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diff --git a/gsl-1.9/specfunc/bessel_zero.c b/gsl-1.9/specfunc/bessel_zero.c
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+/* specfunc/bessel_zero.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_airy.h>
+#include <gsl/gsl_sf_pow_int.h>
+#include <gsl/gsl_sf_bessel.h>
+
+#include "error.h"
+
+#include "bessel_olver.h"
+
+/* For Chebyshev expansions of the roots as functions of nu,
+ * see [G. Nemeth, Mathematical Approximation of Special Functions].
+ * This gives the fits for all nu and s <= 10.
+ * I made the fits for other values of s myself [GJ].
+ */
+
+/* Chebyshev expansion: j_{nu,1} = c_k T_k*(nu/2), nu <= 2 */
+static const double coef_jnu1_a[] = {
+ 3.801775243633476,
+ 1.360704737511120,
+ -0.030707710261106,
+ 0.004526823746202,
+ -0.000808682832134,
+ 0.000159218792489,
+ -0.000033225189761,
+ 0.000007205599763,
+ -0.000001606110397,
+ 0.000000365439424,
+ -0.000000084498039,
+ 0.000000019793815,
+ -0.000000004687054,
+ 0.000000001120052,
+ -0.000000000269767,
+ 0.000000000065420,
+ -0.000000000015961,
+ 0.000000000003914,
+ -0.000000000000965,
+ 0.000000000000239,
+ -0.000000000000059,
+ 0.000000000000015,
+ -0.000000000000004,
+ 0.000000000000001
+};
+
+
+/* Chebyshev expansion: j_{nu,1} = nu c_k T_k*((2/nu)^(2/3)), nu >= 2 */
+static const double coef_jnu1_b[] = {
+ 1.735063412537096,
+ 0.784478100951978,
+ 0.048881473180370,
+ -0.000578279783021,
+ -0.000038984957864,
+ 0.000005758297879,
+ -0.000000327583229,
+ -0.000000003853878,
+ 0.000000002284653,
+ -0.000000000153079,
+ -0.000000000000895,
+ 0.000000000000283,
+ 0.000000000000043,
+ 0.000000000000010,
+ -0.000000000000003
+};
+
+
+/* Chebyshev expansion: j_{nu,2} = c_k T_k*(nu/2), nu <= 2 */
+static const double coef_jnu2_a[] = {
+ 6.992370244046161,
+ 1.446379282056534,
+ -0.023458616207293,
+ 0.002172149448700,
+ -0.000246262775620,
+ 0.000030990180959,
+ -0.000004154183047,
+ 0.000000580766328,
+ -0.000000083648175,
+ 0.000000012317355,
+ -0.000000001844887,
+ 0.000000000280076,
+ -0.000000000042986,
+ 0.000000000006658,
+ -0.000000000001039,
+ 0.000000000000163,
+ -0.000000000000026,
+ 0.000000000000004,
+ -0.000000000000001
+};
+
+
+/* Chebyshev expansion: j_{nu,2} = nu c_k T_k*((2/nu)^(2/3)), nu >= 2 */
+static const double coef_jnu2_b[] = {
+ 2.465611864263400,
+ 1.607952988471069,
+ 0.138758034431497,
+ -0.003687791182054,
+ -0.000051276007868,
+ 0.000045113570749,
+ -0.000007579172152,
+ 0.000000736469208,
+ -0.000000011118527,
+ -0.000000011919884,
+ 0.000000002696788,
+ -0.000000000314488,
+ 0.000000000008124,
+ 0.000000000005211,
+ -0.000000000001292,
+ 0.000000000000158,
+ -0.000000000000004,
+ -0.000000000000003,
+ 0.000000000000001
+};
+
+
+/* Chebyshev expansion: j_{nu,3} = c_k T_k*(nu/3), nu <= 3 */
+static const double coef_jnu3_a[] = {
+ 10.869647065239236,
+ 2.177524286141710,
+ -0.034822817125293,
+ 0.003167249102413,
+ -0.000353960349344,
+ 0.000044039086085,
+ -0.000005851380981,
+ 0.000000812575483,
+ -0.000000116463617,
+ 0.000000017091246,
+ -0.000000002554376,
+ 0.000000000387335,
+ -0.000000000059428,
+ 0.000000000009207,
+ -0.000000000001438,
+ 0.000000000000226,
+ -0.000000000000036,
+ 0.000000000000006,
+ -0.000000000000001
+};
+
+
+/* Chebyshev expansion: j_{nu,3} = nu c_k T_k*((3/nu)^(2/3)), nu >= 3 */
+static const double coef_jnu3_b[] = {
+ 2.522816775173244,
+ 1.673199424973720,
