diff options
| author | Shashank | 2017-05-29 12:40:26 +0530 |
|---|---|---|
| committer | Shashank | 2017-05-29 12:40:26 +0530 |
| commit | 0345245e860375a32c9a437c4a9d9cae807134e9 (patch) | |
| tree | ad51ecbfa7bcd3cc5f09834f1bb8c08feaa526a4 /modules/special_functions/src | |
| download | scilab_for_xcos_on_cloud-0345245e860375a32c9a437c4a9d9cae807134e9.tar.gz scilab_for_xcos_on_cloud-0345245e860375a32c9a437c4a9d9cae807134e9.tar.bz2 scilab_for_xcos_on_cloud-0345245e860375a32c9a437c4a9d9cae807134e9.zip | |
CMSCOPE changed
Diffstat (limited to 'modules/special_functions/src')
60 files changed, 6596 insertions, 0 deletions
diff --git a/modules/special_functions/src/c/.deps/.dirstamp b/modules/special_functions/src/c/.deps/.dirstamp new file mode 100755 index 000000000..e69de29bb --- /dev/null +++ b/modules/special_functions/src/c/.deps/.dirstamp diff --git a/modules/special_functions/src/c/.deps/libscispecial_functions_algo_la-zbeshv.Plo b/modules/special_functions/src/c/.deps/libscispecial_functions_algo_la-zbeshv.Plo new file mode 100755 index 000000000..40d085cf5 --- /dev/null +++ b/modules/special_functions/src/c/.deps/libscispecial_functions_algo_la-zbeshv.Plo @@ -0,0 +1,167 @@ +src/c/libscispecial_functions_algo_la-zbeshv.lo: src/c/zbeshv.c \ + /usr/include/stdc-predef.h /usr/include/string.h /usr/include/features.h \ + /usr/include/x86_64-linux-gnu/sys/cdefs.h \ + /usr/include/x86_64-linux-gnu/bits/wordsize.h \ + /usr/include/x86_64-linux-gnu/gnu/stubs.h \ + /usr/include/x86_64-linux-gnu/gnu/stubs-64.h \ + /usr/lib/gcc/x86_64-linux-gnu/5/include/stddef.h /usr/include/xlocale.h \ + /usr/include/x86_64-linux-gnu/bits/string.h \ + /usr/include/x86_64-linux-gnu/bits/string2.h /usr/include/endian.h \ + /usr/include/x86_64-linux-gnu/bits/endian.h \ + /usr/include/x86_64-linux-gnu/bits/byteswap.h \ + /usr/include/x86_64-linux-gnu/bits/types.h \ + /usr/include/x86_64-linux-gnu/bits/typesizes.h \ + /usr/include/x86_64-linux-gnu/bits/byteswap-16.h /usr/include/stdlib.h \ + /usr/include/x86_64-linux-gnu/bits/string3.h src/c/zbeshv.h \ + ../../modules/core/includes/machine.h \ + ../../modules/core/includes/core_math.h \ + /usr/lib/gcc/x86_64-linux-gnu/5/include-fixed/limits.h \ + /usr/lib/gcc/x86_64-linux-gnu/5/include-fixed/syslimits.h \ + /usr/include/limits.h /usr/include/x86_64-linux-gnu/bits/posix1_lim.h \ + /usr/include/x86_64-linux-gnu/bits/local_lim.h \ + /usr/include/linux/limits.h \ + /usr/include/x86_64-linux-gnu/bits/posix2_lim.h /usr/include/math.h \ + /usr/include/x86_64-linux-gnu/bits/math-vector.h \ + /usr/include/x86_64-linux-gnu/bits/libm-simd-decl-stubs.h \ + /usr/include/x86_64-linux-gnu/bits/huge_val.h \ + /usr/include/x86_64-linux-gnu/bits/huge_valf.h \ + /usr/include/x86_64-linux-gnu/bits/huge_vall.h \ + /usr/include/x86_64-linux-gnu/bits/inf.h \ + /usr/include/x86_64-linux-gnu/bits/nan.h \ + /usr/include/x86_64-linux-gnu/bits/mathdef.h \ + /usr/include/x86_64-linux-gnu/bits/mathcalls.h \ + /usr/include/x86_64-linux-gnu/bits/mathinline.h \ + /usr/include/x86_64-linux-gnu/bits/waitflags.h \ + /usr/include/x86_64-linux-gnu/bits/waitstatus.h \ + /usr/include/x86_64-linux-gnu/sys/types.h /usr/include/time.h \ + /usr/include/x86_64-linux-gnu/sys/select.h \ + /usr/include/x86_64-linux-gnu/bits/select.h \ + /usr/include/x86_64-linux-gnu/bits/sigset.h \ + /usr/include/x86_64-linux-gnu/bits/time.h \ + /usr/include/x86_64-linux-gnu/bits/select2.h \ + /usr/include/x86_64-linux-gnu/sys/sysmacros.h \ + /usr/include/x86_64-linux-gnu/bits/pthreadtypes.h /usr/include/alloca.h \ + /usr/include/x86_64-linux-gnu/bits/stdlib-bsearch.h \ + /usr/include/x86_64-linux-gnu/bits/stdlib-float.h \ + /usr/include/x86_64-linux-gnu/bits/stdlib.h /usr/include/values.h \ + /usr/lib/gcc/x86_64-linux-gnu/5/include/float.h \ + ../../modules/core/includes/returnanan.h \ + ../../modules/core/includes/machine.h + +/usr/include/stdc-predef.h: + +/usr/include/string.h: + +/usr/include/features.h: + +/usr/include/x86_64-linux-gnu/sys/cdefs.h: + +/usr/include/x86_64-linux-gnu/bits/wordsize.h: + +/usr/include/x86_64-linux-gnu/gnu/stubs.h: + +/usr/include/x86_64-linux-gnu/gnu/stubs-64.h: + +/usr/lib/gcc/x86_64-linux-gnu/5/include/stddef.h: + +/usr/include/xlocale.h: + +/usr/include/x86_64-linux-gnu/bits/string.h: + +/usr/include/x86_64-linux-gnu/bits/string2.h: + +/usr/include/endian.h: + +/usr/include/x86_64-linux-gnu/bits/endian.h: + +/usr/include/x86_64-linux-gnu/bits/byteswap.h: + +/usr/include/x86_64-linux-gnu/bits/types.h: + +/usr/include/x86_64-linux-gnu/bits/typesizes.h: + +/usr/include/x86_64-linux-gnu/bits/byteswap-16.h: + +/usr/include/stdlib.h: + +/usr/include/x86_64-linux-gnu/bits/string3.h: + +src/c/zbeshv.h: + +../../modules/core/includes/machine.h: + +../../modules/core/includes/core_math.h: + +/usr/lib/gcc/x86_64-linux-gnu/5/include-fixed/limits.h: + +/usr/lib/gcc/x86_64-linux-gnu/5/include-fixed/syslimits.h: + +/usr/include/limits.h: + +/usr/include/x86_64-linux-gnu/bits/posix1_lim.h: + +/usr/include/x86_64-linux-gnu/bits/local_lim.h: + +/usr/include/linux/limits.h: + +/usr/include/x86_64-linux-gnu/bits/posix2_lim.h: + +/usr/include/math.h: + +/usr/include/x86_64-linux-gnu/bits/math-vector.h: + +/usr/include/x86_64-linux-gnu/bits/libm-simd-decl-stubs.h: + +/usr/include/x86_64-linux-gnu/bits/huge_val.h: + +/usr/include/x86_64-linux-gnu/bits/huge_valf.h: + +/usr/include/x86_64-linux-gnu/bits/huge_vall.h: + +/usr/include/x86_64-linux-gnu/bits/inf.h: + +/usr/include/x86_64-linux-gnu/bits/nan.h: + +/usr/include/x86_64-linux-gnu/bits/mathdef.h: + +/usr/include/x86_64-linux-gnu/bits/mathcalls.h: + +/usr/include/x86_64-linux-gnu/bits/mathinline.h: + +/usr/include/x86_64-linux-gnu/bits/waitflags.h: + +/usr/include/x86_64-linux-gnu/bits/waitstatus.h: + +/usr/include/x86_64-linux-gnu/sys/types.h: + +/usr/include/time.h: + +/usr/include/x86_64-linux-gnu/sys/select.h: + +/usr/include/x86_64-linux-gnu/bits/select.h: + +/usr/include/x86_64-linux-gnu/bits/sigset.h: + +/usr/include/x86_64-linux-gnu/bits/time.h: + +/usr/include/x86_64-linux-gnu/bits/select2.h: + +/usr/include/x86_64-linux-gnu/sys/sysmacros.h: + +/usr/include/x86_64-linux-gnu/bits/pthreadtypes.h: + +/usr/include/alloca.h: + +/usr/include/x86_64-linux-gnu/bits/stdlib-bsearch.h: + +/usr/include/x86_64-linux-gnu/bits/stdlib-float.h: + +/usr/include/x86_64-linux-gnu/bits/stdlib.h: + +/usr/include/values.h: + +/usr/lib/gcc/x86_64-linux-gnu/5/include/float.h: + +../../modules/core/includes/returnanan.h: + +../../modules/core/includes/machine.h: diff --git a/modules/special_functions/src/c/.dirstamp b/modules/special_functions/src/c/.dirstamp new file mode 100755 index 000000000..e69de29bb --- /dev/null +++ b/modules/special_functions/src/c/.dirstamp diff --git a/modules/special_functions/src/c/.libs/libscispecial_functions_algo_la-zbeshv.o b/modules/special_functions/src/c/.libs/libscispecial_functions_algo_la-zbeshv.o Binary files differnew file mode 100755 index 000000000..7a010e5b7 --- /dev/null +++ b/modules/special_functions/src/c/.libs/libscispecial_functions_algo_la-zbeshv.o diff --git a/modules/special_functions/src/c/DllmainSpecial_functions.c b/modules/special_functions/src/c/DllmainSpecial_functions.c new file mode 100755 index 000000000..3fa5526a6 --- /dev/null +++ b/modules/special_functions/src/c/DllmainSpecial_functions.c @@ -0,0 +1,34 @@ + +/* + * Scilab ( http://www.scilab.org/ ) - This file is part of Scilab + * Copyright (C) 2010 - DIGITEO - Allan CORNET + * + * This file must be used under the terms of the CeCILL. + * This source file is licensed as described in the file COPYING, which + * you should have received as part of this distribution. The terms + * are also available at + * http://www.cecill.info/licences/Licence_CeCILL_V2.1-en.txt + * + */ + +#include <windows.h> +/*--------------------------------------------------------------------------*/ +#pragma comment(lib,"../../../../bin/libintl.lib") +/*--------------------------------------------------------------------------*/ +int WINAPI DllMain (HINSTANCE hInstance , DWORD reason, PVOID pvReserved) +{ + switch (reason) + { + case DLL_PROCESS_ATTACH: + break; + case DLL_PROCESS_DETACH: + break; + case DLL_THREAD_ATTACH: + break; + case DLL_THREAD_DETACH: + break; + } + return 1; +} +/*--------------------------------------------------------------------------*/ + diff --git a/modules/special_functions/src/c/core_Import.def b/modules/special_functions/src/c/core_Import.def new file mode 100755 index 000000000..357f1343e --- /dev/null +++ b/modules/special_functions/src/c/core_Import.def @@ -0,0 +1,26 @@ +LIBRARY core.dll + + +EXPORTS +;core +callFunctionFromGateway +com_ +vstk_ +putlhsvar_ +checklhs_ +checkrhs_ +getrhsvar_ +stack_ +getrhscvar_ +intersci_ +createvar_ +errgst_ +check_dim_prop +createcvar_ +check_scalar +check_same_dims +getWarningMode +returnanan_ +overload_ +MyHeapAlloc +MyHeapFree diff --git a/modules/special_functions/src/c/dcd_f_Import.def b/modules/special_functions/src/c/dcd_f_Import.def new file mode 100755 index 000000000..d541d9278 --- /dev/null +++ b/modules/special_functions/src/c/dcd_f_Import.def @@ -0,0 +1,8 @@ +LIBRARY dcd_f.dll + + +EXPORTS +; --------------------------------------- +; dcd_f +; --------------------------------------- +betaln_
