diff options
author | Joel Sherrill <joel.sherrill@OARcorp.com> | 2011-04-08 17:33:11 +0000 |
---|---|---|
committer | Joel Sherrill <joel.sherrill@OARcorp.com> | 2011-04-08 17:33:11 +0000 |
commit | 73f643f3f4a55310b2c8c1a9858906b2dd676e72 (patch) | |
tree | b1df97f18dace4a5702b0bc7aafdfee8a2f25ada /gsl-1.9/specfunc/coulomb.c | |
parent | ee523abdace8337d05ec4a179fcdf5de3fe0f634 (diff) |
2011-04-08 Joel Sherrill <joel.sherrill@oarcorp.com>
* AUTHORS, BUGS, COPYING, ChangeLog, INSTALL, Makefile.am, Makefile.in,
NEWS, README, SUPPORT, THANKS, TODO, acconfig.h, aclocal.m4,
autogen.sh, config.guess, config.h.in, config.sub, configure,
configure.ac, gsl-config.in, gsl-histogram.c, gsl-randist.c, gsl.m4,
gsl.pc.in, gsl.spec.in, gsl_machine.h, gsl_math.h, gsl_mode.h,
gsl_nan.h, gsl_pow_int.h, gsl_precision.h, gsl_types.h,
gsl_version.h.in, install-sh, ltmain.sh, mdate-sh, missing,
mkinstalldirs, templates_off.h, templates_on.h,
test_gsl_histogram.sh, version.c, blas/ChangeLog, blas/Makefile.am,
blas/Makefile.in, blas/TODO, blas/blas.c, blas/gsl_blas.h,
blas/gsl_blas_types.h, block/ChangeLog, block/Makefile.am,
block/Makefile.in, block/block.c, block/block_source.c, block/file.c,
block/fprintf_source.c, block/fwrite_source.c, block/gsl_block.h,
block/gsl_block_char.h, block/gsl_block_complex_double.h,
block/gsl_block_complex_float.h,
block/gsl_block_complex_long_double.h, block/gsl_block_double.h,
block/gsl_block_float.h, block/gsl_block_int.h,
block/gsl_block_long.h, block/gsl_block_long_double.h,
block/gsl_block_short.h, block/gsl_block_uchar.h,
block/gsl_block_uint.h, block/gsl_block_ulong.h,
block/gsl_block_ushort.h, block/gsl_check_range.h, block/init.c,
block/init_source.c, block/test.c, block/test_complex_io.c,
block/test_complex_source.c, block/test_io.c, block/test_source.c,
bspline/ChangeLog, bspline/Makefile.am, bspline/Makefile.in,
bspline/TODO, bspline/bspline.c, bspline/gsl_bspline.h,
bspline/test.c, cblas/ChangeLog, cblas/Makefile.am,
cblas/Makefile.in, cblas/TODO, cblas/caxpy.c, cblas/cblas.h,
cblas/ccopy.c, cblas/cdotc_sub.c, cblas/cdotu_sub.c, cblas/cgbmv.c,
cblas/cgemm.c, cblas/cgemv.c, cblas/cgerc.c, cblas/cgeru.c,
cblas/chbmv.c, cblas/chemm.c, cblas/chemv.c, cblas/cher.c,
cblas/cher2.c, cblas/cher2k.c, cblas/cherk.c, cblas/chpmv.c,
cblas/chpr.c, cblas/chpr2.c, cblas/cscal.c, cblas/csscal.c,
cblas/cswap.c, cblas/csymm.c, cblas/csyr2k.c, cblas/csyrk.c,
cblas/ctbmv.c, cblas/ctbsv.c, cblas/ctpmv.c, cblas/ctpsv.c,
cblas/ctrmm.c, cblas/ctrmv.c, cblas/ctrsm.c, cblas/ctrsv.c,
cblas/dasum.c, cblas/daxpy.c, cblas/dcopy.c, cblas/ddot.c,
cblas/dgbmv.c, cblas/dgemm.c, cblas/dgemv.c, cblas/dger.c,
cblas/dnrm2.c, cblas/drot.c, cblas/drotg.c, cblas/drotm.c,
cblas/drotmg.c, cblas/dsbmv.c, cblas/dscal.c, cblas/dsdot.c,
cblas/dspmv.c, cblas/dspr.c, cblas/dspr2.c, cblas/dswap.c,
cblas/dsymm.c, cblas/dsymv.c, cblas/dsyr.c, cblas/dsyr2.c,
cblas/dsyr2k.c, cblas/dsyrk.c, cblas/dtbmv.c, cblas/dtbsv.c,
cblas/dtpmv.c, cblas/dtpsv.c, cblas/dtrmm.c, cblas/dtrmv.c,
cblas/dtrsm.c, cblas/dtrsv.c, cblas/dzasum.c, cblas/dznrm2.c,
cblas/gsl_cblas.h, cblas/hypot.c, cblas/icamax.c, cblas/idamax.c,
cblas/isamax.c, cblas/izamax.c, cblas/sasum.c, cblas/saxpy.c,
cblas/scasum.c, cblas/scnrm2.c, cblas/scopy.c, cblas/sdot.c,
cblas/sdsdot.c, cblas/sgbmv.c, cblas/sgemm.c, cblas/sgemv.c,
cblas/sger.c, cblas/snrm2.c, cblas/source_asum_c.h,
cblas/source_asum_r.h, cblas/source_axpy_c.h, cblas/source_axpy_r.h,
cblas/source_copy_c.h, cblas/source_copy_r.h, cblas/source_dot_c.h,
cblas/source_dot_r.h, cblas/source_gbmv_c.h, cblas/source_gbmv_r.h,
cblas/source_gemm_c.h, cblas/source_gemm_r.h, cblas/source_gemv_c.h,
cblas/source_gemv_r.h, cblas/source_ger.h, cblas/source_gerc.h,
cblas/source_geru.h, cblas/source_hbmv.h, cblas/source_hemm.h,
cblas/source_hemv.h, cblas/source_her.h, cblas/source_her2.h,
cblas/source_her2k.h, cblas/source_herk.h, cblas/source_hpmv.h,
cblas/source_hpr.h, cblas/source_hpr2.h, cblas/source_iamax_c.h,
cblas/source_iamax_r.h, cblas/source_nrm2_c.h, cblas/source_nrm2_r.h,
cblas/source_rot.h, cblas/source_rotg.h, cblas/source_rotm.h,
cblas/source_rotmg.h, cblas/source_sbmv.h, cblas/source_scal_c.h,
cblas/source_scal_c_s.h, cblas/source_scal_r.h, cblas/source_spmv.h,
cblas/source_spr.h, cblas/source_spr2.h, cblas/source_swap_c.h,
cblas/source_swap_r.h, cblas/source_symm_c.h, cblas/source_symm_r.h,
cblas/source_symv.h, cblas/source_syr.h, cblas/source_syr2.h,
cblas/source_syr2k_c.h, cblas/source_syr2k_r.h,
cblas/source_syrk_c.h, cblas/source_syrk_r.h, cblas/source_tbmv_c.h,
cblas/source_tbmv_r.h, cblas/source_tbsv_c.h, cblas/source_tbsv_r.h,
cblas/source_tpmv_c.h, cblas/source_tpmv_r.h, cblas/source_tpsv_c.h,
cblas/source_tpsv_r.h, cblas/source_trmm_c.h, cblas/source_trmm_r.h,
cblas/source_trmv_c.h, cblas/source_trmv_r.h, cblas/source_trsm_c.h,
cblas/source_trsm_r.h, cblas/source_trsv_c.h, cblas/source_trsv_r.h,
cblas/srot.c, cblas/srotg.c, cblas/srotm.c, cblas/srotmg.c,
cblas/ssbmv.c, cblas/sscal.c, cblas/sspmv.c, cblas/sspr.c,
cblas/sspr2.c, cblas/sswap.c, cblas/ssymm.c, cblas/ssymv.c,
cblas/ssyr.c, cblas/ssyr2.c, cblas/ssyr2k.c, cblas/ssyrk.c,
cblas/stbmv.c, cblas/stbsv.c, cblas/stpmv.c, cblas/stpsv.c,
cblas/strmm.c, cblas/strmv.c, cblas/strsm.c, cblas/strsv.c,
cblas/test.c, cblas/test_amax.c, cblas/test_asum.c,
cblas/test_axpy.c, cblas/test_copy.c, cblas/test_dot.c,
cblas/test_gbmv.c, cblas/test_gemm.c, cblas/test_gemv.c,
cblas/test_ger.c, cblas/test_hbmv.c, cblas/test_hemm.c,
cblas/test_hemv.c, cblas/test_her.c, cblas/test_her2.c,
cblas/test_her2k.c, cblas/test_herk.c, cblas/test_hpmv.c,
cblas/test_hpr.c, cblas/test_hpr2.c, cblas/test_nrm2.c,
cblas/test_rot.c, cblas/test_rotg.c, cblas/test_rotm.c,
cblas/test_rotmg.c, cblas/test_sbmv.c, cblas/test_scal.c,
cblas/test_spmv.c, cblas/test_spr.c, cblas/test_spr2.c,
cblas/test_swap.c, cblas/test_symm.c, cblas/test_symv.c,
cblas/test_syr.c, cblas/test_syr2.c, cblas/test_syr2k.c,
cblas/test_syrk.c, cblas/test_tbmv.c, cblas/test_tbsv.c,
cblas/test_tpmv.c, cblas/test_tpsv.c, cblas/test_trmm.c,
cblas/test_trmv.c, cblas/test_trsm.c, cblas/test_trsv.c,
cblas/tests.c, cblas/tests.h, cblas/xerbla.c, cblas/zaxpy.c,
cblas/zcopy.c, cblas/zdotc_sub.c, cblas/zdotu_sub.c, cblas/zdscal.c,
cblas/zgbmv.c, cblas/zgemm.c, cblas/zgemv.c, cblas/zgerc.c,
cblas/zgeru.c, cblas/zhbmv.c, cblas/zhemm.c, cblas/zhemv.c,
cblas/zher.c, cblas/zher2.c, cblas/zher2k.c, cblas/zherk.c,
cblas/zhpmv.c, cblas/zhpr.c, cblas/zhpr2.c, cblas/zscal.c,
cblas/zswap.c, cblas/zsymm.c, cblas/zsyr2k.c, cblas/zsyrk.c,
cblas/ztbmv.c, cblas/ztbsv.c, cblas/ztpmv.c, cblas/ztpsv.c,
cblas/ztrmm.c, cblas/ztrmv.c, cblas/ztrsm.c, cblas/ztrsv.c,
cdf/ChangeLog, cdf/Makefile.am, cdf/Makefile.in, cdf/beta.c,
cdf/beta_inc.c, cdf/betainv.c, cdf/binomial.c, cdf/cauchy.c,
cdf/cauchyinv.c, cdf/chisq.c, cdf/chisqinv.c, cdf/error.h,
cdf/exponential.c, cdf/exponentialinv.c, cdf/exppow.c, cdf/fdist.c,
cdf/fdistinv.c, cdf/flat.c, cdf/flatinv.c, cdf/gamma.c,
cdf/gammainv.c, cdf/gauss.c, cdf/gaussinv.c, cdf/geometric.c,
cdf/gsl_cdf.h, cdf/gumbel1.c, cdf/gumbel1inv.c, cdf/gumbel2.c,
cdf/gumbel2inv.c, cdf/hypergeometric.c, cdf/laplace.c,
cdf/laplaceinv.c, cdf/logistic.c, cdf/logisticinv.c, cdf/lognormal.c,
cdf/lognormalinv.c, cdf/nbinomial.c, cdf/pareto.c, cdf/paretoinv.c,
cdf/pascal.c, cdf/poisson.c, cdf/rat_eval.h, cdf/rayleigh.c,
cdf/rayleighinv.c, cdf/tdist.c, cdf/tdistinv.c, cdf/test.c,
cdf/test_auto.c, cdf/weibull.c, cdf/weibullinv.c, cheb/ChangeLog,
cheb/Makefile.am, cheb/Makefile.in, cheb/deriv.c, cheb/eval.c,
cheb/gsl_chebyshev.h, cheb/init.c, cheb/integ.c, cheb/test.c,
combination/ChangeLog, combination/Makefile.am,
combination/Makefile.in, combination/combination.c,
combination/file.c, combination/gsl_combination.h,
combination/init.c, combination/test.c, complex/ChangeLog,
complex/Makefile.am, complex/Makefile.in, complex/TODO,
complex/gsl_complex.h, complex/gsl_complex_math.h, complex/math.c,
complex/results.h, complex/results1.h, complex/results_real.h,
complex/test.c, const/ChangeLog, const/Makefile.am,
const/Makefile.in, const/TODO, const/gsl_const.h,
const/gsl_const_cgs.h, const/gsl_const_cgsm.h, const/gsl_const_mks.h,
const/gsl_const_mksa.h, const/gsl_const_num.h, const/test.c,
deriv/ChangeLog, deriv/Makefile.am, deriv/Makefile.in, deriv/deriv.c,
deriv/gsl_deriv.h, deriv/test.c, dht/ChangeLog, dht/Makefile.am,
dht/Makefile.in, dht/dht.c, dht/gsl_dht.h, dht/test.c,
diff/ChangeLog, diff/Makefile.am, diff/Makefile.in, diff/diff.c,
diff/gsl_diff.h, diff/test.c, doc/12-cities.eps, doc/ChangeLog,
doc/Makefile.am, doc/Makefile.in, doc/algorithm.sty,