+ 0.146431617506314,
+ -0.004049001763912,
+ -0.000039517767244,
+ 0.000048781729288,
+ -0.000008729705695,
+ 0.000000928737310,
+ -0.000000028388244,
+ -0.000000012927432,
+ 0.000000003441008,
+ -0.000000000471695,
+ 0.000000000025590,
+ 0.000000000005502,
+ -0.000000000001881,
+ 0.000000000000295,
+ -0.000000000000020,
+ -0.000000000000003,
+ 0.000000000000001
+};
+
+
+/* Chebyshev expansion: j_{nu,4} = c_k T_k*(nu/4), nu <= 4 */
+static const double coef_jnu4_a[] = {
+ 14.750310252773009,
+ 2.908010932941708,
+ -0.046093293420315,
+ 0.004147172321412,
+ -0.000459092310473,
+ 0.000056646951906,
+ -0.000007472351546,
+ 0.000001031210065,
+ -0.000000147008137,
+ 0.000000021475218,
+ -0.000000003197208,
+ 0.000000000483249,
+ -0.000000000073946,
+ 0.000000000011431,
+ -0.000000000001782,
+ 0.000000000000280,
+ -0.000000000000044,
+ 0.000000000000007,
+ -0.000000000000001
+};
+
+
+/* Chebyshev expansion: j_{nu,4} = nu c_k T_k*((4/nu)^(2/3)), nu >= 4 */
+static const double coef_jnu4_b[] = {
+ 2.551681323117914,
+ 1.706177978336572,
+ 0.150357658406131,
+ -0.004234001378590,
+ -0.000033854229898,
+ 0.000050763551485,
+ -0.000009337464057,
+ 0.000001029717834,
+ -0.000000037474196,
+ -0.000000013450153,
+ 0.000000003836180,
+ -0.000000000557404,
+ 0.000000000035748,
+ 0.000000000005487,
+ -0.000000000002187,
+ 0.000000000000374,
+ -0.000000000000031,
+ -0.000000000000003,
+ 0.000000000000001
+};
+
+
+
+/* Chebyshev expansion: j_{nu,5} = c_k T_k*(nu/5), nu <= 5 */
+static const double coef_jnu5_a[] = {
+ 18.632261081028211,
+ 3.638249012596966,
+ -0.057329705998828,
+ 0.005121709126820,
+ -0.000563325259487,
+ 0.000069100826174,
+ -0.000009066603030,
+ 0.000001245181383,
+ -0.000000176737282,
+ 0.000000025716695,
+ -0.000000003815184,
+ 0.000000000574839,
+ -0.000000000087715,
+ 0.000000000013526,
+ -0.000000000002104,
+ 0.000000000000330,
+ -0.000000000000052,
+ 0.000000000000008,
+ -0.000000000000001
+};
+
+
+/* Chebyshev expansion: j_{nu,5} = nu c_k T_k*((5/nu)^(2/3)), nu >= 5 */
+/* FIXME: There is something wrong with this fit, in about the
+ * 9th or 10th decimal place.
+ */
+static const double coef_jnu5_b[] = {
+ 2.569079487591442,
+ 1.726073360882134,
+ 0.152740776809531,
+ -0.004346449660148,
+ -0.000030512461856,
+ 0.000052000821080,
+ -0.000009713343981,
+ 0.000001091997863,
+ -0.000000043061707,
+ -0.000000013779413,
+ 0.000000004082870,
+ -0.000000000611259,
+ 0.000000000042242,
+ 0.000000000005448,
+ -0.000000000002377,
+ 0.000000000000424,
+ -0.000000000000038,
+ -0.000000000000002,
+ 0.000000000000002
+};
+
+
+/* Chebyshev expansion: j_{nu,6} = c_k T_k*(nu/6), nu <= 6 */
+static const double coef_jnu6_a[] = {
+ 22.514836143374042,
+ 4.368367257557198,
+ -0.068550155285562,
+ 0.006093776505822,
+ -0.000667152784957,
+ 0.000081486022398,
+ -0.000010649011647,
+ 0.000001457089679,
+ -0.000000206105082,
+ 0.000000029894724,
+ -0.000000004422012,
+ 0.000000000664471,
+ -0.000000000101140,
+ 0.000000000015561,
+ -0.000000000002416,
+ 0.000000000000378,
+ -0.000000000000060,
+ 0.000000000000009,
+ -0.000000000000002
+};
+
+
+/* Chebyshev expansion: j_{nu,6} = nu c_k T_k*((6/nu)^(2/3)), nu >= 6 */
+static const double coef_jnu6_b[] = {
+ 2.580710285494837,
+ 1.739380728566154,
+ 0.154340696401691,
+ -0.004422028860168,
+ -0.000028305272624,
+ 0.000052845975269,
+ -0.000009968794373,
+ 0.000001134252926,
+ -0.000000046841241,
+ -0.000000014007555,
+ 0.000000004251816,