\ No newline at end of file diff --git a/modules/special_functions/src/c/elementary_functions_f_Import.def b/modules/special_functions/src/c/elementary_functions_f_Import.def new file mode 100755 index 000000000..827558f29 --- /dev/null +++ b/modules/special_functions/src/c/elementary_functions_f_Import.def @@ -0,0 +1,10 @@ +LIBRARY elementary_functions_f.dll + + +EXPORTS +; --------------------------------------- +; elementary_functions_f +; --------------------------------------- +dset_ +wscal_ +dlgama_ diff --git a/modules/special_functions/src/c/libscispecial_functions_algo_la-zbeshv.lo b/modules/special_functions/src/c/libscispecial_functions_algo_la-zbeshv.lo new file mode 100755 index 000000000..5b6c55e1c --- /dev/null +++ b/modules/special_functions/src/c/libscispecial_functions_algo_la-zbeshv.lo @@ -0,0 +1,12 @@ +# src/c/libscispecial_functions_algo_la-zbeshv.lo - a libtool object file +# Generated by libtool (GNU libtool) 2.4.2 +# +# Please DO NOT delete this file! +# It is necessary for linking the library. + +# Name of the PIC object. +pic_object='.libs/libscispecial_functions_algo_la-zbeshv.o' + +# Name of the non-PIC object +non_pic_object=none + diff --git a/modules/special_functions/src/c/slatec_f_Import.def b/modules/special_functions/src/c/slatec_f_Import.def new file mode 100755 index 000000000..426bab0d9 --- /dev/null +++ b/modules/special_functions/src/c/slatec_f_Import.def @@ -0,0 +1,10 @@ +LIBRARY slatec_f.dll + + +EXPORTS +; --------------------------------------- +; slatec_f +; --------------------------------------- +dgammacody_ +dxlegf_ +zbesh_ diff --git a/modules/special_functions/src/c/special_functions.rc b/modules/special_functions/src/c/special_functions.rc new file mode 100755 index 000000000..2c2e81e52 --- /dev/null +++ b/modules/special_functions/src/c/special_functions.rc @@ -0,0 +1,95 @@ +// Microsoft Visual C++ generated resource script. +// + + +#define APSTUDIO_READONLY_SYMBOLS +///////////////////////////////////////////////////////////////////////////// +// +// Generated from the TEXTINCLUDE 2 resource. +// +#define APSTUDIO_HIDDEN_SYMBOLS +#include "windows.h" +///////////////////////////////////////////////////////////////////////////// +#undef APSTUDIO_READONLY_SYMBOLS + +///////////////////////////////////////////////////////////////////////////// +// French (France) resources + +#if !defined(AFX_RESOURCE_DLL) || defined(AFX_TARG_FRA) +#ifdef _WIN32 +LANGUAGE LANG_FRENCH, SUBLANG_FRENCH +#pragma code_page(1252) +#endif //_WIN32 + +#ifdef APSTUDIO_INVOKED +///////////////////////////////////////////////////////////////////////////// +// +// TEXTINCLUDE +// + +1 TEXTINCLUDE +BEGIN + "resource.h\0" +END + +3 TEXTINCLUDE +BEGIN + "\r\n" + "\0" +END + +#endif // APSTUDIO_INVOKED + + +///////////////////////////////////////////////////////////////////////////// +// +// Version +// + +VS_VERSION_INFO VERSIONINFO + FILEVERSION 5,5,2,0 + PRODUCTVERSION 5,5,2,0 + FILEFLAGSMASK 0x17L +#ifdef _DEBUG + FILEFLAGS 0x1L +#else + FILEFLAGS 0x0L +#endif + FILEOS 0x4L + FILETYPE 0x2L + FILESUBTYPE 0x0L +BEGIN + BLOCK "StringFileInfo" + BEGIN + BLOCK "040c04b0" + BEGIN + VALUE "FileDescription", "special_functions module" + VALUE "FileVersion", "5, 5, 2, 0" + VALUE "InternalName", "special_functions module" + VALUE "LegalCopyright", "Copyright (C) 2017" + VALUE "OriginalFilename", "special_functions.dll" + VALUE "ProductName", "special_functions module" + VALUE "ProductVersion", "5, 5, 2, 0" + END + END + BLOCK "VarFileInfo" + BEGIN + VALUE "Translation", 0x40c, 1200 + END +END + +#endif // French (France) resources +///////////////////////////////////////////////////////////////////////////// + + + +#ifndef APSTUDIO_INVOKED +///////////////////////////////////////////////////////////////////////////// +// +// Generated from the TEXTINCLUDE 3 resource. +// + + +///////////////////////////////////////////////////////////////////////////// +#endif // not APSTUDIO_INVOKED + diff --git a/modules/special_functions/src/c/special_functions.vcxproj b/modules/special_functions/src/c/special_functions.vcxproj new file mode 100755 index 000000000..7372d5768 --- /dev/null +++ b/modules/special_functions/src/c/special_functions.vcxproj @@ -0,0 +1,265 @@ +<?xml version="1.0" encoding="utf-8"?> +<Project DefaultTargets="Build" ToolsVersion="4.0" xmlns="http://schemas.microsoft.com/developer/msbuild/2003"> + <ItemGroup Label="ProjectConfigurations"> + <ProjectConfiguration Include="Debug|Win32"> + <Configuration>Debug</Configuration> + <Platform>Win32</Platform> + </ProjectConfiguration> + <ProjectConfiguration Include="Debug|x64"> + <Configuration>Debug</Configuration> + <Platform>x64</Platform> + </ProjectConfiguration> + <ProjectConfiguration Include="Release|Win32"> + <Configuration>Release</Configuration> + <Platform>Win32</Platform> + </ProjectConfiguration> + <ProjectConfiguration Include="Release|x64"> + 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\ No newline at end of file diff --git a/modules/special_functions/src/c/special_functions_f_Import.def b/modules/special_functions/src/c/special_functions_f_Import.def new file mode 100755 index 000000000..63b133e8d --- /dev/null +++ b/modules/special_functions/src/c/special_functions_f_Import.def @@ -0,0 +1,20 @@ +LIBRARY special_functions_f.dll + + +EXPORTS +; --------------------------------------- +; special_functions_f +; --------------------------------------- +intslgamma_ +intsgamma_ +intsdlgamma_ +intscalerf_ +zbesiv_ +dbesiv_ +zbesjv_ +dbesjv_ +zbeskv_ +dbeskv_ +zbesyv_ +dbesyv_ +psi_ diff --git a/modules/special_functions/src/c/zbeshv.c b/modules/special_functions/src/c/zbeshv.c new file mode 100755 index 000000000..62dd7dd61 --- /dev/null +++ b/modules/special_functions/src/c/zbeshv.c @@ -0,0 +1,254 @@ +/* +* Scilab ( http://www.scilab.org/ ) - This file is part of Scilab +* Copyright (C) 2005-2008 - INRIA - Serge STEER +* Copyright (C) 2011 - DIGITEO - Allan CORNET +* +* This file must be used under the terms of the CeCILL. +* This source file is licensed as described in the file COPYING, which +* you should have received as part of this distribution. The terms +* are also available at +* http://www.cecill.info/licences/Licence_CeCILL_V2.1-en.txt +* +*/ +/*--------------------------------------------------------------------------*/ +#include <string.h> +#include "zbeshv.h" +#include "machine.h" +#include "core_math.h" +#include "returnanan.h" +/*--------------------------------------------------------------------------*/ +/* fortran subroutines */ +extern int C2F(dset)(); +extern int C2F(dscal)(); +extern int C2F(wscal)(); +extern int C2F(zbesh)(); +extern int C2F(dcopy)(); +extern double C2F(dlamch)(); +/*--------------------------------------------------------------------------*/ +static int zbeshg(double *x1r, double *x1i, double *alpha, + int *kode, int *k, int *n, double *yr, + double *yi, int *nz, double *wr, + double *wi, int *ierr); +/*--------------------------------------------------------------------------*/ +/* compute besseli function for x and alpha given by vectors */ +/* w : working array of size 2*na (used only if nz>0 and alpha */ +/* contains negative values */ +int zbeshv(double *xr, double *xi, int *nx, + double *alpha, int *na, int *kode, int *k, double + *yr, double *yi, double *wr, double *wi, int *ierr) +{ + double eps = C2F(dlamch)("p", strlen("p")); + int iOne = 1; + int i = 0, j = 0, nz = 0; + + *ierr = 0; + if (*na < 0) + { + /* element wise case x and alpha are supposed to have the same size */ + for (i = 1; i <= *nx; ++i) + { + int ier = 0; + zbeshg(&xr[i - 1], &xi[i - 1], &alpha[i - 1], kode, k, &iOne, &yr[i - 1], + &yi[i - 1], &nz, &wr[1], &wi[0], &ier); + *ierr = Max(*ierr, ier); + } + } + else if (*na == 1) + { + for (i = 1; i <= *nx; ++i) + { + int ier = 0; + zbeshg(&xr[i - 1], &xi[i - 1], &alpha[0], kode, k, &iOne, &yr[i - 1], + &yi[i - 1], &nz, &wr[0], &wi[0], &ier); + *ierr = Max(*ierr, ier); + } + } + else + { + /* compute besselh(x(i),y(j)), i=1,nx,j=1,na */ + double dTmp = 0; + int n = 0; + int l = 1; +L5: + n = 0; +L10: + ++n; + j = l + n; + if (j <= *na && (dTmp = alpha[j] + 1 - alpha[j - 1], fabs(dTmp)) <= eps) + { + goto L10; + } + for (i = 1; i <= *nx; ++i) + { + int ier = 0; + zbeshg(&xr[i - 1], &xi[i - 1], &alpha[l - 1], kode, k, &n, &wr[0], &wi[0], + &nz, &wr[*na], &wi[*na], &ier); + + *ierr = Max(*ierr, ier); + + C2F(dcopy)(&n, &wr[0], &iOne, &yr[(i + (l - 1) * *nx) - 1], nx); + + C2F(dcopy)(&n, &wi[0], &iOne, &yi[(i + (l - 1) * *nx) - 1], nx); + } + + l = j; + + if (l <= *na) + { + goto L5; + } + } + return 0; +} +/*--------------------------------------------------------------------------*/ +static int zbeshg(double *x1r, double *x1i, double *alpha, + int *kode, int *k, int *n, double *yr, + double *yi, int *nz, double *wr, + double *wi, int *ierr) +{ + int iOne = 1; + int iTwo = 2; + double dNegOne = -1.; + + double nan = C2F(returnanan)(); + + int iVal = 0; + int nn = 0; + double xr = *x1r; + double xi = *x1i; + + /* extends cbesi for the case where alpha is negative */ + + if (ISNAN(xr) || ISNAN(xi) || ISNAN(*alpha)) + { + /* NaN case */ + C2F(dset)(n, &nan, &yr[0], &iOne); + C2F(dset)(n, &nan, &yi[0], &iOne); + + *ierr = 4; + } + else if (*alpha >= 0.) + { + C2F(zbesh)(&xr, &xi, alpha, kode, k, n, &yr[0], &yi[0], nz, ierr); + if (*ierr == 1 || *ierr == 2 || *ierr >= 4) + { + C2F(dset)(n, &nan, &yr[0], &iOne); + C2F(dset)(n, &nan, &yi[0], &iOne); + } + } + else if (*alpha == (int)(*alpha)) + { + double a1 = 0.; + /* alpha < 0 and int, */ + /* transform to positive value of alpha */ + if (*alpha - 1 + *n >= 0.) + { + /* 0 is between alpha and alpha+n */ + a1 = 0.; + /* Computing MIN */ + nn = Min(*n, (int) (-(*alpha))); + } + else + { + a1 = -(*alpha - 1 + *n); + nn = *n; + } + C2F(zbesh)(&xr, &xi, &a1, kode, k, n, &wr[0], &wi[0], nz, ierr); + if (*ierr == 1 || *ierr == 2 || *ierr >= 4) + { + C2F(dset)(n, &nan, &yr[0], &iOne); + C2F(dset)(n, &nan, &yi[0], &iOne); + } + else + { + if (*n > nn) + { + /* 0 is between alpha and alpha+n */ + iVal = *n - nn; + C2F(dcopy)(&iVal, &wr[0], &iOne, &yr[nn], &iOne); + C2F(dcopy)(&iVal, &wi[0], &iOne, &yi[nn], &iOne); + C2F(dcopy)(&nn, &wr[1], &iOne, &yr[0], &iOne); + C2F(dcopy)(&nn, &wi[1], &iOne, &yi[0], &iOne); + } + else + { + /* alpha and alpha+n are negative */ + C2F(dcopy)(n, &wr[0], &iOne, &yr[0], &iOne); + C2F(dcopy)(n, &wi[0], &iOne, &yi[0], &iOne); + } + } + iVal = (nn - (((int) fabs(*alpha) + 1) % 2) + 1) / 2; + C2F(dscal)(&iVal, &dNegOne, &yr[((int) fabs(*alpha) + 1) % 2], &iTwo); + C2F(dscal)(&iVal, &dNegOne, &yi[((int) fabs(*alpha) + 1) % 2], &iTwo); + } + else + { + int nz1 = 0; + double a1 = 0.; + /* first alpha is negative non int, transform to positive value of alpha */ + if (*alpha - 1. + *n >= 0.) + { + /* 0 is between alpha and alpha+n */ + nn = (int) (-(*alpha)) + 1; + } + else + { + nn = *n; + } + + /* compute for negative value of alpha+k, transform problem for */ + /* a1:a1+(nn-1) with a1 positive a1+k =abs(alpha+nn-k) */ + a1 = -(*alpha - 1. + nn); + C2F(zbesh)(&xr, &xi, &a1, kode, k, n, &wr[0], &wi[0], &nz1, ierr); + *nz = Max(nz1, 0); + if (*ierr == 0) + { + double a = cos(a1 * M_PI); + double b = sin(a1 * M_PI); + if (*k == 1) + { + C2F(wscal)(&nn, &a, &b, &wr[0], &wi[0], &iOne); + } + else + { + double dNegB = -b; + C2F(wscal)(&nn, &a, &dNegB, &wr[0], &wi[0], &iOne); + } + /* change sign to take into account that sin((a1+k)*pi) and cos((a1+k)*pi) */ + /* changes sign with k */ + if (nn >= 2) + { + iVal = nn / 2; + C2F(dscal)(&iVal, &dNegOne, &wr[1], &iTwo); + C2F(dscal)(&iVal, &dNegOne, &wi[1], &iTwo); + } + } + else if (*ierr == 1 || *ierr == 2 || *ierr >= 4) + { + C2F(dset)(&nn, &nan, &wr[0], &iOne); + C2F(dset)(&nn, &nan, &wi[0], &iOne); + } + + /* store the result in the correct order */ + C2F(dcopy)(&nn, &wr[0], &iOne, &yr[0], &iOne); + C2F(dcopy)(&nn, &wi[0], &iOne, &yi[0], &iOne); + + /* compute for positive value of alpha+k is any */ + if (*n > nn) + { + int ier = 0; + a1 = 1. - a1; + iVal = *n - nn; + C2F(zbesh)(&xr, &xi, &a1, kode, k, &iVal, &yr[nn], &yi[nn], nz, &ier); + if (ier == 1 || ier == 2 || ier >= 