doc/algorithmic.sty, doc/autoconf.texi, doc/blas.texi,
doc/bspline.eps, doc/bspline.texi, doc/calc.sty, doc/cblas.texi,
doc/cheb.eps, doc/cheb.texi, doc/combination.texi, doc/complex.texi,
doc/const.texi, doc/debug.texi, doc/dht.texi, doc/diff.texi,
doc/dwt-orig.eps, doc/dwt-samp.eps, doc/dwt.texi, doc/eigen.texi,
doc/err.texi, doc/fdl.texi, doc/fft-complex-radix2-f.eps,
doc/fft-complex-radix2-t.eps, doc/fft-complex-radix2.eps,
doc/fft-real-mixedradix.eps, doc/fft.texi, doc/fftalgorithms.bib,
doc/fftalgorithms.tex, doc/final-route.eps, doc/fit-exp.eps,
doc/fit-wlinear.eps, doc/fit-wlinear2.eps, doc/fitting.texi,
doc/freemanuals.texi, doc/gpl.texi, doc/gsl-config.1,
doc/gsl-design.texi, doc/gsl-histogram.1, doc/gsl-randist.1,
doc/gsl-ref.info, doc/gsl-ref.info-1, doc/gsl-ref.info-2,
doc/gsl-ref.info-3, doc/gsl-ref.info-4, doc/gsl-ref.info-5,
doc/gsl-ref.info-6, doc/gsl-ref.texi, doc/gsl.3, doc/histogram.eps,
doc/histogram.texi, doc/histogram2d.eps, doc/ieee754.texi,
doc/initial-route.eps, doc/integration.texi, doc/interp.texi,
doc/interp2.eps, doc/interpp2.eps, doc/intro.texi, doc/landau.dat,
doc/linalg.texi, doc/math.texi, doc/mdate-sh, doc/min-interval.eps,
doc/min.texi, doc/montecarlo.texi, doc/multifit.texi,
doc/multimin.eps, doc/multimin.texi, doc/multiroots.texi,
doc/ntuple.eps, doc/ntuple.texi, doc/ode-initval.texi,
doc/permutation.texi, doc/poly.texi, doc/qrng.eps, doc/qrng.texi,
doc/rand-bernoulli.tex, doc/rand-beta.tex, doc/rand-binomial.tex,
doc/rand-bivariate-gaussian.tex, doc/rand-cauchy.tex,
doc/rand-chisq.tex, doc/rand-erlang.tex, doc/rand-exponential.tex,
doc/rand-exppow.tex, doc/rand-fdist.tex, doc/rand-flat.tex,
doc/rand-gamma.tex, doc/rand-gaussian-tail.tex,
doc/rand-gaussian.tex, doc/rand-geometric.tex, doc/rand-gumbel.tex,
doc/rand-gumbel1.tex, doc/rand-gumbel2.tex,
doc/rand-hypergeometric.tex, doc/rand-landau.tex,
doc/rand-laplace.tex, doc/rand-levy.tex, doc/rand-levyskew.tex,
doc/rand-logarithmic.tex, doc/rand-logistic.tex,
doc/rand-lognormal.tex, doc/rand-nbinomial.tex, doc/rand-pareto.tex,
doc/rand-pascal.tex, doc/rand-poisson.tex,
doc/rand-rayleigh-tail.tex, doc/rand-rayleigh.tex,
doc/rand-tdist.tex, doc/rand-weibull.tex, doc/randist.texi,
doc/random-walk.tex, doc/randplots.gnp, doc/rng.texi,
doc/roots-bisection.eps, doc/roots-false-position.eps,
doc/roots-newtons-method.eps, doc/roots-secant-method.eps,
doc/roots.texi, doc/siman-energy.eps, doc/siman-test.eps,
doc/siman.texi, doc/sort.texi, doc/specfunc-airy.texi,
doc/specfunc-bessel.texi, doc/specfunc-clausen.texi,
doc/specfunc-coulomb.texi, doc/specfunc-coupling.texi,
doc/specfunc-dawson.texi, doc/specfunc-debye.texi,
doc/specfunc-dilog.texi, doc/specfunc-elementary.texi,
doc/specfunc-ellint.texi, doc/specfunc-elljac.texi,
doc/specfunc-erf.texi, doc/specfunc-exp.texi,
doc/specfunc-expint.texi, doc/specfunc-fermi-dirac.texi,
doc/specfunc-gamma.texi, doc/specfunc-gegenbauer.texi,
doc/specfunc-hyperg.texi, doc/specfunc-laguerre.texi,
doc/specfunc-lambert.texi, doc/specfunc-legendre.texi,
doc/specfunc-log.texi, doc/specfunc-mathieu.texi,
doc/specfunc-pow-int.texi, doc/specfunc-psi.texi,
doc/specfunc-synchrotron.texi, doc/specfunc-transport.texi,
doc/specfunc-trig.texi, doc/specfunc-zeta.texi, doc/specfunc.texi,
doc/stamp-vti, doc/statistics.texi, doc/sum.texi, doc/texinfo.tex,
doc/usage.texi, doc/vdp.eps, doc/vectors.texi, doc/version-ref.texi,
doc/examples/blas.c, doc/examples/blas.out, doc/examples/block.c,
doc/examples/block.out, doc/examples/bspline.c, doc/examples/cblas.c,
doc/examples/cblas.out, doc/examples/cdf.c, doc/examples/cdf.out,
doc/examples/cheb.c, doc/examples/combination.c,
doc/examples/combination.out, doc/examples/const.c,
doc/examples/const.out, doc/examples/demo_fn.c,
doc/examples/demo_fn.h, doc/examples/diff.c, doc/examples/diff.out,
doc/examples/dwt.c, doc/examples/dwt.dat, doc/examples/ecg.dat,
doc/examples/eigen.c, doc/examples/eigen_nonsymm.c,
doc/examples/expfit.c, doc/examples/fft.c, doc/examples/fftmr.c,
doc/examples/fftreal.c, doc/examples/fitting.c,
doc/examples/fitting2.c, doc/examples/fitting3.c,
doc/examples/histogram.c, doc/examples/histogram2d.c,
doc/examples/ieee.c, doc/examples/ieeeround.c,
doc/examples/integration.c, doc/examples/integration.out,
doc/examples/interp.c, doc/examples/interpp.c, doc/examples/intro.c,
doc/examples/intro.out, doc/examples/linalglu.c,
doc/examples/linalglu.out, doc/examples/matrix.c,
doc/examples/matrixw.c, doc/examples/min.c, doc/examples/min.out,
doc/examples/monte.c, doc/examples/nlfit.c, doc/examples/ntupler.c,
doc/examples/ntuplew.c, doc/examples/ode-initval.c,
doc/examples/odefixed.c, doc/examples/permseq.c,
doc/examples/permshuffle.c, doc/examples/polyroots.c,
doc/examples/polyroots.out, doc/examples/qrng.c,
doc/examples/randpoisson.2.out, doc/examples/randpoisson.c,
doc/examples/randpoisson.out, doc/examples/randwalk.c,
doc/examples/rng.c, doc/examples/rng.out, doc/examples/rngunif.2.out,
doc/examples/rngunif.c, doc/examples/rngunif.out,
doc/examples/rootnewt.c, doc/examples/roots.c, doc/examples/siman.c,
doc/examples/sortsmall.c, doc/examples/sortsmall.out,
doc/examples/specfun.c, doc/examples/specfun.out,
doc/examples/specfun_e.c, doc/examples/specfun_e.out,
doc/examples/stat.c, doc/examples/stat.out, doc/examples/statsort.c,
doc/examples/statsort.out, doc/examples/sum.c, doc/examples/sum.out,
doc/examples/vector.c, doc/examples/vectorr.c,
doc/examples/vectorview.c, doc/examples/vectorview.out,
doc/examples/vectorw.c, eigen/ChangeLog, eigen/Makefile.am,
eigen/Makefile.in, eigen/TODO, eigen/francis.c, eigen/gsl_eigen.h,
eigen/herm.c, eigen/hermv.c, eigen/jacobi.c, eigen/nonsymm.c,
eigen/nonsymmv.c, eigen/qrstep.c, eigen/schur.c, eigen/schur.h,
eigen/sort.c, eigen/symm.c, eigen/symmv.c, eigen/test.c,
err/ChangeLog, err/Makefile.am, err/Makefile.in, err/TODO,
err/error.c, err/gsl_errno.h, err/gsl_message.h, err/message.c,
err/stream.c, err/strerror.c, err/test.c, fft/ChangeLog,
fft/Makefile.am, fft/Makefile.in, fft/TODO, fft/bitreverse.c,
fft/bitreverse.h, fft/c_init.c, fft/c_main.c, fft/c_pass.h,
fft/c_pass_2.c, fft/c_pass_3.c, fft/c_pass_4.c, fft/c_pass_5.c,
fft/c_pass_6.c, fft/c_pass_7.c, fft/c_pass_n.c, fft/c_radix2.c,
fft/compare.h, fft/compare_source.c, fft/complex_internal.h,
fft/dft.c, fft/dft_source.c, fft/factorize.c, fft/factorize.h,
fft/fft.c, fft/gsl_dft_complex.h, fft/gsl_dft_complex_float.h,
fft/gsl_fft.h, fft/gsl_fft_complex.h, fft/gsl_fft_complex_float.h,
fft/gsl_fft_halfcomplex.h, fft/gsl_fft_halfcomplex_float.h,
fft/gsl_fft_real.h, fft/gsl_fft_real_float.h, fft/hc_init.c,
fft/hc_main.c, fft/hc_pass.h, fft/hc_pass_2.c, fft/hc_pass_3.c,
fft/hc_pass_4.c, fft/hc_pass_5.c, fft/hc_pass_n.c, fft/hc_radix2.c,
fft/hc_unpack.c, fft/real_init.c, fft/real_main.c, fft/real_pass.h,
fft/real_pass_2.c, fft/real_pass_3.c, fft/real_pass_4.c,
fft/real_pass_5.c, fft/real_pass_n.c, fft/real_radix2.c,
fft/real_unpack.c, fft/signals.c, fft/signals.h,
fft/signals_source.c, fft/test.c, fft/test_complex_source.c,
fft/test_real_source.c, fft/test_trap_source.c, fft/urand.c,
fit/ChangeLog, fit/Makefile.am, fit/Makefile.in, fit/gsl_fit.h,
fit/linear.c, fit/test.c, gsl/Makefile.am, gsl/Makefile.in,
histogram/ChangeLog, histogram/Makefile.am, histogram/Makefile.in,
histogram/TODO, histogram/add.c, histogram/add2d.c,
histogram/calloc_range.c, histogram/calloc_range2d.c,
histogram/copy.c, histogram/copy2d.c, histogram/file.c,
histogram/file2d.c, histogram/find.c, histogram/find2d.c,
histogram/get.c, histogram/get2d.c, histogram/gsl_histogram.h,
histogram/gsl_histogram2d.h, histogram/init.c, histogram/init2d.c,
histogram/maxval.c, histogram/maxval2d.c, histogram/oper.c,
histogram/oper2d.c, histogram/params.c, histogram/params2d.c,
histogram/pdf.c, histogram/pdf2d.c, histogram/reset.c,
histogram/reset2d.c, histogram/stat.c, histogram/stat2d.c,
histogram/test.c, histogram/test1d.c, histogram/test1d_resample.c,
histogram/test1d_trap.c, histogram/test2d.c,
histogram/test2d_resample.c, histogram/test2d_trap.c,
histogram/urand.c, ieee-utils/ChangeLog, ieee-utils/Makefile.am,
ieee-utils/Makefile.in, ieee-utils/TODO, ieee-utils/endian.c,
ieee-utils/env.c, ieee-utils/fp-aix.c, ieee-utils/fp-darwin.c,
ieee-utils/fp-darwin86.c, ieee-utils/fp-freebsd.c,
ieee-utils/fp-gnuc99.c, ieee-utils/fp-gnum68k.c,
ieee-utils/fp-gnuppc.c, ieee-utils/fp-gnusparc.c,
ieee-utils/fp-gnux86.c, ieee-utils/fp-hpux.c, ieee-utils/fp-hpux11.c,
ieee-utils/fp-irix.c, ieee-utils/fp-netbsd.c,
ieee-utils/fp-openbsd.c, ieee-utils/fp-os2emx.c,
ieee-utils/fp-solaris.c, ieee-utils/fp-sunos4.c,