+ -0.000000000648213,
+ 0.000000000046728,
+ 0.000000000005414,
+ -0.000000000002508,
+ 0.000000000000459,
+ -0.000000000000043,
+ -0.000000000000002,
+ 0.000000000000002
+};
+
+
+/* Chebyshev expansion: j_{nu,7} = c_k T_k*(nu/7), nu <= 7 */
+static const double coef_jnu7_a[] = {
+ 26.397760539730869,
+ 5.098418721711790,
+ -0.079761896398948,
+ 0.007064521280487,
+ -0.000770766522482,
+ 0.000093835449636,
+ -0.000012225308542,
+ 0.000001667939800,
+ -0.000000235288157,
+ 0.000000034040347,
+ -0.000000005023142,
+ 0.000000000753101,
+ -0.000000000114389,
+ 0.000000000017564,
+ -0.000000000002722,
+ 0.000000000000425,
+ -0.000000000000067,
+ 0.000000000000011,
+ -0.000000000000002
+};
+
+
+/* Chebyshev expansion: j_{nu,7} = nu c_k T_k*((7/nu)^(2/3)), nu >= 7 */
+static const double coef_jnu7_b[] = {
+ 2.589033335856773,
+ 1.748907007612678,
+ 0.155488900387653,
+ -0.004476317805688,
+ -0.000026737952924,
+ 0.000053459680946,
+ -0.000010153699240,
+ 0.000001164804272,
+ -0.000000049566917,
+ -0.000000014175403,
+ 0.000000004374840,
+ -0.000000000675135,
+ 0.000000000050004,
+ 0.000000000005387,
+ -0.000000000002603,
+ 0.000000000000485,
+ -0.000000000000047,
+ -0.000000000000002,
+ 0.000000000000002
+};
+
+
+/* Chebyshev expansion: j_{nu,8} = c_k T_k*(nu/8), nu <= 8 */
+static const double coef_jnu8_a[] = {
+ 30.280900001606662,
+ 5.828429205461221,
+ -0.090968381181069,
+ 0.008034479731033,
+ -0.000874254899080,
+ 0.000106164151611,
+ -0.000013798098749,
+ 0.000001878187386,
+ -0.000000264366627,
+ 0.000000038167685,
+ -0.000000005621060,
+ 0.000000000841165,
+ -0.000000000127538,
+ 0.000000000019550,
+ -0.000000000003025,
+ 0.000000000000472,
+ -0.000000000000074,
+ 0.000000000000012,
+ -0.000000000000002
+};
+
+
+/* Chebyshev expansion: j_{nu,8} = nu c_k T_k*((8/nu)^(2/3)), nu >= 8 */
+static const double coef_jnu8_b[] = {
+ 2.595283877150078,
+ 1.756063044986928,
+ 0.156352972371030,
+ -0.004517201896761,
+ -0.000025567187878,
+ 0.000053925472558,
+ -0.000010293734486,
+ 0.000001187923085,
+ -0.000000051625122,
+ -0.000000014304212,
+ 0.000000004468450,
+ -0.000000000695620,
+ 0.000000000052500,
+ 0.000000000005367,
+ -0.000000000002676,
+ 0.000000000000505,
+ -0.000000000000050,
+ -0.000000000000002,
+ 0.000000000000002
+};
+
+
+/* Chebyshev expansion: j_{nu,9} = c_k T_k*(nu/9), nu <= 9 */
+static const double coef_jnu9_a[] = {
+ 34.164181213238386,
+ 6.558412747925228,
+ -0.102171455365016,
+ 0.009003934361201,
+ -0.000977663914535,
+ 0.000118479876579,
+ -0.000015368714220,
+ 0.000002088064285,
+ -0.000000293381154,
+ 0.000000042283900,
+ -0.000000006217033,
+ 0.000000000928887,
+ -0.000000000140627,
+ 0.000000000021526,
+ -0.000000000003326,
+ 0.000000000000518,
+ -0.000000000000081,
+ 0.000000000000013,
+ -0.000000000000002
+};
+
+
+/* Chebyshev expansion: j_{nu,9} = nu c_k T_k*((9/nu)^(2/3)), nu >= 9 */
+static const double coef_jnu9_b[] = {
+ 2.600150240905079,
+ 1.761635491694032,
+ 0.157026743724010,
+ -0.004549100368716,
+ -0.000024659248617,
+ 0.000054291035068,
+ -0.000010403464334,
+ 0.000001206027524,
+ -0.000000053234089,
+ -0.000000014406241,
+ 0.000000004542078,
+ -0.000000000711728,
+ 0.000000000054464,
+ 0.000000000005350,
+ -0.000000000002733,
+ 0.000000000000521,
+ -0.000000000000052,
+ -0.000000000000002,
+ 0.000000000000002
+};
+
+
+/* Chebyshev expansion: j_{nu,10} = c_k T_k*(nu/10), nu <= 10 */
+static const double coef_jnu10_a[] = {