4) + { + iVal = *n - nn; + C2F(dset)(&iVal, &nan, &yr[nn], &iOne); + C2F(dset)(&iVal, &nan, &yi[nn], &iOne); + } + *ierr = Max(*ierr, ier); + } + } + return 0; +} +/*--------------------------------------------------------------------------*/ diff --git a/modules/special_functions/src/c/zbeshv.h b/modules/special_functions/src/c/zbeshv.h new file mode 100755 index 000000000..d10258ff0 --- /dev/null +++ b/modules/special_functions/src/c/zbeshv.h @@ -0,0 +1,21 @@ +/* +* Scilab ( http://www.scilab.org/ ) - This file is part of Scilab +* Copyright (C) 2011 - DIGITEO - Allan CORNET +* +* This file must be used under the terms of the CeCILL. +* This source file is licensed as described in the file COPYING, which +* you should have received as part of this distribution. The terms +* are also available at +* http://www.cecill.info/licences/Licence_CeCILL_V2.1-en.txt +* +*/ +/*--------------------------------------------------------------------------*/ +#ifndef __ZEBSV_H__ +#define __ZEBSV_H__ + +int zbeshv(double *xr, double *xi, int *nx, + double *alpha, int *na, int *kode, int *k, double + *yr, double *yi, double *wr, double *wi, int *ierr); + +#endif /* __ZEBSV_H__ */ +/*--------------------------------------------------------------------------*/ diff --git a/modules/special_functions/src/cpp/.deps/.dirstamp b/modules/special_functions/src/cpp/.deps/.dirstamp new file mode 100755 index 000000000..e69de29bb --- /dev/null +++ b/modules/special_functions/src/cpp/.deps/.dirstamp diff --git a/modules/special_functions/src/cpp/.deps/libscispecial_functions_algo_la-faddeeva.Plo b/modules/special_functions/src/cpp/.deps/libscispecial_functions_algo_la-faddeeva.Plo new file mode 100755 index 000000000..aa7eeb9b2 --- /dev/null +++ b/modules/special_functions/src/cpp/.deps/libscispecial_functions_algo_la-faddeeva.Plo @@ -0,0 +1,327 @@ +src/cpp/libscispecial_functions_algo_la-faddeeva.lo: src/cpp/faddeeva.cpp \ + /usr/include/stdc-predef.h ../../modules/core/includes/machine.h \ + src/cpp/faddeeva.h /usr/include/c++/5/complex \ + /usr/include/x86_64-linux-gnu/c++/5/bits/c++config.h \ + /usr/include/x86_64-linux-gnu/c++/5/bits/os_defines.h \ + /usr/include/features.h /usr/include/x86_64-linux-gnu/sys/cdefs.h \ + /usr/include/x86_64-linux-gnu/bits/wordsize.h \ + /usr/include/x86_64-linux-gnu/gnu/stubs.h \ + /usr/include/x86_64-linux-gnu/gnu/stubs-64.h \ + /usr/include/x86_64-linux-gnu/c++/5/bits/cpu_defines.h \ + /usr/include/c++/5/bits/cpp_type_traits.h \ + /usr/include/c++/5/ext/type_traits.h /usr/include/c++/5/cmath \ + /usr/include/math.h /usr/include/x86_64-linux-gnu/bits/math-vector.h \ + /usr/include/x86_64-linux-gnu/bits/libm-simd-decl-stubs.h \ + /usr/include/x86_64-linux-gnu/bits/huge_val.h \ + /usr/include/x86_64-linux-gnu/bits/huge_valf.h \ + /usr/include/x86_64-linux-gnu/bits/huge_vall.h \ + /usr/include/x86_64-linux-gnu/bits/inf.h \ + /usr/include/x86_64-linux-gnu/bits/nan.h \ + /usr/include/x86_64-linux-gnu/bits/mathdef.h \ + /usr/include/x86_64-linux-gnu/bits/mathcalls.h \ + /usr/include/x86_64-linux-gnu/bits/mathinline.h \ + /usr/include/c++/5/sstream /usr/include/c++/5/istream \ + /usr/include/c++/5/ios /usr/include/c++/5/iosfwd \ + /usr/include/c++/5/bits/stringfwd.h /usr/include/c++/5/bits/memoryfwd.h \ + /usr/include/c++/5/bits/postypes.h /usr/include/c++/5/cwchar \ + /usr/include/wchar.h /usr/include/stdio.h \ + /usr/lib/gcc/x86_64-linux-gnu/5/include/stdarg.h \ + /usr/include/x86_64-linux-gnu/bits/wchar.h \ + /usr/lib/gcc/x86_64-linux-gnu/5/include/stddef.h /usr/include/xlocale.h \ + /usr/include/x86_64-linux-gnu/bits/wchar2.h /usr/include/c++/5/exception \ + /usr/include/c++/5/bits/atomic_lockfree_defines.h \ + /usr/include/c++/5/bits/char_traits.h \ + /usr/include/c++/5/bits/stl_algobase.h \ + /usr/include/c++/5/bits/functexcept.h \ + /usr/include/c++/5/bits/exception_defines.h \ + /usr/include/c++/5/ext/numeric_traits.h \ + /usr/include/c++/5/bits/stl_pair.h /usr/include/c++/5/bits/move.h \ + /usr/include/c++/5/bits/concept_check.h \ + /usr/include/c++/5/bits/stl_iterator_base_types.h \ + /usr/include/c++/5/bits/stl_iterator_base_funcs.h \ + /usr/include/c++/5/debug/debug.h /usr/include/c++/5/bits/stl_iterator.h \ + /usr/include/c++/5/bits/ptr_traits.h \ + /usr/include/c++/5/bits/predefined_ops.h \ + /usr/include/c++/5/bits/localefwd.h \ + /usr/include/x86_64-linux-gnu/c++/5/bits/c++locale.h \ + /usr/include/c++/5/clocale /usr/include/locale.h \ + /usr/include/x86_64-linux-gnu/bits/locale.h /usr/include/c++/5/cctype \ + /usr/include/ctype.h /usr/include/x86_64-linux-gnu/bits/types.h \ + /usr/include/x86_64-linux-gnu/bits/typesizes.h /usr/include/endian.h \ + /usr/include/x86_64-linux-gnu/bits/endian.h \ + /usr/include/x86_64-linux-gnu/bits/byteswap.h \ + /usr/include/x86_64-linux-gnu/bits/byteswap-16.h \ + /usr/include/c++/5/bits/ios_base.h /usr/include/c++/5/ext/atomicity.h \ + /usr/include/x86_64-linux-gnu/c++/5/bits/gthr.h \ + /usr/include/x86_64-linux-gnu/c++/5/bits/gthr-default.h \ + /usr/include/pthread.h /usr/include/sched.h /usr/include/time.h \ + /usr/include/x86_64-linux-gnu/bits/sched.h \ + /usr/include/x86_64-linux-gnu/bits/time.h \ + /usr/include/x86_64-linux-gnu/bits/timex.h \ + /usr/include/x86_64-linux-gnu/bits/pthreadtypes.h \ + /usr/include/x86_64-linux-gnu/bits/setjmp.h \ + /usr/include/x86_64-linux-gnu/c++/5/bits/atomic_word.h \ + /usr/include/c++/5/bits/locale_classes.h /usr/include/c++/5/string \ + /usr/include/c++/5/bits/allocator.h \ + /usr/include/x86_64-linux-gnu/c++/5/bits/c++allocator.h \ + /usr/include/c++/5/ext/new_allocator.h /usr/include/c++/5/new \ + /usr/include/c++/5/bits/ostream_insert.h \ + /usr/include/c++/5/bits/cxxabi_forced.h \ + /usr/include/c++/5/bits/stl_function.h \ + /usr/include/c++/5/backward/binders.h \ + /usr/include/c++/5/bits/range_access.h \ + /usr/include/c++/5/bits/basic_string.h \ + /usr/include/c++/5/ext/alloc_traits.h \ + /usr/include/c++/5/bits/basic_string.tcc \ + /usr/include/c++/5/bits/locale_classes.tcc /usr/include/c++/5/stdexcept \ + /usr/include/c++/5/streambuf /usr/include/c++/5/bits/streambuf.tcc \ + /usr/include/c++/5/bits/basic_ios.h \ + /usr/include/c++/5/bits/locale_facets.h /usr/include/c++/5/cwctype \ + /usr/include/wctype.h \ + /usr/include/x86_64-linux-gnu/c++/5/bits/ctype_base.h \ + /usr/include/c++/5/bits/streambuf_iterator.h \ + /usr/include/x86_64-linux-gnu/c++/5/bits/ctype_inline.h \ + /usr/include/c++/5/bits/locale_facets.tcc \ + /usr/include/c++/5/bits/basic_ios.tcc /usr/include/c++/5/ostream \ + /usr/include/c++/5/bits/ostream.tcc /usr/include/c++/5/bits/istream.tcc \ + /usr/include/c++/5/bits/sstream.tcc \ + ../../modules/elementary_functions/includes/dynlib_elementary_functions.h \ + /usr/include/c++/5/cfloat \ + /usr/lib/gcc/x86_64-linux-gnu/5/include/float.h \ + /usr/include/c++/5/limits + +/usr/include/stdc-predef.h: + +../../modules/core/includes/machine.h: + +src/cpp/faddeeva.h: + +/usr/include/c++/5/complex: + +/usr/include/x86_64-linux-gnu/c++/5/bits/c++config.h: + +/usr/include/x86_64-linux-gnu/c++/5/bits/os_defines.h: + +/usr/include/features.h: + +/usr/include/x86_64-linux-gnu/sys/cdefs.h: + +/usr/include/x86_64-linux-gnu/bits/wordsize.h: + +/usr/include/x86_64-linux-gnu/gnu/stubs.h: + +/usr/include/x86_64-linux-gnu/gnu/stubs-64.h: + +/usr/include/x86_64-linux-gnu/c++/5/bits/cpu_defines.h: + +/usr/include/c++/5/bits/cpp_type_traits.h: + +/usr/include/c++/5/ext/type_traits.h: + +/usr/include/c++/5/cmath: + +/usr/include/math.h: + +/usr/include/x86_64-linux-gnu/bits/math-vector.h: + +/usr/include/x86_64-linux-gnu/bits/libm-simd-decl-stubs.h: + +/usr/include/x86_64-linux-gnu/bits/huge_val.h: + +/usr/include/x86_64-linux-gnu/bits/huge_valf.h: + +/usr/include/x86_64-linux-gnu/bits/huge_vall.h: + +/usr/include/x86_64-linux-gnu/bits/inf.h: + +/usr/include/x86_64-linux-gnu/bits/nan.h: + +/usr/include/x86_64-linux-gnu/bits/mathdef.h: + +/usr/include/x86_64-linux-gnu/bits/mathcalls.h: + +/usr/include/x86_64-linux-gnu/bits/mathinline.h: + +/usr/include/c++/5/sstream: + +/usr/include/c++/5/istream: + +/usr/include/c++/5/ios: + +/usr/include/c++/5/iosfwd: + +/usr/include/c++/5/bits/stringfwd.h: + +/usr/include/c++/5/bits/memoryfwd.h: + +/usr/include/c++/5/bits/postypes.h: + +/usr/include/c++/5/cwchar: + +/usr/include/wchar.h: + +/usr/include/stdio.h: + +/usr/lib/gcc/x86_64-linux-gnu/5/include/stdarg.h: + +/usr/include/x86_64-linux-gnu/bits/wchar.h: + +/usr/lib/gcc/x86_64-linux-gnu/5/include/stddef.h: + +/usr/include/xlocale.h: + +/usr/include/x86_64-linux-gnu/bits/wchar2.h: + +/usr/include/c++/5/exception: + +/usr/include/c++/5/bits/atomic_lockfree_defines.h: + +/usr/include/c++/5/bits/char_traits.h: + +/usr/include/c++/5/bits/stl_algobase.h: + +/usr/include/c++/5/bits/functexcept.h: + +/usr/include/c++/5/bits/exception_defines.h: + +/usr/include/c++/5/ext/numeric_traits.h: + +/usr/include/c++/5/bits/stl_pair.h: + +/usr/include/c++/5/bits/move.h: + +/usr/include/c++/5/bits/concept_check.h: + +/usr/include/c++/5/bits/stl_iterator_base_types.h: + +/usr/include/c++/5/bits/stl_iterator_base_funcs.h: + +/usr/include/c++/5/debug/debug.h: + +/usr/include/c++/5/bits/stl_iterator.h: + +/usr/include/c++/5/bits/ptr_traits.h: + +/usr/include/c++/5/bits/predefined_ops.h: + +/usr/include/c++/5/bits/localefwd.h: + +/usr/include/x86_64-linux-gnu/c++/5/bits/c++locale.h: + +/usr/include/c++/5/clocale: + +/usr/include/locale.h: + +/usr/include/x86_64-linux-gnu/bits/locale.h: + +/usr/include/c++/5/cctype: + +/usr/include/ctype.h: + +/usr/include/x86_64-linux-gnu/bits/types.h: + +/usr/include/x86_64-linux-gnu/bits/typesizes.h: + +/usr/include/endian.h: + +/usr/include/x86_64-linux-gnu/bits/endian.h: + +/usr/include/x86_64-linux-gnu/bits/byteswap.h: + +/usr/include/x86_64-linux-gnu/bits/byteswap-16.h: + +/usr/include/c++/5/bits/ios_base.h: + +/usr/include/c++/5/ext/atomicity.h: + +/usr/include/x86_64-linux-gnu/c++/5/bits/gthr.h: + +/usr/include/x86_64-linux-gnu/c++/5/bits/gthr-default.h: + +/usr/include/pthread.h: + +/usr/include/sched.h: + +/usr/include/time.h: + +/usr/include/x86_64-linux-gnu/bits/sched.h: + +/usr/include/x86_64-linux-gnu/bits/time.h: + +/usr/include/x86_64-linux-gnu/bits/timex.h: + +/usr/include/x86_64-linux-gnu/bits/pthreadtypes.h: + +/usr/include/x86_64-linux-gnu/bits/setjmp.h: + +/usr/include/x86_64-linux-gnu/c++/5/bits/atomic_word.h: + +/usr/include/c++/5/bits/locale_classes.h: + +/usr/include/c++/5/string: + +/usr/include/c++/5/bits/allocator.h: + +/usr/include/x86_64-linux-gnu/c++/5/bits/c++allocator.h: + +/usr/include/c++/5/ext/new_allocator.h: + +/usr/include/c++/5/new: + +/usr/include/c++/5/bits/ostream_insert.h: + +/usr/include/c++/5/bits/cxxabi_forced.h: + +/usr/include/c++/5/bits/stl_function.h: + +/usr/include/c++/5/backward/binders.h: + +/usr/include/c++/5/bits/range_access.h: + +/usr/include/c++/5/bits/basic_string.h: + +/usr/include/c++/5/ext/alloc_traits.h: + +/usr/include/c++/5/bits/basic_string.tcc: + +/usr/include/c++/5/bits/locale_classes.tcc: + +/usr/include/c++/5/stdexcept: + +/usr/include/c++/5/streambuf: + +/usr/include/c++/5/bits/streambuf.tcc: + +/usr/include/c++/5/bits/basic_ios.h: + +/usr/include/c++/5/bits/locale_facets.h: + +/usr/include/c++/5/cwctype: + +/usr/include/wctype.h: + +/usr/include/x86_64-linux-gnu/c++/5/bits/ctype_base.h: + +/usr/include/c++/5/bits/streambuf_iterator.h: + +/usr/include/x86_64-linux-gnu/c++/5/bits/ctype_inline.h: + +/usr/include/c++/5/bits/locale_facets.tcc: + +/usr/include/c++/5/bits/basic_ios.tcc: + +/usr/include/c++/5/ostream: + +/usr/include/c++/5/bits/ostream.tcc: + +/usr/include/c++/5/bits/istream.tcc: + +/usr/include/c++/5/bits/sstream.tcc: + +../../modules/elementary_functions/includes/dynlib_elementary_functions.h: + +/usr/include/c++/5/cfloat: + +/usr/lib/gcc/x86_64-linux-gnu/5/include/float.h: + +/usr/include/c++/5/limits: diff --git a/modules/special_functions/src/cpp/.dirstamp b/modules/special_functions/src/cpp/.dirstamp new file mode 100755 index 000000000..e69de29bb --- /dev/null +++ b/modules/special_functions/src/cpp/.dirstamp diff --git a/modules/special_functions/src/cpp/.libs/libscispecial_functions_algo_la-faddeeva.o b/modules/special_functions/src/cpp/.libs/libscispecial_functions_algo_la-faddeeva.o Binary files differnew file mode 100755 index 000000000..dfb2c2dc6 --- /dev/null +++ b/modules/special_functions/src/cpp/.libs/libscispecial_functions_algo_la-faddeeva.o diff --git a/modules/special_functions/src/cpp/faddeeva.cpp b/modules/special_functions/src/cpp/faddeeva.cpp new file mode 100755 index 000000000..7469bec9d --- /dev/null +++ b/modules/special_functions/src/cpp/faddeeva.cpp @@ -0,0 +1,2895 @@ +// -*- mode:c++; tab-width:2; indent-tabs-mode:nil; -*- + +/* Copyright (c) 2012 Massachusetts Institute of Technology + * + * Permission is hereby granted, free of charge, to any person obtaining + * a copy of this software and associated documentation files (the + * "Software"), to deal in the Software without restriction, including + * without limitation the rights to use, copy, modify, merge, publish, + * distribute, sublicense, and/or sell copies of the Software, and to + * permit persons to whom the Software is furnished to do so, subject to + * the following conditions: + * + * The above copyright notice and this permission notice shall be + * included in all copies or substantial portions of the Software. + * + * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, + * EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF + * MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND + * NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE + * LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION + * OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION + * WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. + */ + +/* (Note that this file can be compiled with either C++, in which + case it uses C++ std::complex<double>, or C, in which case it + uses C99 double complex.) */ + +/* Available at: http://ab-initio.mit.edu/Faddeeva + + Computes various error functions (erf, erfc, erfi, erfcx), + including the Dawson integral, in the complex plane, based + on algorithms for the computation of the Faddeeva function + w(z) = exp(-z^2) * erfc(-i*z). + Given w(z), the error functions are mostly straightforward + to compute, except for certain regions where we have to + switch to Taylor expansions to avoid cancellation errors + [e.g. near the origin for erf(z)]. + + To compute the Faddeeva function, we use a combination of two + algorithms: + + For sufficiently large |z|, we use a continued-fraction expansion + for w(z) similar to those described in: + + Walter Gautschi, "Efficient computation of the complex error + function," SIAM J. Numer. Anal. 7(1), pp. 187-198 (1970) + + G. P. M. Poppe and C. M. J. Wijers, "More efficient computation + of the complex error function," ACM Trans. Math. Soft. 16(1), + pp. 38-46 (1990). + + Unlike those papers, however, we switch to a completely different + algorithm for smaller |z|: + + Mofreh R. Zaghloul and Ahmed N. Ali, "Algorithm 916: Computing the + Faddeyeva and Voigt Functions," ACM Trans. Math. Soft. 38(2), 15 + (2011). + + (I initially used this algorithm for all z, but it turned out to be + significantly slower than the continued-fraction expansion for + larger |z|. On the other hand, it is competitive for smaller |z|, + and is significantly more accurate than the Poppe & Wijers code + in some regions, e.g. in the vicinity of z=1+1i.) + + Note that this is an INDEPENDENT RE-IMPLEMENTATION of these algorithms, + based on the description in the papers ONLY. In particular, I did + not refer to the authors' Fortran or Matlab implementations, respectively, + (which are under restrictive ACM copyright terms and therefore unusable + in free/open-source software). + + Steven G. Johnson, Massachusetts Institute of Technology + http://math.mit.edu/~stevenj + October 2012. + + -- Note that Algorithm 916 assumes that the erfc(x) function, + or rather the scaled function erfcx(x) = exp(x*x)*erfc(x), + is supplied for REAL arguments x. I originally used an + erfcx routine derived from DERFC in SLATEC, but I have + since replaced it with a much faster routine written by + me which uses a combination of continued-fraction expansions + and a lookup table of Chebyshev polynomials. For speed, + I implemented a similar algorithm for Im[w(x)] of real x, + since this comes up frequently in the other error functions. + + A small test program is included the end, which checks + the w(z) etc. results against several known values. To compile + the test function, compile with -DTEST_FADDEEVA (that is, + #define TEST_FADDEEVA). + + If HAVE_CONFIG_H is #defined (e.g. by compiling with -DHAVE_CONFIG_H), + then we #include "config.h", which is assumed to be a GNU autoconf-style + header defining HAVE_* macros to indicate the presence of features. In + particular, if HAVE_ISNAN and HAVE_ISINF are #defined, we use those + functions in math.h instead of defining our own, and if HAVE_ERF and/or + HAVE_ERFC are defined we use those functions from <cmath> for erf and + erfc of real arguments, respectively, instead of defining our own. + + REVISION HISTORY: + 4 October 2012: Initial public release (SGJ) + 5 October 2012: Revised (SGJ) to fix spelling error, + start summation for large x at round(x/a) (> 1) + rather than ceil(x/a) as in the original + paper, which should slightly improve performance + (and, apparently, slightly improves accuracy) + 19 October 2012: Revised (SGJ) to fix bugs for large x, large -y, + and 15<x<26. Performance improvements. Prototype + now supplies default value for relerr. + 24 October 2012: Switch to continued-fraction expansion for + sufficiently large z, for performance reasons. + Also, avoid spurious overflow for |z| > 1e154. + Set relerr argument to min(relerr,0.1). + 27 October 2012: Enhance accuracy in Re[w(z)] taken by itself, + by switching to Alg. 916 in a region near + the real-z axis where continued fractions + have poor relative accuracy in Re[w(z)]. Thanks + to M. Zaghloul for the tip. + 29 October 2012: Replace SLATEC-derived erfcx routine with + completely rewritten code by me, using a very + different algorithm which is much faster. + 30 October 2012: Implemented special-case code for real z + (where real part is exp(-x^2) and imag part is + Dawson integral), using algorithm similar to erfx. + Export ImFaddeeva_w function to make Dawson's + integral directly accessible. + 3 November 2012: Provide implementations of erf, erfc, erfcx, + and Dawson functions in Faddeeva:: namespace, + in addition to Faddeeva::w. Provide header + file Faddeeva.hh. + 4 November 2012: Slightly faster erf for real arguments. + Updated MATLAB and Octave plugins. + 27 November 2012: Support compilation with either C++ or + plain C (using C99 complex numbers). + For real x, use standard-library erf(x) + and erfc(x) if available (for C99 or C++11). + #include "config.h" if HAVE_CONFIG_H is #defined. + 15 December 2012: Portability fixes (copysign, Inf/NaN creation), + use CMPLX/__builtin_complex if available in C, + slight accuracy improvements to erf and dawson + functions near the origin. Use gnulib functions + if GNULIB_NAMESPACE is defined. + 18 December 2012: Slight tweaks (remove recomputation of x*x in Dawson) +*/ + +///////////////////////////////////////////////////////////////////////// +/* If this file is compiled as a part of a larger project, + support using an autoconf-style config.h header file + (with various "HAVE_*" #defines to indicate features) + if HAVE_CONFIG_H is #defined (in GNU autotools style). */ + +#ifdef HAVE_CONFIG_H +# include "machine.h" // CHANGED TO machine.h FOR SCILAB SOURCES +#endif + +///////////////////////////////////////////////////////////////////////// +// macros to allow us to use either C++ or C (with C99 features) + +#ifdef __cplusplus + +# include "faddeeva.h" + +# include <cfloat> +# include <cmath> +# include <limits> +using namespace std; + +// use std::numeric_limits, since 1./0. and 0./0. fail with some compilers (MS) +# define Inf numeric_limits<double>::infinity() +# define NaN numeric_limits<double>::quiet_NaN() + +typedef complex<double> cmplx; + +// Use C-like complex syntax, since the C syntax is more restrictive +# define cexp(z) exp(z) +# define creal(z) real(z) +# define cimag(z) imag(z) +# define cpolar(r,t) polar(r,t) + +# define C(a,b) cmplx(a,b) + +# define FADDEEVA(name) Faddeeva::name +# define FADDEEVA_RE(name) Faddeeva::name + +// isnan/isinf were introduced in C++11 +# if (__cplusplus < 201103L) && (!defined(HAVE_ISNAN) || !defined(HAVE_ISINF)) +static inline bool my_isnan(double x) +{ + return x != x; +} +# define isnan my_isnan +static inline bool my_isinf(double x) +{ + return 1 / x == 0.; +} +# define isinf my_isinf +# elif (__cplusplus >= 201103L) +// g++ gets confused between the C and C++ isnan/isinf functions +# define isnan std::isnan +# define isinf std::isinf +# endif + +// copysign was introduced in C++11 (and is also in POSIX and C99) +# if defined(_WIN32) || defined(__WIN32__) +# define copysign _copysign // of course MS had to be different +# elif defined(GNULIB_NAMESPACE) // we are using using gnulib <cmath> +# define copysign GNULIB_NAMESPACE::copysign +# elif (__cplusplus < 201103L) && !defined(HAVE_COPYSIGN) && !defined(__linux__) && !