ieee-utils/fp-tru64.c, ieee-utils/fp-unknown.c, ieee-utils/fp.c,
ieee-utils/gsl_ieee_utils.h, ieee-utils/make_rep.c,
ieee-utils/print.c, ieee-utils/read.c, ieee-utils/standardize.c,
ieee-utils/test.c, integration/ChangeLog, integration/Makefile.am,
integration/Makefile.in, integration/TODO, integration/append.c,
integration/err.c, integration/gsl_integration.h,
integration/initialise.c, integration/positivity.c,
integration/ptsort.c, integration/qag.c, integration/qagp.c,
integration/qags.c, integration/qawc.c, integration/qawf.c,
integration/qawo.c, integration/qaws.c, integration/qc25c.c,
integration/qc25f.c, integration/qc25s.c, integration/qcheb.c,
integration/qelg.c, integration/qk.c, integration/qk15.c,
integration/qk21.c, integration/qk31.c, integration/qk41.c,
integration/qk51.c, integration/qk61.c, integration/qmomo.c,
integration/qmomof.c, integration/qng.c, integration/qng.h,
integration/qpsrt.c, integration/qpsrt2.c, integration/reset.c,
integration/set_initial.c, integration/test.c, integration/tests.c,
integration/tests.h, integration/util.c, integration/workspace.c,
interpolation/ChangeLog, interpolation/Makefile.am,
interpolation/Makefile.in, interpolation/TODO, interpolation/accel.c,
interpolation/akima.c, interpolation/bsearch.c,
interpolation/bsearch.h, interpolation/cspline.c,
interpolation/gsl_interp.h, interpolation/gsl_spline.h,
interpolation/integ_eval.h, interpolation/interp.c,
interpolation/linear.c, interpolation/poly.c, interpolation/spline.c,
interpolation/test.c, linalg/ChangeLog, linalg/Makefile.am,
linalg/Makefile.in, linalg/TODO, linalg/apply_givens.c,
linalg/balance.c, linalg/balancemat.c, linalg/bidiag.c,
linalg/cholesky.c, linalg/exponential.c, linalg/givens.c,
linalg/gsl_linalg.h, linalg/hermtd.c, linalg/hessenberg.c,
linalg/hh.c, linalg/householder.c, linalg/householdercomplex.c,
linalg/lq.c, linalg/lu.c, linalg/luc.c, linalg/multiply.c,
linalg/ptlq.c, linalg/qr.c, linalg/qrpt.c, linalg/svd.c,
linalg/svdstep.c, linalg/symmtd.c, linalg/test.c, linalg/tridiag.c,
linalg/tridiag.h, matrix/ChangeLog, matrix/Makefile.am,
matrix/Makefile.in, matrix/TODO, matrix/copy.c, matrix/copy_source.c,
matrix/file.c, matrix/file_source.c, matrix/getset.c,
matrix/getset_source.c, matrix/gsl_matrix.h,
matrix/gsl_matrix_char.h, matrix/gsl_matrix_complex_double.h,
matrix/gsl_matrix_complex_float.h,
matrix/gsl_matrix_complex_long_double.h, matrix/gsl_matrix_double.h,
matrix/gsl_matrix_float.h, matrix/gsl_matrix_int.h,
matrix/gsl_matrix_long.h, matrix/gsl_matrix_long_double.h,
matrix/gsl_matrix_short.h, matrix/gsl_matrix_uchar.h,
matrix/gsl_matrix_uint.h, matrix/gsl_matrix_ulong.h,
matrix/gsl_matrix_ushort.h, matrix/init.c, matrix/init_source.c,
matrix/matrix.c, matrix/matrix_source.c, matrix/minmax.c,
matrix/minmax_source.c, matrix/oper.c, matrix/oper_complex_source.c,
matrix/oper_source.c, matrix/prop.c, matrix/prop_source.c,
matrix/rowcol.c, matrix/rowcol_source.c, matrix/submatrix.c,
matrix/submatrix_source.c, matrix/swap.c, matrix/swap_source.c,
matrix/test.c, matrix/test_complex_source.c, matrix/test_source.c,
matrix/test_static.c, matrix/view.c, matrix/view.h,
matrix/view_source.c, min/ChangeLog, min/Makefile.am,
min/Makefile.in, min/bracketing.c, min/brent.c, min/convergence.c,
min/fsolver.c, min/golden.c, min/gsl_min.h, min/min.h, min/test.c,
min/test.h, min/test_funcs.c, monte/ChangeLog, monte/Makefile.am,
monte/Makefile.in, monte/README, monte/TODO, monte/gsl_monte.h,
monte/gsl_monte_miser.h, monte/gsl_monte_plain.h,
monte/gsl_monte_vegas.h, monte/miser.c, monte/plain.c, monte/test.c,
monte/test_main.c, monte/vegas.c, multifit/ChangeLog,
multifit/Makefile.am, multifit/Makefile.in, multifit/TODO,
multifit/convergence.c, multifit/covar.c, multifit/fdfsolver.c,
multifit/fsolver.c, multifit/gradient.c, multifit/gsl_multifit.h,
multifit/gsl_multifit_nlin.h, multifit/lmder.c, multifit/lmiterate.c,
multifit/lmpar.c, multifit/lmset.c, multifit/lmutil.c,
multifit/multilinear.c, multifit/qrsolv.c, multifit/test.c,
multifit/test_brown.c, multifit/test_enso.c,
multifit/test_estimator.c, multifit/test_filip.c, multifit/test_fn.c,
multifit/test_hahn1.c, multifit/test_kirby2.c,
multifit/test_longley.c, multifit/test_nelson.c,
multifit/test_pontius.c, multifit/work.c, multimin/ChangeLog,
multimin/Makefile.am, multimin/Makefile.in, multimin/TODO,
multimin/conjugate_fr.c, multimin/conjugate_pr.c,
multimin/convergence.c, multimin/diff.c,
multimin/directional_minimize.c, multimin/fdfminimizer.c,
multimin/fminimizer.c, multimin/gsl_multimin.h,
multimin/linear_minimize.c, multimin/linear_wrapper.c,
multimin/simplex.c, multimin/steepest_descent.c, multimin/test.c,
multimin/test_funcs.c, multimin/test_funcs.h, multimin/vector_bfgs.c,
multimin/vector_bfgs2.c, multiroots/ChangeLog,
multiroots/Makefile.am, multiroots/Makefile.in, multiroots/broyden.c,
multiroots/convergence.c, multiroots/dnewton.c, multiroots/dogleg.c,
multiroots/enorm.c, multiroots/fdfsolver.c, multiroots/fdjac.c,
multiroots/fsolver.c, multiroots/gnewton.c,
multiroots/gsl_multiroots.h, multiroots/hybrid.c,
multiroots/hybridj.c, multiroots/newton.c, multiroots/test.c,
multiroots/test_funcs.c, multiroots/test_funcs.h, ntuple/ChangeLog,
ntuple/Makefile.am, ntuple/Makefile.in, ntuple/gsl_ntuple.h,
ntuple/ntuple.c, ntuple/test.c, ode-initval/ChangeLog,
ode-initval/Makefile.am, ode-initval/Makefile.in, ode-initval/TODO,
ode-initval/bsimp.c, ode-initval/control.c, ode-initval/cscal.c,
ode-initval/cstd.c, ode-initval/evolve.c, ode-initval/gear1.c,
ode-initval/gear2.c, ode-initval/gsl_odeiv.h,
ode-initval/odeiv_util.h, ode-initval/rk2.c, ode-initval/rk2imp.c,
ode-initval/rk2simp.c, ode-initval/rk4.c, ode-initval/rk4imp.c,
ode-initval/rk8pd.c, ode-initval/rkck.c, ode-initval/rkf45.c,
ode-initval/step.c, ode-initval/test.c, permutation/ChangeLog,
permutation/Makefile.am, permutation/Makefile.in,
permutation/canonical.c, permutation/file.c,
permutation/gsl_permutation.h, permutation/gsl_permute.h,
permutation/gsl_permute_char.h,
permutation/gsl_permute_complex_double.h,
permutation/gsl_permute_complex_float.h,
permutation/gsl_permute_complex_long_double.h,
permutation/gsl_permute_double.h, permutation/gsl_permute_float.h,
permutation/gsl_permute_int.h, permutation/gsl_permute_long.h,
permutation/gsl_permute_long_double.h,
permutation/gsl_permute_short.h, permutation/gsl_permute_uchar.h,
permutation/gsl_permute_uint.h, permutation/gsl_permute_ulong.h,
permutation/gsl_permute_ushort.h, permutation/gsl_permute_vector.h,
permutation/gsl_permute_vector_char.h,
permutation/gsl_permute_vector_complex_double.h,
permutation/gsl_permute_vector_complex_float.h,
permutation/gsl_permute_vector_complex_long_double.h,
permutation/gsl_permute_vector_double.h,
permutation/gsl_permute_vector_float.h,
permutation/gsl_permute_vector_int.h,
permutation/gsl_permute_vector_long.h,
permutation/gsl_permute_vector_long_double.h,
permutation/gsl_permute_vector_short.h,
permutation/gsl_permute_vector_uchar.h,
permutation/gsl_permute_vector_uint.h,
permutation/gsl_permute_vector_ulong.h,
permutation/gsl_permute_vector_ushort.h, permutation/init.c,
permutation/permutation.c, permutation/permute.c,
permutation/permute_source.c, permutation/test.c, poly/ChangeLog,
poly/Makefile.am, poly/Makefile.in, poly/TODO, poly/balance.c,
poly/companion.c, poly/dd.c, poly/eval.c, poly/gsl_poly.h, poly/qr.c,
poly/solve_cubic.c, poly/solve_quadratic.c, poly/test.c,
poly/zsolve.c, poly/zsolve_cubic.c, poly/zsolve_init.c,
poly/zsolve_quadratic.c, qrng/ChangeLog, qrng/Makefile.am,
qrng/Makefile.in, qrng/TODO, qrng/gsl_qrng.h, qrng/niederreiter-2.c,
qrng/qrng.c, qrng/sobol.c, qrng/test.c, randist/ChangeLog,
randist/Makefile.am, randist/Makefile.in, randist/TODO,
randist/bernoulli.c, randist/beta.c, randist/bigauss.c,
randist/binomial.c, randist/binomial_tpe.c, randist/cauchy.c,
randist/chisq.c, randist/dirichlet.c, randist/discrete.c,
randist/erlang.c, randist/exponential.c, randist/exppow.c,
randist/fdist.c, randist/flat.c, randist/gamma.c, randist/gauss.c,
randist/gausstail.c, randist/gausszig.c, randist/geometric.c,
randist/gsl_randist.h, randist/gumbel.c, randist/hyperg.c,
randist/landau.c, randist/laplace.c, randist/levy.c,
randist/logarithmic.c, randist/logistic.c, randist/lognormal.c,
randist/multinomial.c, randist/nbinomial.c, randist/pareto.c,
randist/pascal.c, randist/poisson.c, randist/rayleigh.c,
randist/shuffle.c, randist/sphere.c, randist/tdist.c, randist/test.c,
randist/weibull.c, rng/ChangeLog, rng/Makefile.am, rng/Makefile.in,
