pre2 * sum2; + } + + result->val = sum; + result->err = 2.0 * GSL_DBL_EPSILON * fabs(sum); + + return GSL_SUCCESS; +} + + +/* series of hypergeometric functions for integer j > 0, x > 0 + * [Goano (7)] + */ +static +int +fd_UMseries_int(const int j, const double x, gsl_sf_result * result) +{ + const int nmax = 2000; + double pre; + double lnpre_val; + double lnpre_err; + double sum_even_val = 1.0; + double sum_even_err = 0.0; + double sum_odd_val = 0.0; + double sum_odd_err = 0.0; + int stat_sum; + int stat_e; + int stat_h = GSL_SUCCESS; + int n; + + if(x < 500.0 && j < 80) { + double p = gsl_sf_pow_int(x, j+1); + gsl_sf_result g; + gsl_sf_fact_e(j+1, &g); /* Gamma(j+2) */ + lnpre_val = 0.0; + lnpre_err = 0.0; + pre = p/g.val; + } + else { + double lnx = log(x); + gsl_sf_result lg; + gsl_sf_lngamma_e(j + 2.0, &lg); + lnpre_val = (j+1.0)*lnx - lg.val; + lnpre_err = 2.0 * GSL_DBL_EPSILON * fabs((j+1.0)*lnx) + lg.err; + pre = 1.0; + } + + /* Add up the odd terms of the sum. + */ + for(n=1; n<nmax; n+=2) { + double del_val; + double del_err; + gsl_sf_result U; + gsl_sf_result M; + int stat_h_U = gsl_sf_hyperg_U_int_e(1, j+2, n*x, &U); + int stat_h_F = gsl_sf_hyperg_1F1_int_e(1, j+2, -n*x, &M); + stat_h = GSL_ERROR_SELECT_3(stat_h, stat_h_U, stat_h_F); + del_val = ((j+1.0)*U.val - M.val); + del_err = (fabs(j+1.0)*U.err + M.err); + sum_odd_val += del_val; + sum_odd_err += del_err; + if(fabs(del_val/sum_odd_val) < GSL_DBL_EPSILON) break; + } + + /* Add up the even terms of the sum. + */ + for(n=2; n<nmax; n+=2) { + double del_val; + double del_err; + gsl_sf_result U; + gsl_sf_result M; + int stat_h_U = gsl_sf_hyperg_U_int_e(1, j+2, n*x, &U); + int stat_h_F = gsl_sf_hyperg_1F1_int_e(1, j+2, -n*x, &M); + stat_h = GSL_ERROR_SELECT_3(stat_h, stat_h_U, stat_h_F); + del_val = ((j+1.0)*U.val - M.val); + del_err = (fabs(j+1.0)*U.err + M.err); + sum_even_val -= del_val; + sum_even_err += del_err; + if(fabs(del_val/sum_even_val) < GSL_DBL_EPSILON) break; + } + + stat_sum = ( n >= nmax ? GSL_EMAXITER : GSL_SUCCESS ); + stat_e = gsl_sf_exp_mult_err_e(lnpre_val, lnpre_err, + pre*(sum_even_val + sum_odd_val), + pre*(sum_even_err + sum_odd_err), + result); + result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val); + + return GSL_ERROR_SELECT_3(stat_e, stat_h, stat_sum); +} + + +/*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/ + +/* [Goano (4)] */ +int gsl_sf_fermi_dirac_m1_e(const double x, gsl_sf_result * result) +{ + if(x < GSL_LOG_DBL_MIN) { + UNDERFLOW_ERROR(result); + } + else if(x < 0.0) { + const double ex = exp(x); + result->val = ex/(1.0+ex); + result->err = 2.0 * (fabs(x) + 1.0) * GSL_DBL_EPSILON * fabs(result->val); + return GSL_SUCCESS; + } + else { + double ex = exp(-x); + result->val = 1.0/(1.0 + ex); + result->err = 2.0 * GSL_DBL_EPSILON * (x + 1.0) * ex; + return GSL_SUCCESS; + } +} + + +/* [Goano (3)] */ +int gsl_sf_fermi_dirac_0_e(const double x, gsl_sf_result * result) +{ + if(x < GSL_LOG_DBL_MIN) { + UNDERFLOW_ERROR(result); + } + else if(x < -5.0) { + double ex = exp(x); + double