(defined(__APPLE__) && defined(__MACH__)) && !defined(_AIX) +static inline double my_copysign(double x, double y) +{ + return y < 0 ? -x : x; +} +# define copysign my_copysign +# endif + +// If we are using the gnulib <cmath> (e.g. in the GNU Octave sources), +// gnulib generates a link warning if we use ::floor instead of gnulib::floor. +// This warning is completely innocuous because the only difference between +// gnulib::floor and the system ::floor (and only on ancient OSF systems) +// has to do with floor(-0), which doesn't occur in the usage below, but +// the Octave developers prefer that we silence the warning. +# ifdef GNULIB_NAMESPACE +# define floor GNULIB_NAMESPACE::floor +# endif + +#else // !__cplusplus, i.e. pure C (requires C99 features) + +# include "Faddeeva.h" + +# define _GNU_SOURCE // enable GNU libc NAN extension if possible + +# include <float.h> +# include <math.h> + +typedef double complex cmplx; + +# define FADDEEVA(name) Faddeeva_ ## name +# define FADDEEVA_RE(name) Faddeeva_ ## name ## _re + +/* Constructing complex numbers like 0+i*NaN is problematic in C99 + without the C11 CMPLX macro, because 0.+I*NAN may give NaN+i*NAN if + I is a complex (rather than imaginary) constant. For some reason, + however, it works fine in (pre-4.7) gcc if I define Inf and NaN as + 1/0 and 0/0 (and only if I compile with optimization -O1 or more), + but not if I use the INFINITY or NAN macros. */ + +/* __builtin_complex was introduced in gcc 4.7, but the C11 CMPLX macro + may not be defined unless we are using a recent (2012) version of + glibc and compile with -std=c11... note that icc lies about being + gcc and probably doesn't have this builtin(?), so exclude icc explicitly */ +# if !defined(CMPLX) && (__GNUC__ > 4 || (__GNUC__ == 4 && __GNUC_MINOR__ >= 7)) && !(defined(__ICC) || defined(__INTEL_COMPILER)) +# define CMPLX(a,b) __builtin_complex((double) (a), (double) (b)) +# endif + +# ifdef CMPLX // C11 +# define C(a,b) CMPLX(a,b) +# define Inf INFINITY // C99 infinity +# ifdef NAN // GNU libc extension +# define NaN NAN +# else +# define NaN (0./0.) // NaN +# endif +# else +# define C(a,b) ((a) + I*(b)) +# define Inf (1./0.) +# define NaN (0./0.) +# endif + +static inline cmplx cpolar(double r, double t) +{ + if (r == 0.0 && !isnan(t)) + { + return 0.0; + } + else + { + return C(r * cos(t), r * sin(t)); + } +} + +#endif // !__cplusplus, i.e. pure C (requires C99 features) + +///////////////////////////////////////////////////////////////////////// +// Auxiliary routines to compute other special functions based on w(z) + +// compute erfcx(z) = exp(z^2) erfz(z) +cmplx FADDEEVA(erfcx)(cmplx z, double relerr) +{ + return FADDEEVA(w)(C(-cimag(z), creal(z)), relerr); +} + +// compute the error function erf(x) +double FADDEEVA_RE(erf)(double x) +{ +#if !defined(__cplusplus) + return erf(x); // C99 supplies erf in math.h +#elif (__cplusplus >= 201103L) || defined(HAVE_ERF) + return ::erf(x); // C++11 supplies std::erf in cmath +#else + double mx2 = -x * x; + if (mx2 < -750) // underflow + { + return (x >= 0 ? 1.0 : -1.0); + } + + if (x >= 0) + { + if (x < 8e-2) + { + goto taylor; + } + return 1.0 - exp(mx2) * FADDEEVA_RE(erfcx)(x); + } + else // x < 0 + { + if (x > -8e-2) + { + goto taylor; + } + return exp(mx2) * FADDEEVA_RE(erfcx)(-x) - 1.0; + } + + // Use Taylor series for small |x|, to avoid cancellation inaccuracy + // erf(x) = 2/sqrt(pi) * x * (1 - x^2/3 + x^4/10 - x^6/42 + x^8/216 + ...) +taylor: + return x * (1.1283791670955125739 + + mx2 * (0.37612638903183752464 + + mx2 * (0.11283791670955125739 + + mx2 * (0.026866170645131251760 + + mx2 * 0.0052239776254421878422)))); +#endif +} + +// compute the error function erf(z) +cmplx FADDEEVA(erf)(cmplx z, double relerr) +{ + double x = creal(z), y = cimag(z); + + if (y == 0) + return C(FADDEEVA_RE(erf)(x), + y); // preserve sign of 0 + if (x == 0) // handle separately for speed & handling of y = Inf or NaN + return C(x, // preserve sign of 0 + /* handle y -> Inf limit manually, since + exp(y^2) -> Inf but Im[w(y)] -> 0, so + IEEE will give us a NaN when it should be Inf */ + y * y > 720 ? (y > 0 ? Inf : -Inf) + : exp(y * y) * FADDEEVA(w_im)(y)); + + double mRe_z2 = (y - x) * (x + y); // Re(-z^2), being careful of overflow + double mIm_z2 = -2 * x * y; // Im(-z^2) + if (mRe_z2 < -750) // underflow + { + return (x >= 0 ? 1.0 : -1.0); + } + + /* Handle positive and negative x via different formulas, + using the mirror symmetries of w, to avoid overflow/underflow + problems from multiplying exponentially large and small quantities. */ + if (x >= 0) + { + if (x < 8e-2) + { + if (fabs(y) < 1e-2) + { + goto taylor; + } + else if (fabs(mIm_z2) < 5e-3 && x < 5e-3) + { + goto taylor_erfi; + } + } + /* don't use complex exp function, since that will produce spurious NaN + values when multiplying w in an overflow situation. */ + return 1.0 - exp(mRe_z2) * + (C(cos(mIm_z2), sin(mIm_z2)) + * FADDEEVA(w)(C(-y, x), relerr)); + } + else // x < 0 + { + if (x > -8e-2) // duplicate from above to avoid fabs(x) call + { + if (fabs(y) < 1e-2) + { + goto taylor; + } + else if (fabs(mIm_z2) < 5e-3 && x > -5e-3) + { + goto taylor_erfi; + } + } + else if (isnan(x)) + { + return C(NaN, y == 0 ? 0 : NaN); + } + /* don't use complex exp function, since that will produce spurious NaN + values when multiplying w in an overflow situation. */ + return exp(mRe_z2) * + (C(cos(mIm_z2), sin(mIm_z2)) + * FADDEEVA(w)(C(y, -x), relerr)) - 1.0; + } + + // Use Taylor series for small |z|, to avoid cancellation inaccuracy + // erf(z) = 2/sqrt(pi) * z * (1 - z^2/3 + z^4/10 - z^6/42 + z^8/216 + ...) +taylor: + { + cmplx mz2 = C(mRe_z2, mIm_z2); // -z^2 + return z * (1.1283791670955125739 + + mz2 * (0.37612638903183752464 + + mz2 * (0.11283791670955125739 + + mz2 * (0.026866170645131251760 + + mz2 * 0.0052239776254421878422)))); + } + + /* for small |x| and small |xy|, + use Taylor series to avoid cancellation inaccuracy: + erf(x+iy) = erf(iy) + + 2*exp(y^2)/sqrt(pi) * + [ x * (1 - x^2 * (1+2y^2)/3 + x^4 * (3+12y^2+4y^4)/30 + ... + - i * x^2 * y * (1 - x^2 * (3+2y^2)/6 + ...) ] + where: + erf(iy) = exp(y^2) * Im[w(y)] + */ +taylor_erfi: + { + double x2 = x * x, y2 = y * y; + double expy2 = exp(y2); + return C + (expy2 * x * (1.1283791670955125739 + - x2 * (0.37612638903183752464 + + 0.75225277806367504925 * y2) + + x2 * x2 * (0.11283791670955125739 + + y2 * (0.45135166683820502956 + + 0.15045055561273500986 * y2))), + expy2 * (FADDEEVA(w_im)(y) + - x2 * y * (1.1283791670955125739 + - x2 * (0.56418958354775628695 + + 0.37612638903183752464 * y2)))); + } +} + +// erfi(z) = -i erf(iz) +cmplx FADDEEVA(erfi)(cmplx z, double relerr) +{ + cmplx e = FADDEEVA(erf)(C(-cimag(z), creal(z)), relerr); + return C(cimag(e), -creal(e)); +} + +// erfi(x) = -i erf(ix) +double FADDEEVA_RE(erfi)(double x) +{ + return x * x > 720 ? (x > 0 ? Inf : -Inf) + : exp(x * x) * FADDEEVA(w_im)(x); +} + +// erfc(x) = 1 - erf(x) +double FADDEEVA_RE(erfc)(double x) +{ +#if !defined(__cplusplus) + return erfc(x); // C99 supplies erfc in math.h +#elif (__cplusplus >= 201103L) || defined(HAVE_ERFC) + return ::erfc(x); // C++11 supplies std::erfc in cmath +#else + if (x * x > 750) // underflow + { + return (x >= 0 ? 0.0 : 2.0); + } + return x >= 0 ? exp(-x * x) * FADDEEVA_RE(erfcx)(x) + : 2. - exp(-x * x) * FADDEEVA_RE(erfcx)(-x); +#endif +} + +// erfc(z) = 1 - erf(z) +cmplx FADDEEVA(erfc)(cmplx z, double relerr) +{ + double x = creal(z), y = cimag(z); + + if (x == 0.) + return C(1, + /* handle y -> Inf limit manually, since + exp(y^2) -> Inf but Im[w(y)] -> 0, so + IEEE will give us a NaN when it should be Inf */ + y * y > 720 ? (y > 0 ? -Inf : Inf) + : -exp(y * y) * FADDEEVA(w_im)(y)); + if (y == 0.) + { + if (x * x > 750) // underflow + return C(x >= 0 ? 0.0 : 2.0, + -y); // preserve sign of 0 + return C(x >= 0 ? exp(-x * x) * FADDEEVA_RE(erfcx)(x) + : 2. - exp(-x * x) * FADDEEVA_RE(erfcx)(-x), + -y); // preserve sign of zero + } + + double mRe_z2 = (y - x) * (x + y); // Re(-z^2), being careful of overflow + double mIm_z2 = -2 * x * y; // Im(-z^2) + if (mRe_z2 < -750) // underflow + { + return (x >= 0 ? 0.0 : 2.0); + } + + if (x >= 0) + return cexp(C(mRe_z2, mIm_z2)) + * FADDEEVA(w)(C(-y, x), relerr); + else + return 2.0 - cexp(C(mRe_z2, mIm_z2)) + * FADDEEVA(w)(C(y, -x), relerr); +} + +// compute Dawson(x) = sqrt(pi)/2 * exp(-x^2) * erfi(x) +double FADDEEVA_RE(Dawson)(double x) +{ + const double spi2 = 0.8862269254527580136490837416705725913990; // sqrt(pi)/2 + return spi2 * FADDEEVA(w_im)(x); +} + +// compute Dawson(z) = sqrt(pi)/2 * exp(-z^2) * erfi(z) +cmplx FADDEEVA(Dawson)(cmplx z, double relerr) +{ + const double spi2 = 0.8862269254527580136490837416705725913990; // sqrt(pi)/2 + double x = creal(z), y = cimag(z); + + // handle axes separately for speed & proper handling of x or y = Inf or NaN + if (y == 0) + return C(spi2 * FADDEEVA(w_im)(x), + -y); // preserve sign of 0 + if (x == 0) + { + double y2 = y * y; + if (y2 < 2.5e-5) // Taylor expansion + { + return C(x, // preserve sign of 0 + y * (1. + + y2 * (0.6666666666666666666666666666666666666667 + + y2 * 0.26666666666666666666666666666666666667))); + } + return C(x, // preserve sign of 0 + spi2 * (y >= 0 + ? exp(y2) - FADDEEVA_RE(erfcx)(y) + : FADDEEVA_RE(erfcx)(-y) - exp(y2))); + } + + double mRe_z2 = (y - x) * (x + y); // Re(-z^2), being careful of overflow + double mIm_z2 = -2 * x * y; // Im(-z^2) + cmplx mz2 = C(mRe_z2, mIm_z2); // -z^2 + + /* Handle positive and negative x via different formulas, + using the mirror symmetries of w, to avoid overflow/underflow + problems from multiplying exponentially large and small quantities. */ + if (y >= 0) + { + if (y < 5e-3) + { + if (fabs(x) < 5e-3) + { + goto taylor; + } + else if (fabs(mIm_z2) < 5e-3) + { + goto taylor_realaxis; + } + } + cmplx res = cexp(mz2) - FADDEEVA(w)(z, relerr); + return spi2 * C(-cimag(res), creal(res)); + } + else // y < 0 + { + if (y > -5e-3) // duplicate from above to avoid fabs(x) call + { + if (fabs(x) < 5e-3) + { + goto taylor; + } + else if (fabs(mIm_z2) < 5e-3) + { + goto taylor_realaxis; + } + } + else if (isnan(y)) + { + return C(x == 0 ? 