rng/TODO, rng/borosh13.c, rng/cmrg.c, rng/coveyou.c, rng/default.c,
rng/file.c, rng/fishman18.c, rng/fishman20.c, rng/fishman2x.c,
rng/gfsr4.c, rng/gsl_rng.h, rng/knuthran.c, rng/knuthran2.c,
rng/knuthran2002.c, rng/lecuyer21.c, rng/minstd.c, rng/mrg.c,
rng/mt.c, rng/r250.c, rng/ran0.c, rng/ran1.c, rng/ran2.c, rng/ran3.c,
rng/rand.c, rng/rand48.c, rng/random.c, rng/randu.c, rng/ranf.c,
rng/ranlux.c, rng/ranlxd.c, rng/ranlxs.c, rng/ranmar.c, rng/rng.c,
rng/schrage.c, rng/slatec.c, rng/taus.c, rng/taus113.c, rng/test.c,
rng/transputer.c, rng/tt.c, rng/types.c, rng/uni.c, rng/uni32.c,
rng/vax.c, rng/waterman14.c, rng/zuf.c, roots/ChangeLog,
roots/Makefile.am, roots/Makefile.in, roots/TODO, roots/bisection.c,
roots/brent.c, roots/convergence.c, roots/falsepos.c,
roots/fdfsolver.c, roots/fsolver.c, roots/gsl_roots.h,
roots/newton.c, roots/roots.h, roots/secant.c, roots/steffenson.c,
roots/test.c, roots/test.h, roots/test_funcs.c, siman/ChangeLog,
siman/Makefile.am, siman/Makefile.in, siman/TODO, siman/gsl_siman.h,
siman/siman.c, siman/siman_test_driver.sh, siman/siman_tsp.c,
siman/test.c, sort/ChangeLog, sort/Makefile.am, sort/Makefile.in,
sort/TODO, sort/gsl_heapsort.h, sort/gsl_sort.h,
sort/gsl_sort_char.h, sort/gsl_sort_double.h, sort/gsl_sort_float.h,
sort/gsl_sort_int.h, sort/gsl_sort_long.h,
sort/gsl_sort_long_double.h, sort/gsl_sort_short.h,
sort/gsl_sort_uchar.h, sort/gsl_sort_uint.h, sort/gsl_sort_ulong.h,
sort/gsl_sort_ushort.h, sort/gsl_sort_vector.h,
sort/gsl_sort_vector_char.h, sort/gsl_sort_vector_double.h,
sort/gsl_sort_vector_float.h, sort/gsl_sort_vector_int.h,
sort/gsl_sort_vector_long.h, sort/gsl_sort_vector_long_double.h,
sort/gsl_sort_vector_short.h, sort/gsl_sort_vector_uchar.h,
sort/gsl_sort_vector_uint.h, sort/gsl_sort_vector_ulong.h,
sort/gsl_sort_vector_ushort.h, sort/sort.c, sort/sortind.c,
sort/sortvec.c, sort/sortvec_source.c, sort/sortvecind.c,
sort/sortvecind_source.c, sort/subset.c, sort/subset_source.c,
sort/subsetind.c, sort/subsetind_source.c, sort/test.c,
sort/test_heapsort.c, sort/test_source.c, specfunc/ChangeLog,
specfunc/Makefile.am, specfunc/Makefile.in, specfunc/TODO,
specfunc/airy.c, specfunc/airy_der.c, specfunc/airy_zero.c,
specfunc/atanint.c, specfunc/bessel.c, specfunc/bessel.h,
specfunc/bessel_I0.c, specfunc/bessel_I1.c, specfunc/bessel_In.c,
specfunc/bessel_Inu.c, specfunc/bessel_J0.c, specfunc/bessel_J1.c,
specfunc/bessel_Jn.c, specfunc/bessel_Jnu.c, specfunc/bessel_K0.c,
specfunc/bessel_K1.c, specfunc/bessel_Kn.c, specfunc/bessel_Knu.c,
specfunc/bessel_Y0.c, specfunc/bessel_Y1.c, specfunc/bessel_Yn.c,
specfunc/bessel_Ynu.c, specfunc/bessel_amp_phase.c,
specfunc/bessel_amp_phase.h, specfunc/bessel_i.c,
specfunc/bessel_j.c, specfunc/bessel_k.c, specfunc/bessel_olver.c,
specfunc/bessel_olver.h, specfunc/bessel_sequence.c,
specfunc/bessel_temme.c, specfunc/bessel_temme.h,
specfunc/bessel_y.c, specfunc/bessel_zero.c, specfunc/beta.c,
specfunc/beta_inc.c, specfunc/cheb_eval.c, specfunc/cheb_eval_mode.c,
specfunc/chebyshev.h, specfunc/check.h, specfunc/clausen.c,
specfunc/coulomb.c, specfunc/coulomb_bound.c, specfunc/coupling.c,
specfunc/dawson.c, specfunc/debye.c, specfunc/dilog.c,
specfunc/elementary.c, specfunc/ellint.c, specfunc/elljac.c,
specfunc/erfc.c, specfunc/error.h, specfunc/eval.h, specfunc/exp.c,
specfunc/expint.c, specfunc/expint3.c, specfunc/fermi_dirac.c,
specfunc/gamma.c, specfunc/gamma_inc.c, specfunc/gegenbauer.c,
specfunc/gsl_sf.h, specfunc/gsl_sf_airy.h, specfunc/gsl_sf_bessel.h,
specfunc/gsl_sf_clausen.h, specfunc/gsl_sf_coulomb.h,
specfunc/gsl_sf_coupling.h, specfunc/gsl_sf_dawson.h,
specfunc/gsl_sf_debye.h, specfunc/gsl_sf_dilog.h,
specfunc/gsl_sf_elementary.h, specfunc/gsl_sf_ellint.h,
specfunc/gsl_sf_elljac.h, specfunc/gsl_sf_erf.h,
specfunc/gsl_sf_exp.h, specfunc/gsl_sf_expint.h,
specfunc/gsl_sf_fermi_dirac.h, specfunc/gsl_sf_gamma.h,
specfunc/gsl_sf_gegenbauer.h, specfunc/gsl_sf_hyperg.h,
specfunc/gsl_sf_laguerre.h, specfunc/gsl_sf_lambert.h,
specfunc/gsl_sf_legendre.h, specfunc/gsl_sf_log.h,
specfunc/gsl_sf_mathieu.h, specfunc/gsl_sf_pow_int.h,
specfunc/gsl_sf_psi.h, specfunc/gsl_sf_result.h,
specfunc/gsl_sf_synchrotron.h, specfunc/gsl_sf_transport.h,
specfunc/gsl_sf_trig.h, specfunc/gsl_sf_zeta.h,
specfunc/gsl_specfunc.h, specfunc/hyperg.c, specfunc/hyperg.h,
specfunc/hyperg_0F1.c, specfunc/hyperg_1F1.c, specfunc/hyperg_2F0.c,
specfunc/hyperg_2F1.c, specfunc/hyperg_U.c, specfunc/laguerre.c,
specfunc/lambert.c, specfunc/legendre.h, specfunc/legendre_H3d.c,
specfunc/legendre_Qn.c, specfunc/legendre_con.c,
specfunc/legendre_poly.c, specfunc/log.c, specfunc/mathieu_angfunc.c,
specfunc/mathieu_charv.c, specfunc/mathieu_coeff.c,
specfunc/mathieu_radfunc.c, specfunc/mathieu_workspace.c,
specfunc/poch.c, specfunc/pow_int.c, specfunc/psi.c,
specfunc/recurse.h, specfunc/result.c, specfunc/shint.c,
specfunc/sinint.c, specfunc/synchrotron.c, specfunc/test_airy.c,
specfunc/test_bessel.c, specfunc/test_coulomb.c,
specfunc/test_dilog.c, specfunc/test_gamma.c, specfunc/test_hyperg.c,
specfunc/test_legendre.c, specfunc/test_mathieu.c,
specfunc/test_sf.c, specfunc/test_sf.h, specfunc/transport.c,
specfunc/trig.c, specfunc/zeta.c, statistics/ChangeLog,
statistics/Makefile.am, statistics/Makefile.in, statistics/TODO,
statistics/absdev.c, statistics/absdev_source.c,
statistics/covariance.c, statistics/covariance_source.c,
statistics/gsl_statistics.h, statistics/gsl_statistics_char.h,
statistics/gsl_statistics_double.h,
statistics/gsl_statistics_float.h, statistics/gsl_statistics_int.h,
statistics/gsl_statistics_long.h,
statistics/gsl_statistics_long_double.h,
statistics/gsl_statistics_short.h, statistics/gsl_statistics_uchar.h,
statistics/gsl_statistics_uint.h, statistics/gsl_statistics_ulong.h,
statistics/gsl_statistics_ushort.h, statistics/kurtosis.c,
statistics/kurtosis_source.c, statistics/lag1.c,
statistics/lag1_source.c, statistics/mean.c,
statistics/mean_source.c, statistics/median.c,
statistics/median_source.c, statistics/minmax.c,
statistics/minmax_source.c, statistics/p_variance.c,
statistics/p_variance_source.c, statistics/quantiles.c,
statistics/quantiles_source.c, statistics/skew.c,
statistics/skew_source.c, statistics/test.c,
statistics/test_float_source.c, statistics/test_int_source.c,
statistics/test_nist.c, statistics/ttest.c,
statistics/ttest_source.c, statistics/variance.c,
statistics/variance_source.c, statistics/wabsdev.c,
statistics/wabsdev_source.c, statistics/wkurtosis.c,
statistics/wkurtosis_source.c, statistics/wmean.c,
statistics/wmean_source.c, statistics/wskew.c,
statistics/wskew_source.c, statistics/wvariance.c,
statistics/wvariance_source.c, sum/ChangeLog, sum/Makefile.am,
sum/Makefile.in, sum/gsl_sum.h, sum/levin_u.c, sum/levin_utrunc.c,
sum/test.c, sum/work_u.c, sum/work_utrunc.c, sys/ChangeLog,
sys/Makefile.am, sys/Makefile.in, sys/coerce.c, sys/expm1.c,
sys/fcmp.c, sys/fdiv.c, sys/gsl_sys.h, sys/hypot.c, sys/infnan.c,
sys/invhyp.c, sys/ldfrexp.c, sys/log1p.c, sys/minmax.c,
sys/pow_int.c, sys/prec.c, sys/test.c, test/ChangeLog,
test/Makefile.am, test/Makefile.in, test/gsl_test.h, test/results.c,
utils/Makefile.am, utils/Makefile.in, utils/README, utils/memcpy.c,
utils/memmove.c, utils/placeholder.c, utils/strdup.c, utils/strtol.c,
utils/strtoul.c, utils/system.h, vector/ChangeLog,
vector/Makefile.am, vector/Makefile.in, vector/TODO, vector/copy.c,
vector/copy_source.c, vector/file.c, vector/file_source.c,
vector/gsl_vector.h, vector/gsl_vector_char.h,
vector/gsl_vector_complex.h, vector/gsl_vector_complex_double.h,
vector/gsl_vector_complex_float.h,
vector/gsl_vector_complex_long_double.h, vector/gsl_vector_double.h,
vector/gsl_vector_float.h, vector/gsl_vector_int.h,
vector/gsl_vector_long.h, vector/gsl_vector_long_double.h,
vector/gsl_vector_short.h, vector/gsl_vector_uchar.h,
vector/gsl_vector_uint.h, vector/gsl_vector_ulong.h,
vector/gsl_vector_ushort.h, vector/init.c, vector/init_source.c,
vector/minmax.c, vector/minmax_source.c, vector/oper.c,
vector/oper_source.c, vector/prop.c, vector/prop_source.c,
vector/reim.c, vector/reim_source.c, vector/subvector.c,
vector/subvector_source.c, vector/swap.c, vector/swap_source.c,
vector/test.c, vector/test_complex_source.c, vector/test_source.c,
vector/test_static.c, vector/vector.c, vector/vector_source.c,
vector/view.c, vector/view.h, vector/view_source.c,
wavelet/ChangeLog, wavelet/Makefile.am, wavelet/Makefile.in,
wavelet/TODO, wavelet/bspline.c, wavelet/daubechies.c, wavelet/dwt.c,
wavelet/gsl_wavelet.h, wavelet/gsl_wavelet2d.h, wavelet/haar.c,
wavelet/test.c, wavelet/wavelet.c: New files.