+ 38.047560766184647,
+ 7.288377637926008,
+ -0.113372193277897,
+ 0.009973047509098,
+ -0.001081019701335,
+ 0.000130786983847,
+ -0.000016937898538,
+ 0.000002297699179,
+ -0.000000322354218,
+ 0.000000046392941,
+ -0.000000006811759,
+ 0.000000001016395,
+ -0.000000000153677,
+ 0.000000000023486,
+ -0.000000000003616,
+ 0.000000000000561,
+ -0.000000000000095,
+ 0.000000000000027,
+ -0.000000000000013,
+ 0.000000000000005
+};
+
+
+/* Chebyshev expansion: j_{nu,10} = nu c_k T_k*((10/nu)^(2/3)), nu >= 10 */
+static const double coef_jnu10_b[] = {
+ 2.604046346867949,
+ 1.766097596481182,
+ 0.157566834446511,
+ -0.004574682244089,
+ -0.000023934500688,
+ 0.000054585558231,
+ -0.000010491765415,
+ 0.000001220589364,
+ -0.000000054526331,
+ -0.000000014489078,
+ 0.000000004601510,
+ -0.000000000724727,
+ 0.000000000056049,
+ 0.000000000005337,
+ -0.000000000002779,
+ 0.000000000000533,
+ -0.000000000000054,
+ -0.000000000000002,
+ 0.000000000000002
+};
+
+
+/* Chebyshev expansion: j_{nu,11} = c_k T_k*(nu/22), nu <= 22 */
+static const double coef_jnu11_a[] = {
+ 49.5054081076848637,
+ 15.33692279367165101,
+ -0.33677234163517130,
+ 0.04623235772920729,
+ -0.00781084960665093,
+ 0.00147217395434708,
+ -0.00029695043846867,
+ 0.00006273356860235,
+ -0.00001370575125628,
+ 3.07171282012e-6,
+ -7.0235041249e-7,
+ 1.6320559339e-7,
+ -3.843117306e-8,
+ 9.15083800e-9,
+ -2.19957642e-9,
+ 5.3301703e-10,
+ -1.3007541e-10,
+ 3.193827e-11,
+ -7.88605e-12,
+ 1.95918e-12,
+ -4.9020e-13,
+ 1.2207e-13,
+ -2.820e-14,
+ 5.25e-15,
+ -1.88e-15,
+ 2.80e-15,
+ -2.45e-15
+};
+
+
+/* Chebyshev expansion: j_{nu,12} = c_k T_k*(nu/24), nu <= 24 */
+static const double coef_jnu12_a[] = {
+ 54.0787833216641519,
+ 16.7336367772863598,
+ -0.36718411124537953,
+ 0.05035523375053820,
+ -0.00849884978867533,
+ 0.00160027692813434,
+ -0.00032248114889921,
+ 0.00006806354127199,
+ -0.00001485665901339,
+ 3.32668783672e-6,
+ -7.5998952729e-7,
+ 1.7644939709e-7,
+ -4.151538210e-8,
+ 9.87722772e-9,
+ -2.37230133e-9,
+ 5.7442875e-10,
+ -1.4007767e-10,
+ 3.437166e-11,
+ -8.48215e-12,
+ 2.10554e-12,
+ -5.2623e-13,
+ 1.3189e-13,
+ -3.175e-14,
+ 5.73e-15,
+ 5.6e-16,
+ -8.7e-16,
+ -6.5e-16
+};
+
+
+/* Chebyshev expansion: j_{nu,13} = c_k T_k*(nu/26), nu <= 26 */
+static const double coef_jnu13_a[] = {
+ 58.6521941921708890,
+ 18.1303398137970284,
+ -0.39759381380126650,
+ 0.05447765240465494,
+ -0.00918674227679980,
+ 0.00172835361420579,
+ -0.00034800528297612,
+ 0.00007339183835188,
+ -0.00001600713368099,
+ 3.58154960392e-6,
+ -8.1759873497e-7,
+ 1.8968523220e-7,
+ -4.459745253e-8,
+ 1.060304419e-8,
+ -2.54487624e-9,
+ 6.1580214e-10,
+ -1.5006751e-10,
+ 3.679707e-11,
+ -9.07159e-12,
+ 2.24713e-12,
+ -5.5943e-13,
+ 1.4069e-13,
+ -3.679e-14,
+ 1.119e-14,
+ -4.99e-15,
+ 3.43e-15,
+ -2.85e-15,
+ 2.3e-15,
+ -1.7e-15,
+ 8.7e-16
+};
+
+
+/* Chebyshev expansion: j_{nu,14} = c_k T_k*(nu/28), nu <= 28 */
+static const double coef_jnu14_a[] = {
+ 63.2256329577315566,
+ 19.5270342832914901,
+ -0.42800190567884337,
+ 0.05859971627729398,
+ -0.00987455163523582,
+ 0.00185641011402081,
+ -0.00037352439419968,
+ 0.00007871886257265,
+ -0.00001715728110045,
+ 3.83632624437e-6,
+ -8.7518558668e-7,
+ 2.0291515353e-7,
+ -4.767795233e-8,
+ 1.132844415e-8,
+ -2.71734219e-9,
+ 6.5714886e-10,
+ -1.6005342e-10,
+ 3.922557e-11,
+ -9.66637e-12,
+ 2.39379e-12,
+ -5.9541e-13,
+ 1.4868e-13,
+ -3.726e-14,
+ 9.37e-15,
+ -2.36e-15,