ser = 1.0 - ex*(0.5 - ex*(1.0/3.0 - ex*(1.0/4.0 - ex*(1.0/5.0 - ex/6.0)))); + result->val = ex * ser; + result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val); + return GSL_SUCCESS; + } + else if(x < 10.0) { + result->val = log(1.0 + exp(x)); + result->err = fabs(x * GSL_DBL_EPSILON); + return GSL_SUCCESS; + } + else { + double ex = exp(-x); + result->val = x + ex * (1.0 - 0.5*ex + ex*ex/3.0 - ex*ex*ex/4.0); + result->err = (x + ex) * GSL_DBL_EPSILON; + return GSL_SUCCESS; + } +} + + +int gsl_sf_fermi_dirac_1_e(const double x, gsl_sf_result * result) +{ + if(x < GSL_LOG_DBL_MIN) { + UNDERFLOW_ERROR(result); + } + else if(x < -1.0) { + /* series [Goano (6)] + */ + double ex = exp(x); + double term = ex; + double sum = term; + int n; + for(n=2; n<100 ; n++) { + double rat = (n-1.0)/n; + term *= -ex * rat * rat; + sum += term; + if(fabs(term/sum) < GSL_DBL_EPSILON) break; + } + result->val = sum; + result->err = 2.0 * fabs(sum) * GSL_DBL_EPSILON; + return GSL_SUCCESS; + } + else if(x < 1.0) { + return cheb_eval_e(&fd_1_a_cs, x, result); + } + else if(x < 4.0) { + double t = 2.0/3.0*(x-1.0) - 1.0; + return cheb_eval_e(&fd_1_b_cs, t, result); + } + else if(x < 10.0) { + double t = 1.0/3.0*(x-4.0) - 1.0; + return cheb_eval_e(&fd_1_c_cs, t, result); + } + else if(x < 30.0) { + double t = 0.1*x - 2.0; + gsl_sf_result c; + cheb_eval_e(&fd_1_d_cs, t, &c); + result->val = c.val * x*x; + result->err = c.err * x*x + GSL_DBL_EPSILON * fabs(result->val); + return GSL_SUCCESS; + } + else if(x < 1.0/GSL_SQRT_DBL_EPSILON) { + double t = 60.0/x - 1.0; + gsl_sf_result c; + cheb_eval_e(&fd_1_e_cs, t, &c); + result->val = c.val * x*x; + result->err = c.err * x*x + GSL_DBL_EPSILON * fabs(result->val); + return GSL_SUCCESS; + } + else if(x < GSL_SQRT_DBL_MAX) { + result->val = 0.5 * x*x; + result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val); + return GSL_SUCCESS; + } + else { + OVERFLOW_ERROR(result); + } +} + + +int gsl_sf_fermi_dirac_2_e(const double x, gsl_sf_result * result) +{ + if(x < GSL_LOG_DBL_MIN) { + UNDERFLOW_ERROR(result); + } + else if(x < -1.0) { + /* series [Goano (6)] + */ + double ex = exp(x); + double term = ex; + double sum = term; + int n; + for(n=2; n<100 ; n++) { + double rat = (n-1.0)/n; + term *= -ex * rat * rat * rat; + sum += term; + if(fabs(term/sum) < GSL_DBL_EPSILON) break; + } + result->val = sum; + result->err = 2.0 * GSL_DBL_EPSILON * fabs(sum); + return GSL_SUCCESS; + } + else if(x < 1.0) { + return cheb_eval_e(&fd_2_a_cs, x, result); + } + else if(x < 4.0) { + double t = 2.0/3.0*(x-1.0) - 1.0; + return cheb_eval_e(&fd_2_b_cs, t, result); + } + else if(x < 10.0) { + double t = 1.0/3.0*(x-4.0) - 1.0; + return cheb_eval_e(&fd_2_c_cs, t, result); + } + else if(x < 30.0) { + double t = 0.1*x - 2.0; + gsl_sf_result c; + cheb_eval_e(&fd_2_d_cs, t, &c); + result->val = c.val * x*x*x; + result->err = c.err * x*x*x + 3.0 * GSL_DBL_EPSILON * fabs(result->val); + return GSL_SUCCESS; + } + else if(x < 1.0/GSL_ROOT3_DBL_EPSILON) { + double t = 60.0/x - 1.0; + gsl_sf_result