Diffstat (limited to 'gsl-1.9/specfunc/coulomb.c')
-rw-r--r-- | gsl-1.9/specfunc/coulomb.c | 1417 |
1 files changed, 1417 insertions, 0 deletions
diff --git a/gsl-1.9/specfunc/coulomb.c b/gsl-1.9/specfunc/coulomb.c new file mode 100644 index 0000000..7c68076 --- /dev/null +++ b/gsl-1.9/specfunc/coulomb.c @@ -0,0 +1,1417 @@ +/* specfunc/coulomb.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 */ + +/* Evaluation of Coulomb wave functions F_L(eta, x), G_L(eta, x), + * and their derivatives. A combination of Steed's method, asymptotic + * results, and power series. + * + * Steed's method: + * [Barnett, CPC 21, 297 (1981)] + * Power series and other methods: + * [Biedenharn et al., PR 97, 542 (1954)] + * [Bardin et al., CPC 3, 73 (1972)] + * [Abad+Sesma, CPC 71, 110 (1992)] + */ +#include <config.h> +#include <gsl/gsl_math.h> +#include <gsl/gsl_errno.h> +#include <gsl/gsl_sf_exp.h> +#include <gsl/gsl_sf_psi.h> +#include <gsl/gsl_sf_airy.h> +#include <gsl/gsl_sf_pow_int.h> +#include <gsl/gsl_sf_gamma.h> +#include <gsl/gsl_sf_coulomb.h> + +#include "error.h" + +/* the L=0 normalization constant + * [Abramowitz+Stegun 14.1.8] + */ +static +double +C0sq(double eta) +{ + double twopieta = 2.0*M_PI*eta; + + if(fabs(eta) < GSL_DBL_EPSILON) { + return 1.0; + } + else if(twopieta > GSL_LOG_DBL_MAX) { + return 0.0; + } + else { + gsl_sf_result scale; + gsl_sf_expm1_e(twopieta, &scale); + return twopieta/scale.val; + } +} + + +/* the full definition of C_L(eta) for any valid L and eta + * [Abramowitz and Stegun 14.1.7] + * This depends on the complex gamma function. For large + * arguments the phase of the complex gamma function is not + * very accurately determined. However the modulus is, and that + * is all that we need to calculate C_L. + * + * This is not valid for L <= -3/2 or L = -1. + */ +static +int +CLeta(double L, double eta, gsl_sf_result * result) +{ + gsl_sf_result ln1; /* log of numerator Gamma function */ + gsl_sf_result ln2; /* log of denominator Gamma function */ + double sgn = 1.0; + double arg_val, arg_err; + + if(fabs(eta/(L+1.0)) < GSL_DBL_EPSILON) { + gsl_sf_lngamma_e(L+1.0, &ln1); + } + else { + gsl_sf_result p1; /* phase of numerator Gamma -- not used */ + gsl_sf_lngamma_complex_e(L+1.0, eta, &ln1, &p1); /* should be ok */ + } + + gsl_sf_lngamma_e(2.0*(L+1.0), &ln2); + if(L < -1.0) sgn = -sgn; + + arg_val = L*M_LN2 - 0.5*eta*M_PI + ln1.val - ln2.val; + arg_err = ln1.err + ln2.err; + arg_err += GSL_DBL_EPSILON * (fabs(L*M_LN2) + fabs(0.5*eta*M_PI)); + return gsl_sf_exp_err_e(arg_val, arg_err, result); +} + + +int +gsl_sf_coulomb_CL_e(double lam, double eta, gsl_sf_result * result) +{ + /* CHECK_POINTER(result) */ + + if(lam <= -1.0) { + DOMAIN_ERROR(result); + } + else if(fabs(lam) < GSL_DBL_EPSILON) { + /* saves a calculation of complex_lngamma(), otherwise not necessary */ + result->val = sqrt(C0sq(eta)); + result->err = 2.0 * GSL_DBL_EPSILON * result->val; + return GSL_SUCCESS; + } + else { + return CLeta(lam, eta, result); + } +} + + +/* cl[0] .. cl[kmax] = C_{lam_min}(eta) .. C_{lam_min+kmax}(eta) + */ +int +gsl_sf_coulomb_CL_array(double lam_min, int kmax, double eta, double * cl) +{ + int k; + gsl_sf_result cl_0; + gsl_sf_coulomb_CL_e(lam_min, eta, &cl_0); + cl[0] = cl_0.val; + + for(k=1; k<=kmax; k++) { + double L = lam_min + k; + cl[k] = cl[k-1] * hypot(L, eta)/(L*(2.0*L+1.0)); + } + + return GSL_SUCCESS; +} + + +/* Evaluate the series for Phi_L(eta,x) and Phi_L*(eta,x) + * [Abramowitz+Stegun 14.1.5] + * [Abramowitz+Stegun 14.1.13] + * + * The sequence of coefficients A_k^L is + * manifestly well-controlled for L >= -1/2 + * and eta < 10. + * + * This makes sense since this is the region + * away from threshold, and you expect + * the evaluation to become easier as you + * get farther from threshold. + * + * Empirically, this is quite well-behaved for + * L >= -1/2 + * eta < 10 + * x < 10 + */ +#if 0 +static +int +coulomb_Phi_series(const double lam, const double eta, const double x, + double * result, double * result_star) +{ + int kmin = 5; + int kmax = 200; + int k; + double Akm2 = 1.0; + double Akm1 = eta/(lam+1.0); + double Ak; + + double xpow = x; + double sum = Akm2 + Akm1*x; + double sump = (lam+1.0)*Akm2 + (lam+2.0)*Akm1*x; + double prev_abs_del = fabs(Akm1*x); + double prev_abs_del_p = (lam+2.0) * prev_abs_del; + + for(k=2; k<kmax; k++) { + double del; + double del_p; + double abs_del; + double abs_del_p; + + Ak = (2.0*eta*Akm1 - Akm2)/(k*(2.0*lam + 1.0 + k)); + + xpow *= x; + del = Ak*xpow; + del_p = (k+lam+1.0)*del; + sum += del; + sump += del_p; + + abs_del = fabs(del); + abs_del_p = fabs(del_p); + + if( abs_del/(fabs(sum)+abs_del) < GSL_DBL_EPSILON + && prev_abs_del/(fabs(sum)+prev_abs_del) < GSL_DBL_EPSILON + && abs_del_p/(fabs(sump)+abs_del_p) < GSL_DBL_EPSILON + && prev_abs_del_p/(fabs(sump)+prev_abs_del_p) < GSL_DBL_EPSILON + && k > kmin + ) break; + + /* We need to keep track of the previous delta because when + * eta is near zero the odd terms of the sum are very small + * and this could lead to premature termination. + */ + prev_abs_del = abs_del; + prev_abs_del_p = abs_del_p; + + Akm2 = Akm1; + Akm1 = Ak; + } + + *result = sum; + *result_star = sump; + + if(k==kmax) { + GSL_ERROR ("error", GSL_EMAXITER); + } + else { + return GSL_SUCCESS; + } +} +#endif /* 0 */ + + +/* Determine the connection phase, phi_lambda. + * See coulomb_FG_series() below. We have + * to be careful about sin(phi)->0. Note that + * there is an underflow condition for large + * positive eta in any case. + */ +static +int +coulomb_connection(const double lam, const double eta, + double * cos_phi, double * sin_phi) +{ + if(eta > -GSL_LOG_DBL_MIN/2.0*M_PI-1.0) { + *cos_phi = 1.0; + *sin_phi = 0.0; + GSL_ERROR ("error", GSL_EUNDRFLW); + } + else if(eta > -GSL_LOG_DBL_EPSILON/(4.0*M_PI)) { + const double eps = 2.0 * exp(-2.0*M_PI*eta); + const double tpl = tan(M_PI * lam); + const double dth = eps * tpl / (tpl*tpl + 1.0); + *cos_phi = -1.0 + 0.5 * dth*dth; + *sin_phi = -dth; + return GSL_SUCCESS; + } + else { + double X = tanh(M_PI * eta) / tan(M_PI * lam); + double phi = -atan(X) - (lam + 0.5) * M_PI; + *cos_phi = cos(phi); + *sin_phi = sin(phi); + return GSL_SUCCESS; + } +} + + +/* Evaluate the Frobenius series for F_lam(eta,x) and G_lam(eta,x). + * Homegrown algebra. Evaluates the series for F_{lam} and + * F_{-lam-1}, then uses + * G_{lam} = (F_{lam} cos(phi) - F_{-lam-1}) / sin(phi) + * where + * phi = Arg[Gamma[1+lam+I eta]] - Arg[Gamma[-lam + I eta]] - (lam+1/2)Pi + * = Arg[Sin[Pi(-lam+I eta)] - (lam+1/2)Pi + * = atan2(-cos(lam Pi)sinh(eta Pi), -sin(lam Pi)cosh(eta Pi)) - (lam+1/2)Pi + * + * = -atan(X) - (lam+1/2) Pi, X = tanh(eta Pi)/tan(lam Pi) + * + * Not appropriate for lam <= -1/2, lam = 0, or lam >= 1/2. + */ +static +int +coulomb_FG_series(const double lam, const double eta, const double x, + gsl_sf_result * F, gsl_sf_result * G) +{ + const int max_iter = 800; + gsl_sf_result ClamA; + gsl_sf_result ClamB; + int stat_A = CLeta(lam, eta, &ClamA); + int stat_B = CLeta(-lam-1.0, eta, &ClamB); + const double tlp1 = 2.0*lam + 1.0; + const double pow_x = pow(x, lam); + double cos_phi_lam; + double sin_phi_lam; + + double uA_mm2 = 1.0; /* uA sum is for F_{lam} */ + double uA_mm1 = x*eta/(lam+1.0); + double uA_m; + double uB_mm2 = 1.0; /* uB sum is for F_{-lam-1} */ + double uB_mm1 = -x*eta/lam; + double uB_m; + double A_sum = uA_mm2 + uA_mm1; + double B_sum = uB_mm2 + uB_mm1; + double A_abs_del_prev = fabs(A_sum); + double B_abs_del_prev = fabs(B_sum); + gsl_sf_result FA, FB; + int m = 2; + + int stat_conn = coulomb_connection(lam, eta, &cos_phi_lam, &sin_phi_lam); + + if(stat_conn == GSL_EUNDRFLW) { + F->val = 0.0; /* FIXME: should this be set to Inf too like G? */ + F->err = 0.0; + OVERFLOW_ERROR(G); + } + + while(m < max_iter) { + double abs_dA; + double abs_dB; + uA_m = x*(2.0*eta*uA_mm1 - x*uA_mm2)/(m*(m+tlp1)); + uB_m = x*(2.0*eta*uB_mm1 - x*uB_mm2)/(m*(m-tlp1)); + A_sum += uA_m; + B_sum += uB_m; + abs_dA = fabs(uA_m); + abs_dB = fabs(uB_m); + if(m > 15) { + /* Don't bother checking until we have gone out a little ways; + * a minor optimization. Also make sure to check both the + * current and the previous increment because the odd and even + * terms of the sum can have very different behaviour, depending + * on the value of eta. + */ + double max_abs_dA = GSL_MAX(abs_dA, A_abs_del_prev); + double max_abs_dB = GSL_MAX(abs_dB, B_abs_del_prev); + double abs_A = fabs(A_sum); + double abs_B = fabs(B_sum); + if( max_abs_dA/(max_abs_dA + abs_A) < 4.0*GSL_DBL_EPSILON + && max_abs_dB/(max_abs_dB + abs_B) < 4.0*GSL_DBL_EPSILON + ) break; + } + A_abs_del_prev = abs_dA; + B_abs_del_prev = abs_dB; + uA_mm2 = uA_mm1; + uA_mm1 = uA_m; + uB_mm2 = uB_mm1; + uB_mm1 = uB_m; + m++; + } + + FA.val = A_sum * ClamA.val * pow_x * x; + FA.err = fabs(A_sum) * ClamA.err * pow_x * x + 2.0*GSL_DBL_EPSILON*fabs(FA.val); + FB.val = B_sum * ClamB.val / pow_x; + FB.err = fabs(B_sum) * ClamB.err / pow_x + 