+ 6.0e-16
+};
+
+
+/* Chebyshev expansion: j_{nu,15} = c_k T_k*(nu/30), nu <= 30 */
+static const double coef_jnu15_a[] = {
+ 67.7990939565631635,
+ 20.9237219226859859,
+ -0.45840871823085836,
+ 0.06272149946755639,
+ -0.01056229551143042,
+ 0.00198445078693100,
+ -0.00039903958650729,
+ 0.00008404489865469,
+ -0.00001830717574922,
+ 4.09103745566e-6,
+ -9.3275533309e-7,
+ 2.1614056403e-7,
+ -5.075725222e-8,
+ 1.205352081e-8,
+ -2.88971837e-9,
+ 6.9846848e-10,
+ -1.7002946e-10,
+ 4.164941e-11,
+ -1.025859e-11,
+ 2.53921e-12,
+ -6.3128e-13,
+ 1.5757e-13,
+ -3.947e-14,
+ 9.92e-15,
+ -2.50e-15,
+ 6.3e-16
+};
+
+
+/* Chebyshev expansion: j_{nu,16} = c_k T_k*(nu/32), nu <= 32 */
+static const double coef_jnu16_a[] = {
+ 72.3725729616724770,
+ 22.32040402918608585,
+ -0.48881449782358690,
+ 0.06684305681828766,
+ -0.01124998690363398,
+ 0.00211247882775445,
+ -0.00042455166484632,
+ 0.00008937015316346,
+ -0.00001945687139551,
+ 4.34569739281e-6,
+ -9.9031173548e-7,
+ 2.2936247195e-7,
+ -5.383562595e-8,
+ 1.277835103e-8,
+ -3.06202860e-9,
+ 7.3977037e-10,
+ -1.8000071e-10,
+ 4.407196e-11,
+ -1.085046e-11,
+ 2.68453e-12,
+ -6.6712e-13,
+ 1.6644e-13,
+ -4.168e-14,
+ 1.047e-14,
+ -2.64e-15,
+ 6.7e-16
+};
+
+
+/* Chebyshev expansion: j_{nu,17} = c_k T_k*(nu/34), nu <= 34 */
+static const double coef_jnu17_a[] = {
+ 76.9460667535209549,
+ 23.71708159112252670,
+ -0.51921943142405352,
+ 0.07096442978067622,
+ -0.01193763559341369,
+ 0.00224049662974902,
+ -0.00045006122941781,
+ 0.00009469477941684,
+ -0.00002060640777107,
+ 4.60031647195e-6,
+ -1.04785755046e-6,
+ 2.4258161247e-7,
+ -5.691327087e-8,
+ 1.350298805e-8,
+ -3.23428733e-9,
+ 7.8105847e-10,
+ -1.8996825e-10,
+ 4.649350e-11,
+ -1.144205e-11,
+ 2.82979e-12,
+ -7.0294e-13,
+ 1.7531e-13,
+ -4.388e-14,
+ 1.102e-14,
+ -2.78e-15,
+ 7.0e-16
+};
+
+
+/* Chebyshev expansion: j_{nu,18} = c_k T_k*(nu/36), nu <= 36 */
+static const double coef_jnu18_a[] = {
+ 81.5195728368096659,
+ 25.11375537470259305,
+ -0.54962366347317668,
+ 0.07508565026117689,
+ -0.01262524908033818,
+ 0.00236850602019778,
+ -0.00047556873651929,
+ 0.00010001889347161,
+ -0.00002175581482429,
+ 4.85490251239e-6,
+ -1.10539483940e-6,
+ 2.5579853343e-7,
+ -5.999033352e-8,
+ 1.422747129e-8,
+ -3.40650521e-9,
+ 8.2233565e-10,
+ -1.9993286e-10,
+ 4.891426e-11,
+ -1.203343e-11,
+ 2.97498e-12,
+ -7.3875e-13,
+ 1.8418e-13,
+ -4.608e-14,
+ 1.157e-14,
+ -2.91e-15,
+ 7.4e-16
+};
+
+
+/* Chebyshev expansion: j_{nu,19} = c_k T_k*(nu/38), nu <= 38 */
+static const double coef_jnu19_a[] = {
+ 86.0930892477047512,
+ 26.51042598308271729,
+ -0.58002730731948358,
+ 0.07920674321589394,
+ -0.01331283320930301,
+ 0.00249650841778073,
+ -0.00050107453900793,
+ 0.00010534258471335,
+ -0.00002290511552874,
+ 5.10946148897e-6,
+ -1.16292517157e-6,
+ 2.6901365037e-7,
+ -6.306692473e-8,
+ 1.495183048e-8,
+ -3.57869025e-9,
+ 8.6360410e-10,
+ -2.0989514e-10,
+ 5.133439e-11,
+ -1.262465e-11,
+ 3.12013e-12,
+ -7.7455e-13,
+ 1.9304e-13,
+ -4.829e-14,
+ 1.212e-14,
+ -3.05e-15,
+ 7.7e-16
+};
+
+
+/* Chebyshev expansion: j_{nu,20} = c_k T_k*(nu/40), nu <= 40 */
+static const double coef_jnu20_a[] = {
+ 90.6666144195163770,
+ 27.9070938975436823,
+ -0.61043045315390591,
+ 0.08332772844325554,
+ -0.01400039260208282,
+ 0.00262450494035660,
+ -0.00052657891389470,
+ 0.00011066592304919,
+ -0.00002405432778364,
+ 5.36399803946e-6,
+ -1.22044976064e-6,
+ 2.8222728362e-7,