c; + cheb_eval_e(&fd_2_e_cs, t, &c); + result->val = c.val * x*x*x; + result->err = c.err * x*x*x + 3.0 * GSL_DBL_EPSILON * fabs(result->val); + return GSL_SUCCESS; + } + else if(x < GSL_ROOT3_DBL_MAX) { + result->val = 1.0/6.0 * x*x*x; + result->err = 3.0 * GSL_DBL_EPSILON * fabs(result->val); + return GSL_SUCCESS; + } + else { + OVERFLOW_ERROR(result); + } +} + + +int gsl_sf_fermi_dirac_int_e(const int j, const double x, gsl_sf_result * result) +{ + if(j < -1) { + return fd_nint(j, x, result); + } + else if (j == -1) { + return gsl_sf_fermi_dirac_m1_e(x, result); + } + else if(j == 0) { + return gsl_sf_fermi_dirac_0_e(x, result); + } + else if(j == 1) { + return gsl_sf_fermi_dirac_1_e(x, result); + } + else if(j == 2) { + return gsl_sf_fermi_dirac_2_e(x, result); + } + else if(x < 0.0) { + return fd_neg(j, x, result); + } + else if(x == 0.0) { + return gsl_sf_eta_int_e(j+1, result); + } + else if(x < 1.5) { + return fd_series_int(j, x, result); + } + else { + gsl_sf_result fasymp; + int stat_asymp = fd_asymp(j, x, &fasymp); + + if(stat_asymp == GSL_SUCCESS) { + result->val = fasymp.val; + result->err = fasymp.err; + result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val); + return stat_asymp; + } + else { + return fd_UMseries_int(j, x, result); + } + } +} + + +int gsl_sf_fermi_dirac_mhalf_e(const double x, gsl_sf_result * result) +{ + if(x < GSL_LOG_DBL_MIN) { + UNDERFLOW_ERROR(result); + } + else if(x < -1.0) { + /* series [Goano (6)] + */ + double ex = exp(x); + double term = ex; + double sum = term; + int n; + for(n=2; n<200 ; n++) { + double rat = (n-1.0)/n; + term *= -ex * sqrt(rat); + sum += term; + if(fabs(term/sum) < GSL_DBL_EPSILON) break; + } + result->val = sum; + result->err = 2.0 * fabs(sum) * GSL_DBL_EPSILON; + return GSL_SUCCESS; + } + else if(x < 1.0) { + return cheb_eval_e(&fd_mhalf_a_cs, x, result); + } + else if(x < 4.0) { + double t = 2.0/3.0*(x-1.0) - 1.0; + return cheb_eval_e(&fd_mhalf_b_cs, t, result); + } + else if(x < 10.0) { + double t = 1.0/3.0*(x-4.0) - 1.0; + return cheb_eval_e(&fd_mhalf_c_cs, t, result); + } + else if(x < 30.0) { + double rtx = sqrt(x); + double t = 0.1*x - 2.0; + gsl_sf_result c; + cheb_eval_e(&fd_mhalf_d_cs, t, &c); + result->val = c.val * rtx; + result->err = c.err * rtx + 0.5 * GSL_DBL_EPSILON * fabs(result->val); + return GSL_SUCCESS; + } + else { + return fd_asymp(-0.5, x, result); + } +} + + +int gsl_sf_fermi_dirac_half_e(const double x, gsl_sf_result * result) +{ + if(x < GSL_LOG_DBL_MIN) { + UNDERFLOW_ERROR(result); + } + else if(x < -1.0) { + /* series [Goano (6)] + */ + double ex = exp(x); + double term = ex; + double sum = term; + int n; + for(n=2; n<100 ; n++) { + double rat = (n-1.0)/n; + term *= -ex * rat * sqrt(rat); + sum += term; + if(fabs(term/sum) < GSL_DBL_EPSILON) break; + } + result->val = sum; + result->err = 2.0 * fabs(sum) * GSL_DBL_EPSILON; + return GSL_SUCCESS; + } + else if(x < 1.0) { + return cheb_eval_e(&fd_half_a_cs, x, result); + } + else if(x < 4.0) { + double t = 2.0/3.0*(x-1.0) - 1.0; + return