2.0*GSL_DBL_EPSILON*fabs(FB.val); + + F->val = FA.val; + F->err = FA.err; + + G->val = (FA.val * cos_phi_lam - FB.val)/sin_phi_lam; + G->err = (FA.err * fabs(cos_phi_lam) + FB.err)/fabs(sin_phi_lam); + + if(m >= max_iter) + GSL_ERROR ("error", GSL_EMAXITER); + else + return GSL_ERROR_SELECT_2(stat_A, stat_B); +} + + +/* Evaluate the Frobenius series for F_0(eta,x) and G_0(eta,x). + * See [Bardin et al., CPC 3, 73 (1972), (14)-(17)]; + * note the misprint in (17): nu_0=1 is correct, not nu_0=0. + */ +static +int +coulomb_FG0_series(const double eta, const double x, + gsl_sf_result * F, gsl_sf_result * G) +{ + const int max_iter = 800; + const double x2 = x*x; + const double tex = 2.0*eta*x; + gsl_sf_result C0; + int stat_CL = CLeta(0.0, eta, &C0); + gsl_sf_result r1pie; + int psi_stat = gsl_sf_psi_1piy_e(eta, &r1pie); + double u_mm2 = 0.0; /* u_0 */ + double u_mm1 = x; /* u_1 */ + double u_m; + double v_mm2 = 1.0; /* nu_0 */ + double v_mm1 = tex*(2.0*M_EULER-1.0+r1pie.val); /* nu_1 */ + double v_m; + double u_sum = u_mm2 + u_mm1; + double v_sum = v_mm2 + v_mm1; + double u_abs_del_prev = fabs(u_sum); + double v_abs_del_prev = fabs(v_sum); + int m = 2; + double u_sum_err = 2.0 * GSL_DBL_EPSILON * fabs(u_sum); + double v_sum_err = 2.0 * GSL_DBL_EPSILON * fabs(v_sum); + double ln2x = log(2.0*x); + + while(m < max_iter) { + double abs_du; + double abs_dv; + double m_mm1 = m*(m-1.0); + u_m = (tex*u_mm1 - x2*u_mm2)/m_mm1; + v_m = (tex*v_mm1 - x2*v_mm2 - 2.0*eta*(2*m-1)*u_m)/m_mm1; + u_sum += u_m; + v_sum += v_m; + abs_du = fabs(u_m); + abs_dv = fabs(v_m); + u_sum_err += 2.0 * GSL_DBL_EPSILON * abs_du; + v_sum_err += 2.0 * GSL_DBL_EPSILON * abs_dv; + if(m > 15) { + /* Don't bother checking until we have gone out a little ways; + * a minor optimization. Also make sure to check both the + * current and the previous increment because the odd and even + * terms of the sum can have very different behaviour, depending + * on the value of eta. + */ + double max_abs_du = GSL_MAX(abs_du, u_abs_del_prev); + double max_abs_dv = GSL_MAX(abs_dv, v_abs_del_prev); + double abs_u = fabs(u_sum); + double abs_v = fabs(v_sum); + if( max_abs_du/(max_abs_du + abs_u) < 40.0*GSL_DBL_EPSILON + && max_abs_dv/(max_abs_dv + abs_v) < 40.0*GSL_DBL_EPSILON + ) break; + } + u_abs_del_prev = abs_du; + v_abs_del_prev = abs_dv; + u_mm2 = u_mm1; + u_mm1 = u_m; + v_mm2 = v_mm1; + v_mm1 = v_m; + m++; + } + + F->val = C0.val * u_sum; + F->err = C0.err * fabs(u_sum); + F->err += fabs(C0.val) * u_sum_err; + F->err += 2.0 * GSL_DBL_EPSILON * fabs(F->val); + + G->val = (v_sum + 2.0*eta*u_sum * ln2x) / C0.val; + G->err = (fabs(v_sum) + fabs(2.0*eta*u_sum * ln2x)) / fabs(C0.val) * fabs(C0.err/C0.val); + G->err += (v_sum_err + fabs(2.0*eta*u_sum_err*ln2x)) / fabs(C0.val); + G->err += 2.0 * GSL_DBL_EPSILON * fabs(G->val); + + if(m == max_iter) + GSL_ERROR ("error", GSL_EMAXITER); + else + return GSL_ERROR_SELECT_2(psi_stat, stat_CL); +} + + +/* Evaluate the Frobenius series for F_{-1/2}(eta,x) and G_{-1/2}(eta,x). + * Homegrown algebra. + */ +static +int +coulomb_FGmhalf_series(const double eta, const double x, + gsl_sf_result * F, gsl_sf_result * G) +{ + const int max_iter = 800; + const double rx = sqrt(x); + const double x2 = x*x; + const double tex = 2.0*eta*x; + gsl_sf_result Cmhalf; + int stat_CL = CLeta(-0.5, eta, &Cmhalf); + double u_mm2 = 1.0; /* u_0 */ + double u_mm1 = tex * u_mm2; /* u_1 */ + double u_m; + double v_mm2, v_mm1, v_m; + double f_sum, g_sum; + double tmp1; + gsl_sf_result rpsi_1pe; + gsl_sf_result rpsi_1p2e; + int m = 2; + + gsl_sf_psi_1piy_e(eta, &rpsi_1pe); + gsl_sf_psi_1piy_e(2.0*eta, &rpsi_1p2e); + + v_mm2 = 2.0*M_EULER - M_LN2 - rpsi_1pe.val + 2.0*rpsi_1p2e.val; + v_mm1 = tex*(v_mm2 - 2.0*u_mm2); + + f_sum = u_mm2 + u_mm1; + g_sum = v_mm2 + v_mm1; + + while(m < max_iter) { + double m2 = m*m; + u_m = (tex*u_mm1 - x2*u_mm2)/m2; + v_m = (tex*v_mm1 - x2*v_mm2 - 2.0*m*u_m)/m2; + f_sum += u_m; + g_sum += v_m; + if( f_sum != 0.0 + && g_sum != 0.0 + && (fabs(u_m/f_sum) + fabs(v_m/g_sum) < 10.0*GSL_DBL_EPSILON)) break; + u_mm2 = u_mm1; + u_mm1 = u_m; + v_mm2 = v_mm1; + v_mm1 = v_m; + m++; + } + + F->val = Cmhalf.val * rx * f_sum; + F->err = Cmhalf.err * fabs(rx * f_sum) + 2.0*GSL_DBL_EPSILON*fabs(F->val); + + tmp1 = f_sum*log(x); + G->val = -rx*(tmp1 + g_sum)/Cmhalf.val; + G->err = fabs(rx)*(fabs(tmp1) + fabs(g_sum))/fabs(Cmhalf.val) * fabs(Cmhalf.err/Cmhalf.val); + + if(m == max_iter) + GSL_ERROR ("error", GSL_EMAXITER); + else + return stat_CL; +} + + +/* Evolve the backwards recurrence for F,F'. + * + * F_{lam-1} = (S_lam F_lam + F_lam') / R_lam + * F_{lam-1}' = (S_lam F_{lam-1} - R_lam F_lam) + * where + * R_lam = sqrt(1 + (eta/lam)^2) + * S_lam = lam/x + eta/lam + * + */ +static +int +coulomb_F_recur(double lam_min, int kmax, + double eta, double x, + double F_lam_max, double Fp_lam_max, + double * F_lam_min, double * Fp_lam_min + ) +{ + double x_inv = 1.0/x; + double fcl = F_lam_max; + double fpl = Fp_lam_max; + double lam_max = lam_min + kmax; + double lam = lam_max; + int k; + + for(k=kmax-1; k>=0; k--) { + double el = eta/lam; + double rl = hypot(1.0, el); + double sl = el + lam*x_inv; + double fc_lm1; + fc_lm1 = (fcl*sl + fpl)/rl; + fpl = fc_lm1*sl - fcl*rl; + fcl = fc_lm1; + lam -= 1.0; + } + + *F_lam_min = fcl; + *Fp_lam_min = fpl; + return GSL_SUCCESS; +} + + +/* Evolve the forward recurrence for G,G'. + * + * G_{lam+1} = (S_lam G_lam - G_lam')/R_lam + * G_{lam+1}' = R_{lam+1} G_lam - S_lam G_{lam+1} + * + * where S_lam and R_lam are as above in the F recursion. + */ +static +int +coulomb_G_recur(const double lam_min, const int kmax, + const double eta, const double x, + const double G_lam_min, const double Gp_lam_min, + double * G_lam_max, double * Gp_lam_max + ) +{ + double x_inv = 1.0/x; + double gcl = G_lam_min; + double gpl = Gp_lam_min; + double lam = lam_min + 1.0; + int k; + + for(k=1; k<=kmax; k++) { + double el = eta/lam; + double rl = hypot(1.0, el); + double sl = el + lam*x_inv; + double gcl1 = (sl*gcl - gpl)/rl; + gpl = rl*gcl - sl*gcl1; + gcl = gcl1; + lam += 1.0; + } + + *G_lam_max = gcl; + *Gp_lam_max = gpl; + return GSL_SUCCESS; +} + + +/* Evaluate the first continued fraction, giving + * the ratio F'/F at the upper lambda value. + * We also determine the sign of F at that point, + * since it is the sign of the last denominator + * in the continued fraction. + */ +static +int +coulomb_CF1(double lambda, + double eta, double x, + double * fcl_sign, + double * result, + int * count + ) +{ + const double CF1_small = 1.e-30; + const double CF1_abort = 1.0e+05; + const double CF1_acc = 2.0*GSL_DBL_EPSILON; + const double x_inv = 1.0/x; + const double px = lambda + 1.0 + CF1_abort; + + double pk = lambda + 1.0; + double F = eta/pk + pk*x_inv; + double D, C; + double df; + + *fcl_sign = 1.0; + *count = 0; + + if(fabs(F) < CF1_small) F = CF1_small; + D = 0.0; + C = F; + + do { + double pk1 = pk + 1.0; + double ek = eta / pk; + double rk2 = 1.0 + ek*ek; + double tk = (pk + pk1)*(x_inv + ek/pk1); + D = tk - rk2 * D; + C = tk - rk2 / C; + if(fabs(C) < CF1_small) C = CF1_small; + if(fabs(D) < CF1_small) D = CF1_small; + D = 1.0/D; + df = D * C; + F = F * df; + if(D < 0.0) { + /* sign of result depends on sign of denominator */ + *fcl_sign = - *fcl_sign; + } + pk = pk1; + if( pk > px ) { + *result = F; + GSL_ERROR ("error", GSL_ERUNAWAY); + } + ++(*count); + } + while(fabs(df-1.0) > CF1_acc); + + *result = F; + return GSL_SUCCESS; +} + + +#if 0 +static +int +old_coulomb_CF1(const double lambda, + double eta, double x, + double * fcl_sign, + double * result + ) +{ + const double CF1_abort = 1.e5; + const double CF1_acc = 10.0*GSL_DBL_EPSILON; + const double x_inv = 1.0/x; + const double px = lambda + 1.0 + CF1_abort; + + double pk = lambda + 1.0; + + double D; + double df; + + double F; + double p; + double pk1; + double ek; + + double fcl = 1.0; + + double tk; + + while(1) { + ek = eta/pk; + F = (ek + pk*x_inv)*fcl + (fcl - 1.0)*x_inv; + pk1 = pk + 1.0; + if(fabs(eta*x + pk*pk1) > CF1_acc) break; + fcl = (1.0 + ek*ek)/(1.0 + eta*eta/(pk1*pk1)); + pk = 2.0 + pk; + } + + D = 1.0/((pk + pk1)*(x_inv + ek/pk1)); + df = -fcl*(1.0 + ek*ek)*D; + + if(fcl != 1.0) fcl = -1.0; + if(D < 0.0) fcl = -fcl; + + F = F + df; + + p = 1.0; + do { + pk = pk1; + pk1 = pk + 1.0; + ek = eta / pk; + tk = (pk + pk1)*(x_inv + ek/pk1); + D = tk - D*(1.0+ek*ek); + if(fabs(D) < sqrt(CF1_acc)) { + p += 1.0; + if(p > 2.0) { + printf("HELP............