+ -6.614312964e-8,
+ 1.567608839e-8,
+ -3.75084856e-9,
+ 9.0486546e-10,
+ -2.1985553e-10,
+ 5.375401e-11,
+ -1.321572e-11,
+ 3.26524e-12,
+ -8.1033e-13,
+ 2.0190e-13,
+ -5.049e-14,
+ 1.267e-14,
+ -3.19e-15,
+ 8.0e-16,
+ -2.0e-16
+};
+
+
+static const double * coef_jnu_a[] = {
+ 0,
+ coef_jnu1_a,
+ coef_jnu2_a,
+ coef_jnu3_a,
+ coef_jnu4_a,
+ coef_jnu5_a,
+ coef_jnu6_a,
+ coef_jnu7_a,
+ coef_jnu8_a,
+ coef_jnu9_a,
+ coef_jnu10_a,
+ coef_jnu11_a,
+ coef_jnu12_a,
+ coef_jnu13_a,
+ coef_jnu14_a,
+ coef_jnu15_a,
+ coef_jnu16_a,
+ coef_jnu17_a,
+ coef_jnu18_a,
+ coef_jnu19_a,
+ coef_jnu20_a
+};
+
+static const size_t size_jnu_a[] = {
+ 0,
+ sizeof(coef_jnu1_a)/sizeof(double),
+ sizeof(coef_jnu2_a)/sizeof(double),
+ sizeof(coef_jnu3_a)/sizeof(double),
+ sizeof(coef_jnu4_a)/sizeof(double),
+ sizeof(coef_jnu5_a)/sizeof(double),
+ sizeof(coef_jnu6_a)/sizeof(double),
+ sizeof(coef_jnu7_a)/sizeof(double),
+ sizeof(coef_jnu8_a)/sizeof(double),
+ sizeof(coef_jnu9_a)/sizeof(double),
+ sizeof(coef_jnu10_a)/sizeof(double),
+ sizeof(coef_jnu11_a)/sizeof(double),
+ sizeof(coef_jnu12_a)/sizeof(double),
+ sizeof(coef_jnu13_a)/sizeof(double),
+ sizeof(coef_jnu14_a)/sizeof(double),
+ sizeof(coef_jnu15_a)/sizeof(double),
+ sizeof(coef_jnu16_a)/sizeof(double),
+ sizeof(coef_jnu17_a)/sizeof(double),
+ sizeof(coef_jnu18_a)/sizeof(double),
+ sizeof(coef_jnu19_a)/sizeof(double),
+ sizeof(coef_jnu20_a)/sizeof(double)
+};
+
+
+static const double * coef_jnu_b[] = {
+ 0,
+ coef_jnu1_b,
+ coef_jnu2_b,
+ coef_jnu3_b,
+ coef_jnu4_b,
+ coef_jnu5_b,
+ coef_jnu6_b,
+ coef_jnu7_b,
+ coef_jnu8_b,
+ coef_jnu9_b,
+ coef_jnu10_b
+};
+
+static const size_t size_jnu_b[] = {
+ 0,
+ sizeof(coef_jnu1_b)/sizeof(double),
+ sizeof(coef_jnu2_b)/sizeof(double),
+ sizeof(coef_jnu3_b)/sizeof(double),
+ sizeof(coef_jnu4_b)/sizeof(double),
+ sizeof(coef_jnu5_b)/sizeof(double),
+ sizeof(coef_jnu6_b)/sizeof(double),
+ sizeof(coef_jnu7_b)/sizeof(double),
+ sizeof(coef_jnu8_b)/sizeof(double),
+ sizeof(coef_jnu9_b)/sizeof(double),
+ sizeof(coef_jnu10_b)/sizeof(double)
+};
+
+
+
+/* Evaluate Clenshaw recurrence for
+ * a T* Chebyshev series.
+ * sizeof(c) = N+1
+ */
+static double
+clenshaw(const double * c, int N, double u)
+{
+ double B_np1 = 0.0;
+ double B_n = c[N];
+ double B_nm1;
+ int n;
+ for(n=N; n>0; n--) {
+ B_nm1 = 2.0*(2.0*u-1.0) * B_n - B_np1 + c[n-1];
+ B_np1 = B_n;
+ B_n = B_nm1;
+ }
+ return B_n - (2.0*u-1.0)*B_np1;
+}
+
+
+
+/* correction terms to leading McMahon expansion
+ * [Abramowitz+Stegun 9.5.12]
+ * [Olver, Royal Society Math. Tables, v. 7]
+ * We factor out a beta, so that this is a multiplicative
+ * correction:
+ * j_{nu,s} = beta(s,nu) * mcmahon_correction(nu, beta(s,nu))
+ * macmahon_correction --> 1 as s --> Inf
+ */
+static double
+mcmahon_correction(const double mu, const double beta)
+{
+ const double eb = 8.0*beta;
+ const double ebsq = eb*eb;
+
+ if(mu < GSL_DBL_EPSILON) {
+ /* Prevent division by zero below. */
+ const double term1 = 1.0/ebsq;
+ const double term2 = -4.0*31.0/(3*ebsq*ebsq);
+ const double term3 = 32.0*3779.0/(15.0*ebsq*ebsq*ebsq);
+ const double term4 = -64.0*6277237.0/(105.0*ebsq*ebsq*ebsq*ebsq);
+ const double term5 = 512.0*2092163573.0/(315.0*ebsq*ebsq*ebsq*ebsq*ebsq);
+ return 1.0 + 8.0*(term1 + term2 + term3 + term4 + term5);
+ }
+ else {
+ /* Here we do things in terms of 1/mu, which
+ * is purely to prevent overflow in the very
+ * unlikely case that mu is really big.