cheb_eval_e(&fd_half_b_cs, t, result); + } + else if(x < 10.0) { + double t = 1.0/3.0*(x-4.0) - 1.0; + return cheb_eval_e(&fd_half_c_cs, t, result); + } + else if(x < 30.0) { + double x32 = x*sqrt(x); + double t = 0.1*x - 2.0; + gsl_sf_result c; + cheb_eval_e(&fd_half_d_cs, t, &c); + result->val = c.val * x32; + result->err = c.err * x32 + 1.5 * GSL_DBL_EPSILON * fabs(result->val); + return GSL_SUCCESS; + } + else { + return fd_asymp(0.5, x, result); + } +} + + +int gsl_sf_fermi_dirac_3half_e(const double x, gsl_sf_result * result) +{ + if(x < GSL_LOG_DBL_MIN) { + UNDERFLOW_ERROR(result); + } + else if(x < -1.0) { + /* series [Goano (6)] + */ + double ex = exp(x); + double term = ex; + double sum = term; + int n; + for(n=2; n<100 ; n++) { + double rat = (n-1.0)/n; + term *= -ex * rat * rat * sqrt(rat); + sum += term; + if(fabs(term/sum) < GSL_DBL_EPSILON) break; + } + result->val = sum; + result->err = 2.0 * fabs(sum) * GSL_DBL_EPSILON; + return GSL_SUCCESS; + } + else if(x < 1.0) { + return cheb_eval_e(&fd_3half_a_cs, x, result); + } + else if(x < 4.0) { + double t = 2.0/3.0*(x-1.0) - 1.0; + return cheb_eval_e(&fd_3half_b_cs, t, result); + } + else if(x < 10.0) { + double t = 1.0/3.0*(x-4.0) - 1.0; + return cheb_eval_e(&fd_3half_c_cs, t, result); + } + else if(x < 30.0) { + double x52 = x*x*sqrt(x); + double t = 0.1*x - 2.0; + gsl_sf_result c; + cheb_eval_e(&fd_3half_d_cs, t, &c); + result->val = c.val * x52; + result->err = c.err * x52 + 2.5 * GSL_DBL_EPSILON * fabs(result->val); + return GSL_SUCCESS; + } + else { + return fd_asymp(1.5, x, result); + } +} + +/* [Goano p. 222] */ +int gsl_sf_fermi_dirac_inc_0_e(const double x, const double b, gsl_sf_result * result) +{ + if(b < 0.0) { + DOMAIN_ERROR(result); + } + else { + double arg = b - x; + gsl_sf_result f0; + int status = gsl_sf_fermi_dirac_0_e(arg, &f0); + result->val = f0.val - arg; + result->err = f0.err + GSL_DBL_EPSILON * (fabs(x) + fabs(b)); + return status; + } +} + + + +/*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/ + +#include "eval.h" + +double gsl_sf_fermi_dirac_m1(const double x) +{ + EVAL_RESULT(gsl_sf_fermi_dirac_m1_e(x, &result)); +} + +double gsl_sf_fermi_dirac_0(const double x) +{ + EVAL_RESULT(gsl_sf_fermi_dirac_0_e(x, &result)); +} + +double gsl_sf_fermi_dirac_1(const double x) +{ + EVAL_RESULT(gsl_sf_fermi_dirac_1_e(x, &result)); +} + +double gsl_sf_fermi_dirac_2(const double x) +{ + EVAL_RESULT(gsl_sf_fermi_dirac_2_e(x, &result)); +} + +double gsl_sf_fermi_dirac_int(const int j, const double x) +{ + EVAL_RESULT(gsl_sf_fermi_dirac_int_e(j, x, &result)); +} + +double gsl_sf_fermi_dirac_mhalf(const double x) +{ + EVAL_RESULT(gsl_sf_fermi_dirac_mhalf_e(x, &result)); +} + +double gsl_sf_fermi_dirac_half(const double x) +{ + EVAL_RESULT(gsl_sf_fermi_dirac_half_e(x, &result)); +} + +double gsl_sf_fermi_dirac_3half(const double x) +{ + EVAL_RESULT(gsl_sf_fermi_dirac_3half_e(x, &result)); +} + +double gsl_sf_fermi_dirac_inc_0(const double x, const double b) +{ + EVAL_RESULT(gsl_sf_fermi_dirac_inc_0_e(x, b, &result)); +} |