\n"); + } + } + D = 1.0/D; + if(D < 0.0) { + /* sign of result depends on sign of denominator */ + fcl = -fcl; + } + df = df*(D*tk - 1.0); + F = F + df; + if( pk > px ) { + GSL_ERROR ("error", GSL_ERUNAWAY); + } + } + while(fabs(df) > fabs(F)*CF1_acc); + + *fcl_sign = fcl; + *result = F; + return GSL_SUCCESS; +} +#endif /* 0 */ + + +/* Evaluate the second continued fraction to + * obtain the ratio + * (G' + i F')/(G + i F) := P + i Q + * at the specified lambda value. + */ +static +int +coulomb_CF2(const double lambda, const double eta, const double x, + double * result_P, double * result_Q, int * count + ) +{ + int status = GSL_SUCCESS; + + const double CF2_acc = 4.0*GSL_DBL_EPSILON; + const double CF2_abort = 2.0e+05; + + const double wi = 2.0*eta; + const double x_inv = 1.0/x; + const double e2mm1 = eta*eta + lambda*(lambda + 1.0); + + double ar = -e2mm1; + double ai = eta; + + double br = 2.0*(x - eta); + double bi = 2.0; + + double dr = br/(br*br + bi*bi); + double di = -bi/(br*br + bi*bi); + + double dp = -x_inv*(ar*di + ai*dr); + double dq = x_inv*(ar*dr - ai*di); + + double A, B, C, D; + + double pk = 0.0; + double P = 0.0; + double Q = 1.0 - eta*x_inv; + + *count = 0; + + do { + P += dp; + Q += dq; + pk += 2.0; + ar += pk; + ai += wi; + bi += 2.0; + D = ar*dr - ai*di + br; + di = ai*dr + ar*di + bi; + C = 1.0/(D*D + di*di); + dr = C*D; + di = -C*di; + A = br*dr - bi*di - 1.; + B = bi*dr + br*di; + C = dp*A - dq*B; + dq = dp*B + dq*A; + dp = C; + if(pk > CF2_abort) { + status = GSL_ERUNAWAY; + break; + } + ++(*count); + } + while(fabs(dp)+fabs(dq) > (fabs(P)+fabs(Q))*CF2_acc); + + if(Q < CF2_abort*GSL_DBL_EPSILON*fabs(P)) { + status = GSL_ELOSS; + } + + *result_P = P; + *result_Q = Q; + return status; +} + + +/* WKB evaluation of F, G. Assumes 0 < x < turning point. + * Overflows are trapped, GSL_EOVRFLW is signalled, + * and an exponent is returned such that: + * + * result_F = fjwkb * exp(-exponent) + * result_G = gjwkb * exp( exponent) + * + * See [Biedenharn et al. Phys. Rev. 97, 542-554 (1955), Section IV] + * + * Unfortunately, this is not very accurate in general. The + * test cases typically have 3-4 digits of precision. One could + * argue that this is ok for general use because, for instance, + * F is exponentially small in this region and so the absolute + * accuracy is still roughly acceptable. But it would be better + * to have a systematic method for improving the precision. See + * the Abad+Sesma method discussion below. + */ +static +int +coulomb_jwkb(const double lam, const double eta, const double x, + gsl_sf_result * fjwkb, gsl_sf_result * gjwkb, + double * exponent) +{ + const double llp1 = lam*(lam+1.0) + 6.0/35.0; + const double llp1_eff = GSL_MAX(llp1, 0.0); + const double rho_ghalf = sqrt(x*(2.0*eta - x) + llp1_eff); + const double sinh_arg = sqrt(llp1_eff/(eta*eta+llp1_eff)) * rho_ghalf / x; + const double sinh_inv = log(sinh_arg + hypot(1.0,sinh_arg)); + + const double phi = fabs(rho_ghalf - eta*atan2(rho_ghalf,x-eta) - sqrt(llp1_eff) * sinh_inv); + + const double zeta_half = pow(3.0*phi/2.0, 1.0/3.0); + const double prefactor = sqrt(M_PI*phi*x/(6.0 * rho_ghalf)); + + double F = prefactor * 3.0/zeta_half; + double G = prefactor * 3.0/zeta_half; /* Note the sqrt(3) from Bi normalization */ + double F_exp; + double G_exp; + + const double airy_scale_exp = phi; + gsl_sf_result ai; + gsl_sf_result bi; + gsl_sf_airy_Ai_scaled_e(zeta_half*zeta_half, GSL_MODE_DEFAULT, &ai); + gsl_sf_airy_Bi_scaled_e(zeta_half*zeta_half, GSL_MODE_DEFAULT, &bi); + F *= ai.val; + G *= bi.val; + F_exp = log(F) - airy_scale_exp; + G_exp = log(G) + airy_scale_exp; + + if(G_exp >= GSL_LOG_DBL_MAX) { + fjwkb->val = F; + gjwkb->val = G; + fjwkb->err = 1.0e-3 * fabs(F); /* FIXME: real error here ... could be smaller */ + gjwkb->err = 1.0e-3 * fabs(G); + *exponent = airy_scale_exp; + GSL_ERROR ("error", GSL_EOVRFLW); + } + else { + fjwkb->val = exp(F_exp); + gjwkb->val = exp(G_exp); + fjwkb->err = 1.0e-3 * fabs(fjwkb->val); + gjwkb->err = 1.0e-3 * fabs(gjwkb->val); + *exponent = 0.0; + return GSL_SUCCESS; + } +} + + +/* Asymptotic evaluation of F and G below the minimal turning point. + * + * This is meant to be a drop-in replacement for coulomb_jwkb(). + * It uses the expressions in [Abad+Sesma]. This requires some + * work because I am not sure where it is valid. They mumble + * something about |x| < |lam|^(-1/2) or 8|eta x| > lam when |x| < 1. + * This seems true, but I thought the result was based on a uniform + * expansion and could be controlled by simply using more terms. + */ +#if 0 +static +int +coulomb_AS_xlt2eta(const double lam, const double eta, const double x, + gsl_sf_result * f_AS, gsl_sf_result * g_AS, + double * exponent) +{ + /* no time to do this now... */ +} +#endif /* 0 */ + + + +/*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/ + +int +gsl_sf_coulomb_wave_FG_e(const double eta, const double x, + const double lam_F, + const int k_lam_G, /* lam_G = lam_F - k_lam_G */ + gsl_sf_result * F, gsl_sf_result * Fp, + gsl_sf_result * G, gsl_sf_result * Gp, + double * exp_F, double * exp_G) +{ + const double lam_G = lam_F - k_lam_G; + + if(x < 0.0 || lam_F <= -0.5 || lam_G <= -0.5) { + GSL_SF_RESULT_SET(F, 0.0, 0.0); + GSL_SF_RESULT_SET(Fp, 0.0, 0.0); + GSL_SF_RESULT_SET(G, 0.0, 0.0); + GSL_SF_RESULT_SET(Gp, 0.0, 0.0); + *exp_F = 0.0; + *exp_G = 0.0; + GSL_ERROR ("domain error", GSL_EDOM); + } + else if(x == 0.0) { + gsl_sf_result C0; + CLeta(0.0, eta, &C0); + GSL_SF_RESULT_SET(F, 0.0, 0.0); + GSL_SF_RESULT_SET(Fp, 0.0, 0.0); + GSL_SF_RESULT_SET(G, 0.0, 0.0); /* FIXME: should be Inf */ + GSL_SF_RESULT_SET(Gp, 0.0, 0.0); /* FIXME: should be Inf */ + *exp_F = 0.0; + *exp_G = 0.0; + if(lam_F == 0.0){ + GSL_SF_RESULT_SET(Fp, C0.val, C0.err); + } + if(lam_G == 0.0) { + GSL_SF_RESULT_SET(Gp, 1.0/C0.val, fabs(C0.err/C0.val)/fabs(C0.val)); + } + GSL_ERROR ("domain error", GSL_EDOM); + /* After all, since we are asking for G, this is a domain error... */ + } + else if(x < 1.2 && 2.0*M_PI*eta < 0.9*(-GSL_LOG_DBL_MIN) && fabs(eta*x) < 10.0) { + /* Reduce to a small lambda value and use the series + * representations for F and G. We cannot allow eta to + * be large and positive because the connection formula + * for G_lam is badly behaved due to an underflow in sin(phi_lam) + * [see coulomb_FG_series() and coulomb_connection() above]. + * Note that large negative eta is ok however. + */ + const double SMALL = GSL_SQRT_DBL_EPSILON; + const int N = (int)(lam_F + 0.5); + const int span = GSL_MAX(k_lam_G, N); + const double lam_min = lam_F - N; /* -1/2 <= lam_min < 1/2 */ + double F_lam_F, Fp_lam_F; + double G_lam_G, Gp_lam_G; + double F_lam_F_err, Fp_lam_F_err; + double Fp_over_F_lam_F; + double F_sign_lam_F; + double F_lam_min_unnorm, Fp_lam_min_unnorm; + double Fp_over_F_lam_min; + gsl_sf_result F_lam_min; + gsl_sf_result G_lam_min, Gp_lam_min; + double F_scale; + double Gerr_frac; + double F_scale_frac_err; + double F_unnorm_frac_err; + + /* Determine F'/F at lam_F. */ + int CF1_count; + int stat_CF1 = coulomb_CF1(lam_F, eta, x, &F_sign_lam_F, &Fp_over_F_lam_F, &CF1_count); + + int stat_ser; + int stat_Fr; + int stat_Gr; + + /* Recurse down with unnormalized F,F' values. */ + F_lam_F = SMALL; + Fp_lam_F = Fp_over_F_lam_F * F_lam_F; + if(span != 0) { + stat_Fr = coulomb_F_recur(lam_min, span, eta, x, + F_lam_F, Fp_lam_F, + &F_lam_min_unnorm, &Fp_lam_min_unnorm + ); + } + else { + F_lam_min_unnorm = F_lam_F; + Fp_lam_min_unnorm = Fp_lam_F; + stat_Fr = GSL_SUCCESS; + } + + /* Determine F and G at lam_min. */ + if(lam_min == -0.5) { + stat_ser = coulomb_FGmhalf_series(eta, x, &F_lam_min, &G_lam_min); + } + else if(lam_min == 0.0) { + stat_ser = coulomb_FG0_series(eta, x, &F_lam_min, &G_lam_min); + } + else if(lam_min == 0.5) { + /* This cannot happen. */ + F->val = F_lam_F; + F->err = 2.0 * GSL_DBL_EPSILON * fabs(F->val); + Fp->val = Fp_lam_F; + Fp->err = 2.0 * GSL_DBL_EPSILON * fabs(Fp->val); + G->val = G_lam_G; + G->err = 2.0 * GSL_DBL_EPSILON * fabs(G->val); + Gp->val = Gp_lam_G; + Gp->err = 2.0 * GSL_DBL_EPSILON * fabs(Gp->val); + *exp_F = 0.0; + *exp_G = 0.0; + GSL_ERROR ("error", GSL_ESANITY); + } + else { + stat_ser = coulomb_FG_series(lam_min, eta, x, &F_lam_min, &G_lam_min); + } + + /* Determine remaining quantities. */ + Fp_over_F_lam_min = Fp_lam_min_unnorm / F_lam_min_unnorm; + Gp_lam_min.val = Fp_over_F_lam_min*G_lam_min.val - 1.0/F_lam_min.val; + Gp_lam_min.err = fabs(Fp_over_F_lam_min)*G_lam_min.err; + Gp_lam_min.err += fabs(1.0/F_lam_min.val) * fabs(F_lam_min.err/F_lam_min.val); + F_scale = F_lam_min.val / F_lam_min_unnorm; + + /* Apply scale to the original F,F' values. */ + F_scale_frac_err = fabs(F_lam_min.err/F_lam_min.val); + F_unnorm_frac_err = 2.0*GSL_DBL_EPSILON*(CF1_count+span+1); + F_lam_F *= F_scale; + F_lam_F_err = fabs(F_lam_F) * (F_unnorm_frac_err + F_scale_frac_err); + Fp_lam_F *= F_scale; + Fp_lam_F_err = fabs(Fp_lam_F) * (F_unnorm_frac_err + F_scale_frac_err); + + /* Recurse up to get the required G,G' values. */ + stat_Gr = coulomb_G_recur(lam_min, GSL_MAX(N-k_lam_G,0), eta, x, + G_lam_min.val, Gp_lam_min.val, + &G_lam_G, &Gp_lam_G + ); + + F->val = F_lam_F; + F->err = F_lam_F_err; + F->err += 2.0 * GSL_DBL_EPSILON * fabs(F_lam_F); + + Fp->val = Fp_lam_F; + Fp->err = Fp_lam_F_err; + Fp->err += 2.0 * GSL_DBL_EPSILON * fabs(Fp_lam_F); + + Gerr_frac = fabs(G_lam_min.err/G_lam_min.val) + fabs(Gp_lam_min.err/Gp_lam_min.val); + + G->val = G_lam_G; + G->err = Gerr_frac * fabs(G_lam_G); + G->err += 2.0 * (CF1_count+1) * GSL_DBL_EPSILON * fabs(G->val); + + Gp->val = Gp_lam_G; + Gp->err = Gerr_frac * fabs(Gp->val); + Gp->err += 2.0 * (CF1_count+1) * GSL_DBL_EPSILON * fabs(Gp->val); + + *exp_F = 0.0; + *exp_G = 0.0; + + return GSL_ERROR_SELECT_4(stat_ser, stat_CF1, stat_Fr, stat_Gr); + } + else if(x < 2.0*eta) { + /* Use WKB approximation to obtain F and G at the two + * lambda values, and use the Wronskian and the + * continued fractions for F'/F to obtain F' and G'. + */ + gsl_sf_result F_lam_F, G_lam_F; + gsl_sf_result F_lam_G, G_lam_G; + double exp_lam_F, exp_lam_G; + int stat_lam_F; + int stat_lam_G; + int stat_CF1_lam_F; + int stat_CF1_lam_G; + int CF1_count; + double Fp_over_F_lam_F; + double Fp_over_F_lam_G; + double F_sign_lam_F; + double F_sign_lam_G; + + stat_lam_F = coulomb_jwkb(lam_F, eta, x, &F_lam_F, &G_lam_F, &exp_lam_F); + if(k_lam_G == 0) { + stat_lam_G = stat_lam_F; + F_lam_G = F_lam_F; + G_lam_G = G_lam_F; + exp_lam_G = exp_lam_F; + } + else { + stat_lam_G = coulomb_jwkb(lam_G, eta, x, &F_lam_G, &G_lam_G, &exp_lam_G); + } + + stat_CF1_lam_F = coulomb_CF1(lam_F, eta, x, &F_sign_lam_F, &Fp_over_F_lam_F, &CF1_count); + if(k_lam_G == 0) { + stat_CF1_lam_G = stat_CF1_lam_F; + F_sign_lam_G = F_sign_lam_F; + Fp_over_F_lam_G = Fp_over_F_lam_F; + } + else { + stat_CF1_lam_G = coulomb_CF1(lam_G, eta, x, &F_sign_lam_G, &Fp_over_F_lam_G, &CF1_count); + } + + F->val = F_lam_F.val; + F->err = F_lam_F.err; + + G->val = G_lam_G.val; + G->err = G_lam_G.err; + + Fp->val = Fp_over_F_lam_F * F_lam_F.val; + Fp->err = fabs(Fp_over_F_lam_F) * F_lam_F.err; + Fp->err += 2.0*GSL_DBL_EPSILON*fabs(Fp->val); + + Gp->val = Fp_over_F_lam_G * G_lam_G.val - 1.0/F_lam_G.val; + Gp->err = fabs(Fp_over_F_lam_G) * G_lam_G.err; + Gp->err += fabs(1.0/F_lam_G.val) * fabs(F_lam_G.err/F_lam_G.val); + + *exp_F = exp_lam_F; + *exp_G = exp_lam_G; + + if(stat_lam_F == GSL_EOVRFLW || stat_lam_G == GSL_EOVRFLW) { + GSL_ERROR ("overflow", GSL_EOVRFLW); + } + else { + return GSL_ERROR_SELECT_2(stat_lam_F, stat_lam_G); + } + } + else { + /* x > 2 eta, so we know that we can find a lambda value such + * that x is above the turning point. We do this, evaluate + * using Steed's method at that oscillatory point, then + * use recursion on F and G to obtain the required values. + * + * lam_0 = a value of lambda such that x is below the turning point + * lam_min = minimum of lam_0 and the requested lam_G, since + * we must go at least as low as lam_G + */ + const double SMALL = GSL_SQRT_DBL_EPSILON; + const double C = sqrt(1.0 + 4.0*x*(x-2.0*eta)); + const int N = ceil(lam_F - C + 0.5); + const double lam_0 = lam_F - GSL_MAX(N, 0); + const double lam_min = GSL_MIN(lam_0, lam_G); + double F_lam_F, Fp_lam_F; + double G_lam_G, Gp_lam_G; + double F_lam_min_unnorm, Fp_lam_min_unnorm; + double F_lam_min, Fp_lam_min; + double G_lam_min, Gp_lam_min; + double Fp_over_F_lam_F; + double Fp_over_F_lam_min; + double F_sign_lam_F, F_sign_lam_min; + double P_lam_min, Q_lam_min; + double alpha; + double gamma; + double F_scale; + + int CF1_count; + int CF2_count; + int stat_CF1 = coulomb_CF1(lam_F, eta, x, &F_sign_lam_F, &Fp_over_F_lam_F, &CF1_count); + int stat_CF2; + int stat_Fr; + int stat_Gr; + + int F_recur_count; + int G_recur_count; + + double err_amplify; + + F_lam_F = F_sign_lam_F * SMALL; /* unnormalized */ + Fp_lam_F = Fp_over_F_lam_F * F_lam_F; + + /* Backward recurrence to get F,Fp at lam_min */ + F_recur_count = GSL_MAX(k_lam_G, N); + stat_Fr = coulomb_F_recur(lam_min, F_recur_count, eta, x, + F_lam_F, Fp_lam_F, + &F_lam_min_unnorm, &Fp_lam_min_unnorm + ); + Fp_over_F_lam_min = Fp_lam_min_unnorm / F_lam_min_unnorm; + + /* Steed evaluation to complete evaluation of F,Fp,G,Gp at lam_min */ + stat_CF2 = coulomb_CF2(lam_min, eta, x, &P_lam_min, &Q_lam_min, &CF2_count); + alpha = Fp_over_F_lam_min - P_lam_min; + gamma = alpha/Q_lam_min; + + F_sign_lam_min = GSL_SIGN(F_lam_min_unnorm) ; + + F_lam_min = F_sign_lam_min / sqrt(alpha*alpha/Q_lam_min + Q_lam_min); + Fp_lam_min = Fp_over_F_lam_min * F_lam_min; + G_lam_min = gamma * F_lam_min; + Gp_lam_min = (P_lam_min * gamma - Q_lam_min) * F_lam_min; + + /* Apply scale to values of F,Fp at lam_F (the top). */ + F_scale = F_lam_min / F_lam_min_unnorm; + F_lam_F *= F_scale; + Fp_lam_F *= F_scale; + + /* Forward recurrence to get G,Gp at lam_G (the top). */ + G_recur_count = GSL_MAX(N-k_lam_G,0); + stat_Gr = coulomb_G_recur(lam_min, G_recur_count, eta, x, + G_lam_min, Gp_lam_min, + &G_lam_G, &Gp_lam_G + ); + + err_amplify = CF1_count + CF2_count + F_recur_count + G_recur_count + 1; + + F->val = F_lam_F; + F->err = 8.0*err_amplify*GSL_DBL_EPSILON * fabs(F->val); + + Fp->val = Fp_lam_F; + Fp->err = 8.0*err_amplify*GSL_DBL_EPSILON * fabs(Fp->val); + + G->val = G_lam_G; + G->err = 8.0*err_amplify*GSL_DBL_EPSILON * fabs(G->val); + + Gp->val = Gp_lam_G; + Gp->err = 8.0*err_amplify*GSL_DBL_EPSILON * fabs(Gp->val); + + *exp_F = 0.0; + *exp_G = 0.0; + + return GSL_ERROR_SELECT_4(stat_CF1, stat_CF2, stat_Fr, stat_Gr); + } +} + + +int +gsl_sf_coulomb_wave_F_array(double lam_min, int kmax, + double eta, double x, + double * fc_array, + double * F_exp) +{ + if(x == 0.0) { + int k; + *F_exp = 0.0; + for(k=0; k<=kmax; k++) { + fc_array[k] = 0.0; + } + if(lam_min == 0.0){ + gsl_sf_result f_0; + CLeta(0.0, eta, &f_0); + fc_array[0] = f_0.val; + } + return GSL_SUCCESS; + } + else { + const double x_inv = 1.0/x; + const double lam_max = lam_min + kmax; + gsl_sf_result F, Fp; + gsl_sf_result G, Gp; + double G_exp; + + int stat_FG = gsl_sf_coulomb_wave_FG_e(eta, x, lam_max, 0, + &F, &Fp, &G, &Gp, F_exp, &G_exp); + + double fcl = F.val; + double fpl = Fp.val; + double lam = lam_max; + int k; + + fc_array[kmax] = F.val; + + for(k=kmax-1; k>=0; k--) { + double el = eta/lam; + double rl = hypot(1.0, el); + double sl = el + lam*x_inv; + double fc_lm1 = (fcl*sl + fpl)/rl; + fc_array[k] = fc_lm1; + fpl = fc_lm1*sl - fcl*rl; + fcl = fc_lm1; + lam -= 1.0; + } + + return stat_FG; + } +} + + +int +gsl_sf_coulomb_wave_FG_array(double lam_min, int kmax, + double eta, double x, + double * fc_array, double * gc_array, + double * F_exp, double * G_exp) +{ + const double x_inv = 1.0/x; + const double lam_max = lam_min + kmax; + gsl_sf_result F, Fp; + gsl_sf_result G, Gp; + + int stat_FG = gsl_sf_coulomb_wave_FG_e(eta, x, lam_max, kmax, + &F, &Fp, &G, &Gp, F_exp, G_exp); + + double fcl = F.val; + double fpl = Fp.val; + double lam = lam_max; + int k; + + double gcl, gpl; + + fc_array[kmax] = F.val; + + for(k=kmax-1; k>=0; k--) { + double el = eta/lam; + double rl = hypot(1.0, el); + double sl = el + lam*x_inv; + double fc_lm1; + fc_lm1 = (fcl*sl + fpl)/rl; + fc_array[k] = fc_lm1; + fpl = fc_lm1*sl - fcl*rl; + fcl = fc_lm1; + lam -= 1.0; + } + + gcl = G.val; + gpl = Gp.val; + lam = lam_min + 1.0; + + gc_array[0] = G.val; + + for(k=1; k<=kmax; k++) { + double el = eta/lam; + double rl = hypot(1.0, el); + double sl = el + lam*x_inv; + double gcl1 = (sl*gcl - gpl)/rl; + gc_array[k] = gcl1; + gpl = rl*gcl - sl*gcl1; + gcl = gcl1; + lam += 1.0; + } + + return stat_FG; +} + + +int +gsl_sf_coulomb_wave_FGp_array(double lam_min, int kmax, + double eta, double x, + double * fc_array, double * fcp_array, + double * gc_array, double * gcp_array, + double * F_exp, double * G_exp) + +{ + const double x_inv = 1.0/x; + const double lam_max = lam_min + kmax; + gsl_sf_result F, Fp; + gsl_sf_result G, Gp; + + int stat_FG = gsl_sf_coulomb_wave_FG_e(eta, x, lam_max, kmax, + &F, &Fp, &G, &Gp, F_exp, G_exp); + + double fcl = F.val; + double fpl = Fp.val; + double lam = lam_max; + int k; + + double gcl, gpl; + + fc_array[kmax] = F.val; + fcp_array[kmax] = Fp.val; + + for(k=kmax-1; k>=0; k--) { + double el = eta/lam; + double rl = hypot(1.0, el); + double sl = el + lam*x_inv; + double fc_lm1; + fc_lm1 = (fcl*sl + fpl)/rl; + fc_array[k] = fc_lm1; + fpl = fc_lm1*sl - fcl*rl; + fcp_array[k] = fpl; + fcl = fc_lm1; + lam -= 1.0; + } + + gcl = G.val; + gpl = Gp.val; + lam = lam_min + 1.0; + + gc_array[0] = G.val; + gcp_array[0] = Gp.val; + + for(k=1; k<=kmax; k++) { + double el = eta/lam; + double rl = hypot(1.0, el); + double sl = el + lam*x_inv; + double gcl1 = (sl*gcl - gpl)/rl; + gc_array[k] = gcl1; + gpl = rl*gcl - sl*gcl1; + gcp_array[k] = gpl; + gcl = gcl1; + lam += 1.0; + } + + return stat_FG; +} + + +int +gsl_sf_coulomb_wave_sphF_array(double lam_min, int kmax, + double eta, double x, + double * fc_array, + double * F_exp) +{ + if(x < 0.0 || lam_min < -0.5) { + GSL_ERROR ("domain error", GSL_EDOM); + } + else if(x < 10.0/GSL_DBL_MAX) { + int k; + for(k=0; k<=kmax; k++) { + fc_array[k] = 0.0; + } + if(lam_min == 0.0) { + fc_array[0] = sqrt(C0sq(eta)); + } + *F_exp = 0.0; + if(x == 0.0) + return GSL_SUCCESS; + else + GSL_ERROR ("underflow", GSL_EUNDRFLW); + } + else { + int k; + int stat_F = gsl_sf_coulomb_wave_F_array(lam_min, kmax, + eta, x, + fc_array, + F_exp); + + for(k=0; k<=kmax; k++) { + fc_array[k] = fc_array[k] / x; + } + return stat_F; + } +} + + |