+ */
+ const double mi = 1.0/mu;
+ const double r = mu/ebsq;
+ const double n2 = 4.0/3.0 * (7.0 - 31.0*mi);
+ const double n3 = 32.0/15.0 * (83.0 + (-982.0 + 3779.0*mi)*mi);
+ const double n4 = 64.0/105.0 * (6949.0 + (-153855.0 + (1585743.0 - 6277237.0*mi)*mi)*mi);
+ const double n5 = 512.0/315.0 * (70197.0 + (-2479316.0 + (48010494.0 + (-512062548.0 + 2092163573.0*mi)*mi)*mi)*mi);
+ const double n6 = 2048.0/3465.0 * (5592657.0 + (-287149133.0 + (8903961290.0 + (-179289628602.0 + (1982611456181.0 - 8249725736393.0*mi)*mi)*mi)*mi)*mi);
+ const double term1 = (1.0 - mi) * r;
+ const double term2 = term1 * n2 * r;
+ const double term3 = term1 * n3 * r*r;
+ const double term4 = term1 * n4 * r*r*r;
+ const double term5 = term1 * n5 * r*r*r*r;
+ const double term6 = term1 * n6 * r*r*r*r*r;
+ return 1.0 - 8.0*(term1 + term2 + term3 + term4 + term5 + term6);
+ }
+}
+
+
+/* Assumes z >= 1.0 */
+static double
+olver_b0(double z, double minus_zeta)
+{
+ if(z < 1.02) {
+ const double a = 1.0-z;
+ const double c0 = 0.0179988721413553309252458658183;
+ const double c1 = 0.0111992982212877614645974276203;
+ const double c2 = 0.0059404069786014304317781160605;
+ const double c3 = 0.0028676724516390040844556450173;
+ const double c4 = 0.0012339189052567271708525111185;
+ const double c5 = 0.0004169250674535178764734660248;
+ const double c6 = 0.0000330173385085949806952777365;
+ const double c7 = -0.0001318076238578203009990106425;
+ const double c8 = -0.0001906870370050847239813945647;
+ return c0 + a*(c1 + a*(c2 + a*(c3 + a*(c4 + a*(c5 + a*(c6 + a*(c7 + a*c8)))))));
+ }
+ else {
+ const double abs_zeta = minus_zeta;
+ const double t = 1.0/(z*sqrt(1.0 - 1.0/(z*z)));
+ return -5.0/(48.0*abs_zeta*abs_zeta) + t*(3.0 + 5.0*t*t)/(24.0*sqrt(abs_zeta));
+ }
+}
+
+
+inline
+static double
+olver_f1(double z, double minus_zeta)
+{
+ const double b0 = olver_b0(z, minus_zeta);
+ const double h2 = sqrt(4.0*minus_zeta/(z*z-1.0)); /* FIXME */
+ return 0.5 * z * h2 * b0;
+}
+
+
+int
+gsl_sf_bessel_zero_J0_e(unsigned int s, gsl_sf_result * result)
+{
+ /* CHECK_POINTER(result) */
+
+ if(s == 0){
+ result->val = 0.0;
+ result->err = 0.0;
+ GSL_ERROR ("error", GSL_EINVAL);
+ }
+ else {
+ /* See [F. Lether, J. Comp. Appl .Math. 67, 167 (1996)]. */
+
+ static const double P[] = { 1567450796.0/12539606369.0,
+ 8903660.0/2365861.0,
+ 10747040.0/536751.0,
+ 17590991.0/1696654.0
+ };
+ static const double Q[] = { 1.0,
+ 29354255.0/954518.0,
+ 76900001.0/431847.0,
+ 67237052.0/442411.0
+ };
+
+ const double beta = (s - 0.25) * M_PI;
+ const double bi2 = 1.0/(beta*beta);
+ const double R33num = P[0] + bi2 * (P[1] + bi2 * (P[2] + P[3] * bi2));
+ const double R33den = Q[0] + bi2 * (Q[1] + bi2 * (Q[2] + Q[3] * bi2));
+ const double R33 = R33num/R33den;
+ result->val = beta + R33/beta;
+ result->err = fabs(3.0e-15 * result->val);
+ return GSL_SUCCESS;
+ }
+}
+
+
+int
+gsl_sf_bessel_zero_J1_e(unsigned int s, gsl_sf_result * result)
+{
+ /* CHECK_POINTER(result) */
+
+ if(s == 0) {
+ result->val = 0.0;
+ result->err = 0.0;
+ return GSL_SUCCESS;
+ }
+ else {
+ /* See [M. Branders et al., J. Comp. Phys. 42, 403 (1981)]. */
+
+ static const double a[] = { -0.362804405737084,
+ 0.120341279038597,
+ 0.439454547101171e-01,
+ 0.159340088474713e-02
+ };
+ static const double b[] = { 1.0,
+ -0.325641790801361,
+ -0.117453445968927,
+ -0.424906902601794e-02
+ };
+
+ const double beta = (s + 0.25) * M_PI;
+ const double bi2 = 1.0/(beta*beta);
+ const double Rnum = a[3] + bi2 * (a[2] + bi2 * (a[1] + bi2 * a[0]));
+ const double Rden = b[3] + bi2 * (b[2] + bi2 * (b[1] + bi2 * b[0]));
+ const double R = Rnum/Rden;
+ result->val = beta * (1.0 + R*bi2);
+ result->err = fabs(2.0e-14 * result->val);
+ return GSL_SUCCESS;
+ }
+}
+
+
+int
+gsl_sf_bessel_zero_Jnu_e(double nu, unsigned int s, gsl_sf_result * result)
+{
+ /* CHECK_POINTER(result) */
+
+ if(nu <= -1.0) {
+ DOMAIN_ERROR(result);
+ }
+ else if(s == 0) {
+ result->val = 0.0;
+ result->err = 0.0;
+ if (nu == 0.0) {
+ GSL_ERROR ("no zero-th root for nu = 0.0", GSL_EINVAL);
+ }
+ return GSL_SUCCESS;
+ }
+ else if(nu < 0.0) {
+ /* This can be done, I'm just lazy now. */
+ result->val = 0.0;
+ result->err = 0.0;
+ GSL_ERROR("unimplemented", GSL_EUNIMPL);
+ }
+ else if(s == 1) {
+ /* Chebyshev fits for the first positive zero.
+ * For some reason Nemeth made this different from the others.
+ */
+ if(nu < 2.0) {
+ const double * c = coef_jnu_a[s];
+ const size_t L = size_jnu_a[s];
+ const double arg = nu/2.0;
+ const double chb = clenshaw(c, L-1, arg);
+ result->val = chb;
+ result->err = 2.0e-15 * result->val;
+ }
+ else {
+ const double * c = coef_jnu_b[s];
+ const size_t L = size_jnu_b[s];
+ const double arg = pow(2.0/nu, 2.0/3.0);
+ const double chb = clenshaw(c, L-1, arg);
+ result->val = nu * chb;
+ result->err = 2.0e-15 * result->val;
+ }
+ return GSL_SUCCESS;
+ }
+ else if(s <= 10) {
+ /* Chebyshev fits for the first 10 positive zeros. */
+ if(nu < s) {
+ const double * c = coef_jnu_a[s];
+ const size_t L = size_jnu_a[s];
+ const double arg = nu/s;
+ const double chb = clenshaw(c, L-1, arg);
+ result->val = chb;
+ result->err = 2.0e-15 * result->val;
+ }
+ else {
+ const double * c = coef_jnu_b[s];
+ const size_t L = size_jnu_b[s];
+ const double arg = pow(s/nu, 2.0/3.0);
+ const double chb = clenshaw(c, L-1, arg);
+ result->val = nu * chb;
+ result->err = 2.0e-15 * result->val;
+
+ /* FIXME: truth in advertising for the screwed up
+ * s = 5 fit. Need to fix that.
+ */
+ if(s == 5) {
+ result->err *= 5.0e+06;
+ }
+ }
+ return GSL_SUCCESS;
+ }
+ else if(s > 0.5*nu && s <= 20) {
+ /* Chebyshev fits for 10 < s <= 20. */
+ const double * c = coef_jnu_a[s];
+ const size_t L = size_jnu_a[s];
+ const double arg = nu/(2.0*s);
+ const double chb = clenshaw(c, L-1, arg);
+ result->val = chb;
+ result->err = 4.0e-15 * chb;
+ return GSL_SUCCESS;
+ }
+ else if(s > 2.0 * nu) {
+ /* McMahon expansion if s is large compared to nu. */
+ const double beta = (s + 0.5*nu - 0.25) * M_PI;
+ const double mc = mcmahon_correction(4.0*nu*nu, beta);
+ gsl_sf_result rat12;
+ gsl_sf_pow_int_e(nu/beta, 14, &rat12);
+ result->val = beta * mc;
+ result->err = 4.0 * fabs(beta) * rat12.val;
+ result->err += 4.0 * fabs(GSL_DBL_EPSILON * result->val);
+ return GSL_SUCCESS;
+ }
+ else {
+ /* Olver uniform asymptotic. */
+ gsl_sf_result as;
+ const int stat_as = gsl_sf_airy_zero_Ai_e(s, &as);
+ const double minus_zeta = -pow(nu,-2.0/3.0) * as.val;
+ const double z = gsl_sf_bessel_Olver_zofmzeta(minus_zeta);
+ const double f1 = olver_f1(z, minus_zeta);
+ result->val = nu * (z + f1/(nu*nu));
+ result->err = 0.001/(nu*nu*nu);
+ result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
+ return stat_as;
+ }
+}
+
+
+/*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/
+
+#include "eval.h"
+
+double gsl_sf_bessel_zero_J0(unsigned int s)
+{
+ EVAL_RESULT(gsl_sf_bessel_zero_J0_e(s, &result));
+}
+
+double gsl_sf_bessel_zero_J1(unsigned int s)
+{
+ EVAL_RESULT(gsl_sf_bessel_zero_J1_e(s, &result));
+}
+
+double gsl_sf_bessel_zero_Jnu(double nu, unsigned int s)
+{
+ EVAL_RESULT(gsl_sf_bessel_zero_Jnu_e(nu, s, &result));
+}
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