0 : NaN, NaN); + } + cmplx res = FADDEEVA(w)(-z, relerr) - cexp(mz2); + return spi2 * C(-cimag(res), creal(res)); + } + + // Use Taylor series for small |z|, to avoid cancellation inaccuracy + // dawson(z) = z - 2/3 z^3 + 4/15 z^5 + ... +taylor: + return z * (1. + + mz2 * (0.6666666666666666666666666666666666666667 + + mz2 * 0.2666666666666666666666666666666666666667)); + + /* for small |y| and small |xy|, + use Taylor series to avoid cancellation inaccuracy: + dawson(x + iy) + = D + y^2 (D + x - 2Dx^2) + + y^4 (D/2 + 5x/6 - 2Dx^2 - x^3/3 + 2Dx^4/3) + + iy [ (1-2Dx) + 2/3 y^2 (1 - 3Dx - x^2 + 2Dx^3) + + y^4/15 (4 - 15Dx - 9x^2 + 20Dx^3 + 2x^4 - 4Dx^5) ] + ... + where D = dawson(x) + + However, for large |x|, 2Dx -> 1 which gives cancellation problems in + this series (many of the leading terms cancel). So, for large |x|, + we need to substitute a continued-fraction expansion for D. + + dawson(x) = 0.5 / (x-0.5/(x-1/(x-1.5/(x-2/(x-2.5/(x...)))))) + + The 6 terms shown here seems to be the minimum needed to be + accurate as soon as the simpler Taylor expansion above starts + breaking down. Using this 6-term expansion, factoring out the + denominator, and simplifying with Maple, we obtain: + + Re dawson(x + iy) * (-15 + 90x^2 - 60x^4 + 8x^6) / x + = 33 - 28x^2 + 4x^4 + y^2 (18 - 4x^2) + 4 y^4 + Im dawson(x + iy) * (-15 + 90x^2 - 60x^4 + 8x^6) / y + = -15 + 24x^2 - 4x^4 + 2/3 y^2 (6x^2 - 15) - 4 y^4 + + Finally, for |x| > 5e7, we can use a simpler 1-term continued-fraction + expansion for the real part, and a 2-term expansion for the imaginary + part. (This avoids overflow problems for huge |x|.) This yields: + + Re dawson(x + iy) = [1 + y^2 (1 + y^2/2 - (xy)^2/3)] / (2x) + Im dawson(x + iy) = y [ -1 - 2/3 y^2 + y^4/15 (2x^2 - 4) ] / (2x^2 - 1) + + */ +taylor_realaxis: + { + double x2 = x * x; + if (x2 > 1600) // |x| > 40 + { + double y2 = y * y; + if (x2 > 25e14) // |x| > 5e7 + { + double xy2 = (x * y) * (x * y); + return C((0.5 + y2 * (0.5 + 0.25 * y2 + - 0.16666666666666666667 * xy2)) / x, + y * (-1 + y2 * (-0.66666666666666666667 + + 0.13333333333333333333 * xy2 + - 0.26666666666666666667 * y2)) + / (2 * x2 - 1)); + } + return (1. / (-15 + x2 * (90 + x2 * (-60 + 8 * x2)))) * + C(x * (33 + x2 * (-28 + 4 * x2) + + y2 * (18 - 4 * x2 + 4 * y2)), + y * (-15 + x2 * (24 - 4 * x2) + + y2 * (4 * x2 - 10 - 4 * y2))); + } + else + { + double D = spi2 * FADDEEVA(w_im)(x); + double y2 = y * y; + return C + (D + y2 * (D + x - 2 * D * x2) + + y2 * y2 * (D * (0.5 - x2 * (2 - 0.66666666666666666667 * x2)) + + x * (0.83333333333333333333 + - 0.33333333333333333333 * x2)), + y * (1 - 2 * D * x + + y2 * 0.66666666666666666667 * (1 - x2 - D * x * (3 - 2 * x2)) + + y2 * y2 * (0.26666666666666666667 - + x2 * (0.6 - 0.13333333333333333333 * x2) + - D * x * (1 - x2 * (1.3333333333333333333 + - 0.26666666666666666667 * x2))))); + } + } +} + +///////////////////////////////////////////////////////////////////////// + +// return sinc(x) = sin(x)/x, given both x and sin(x) +// [since we only use this in cases where sin(x) has already been computed] +static inline double sinc(double x, double sinx) +{ + return fabs(x) < 1e-4 ? 1 - (0.1666666666666666666667) * x * x : sinx / x; +} + +// sinh(x) via Taylor series, accurate to machine precision for |x| < 1e-2 +static inline double sinh_taylor(double x) +{ + return x * (1 + (x * x) * (0.1666666666666666666667 + + 0.00833333333333333333333 * (x * x))); +} + +static inline double sqr(double x) +{ + return x * x; +} + +// precomputed table of expa2n2[n-1] = exp(-a2*n*n) +// for double-precision a2 = 0.26865... in FADDEEVA(w), below. +static const double expa2n2[] = +{ + 7.64405281671221563e-01, + 3.41424527166548425e-01, + 8.91072646929412548e-02, + 1.35887299055460086e-02, + 1.21085455253437481e-03, + 6.30452613933449404e-05, + 1.91805156577114683e-06, + 3.40969447714832381e-08, + 3.54175089099469393e-10, + 2.14965079583260682e-12, + 7.62368911833724354e-15, + 1.57982797110681093e-17, + 1.91294189103582677e-20, + 1.35344656764205340e-23, + 5.59535712428588720e-27, + 1.35164257972401769e-30, + 1.90784582843501167e-34, + 1.57351920291442930e-38, + 7.58312432328032845e-43, + 2.13536275438697082e-47, + 3.51352063787195769e-52, + 3.37800830266396920e-57, + 1.89769439468301000e-62, + 6.22929926072668851e-68, + 1.19481172006938722e-73, + 1.33908181133005953e-79, + 8.76924303483223939e-86, + 3.35555576166254986e-92, + 7.50264110688173024e-99, + 9.80192200745410268e-106, + 7.48265412822268959e-113, + 3.33770122566809425e-120, + 8.69934598159861140e-128, + 1.32486951484088852e-135, + 1.17898144201315253e-143, + 6.13039120236180012e-152, + 1.86258785950822098e-160, + 3.30668408201432783e-169, + 3.43017280887946235e-178, + 2.07915397775808219e-187, + 7.36384545323984966e-197, + 1.52394760394085741e-206, + 1.84281935046532100e-216, + 1.30209553802992923e-226, + 5.37588903521080531e-237, + 1.29689584599763145e-247, + 1.82813078022866562e-258, + 1.50576355348684241e-269, + 7.24692320799294194e-281, + 2.03797051314726829e-292, + 3.34880215927873807e-304, + 0.0 // underflow (also prevents reads past array end, below) +}; + +///////////////////////////////////////////////////////////////////////// + +cmplx FADDEEVA(w)(cmplx z, double relerr) +{ + if (creal(z) == 0.0) + return C(FADDEEVA_RE(erfcx)(cimag(z)), + creal(z)); // give correct sign of 0 in cimag(w) + else if (cimag(z) == 0) + return C(exp(-sqr(creal(z))), + FADDEEVA(w_im)(creal(z))); + + double a, a2, c; + if (relerr <= DBL_EPSILON) + { + relerr = DBL_EPSILON; + a = 0.518321480430085929872; // pi / sqrt(-log(eps*0.5)) + c = 0.329973702884629072537; // (2/pi) * a; + a2 = 0.268657157075235951582; // a^2 + } + else + { + const double pi = 3.14159265358979323846264338327950288419716939937510582; + if (relerr > 0.1) + { + relerr = 0.1; // not sensible to compute < 1 digit + } + a = pi / sqrt(-log(relerr * 0.5)); + c = (2 / pi) * a; + a2 = a * a; + } + const double x = fabs(creal(z)); + const double y = cimag(z), ya = fabs(y); + + cmplx ret = 0.; // return value + + double sum1 = 0, sum2 = 0, sum3 = 0, sum4 = 0, sum5 = 0; + +#define USE_CONTINUED_FRACTION 1 // 1 to use continued fraction for large |z| + +#if USE_CONTINUED_FRACTION + if (ya > 7 || (x > 6 // continued fraction is faster + /* As pointed out by M. Zaghloul, the continued + fraction seems to give a large relative error in + Re w(z) for |x| ~ 6 and small |y|, so use + algorithm 816 in this region: */ + && (ya > 0.1 || (x > 8 && ya > 1e-10) || x > 28))) + { + + /* Poppe & Wijers suggest using a number of terms + nu = 3 + 1442 / (26*rho + 77) + where rho = sqrt((x/x0)^2 + (y/y0)^2) where x0=6.3, y0=4.4. + (They only use this expansion for rho >= 1, but rho a little less + than 1 seems okay too.) + Instead, I did my own fit to a slightly different function + that avoids the hypotenuse calculation, using NLopt to minimize + the sum of the squares of the errors in nu with the constraint + that the estimated nu be >= minimum nu to attain machine precision. + I also separate the regions where nu == 2 and nu == 1. */ + const double ispi = 0.56418958354775628694807945156; // 1 / sqrt(pi) + double xs = y < 0 ? -creal(z) : creal(z); // compute for -z if y < 0 + if (x + ya > 4000) // nu <= 2 + { + if (x + ya > 1e7) // nu == 1, w(z) = i/sqrt(pi) / z + { + // scale to avoid overflow + if (x > ya) + { + double yax = ya / xs; + double denom = ispi / (xs + yax * ya); + ret = C(denom * yax, denom); + } + else if (isinf(ya)) + return ((isnan(x) || y < 0) + ? C(NaN, NaN) : C(0, 0)); + else + { + double xya = xs / ya; + double denom = ispi / (xya * xs + ya); + ret = C(denom, denom * xya); + } + } + else // nu == 2, w(z) = i/sqrt(pi) * z / (z*z - 0.5) + { + double dr = xs * xs - ya * ya - 0.5, di = 2 * xs * ya; + double denom = ispi / (dr * dr + di * di); + ret = C(denom * (xs * di - ya * dr), denom * (xs * dr + ya * di)); + } + } + else // compute nu(z) estimate and do general continued fraction + { + const double c0 = 3.9, c1 = 11.398, c2 = 0.08254, c3 = 0.1421, c4 = 0.2023; // fit + double nu = floor(c0 + c1 / (c2 * x + c3 * ya + c4)); + double wr = xs, wi = ya; + for (nu = 0.5 * (nu - 1); nu > 0.4; nu -= 0.5) + { + // w <- z - nu/w: + double denom = nu / (wr * wr + wi * wi); + wr = xs - wr * denom; + wi = ya + wi * denom; + } + { + // w(z) = i/sqrt(pi) / w: + double denom = ispi / (wr * wr + wi * wi); + ret = C(denom * wi, denom * wr); + } + } + if (y < 0) + { + // use w(z) = 2.0*exp(-z*z) - w(-z), + // but be careful of overflow in exp(-z*z) + // = exp(-(xs*xs-ya*ya) -2*i*xs*ya) + return 2.0 * cexp(C((ya - xs) * (xs + ya), 2 * xs * y)) - ret; + } + else + { + return ret; + } + } +#else // !USE_CONTINUED_FRACTION + if (x + ya > 1e7) // w(z) = i/sqrt(pi) / z, to machine precision + { + const double ispi = 0.56418958354775628694807945156; // 1 / sqrt(pi) + double xs = y < 0 ? -creal(z) : creal(z); // compute for -z if y < 0 + // scale to avoid overflow + if (x > ya) + { + double yax = ya / xs; + double denom = ispi / (xs + yax * ya); + ret = C(denom * yax, denom); + } + else + { + double xya = xs / ya; + double denom = ispi / (xya * xs + ya); + ret = C(denom, denom * xya); + } + if (y < 0) + { + // use w(z) = 2.0*exp(-z*z) - w(-z), + // but be careful of overflow in exp(-z*z) + // = exp(-(xs*xs-ya*ya) -2*i*xs*ya) + return 2.0 * cexp(C((ya - xs) * (xs + ya), 2 * xs * y)) - ret; + } + else + { + return ret; + } + } +#endif // !USE_CONTINUED_FRACTION + + /* Note: The test that seems to be suggested in the paper is x < + sqrt(-log(DBL_MIN)), about 26.6, since otherwise exp(-x^2) + underflows to zero and sum1,sum2,sum4 are zero. However, long + before this occurs, the sum1,sum2,sum4 contributions are + negligible in double precision; I find that this happens for x > + about 6, for all y. On the other hand, I find that the case + where we compute all of the sums is faster (at least with the + precomputed expa2n2 table) until about x=10. Furthermore, if we + try to compute all of the sums for x > 20, I find that we + sometimes run into numerical problems because underflow/overflow + problems start to appear in the various coefficients of the sums, + below. Therefore, we use x < 10 here. */ + else if (x < 10) + { + double prod2ax = 1, prodm2ax = 1; + double expx2; + + if (isnan(y)) + { + return C(y, y); + } + + /* Somewhat ugly copy-and-paste duplication here, but I see significant + speedups from using the special-case code with the precomputed + exponential, and the x < 5e-4 special case is needed for accuracy. */ + + if (relerr == DBL_EPSILON) // use precomputed exp(-a2*(n*n)) table + { + if (x < 5e-4) // compute sum4 and sum5 together as sum5-sum4 + { + const double x2 = x * x; + expx2 = 1 - x2 * (1 - 0.5 * x2); // exp(-x*x) via Taylor + // compute exp(2*a*x) and exp(-2*a*x) via Taylor, to double precision + const double ax2 = 1.036642960860171859744 * x; // 2*a*x + const double exp2ax = + 1 + ax2 * (1 + ax2 * (0.5 + 0.166666666666666666667 * ax2)); + const double expm2ax = + 1 - ax2 * (1 - ax2 * (0.5 - 0.166666666666666666667 * ax2)); + for (int n = 1; 1; ++n) + { + const double coef = expa2n2[n - 1] * expx2 / (a2 * (n * n) + y * y); + prod2ax *= exp2ax; + prodm2ax *= expm2ax; + sum1 += coef; + sum2 += coef * prodm2ax; + sum3 += coef * prod2ax; + + // really = sum5 - sum4 + sum5 += coef * (2 * a) * n * sinh_taylor((2 * a) * n * x); + + // test convergence via sum3 + if (coef * prod2ax < relerr * sum3) + { + break; + } + } + } + else // x > 5e-4, compute sum4 and sum5 separately + { + expx2 = exp(-x * x); + const double exp2ax = exp((2 * a) * x), expm2ax = 1 / exp2ax; + for (int n = 1; 1; ++n) + { + const double coef = expa2n2[n - 1] * expx2 / (a2 * (n * n) + y * y); + prod2ax *= exp2ax; + prodm2ax *= expm2ax; + sum1 += coef; + sum2 += coef * prodm2ax; + sum4 += (coef * prodm2ax) * (a * n); + sum3 += coef * prod2ax; + sum5 += (coef * prod2ax) * (a * n); + // test convergence via sum5, since this sum has the slowest decay + if ((coef * prod2ax) * (a * n) < relerr * sum5) + { + break; + } + } + } + } + else // relerr != DBL_EPSILON, compute exp(-a2*(n*n)) on the fly + { + const double exp2ax = exp((2 * a) * x), expm2ax = 1 / exp2ax; + if (x < 5e-4) // compute sum4 and sum5 together as sum5-sum4 + { + const double x2 = x * x; + expx2 = 1 - x2 * (1 - 0.5 * x2); // exp(-x*x) via Taylor + for (int n = 1; 1; ++n) + { + const double coef = exp(-a2 * (n * n)) * expx2 / (a2 * (n * n) + y * y); + prod2ax *= exp2ax; + prodm2ax *= expm2ax; + sum1 += coef; + sum2 += coef * prodm2ax; + sum3 += coef * prod2ax; + + // really = sum5 - sum4 + sum5 += coef * (2 * a) * n * sinh_taylor((2 * a) * n * x); + + // test convergence via sum3 + if (coef * prod2ax < relerr * sum3) + { + break; + } + } + } + else // x > 5e-4, compute sum4 and sum5 separately + { + expx2 = exp(-x * x); + for (int n = 1; 1; ++n) + { + const double coef = exp(-a2 * (n * n)) * expx2 / (a2 * (n * n) + y * y); + prod2ax *= exp2ax; + prodm2ax *= expm2ax; + sum1 += coef; + sum2 += coef * prodm2ax; + sum4 += (coef * prodm2ax) * (a * n); + sum3 += coef * prod2ax; + sum5 += (coef * prod2ax) * (a * n); + // test convergence via sum5, since this sum has the slowest decay + if ((coef * prod2ax) * (a * n) < relerr * sum5) + { + break; + } + } + } + } + const double expx2erfcxy = // avoid spurious overflow for large negative y + y > -6 // for y < -6, erfcx(y) = 2*exp(y*y) to double precision + ? expx2 * FADDEEVA_RE(erfcx)(y) : 2 * exp(y * y - x * x); + if (y > 5) // imaginary terms cancel + { + const double sinxy = sin(x * y); + ret = (expx2erfcxy - c * y * sum1) * cos(2 * x * y) + + (c * x * expx2) * sinxy * sinc(x * y, sinxy); + } + else + { + double xs = creal(z); + const double sinxy = sin(xs * y); + const double sin2xy = sin(2 * xs * y), cos2xy = cos(2 * xs * y); + const double coef1 = expx2erfcxy - c * y * sum1; + const double coef2 = c * xs * expx2; + ret = C(coef1 * cos2xy + coef2 * sinxy * sinc(xs * y, sinxy), + coef2 * sinc(2 * xs * y, sin2xy) - coef1 * sin2xy); + } + } + else // x large: only sum3 & sum5 contribute (see above note) + { + if (isnan(x)) + { + return C(x, x); + } + if (isnan(y)) + { + return C(y, y); + } + +#if USE_CONTINUED_FRACTION + ret = exp(-x * x); // |y| < 1e-10, so we only need exp(-x*x) term +#else + if (y < 0) + { + /* erfcx(y) ~ 2*exp(y*y) + (< 1) if y < 0, so + erfcx(y)*exp(-x*x) ~ 2*exp(y*y-x*x) term may not be negligible + if y*y - x*x > -36 or so. So, compute this term just in case. + We also need the -exp(-x*x) term to compute Re[w] accurately + in the case where y is very small. */ + ret = cpolar(2 * exp(y * y - x * x) - exp(-x * x), -2 * creal(z) * y); + } + else + { + ret = exp(-x * x); // not negligible in real part if y very small + } +#endif + // (round instead of ceil as in original paper; note that x/a > 1 here) + double n0 = floor(x / a + 0.5); // sum in both directions, starting at n0 + double dx = a * n0 - x; + sum3 = exp(-dx * dx) / (a2 * (n0 * n0) + y * y); + sum5 = a * n0 * sum3; + double exp1 = exp(4 * a * dx), exp1dn = 1; + int dn; + for (dn = 1; n0 - dn > 0; ++dn) // loop over n0-dn and n0+dn terms + { + double np = n0 + dn, nm = n0 - dn; + double tp = exp(-sqr(a * dn + dx)); + double tm = tp * (exp1dn *= exp1); // trick to get tm from tp + tp /= (a2 * (np * np) + y * y); + tm /= (a2 * (nm * nm) + y * y); + sum3 += tp + tm; + sum5 += a * (np * tp + nm * tm); + if (a * (np * tp + nm * tm) < relerr * sum5) + { + goto finish; + } + } + while (1) // loop over n0+dn terms only (since n0-dn <= 0) + { + double np = n0 + dn++; + double tp = exp(-sqr(a * dn + dx)) / (a2 * (np * np) + y * y); + sum3 += tp; + sum5 += a * np * tp; + if (a * np * tp < relerr * sum5) + { + goto finish; + } + } + } +finish: + return ret + C((0.5 * c) * y * (sum2 + sum3), + (0.5 * c) * copysign(sum5 - sum4, creal(z))); +} + +///////////////////////////////////////////////////////////////////////// + +/* erfcx(x) = exp(x^2) erfc(x) function, for real x, written by + Steven G. Johnson, October 2012. + + This function combines a few different ideas. + + First, for x > 50, it uses a continued-fraction expansion (same as + for the Faddeeva function, but with algebraic simplifications for z=i*x). + + Second, for 0 <= x <= 50, it uses Chebyshev polynomial approximations, + but with two twists: + + a) It maps x to y = 4 / (4+x) in [0,1]. This simple transformation, + inspired by a similar transformation in the octave-forge/specfun + erfcx by Soren Hauberg, results in much faster Chebyshev convergence + than other simple transformations I have examined. + + b) Instead of using a single Chebyshev polynomial for the entire + [0,1] y interval, we break the interval up into 100 equal + subintervals, with a switch/lookup table, and use much lower + degree Chebyshev polynomials in each subinterval. This greatly + improves performance in my tests. + + For x < 0, we use the relationship erfcx(-x) = 2 exp(x^2) - erfc(x), + with the usual checks for overflow etcetera. + + Performance-wise, it seems to be substantially faster than either + the SLATEC DERFC function [or an erfcx function derived therefrom] + or Cody's CALERF function (from netlib.org/specfun), while + retaining near machine precision in accuracy. */ + +/* Given y100=100*y, where y = 4/(4+x) for x >= 0, compute erfc(x). + + Uses a look-up table of 100 different Chebyshev polynomials + for y intervals [0,0.01], [0.01,0.02], ...., [0.99,1], generated + with the help of Maple and a little shell script. This allows + the Chebyshev polynomials to be of significantly lower degree (about 1/4) + compared to fitting the whole [0,1] interval with a single polynomial. */ +static double erfcx_y100(double y100) +{ + switch ((int) y100) + { + case 0: + { + double t = 2 * y100 - 1; + return 0.70878032454106438663e-3 + (0.71234091047026302958e-3 + (0.35779077297597742384e-5 + (0.17403143962587937815e-7 + (0.81710660047307788845e-10 + (0.36885022360434957634e-12 + 0.15917038551111111111e-14 * t) * t) * t) * t) * t) * t; + } + case 1: + { + double t = 2 * y100 - 3; + return 0.21479143208285144230e-2 + (0.72686402367379996033e-3 + (0.36843175430938995552e-5 + (0.18071841272149201685e-7 + (0.85496449296040325555e-10 + (0.38852037518534291510e-12 + 0.16868473576888888889e-14 * t) * t) * t) * t) * t) * t; + } + case 2: + { + double t = 2 * y100 - 5; + return 0.36165255935630175090e-2 + (0.74182092323555510862e-3 + (0.37948319957528242260e-5 + (0.18771627021793087350e-7 + (0.89484715122415089123e-10 + (0.40935858517772440862e-12 + 0.17872061464888888889e-14 * t) * t) * t) * t) * t) * t; + } + case 3: + { + double t = 2 * y100 - 7; + return 0.51154983860031979264e-2 + (0.75722840734791660540e-3 + (0.39096425726735703941e-5 + (0.19504168704300468210e-7 + (0.93687503063178993915e-10 + (0.43143925959079664747e-12 + 0.18939926435555555556e-14 * t) * t) * t) * t) * t) * t; + } + case 4: + { + double t = 2 * y100 - 9; + return 0.66457513172673049824e-2 + (0.77310406054447454920e-3 + (0.40289510589399439385e-5 + (0.20271233238288381092e-7 + (0.98117631321709100264e-10 + (0.45484207406017752971e-12 + 0.20076352213333333333e-14 * t) * t) * t) * t) * t) * t; + } + case 5: + { + double t = 2 * y100 - 11; + return 0.82082389970241207883e-2 + (0.78946629611881710721e-3 + (0.41529701552622656574e-5 + (0.21074693344544655714e-7 + (0.10278874108587317989e-9 + (0.47965201390613339638e-12 + 0.21285907413333333333e-14 * t) * t) * t) * t) * t) * t; + } + case 6: + { + double t = 2 * y100 - 13; + return 0.98039537275352193165e-2 + (0.80633440108342840956e-3 + (0.42819241329736982942e-5 + (0.21916534346907168612e-7 + (0.10771535136565470914e-9 + (0.50595972623692822410e-12 + 0.22573462684444444444e-14 * t) * t) * t) * t) * t) * t; + } + case 7: + { + double t = 2 * y100 - 15; + return 0.11433927298290302370e-1 + (0.82372858383196561209e-3 + (0.44160495311765438816e-5 + (0.22798861426211986056e-7 + (0.11291291745879239736e-9 + (0.53386189365816880454e-12 + 0.23944209546666666667e-14 * t) * t) * t) * t) * t) * t; + } + case 8: + { + double t = 2 * y100 - 17; + return 0.13099232878814653979e-1 + (0.84167002467906968214e-3 + (0.45555958988457506002e-5 + (0.23723907357214175198e-7 + (0.11839789326602695603e-9 + (0.56346163067550237877e-12 + 0.25403679644444444444e-14 * t) * t) * t) * t) * t) * t; + } + case 9: + { + double t = 2 * y100 - 19; + return 0.14800987015587535621e-1 + (0.86018092946345943214e-3 + (0.47008265848816866105e-5 + (0.24694040760197315333e-7 + (0.12418779768752299093e-9 + (0.59486890370320261949e-12 + 0.26957764568888888889e-14 * t) * t) * t) * t) * t) * t; + } + case 10: + { + double t = 2 * y100 - 21; + return 0.16540351739394069380e-1 + (0.87928458641241463952e-3 + (0.48520195793001753903e-5 + (0.25711774900881709176e-7 + (0.13030128534230822419e-9 + (0.62820097586874779402e-12 + 0.28612737351111111111e-14 * t) * t) * t) * t) * t) * t; + } + case 11: + { + double t = 2 * y100 - 23; + return 0.18318536789842392647e-1 + (0.89900542647891721692e-3 + (0.50094684089553365810e-5 + (0.26779777074218070482e-7 + (0.13675822186304615566e-9 + (0.66358287745352705725e-12 + 0.30375273884444444444e-14 * t) * t) * t) * t) * t) * t; + } + case 12: + { + double t = 2 * y100 - 25; + return 0.20136801964214276775e-1 + (0.91936908737673676012e-3 + (0.51734830914104276820e-5 + (0.27900878609710432673e-7 + (0.14357976402809042257e-9 + (0.70114790311043728387e-12 + 0.32252476000000000000e-14 * t) * t) * t) * t) * t) * t; + } + case 13: + { + double t = 2 * y100 - 27; + return 0.21996459598282740954e-1 + (0.94040248155366777784e-3 + (0.53443911508041164739e-5 + (0.29078085538049374673e-7 + (0.15078844500329731137e-9 + (0.74103813647499204269e-12 + 0.34251892320000000000e-14 * t) * t) * t) * t) * t) * t; + } + case 14: + { + double t = 2 * y100 - 29; + return 0.23898877187226319502e-1 + (0.96213386835900177540e-3 + (0.55225386998049012752e-5 + (0.30314589961047687059e-7 + (0.15840826497296335264e-9 + (0.78340500472414454395e-12 + 0.36381553564444444445e-14 * t) * t) * t) * t) * t) * t; + } + case 15: + { + double t = 2 * y100 - 31; + return 0.25845480155298518485e-1 + (0.98459293067820123389e-3 + (0.57082915920051843672e-5 + (0.31613782169164830118e-7 + (0.16646478745529630813e-9 + (0.82840985928785407942e-12 + 0.38649975768888888890e-14 * t) * t) * t) * t) * t) * t; + } + case 16: + { + double t = 2 * y100 - 33; + return 0.27837754783474696598e-1 + (0.10078108563256892757e-2 + (0.59020366493792212221e-5 + (0.32979263553246520417e-7 + (0.17498524159268458073e-9 + (0.87622459124842525110e-12 + 0.41066206488888888890e-14 * t) * t) * t) * t) * t) * t; + } + case 17: + { + double t = 2 * y100 - 35; + return 0.29877251304899307550e-1 + (0.10318204245057349310e-2 + (0.61041829697162055093e-5 + (0.34414860359542720579e-7 + (0.18399863072934089607e-9 + (0.92703227366365046533e-12 + 0.43639844053333333334e-14 * t) * t) * t) * t) * t) * t; + } + case 18: + { + double t = 2 * y100 - 37; + return 0.31965587178596443475e-1 + (0.10566560976716574401e-2 + (0.63151633192414586770e-5 + (0.35924638339521924242e-7 + (0.19353584758781174038e-9 + (0.98102783859889264382e-12 + 0.46381060817777777779e-14 * t) * t) * t) * t) * t) * t; + } + case 19: + { + double t = 2 * y100 - 39; + return 0.34104450552588334840e-1 + (0.10823541191350532574e-2 + (0.65354356159553934436e-5 + (0.37512918348533521149e-7 + (0.20362979635817883229e-9 + (0.10384187833037282363e-11 + 0.49300625262222222221e-14 * t) * t) * t) * t) * t) * t; + } + case 20: + { + double t = 2 * y100 - 41; + return 0.36295603928292425716e-1 + (0.11089526167995268200e-2 + (0.67654845095518363577e-5 + (0.39184292949913591646e-7 + (0.21431552202133775150e-9 + (0.10994259106646731797e-11 + 0.52409949102222222221e-14 * t) * t) * t) * t) * t) * t; + } + case 21: + { + double t = 2 * y100 - 43; + return 0.38540888038840509795e-1 + (0.11364917134175420009e-2 + (0.70058230641246312003e-5 + (0.40943644083718586939e-7 + (0.22563034723692881631e-9 + (0.11642841011361992885e-11 + 0.55721092871111111110e-14 * t) * t) * t) * t) * t) * t; + } + case 22: + { + double t = 2 * y100 - 45; + return 0.40842225954785960651e-1 + (0.11650136437945673891e-2 + (0.72569945502343006619e-5 + (0.42796161861855042273e-7 + (0.23761401711005024162e-9 + (0.12332431172381557035e-11 + 0.59246802364444444445e-14 * t) * t) * t) * t) * t) * t; + } + case 23: + { + double t = 2 * y100 - 47; + return 0.43201627431540222422e-1 + (0.11945628793917272199e-2 + (0.75195743532849206263e-5 + (0.44747364553960993492e-7 + (0.25030885216472953674e-9 + (0.13065684400300476484e-11 + 0.63000532853333333334e-14 * t) * t) * t) * t) * t) * t; + } + case 24: + { + double t = 2 * y100 - 49; + return 0.45621193513810471438e-1 + (0.12251862608067529503e-2 + (0.77941720055551920319e-5 + (0.46803119830954460212e-7 + (0.26375990983978426273e-9 + (0.13845421370977119765e-11 + 0.66996477404444444445e-14 * t) * t) * t) * t) * t) * t; + } + case 25: + { + double t = 2 * y100 - 51; + return 0.48103121413299865517e-1 + (0.12569331386432195113e-2 + (0.80814333496367673980e-5 + (0.48969667335682018324e-7 + (0.27801515481905748484e-9 + (0.14674637611609884208e-11 + 0.71249589351111111110e-14 * t) * t) * t) * t) * t) * t; + } + case 26: + { + double t = 2 * y100 - 53; + return 0.50649709676983338501e-1 + (0.12898555233099055810e-2 + (0.83820428414568799654e-5 + (0.51253642652551838659e-7 + (0.29312563849675507232e-9 + (0.15556512782814827846e-11 + 0.75775607822222222221e-14 * t) * t) * t) * t) * t) * t; + } + case 27: + { + double t = 2 * y100 - 55; + return 0.53263363664388864181e-1 + (0.13240082443256975769e-2 + (0.86967260015007658418e-5 + (0.53662102750396795566e-7 + (0.30914568786634796807e-9 + (0.16494420240828493176e-11 + 0.80591079644444444445e-14 * t) * t) * t) * t) * t) * t; + } + case 28: + { + double t = 2 * y100 - 57; + return 0.55946601353500013794e-1 + (0.13594491197408190706e-2 + (0.90262520233016380987e-5 + (0.56202552975056695376e-7 + (0.32613310410503135996e-9 + (0.17491936862246367398e-11 + 0.85713381688888888890e-14 * t) * t) * t) * t) * t) * t; + } + case 29: + { + double t = 2 * y100 - 59; + return 0.58702059496154081813e-1 + (0.13962391363223647892e-2 + (0.93714365487312784270e-5 + (0.58882975670265286526e-7 + (0.34414937110591753387e-9 + (0.18552853109751857859e-11 + 0.91160736711111111110e-14 * t) * t) * t) * t) * t) * t; + } + case 30: + { + double t = 2 * y100 - 61; + return 0.61532500145144778048e-1 + (0.14344426411912015247e-2 + (0.97331446201016809696e-5 + (0.61711860507347175097e-7 + (0.36325987418295300221e-9 + (0.19681183310134518232e-11 + 0.96952238400000000000e-14 * t) * t) * t) * t) * t) * t; + } + case 31: + { + double t = 2 * y100 - 63; + return 0.64440817576653297993e-1 + (0.14741275456383131151e-2 + (0.10112293819576437838e-4 + (0.64698236605933246196e-7 + (0.38353412915303665586e-9 + (0.20881176114385120186e-11 + 0.10310784480000000000e-13 * t) * t) * t) * t) * t) * t; + } + case 32: + { + double t = 2 * y100 - 65; + return 0.67430045633130393282e-1 + (0.15153655418916540370e-2 + (0.10509857606888328667e-4 + (0.67851706529363332855e-7 + (0.40504602194811140006e-9 + (0.22157325110542534469e-11 + 0.10964842115555555556e-13 * t) * t) * t) * t) * t) * t; + } + case 33: + { + double t = 2 * y100 - 67; + return 0.70503365513338850709e-1 + (0.15582323336495709827e-2 + (0.10926868866865231089e-4 + (0.71182482239613507542e-7 + (0.42787405890153386710e-9 + (0.23514379522274416437e-11 + 0.11659571751111111111e-13 * t) * t) * t) * t) * t) * t; + } + case 34: + { + double t = 2 * y100 - 69; + return 0.73664114037944596353e-1 + (0.16028078812438820413e-2 + (0.11364423678778207991e-4 + (0.74701423097423182009e-7 + (0.45210162777476488324e-9 + (0.24957355004088569134e-11 + 0.12397238257777777778e-13 * t) * t) * t) * t) * t) * t; + } + case 35: + { + double t = 2 * y100 - 71; + return 0.76915792420819562379e-1 + (0.16491766623447889354e-2 + (0.11823685320041302169e-4 + (0.78420075993781544386e-7 + (0.47781726956916478925e-9 + (0.26491544403815724749e-11 + 0.13180196462222222222e-13 * t) * t) * t) * t) * t) * t; + } + case 36: + { + double t = 2 * y100 - 73; + return 0.80262075578094612819e-1 + (0.16974279491709504117e-2 + (0.12305888517309891674e-4 + (0.82350717698979042290e-7 + (0.50511496109857113929e-9 + (0.28122528497626897696e-11 + 0.14010889635555555556e-13 * t) * t) * t) * t) * t) * t; + } + case 37: + { + double t = 2 * y100 - 75; + return 0.83706822008980357446e-1 + (0.17476561032212656962e-2 + (0.12812343958540763368e-4 + (0.86506399515036435592e-7 + (0.53409440823869467453e-9 + (0.29856186620887555043e-11 + 0.14891851591111111111e-13 * t) * t) * t) * t) * t) * t; + } + case 38: + { + double t = 2 * y100 - 77; + return 0.87254084284461718231e-1 + (0.17999608886001962327e-2 + (0.13344443080089492218e-4 + (0.90900994316429008631e-7 + (0.56486134972616465316e-9 + (0.31698707080033956934e-11 + 0.15825697795555555556e-13 * t) * t) * t) * t) * t) * t; + } + case 39: + { + double t = 2 * y100 - 79; + return 0.90908120182172748487e-1 + (0.18544478050657699758e-2 + (0.13903663143426120077e-4 + (0.95549246062549906177e-7 + (0.59752787125242054315e-9 + (0.33656597366099099413e-11 + 0.16815130613333333333e-13 * t) * t) * t) * t) * t) * t; + } + case 40: + { + double t = 2 * y100 - 81; + return 0.94673404508075481121e-1 + (0.19112284419887303347e-2 + (0.14491572616545004930e-4 + (0.10046682186333613697e-6 + (0.63221272959791000515e-9 + (0.35736693975589130818e-11 + 0.17862931591111111111e-13 * t) * t) * t) * t) * t) * t; + } + case 41: + { + double t = 2 * y100 - 83; + return 0.98554641648004456555e-1 + (0.19704208544725622126e-2 + (0.15109836875625443935e-4 + (0.10567036667675984067e-6 + (0.66904168640019354565e-9 + (0.37946171850824333014e-11 + 0.18971959040000000000e-13 * t) * t) * t) * t) * t) * t; + } + case 42: + { + double t = 2 * y100 - 85; + return 0.10255677889470089531e0 + (0.20321499629472857418e-2 + (0.15760224242962179564e-4 + (0.11117756071353507391e-6 + (0.70814785110097658502e-9 + (0.40292553276632563925e-11 + 0.20145143075555555556e-13 * t) * t) * t) * t) * t) * t; + } + case 43: + { + double t = 2 * y100 - 87; + return 0.10668502059865093318e0 + (0.20965479776148731610e-2 + (0.16444612377624983565e-4 + (0.11700717962026152749e-6 + (0.74967203250938418991e-9 + (0.42783716186085922176e-11 + 0.21385479360000000000e-13 * t) * t) * t) * t) * t) * t; + } + case 44: + { + double t = 2 * y100 - 89; + return 0.11094484319386444474e0 + (0.21637548491908170841e-2 + (0.17164995035719657111e-4 + (0.12317915750735938089e-6 + (0.79376309831499633734e-9 + (0.45427901763106353914e-11 + 0.22696025653333333333e-13 * t) * t) * t) * t) * t) * t; + } + case 45: + { + double t = 2 * y100 - 91; + return 0.11534201115268804714e0 + (0.22339187474546420375e-2 + (0.17923489217504226813e-4 + (0.12971465288245997681e-6 + (0.84057834180389073587e-9 + (0.48233721206418027227e-11 + 0.24079890062222222222e-13 * t) * t) * t) * t) * t) * t; + } + case 46: + { + double t = 2 * y100 - 93; + return 0.11988259392684094740e0 + (0.23071965691918689601e-2 + (0.18722342718958935446e-4 + (0.13663611754337957520e-6 + (0.89028385488493287005e-9 + (0.51210161569225846701e-11 + 0.25540227111111111111e-13 * t) * t) * t) * t) * t) * t; + } + case 47: + { + double t = 2 * y100 - 95; + return 0.12457298393509812907e0 + (0.23837544771809575380e-2 + (0.19563942105711612475e-4 + (0.14396736847739470782e-6 + (0.94305490646459247016e-9 + (0.54366590583134218096e-11 + 0.27080225920000000000e-13 * t) * t) * t) * t) * t) * t; + } + case 48: + { + double t = 2 * y100 - 97; + return 0.12941991566142438816e0 + (0.24637684719508859484e-2 + (0.20450821127475879816e-4 + (0.15173366280523906622e-6 + (0.99907632506389027739e-9 + (0.57712760311351625221e-11 + 0.28703099555555555556e-13 * t) * t) * t) * t) * t) * t; + } + case 49: + { + double t = 2 * y100 - 99; + return 0.13443048593088696613e0 + (0.25474249981080823877e-2 + (0.21385669591362915223e-4 + (0.15996177579900443030e-6 + (0.10585428844575134013e-8 + (0.61258809536787882989e-11 + 0.30412080142222222222e-13 * t) * t) * t) * t) * t) * t; + } + case 50: + { + double t = 2 * y100 - 101; + return 0.13961217543434561353e0 + (0.26349215871051761416e-2 + (0.22371342712572567744e-4 + (0.16868008199296822247e-6 + (0.11216596910444996246e-8 + (0.65015264753090890662e-11 + 0.32210394506666666666e-13 * t) * t) * t) * t) * t) * t; + } + case 51: + { + double t = 2 * y100 - 103; + return 0.14497287157673800690e0 + (0.27264675383982439814e-2 + (0.23410870961050950197e-4 + (0.17791863939526376477e-6 + (0.11886425714330958106e-8 + (0.68993039665054288034e-11 + 0.34101266222222222221e-13 * t) * t) * t) * t) * t) * t; + } + case 52: + { + double t = 2 * y100 - 105; + return 0.15052089272774618151e0 + (0.28222846410136238008e-2 + (0.24507470422713397006e-4 + (0.18770927679626136909e-6 + (0.12597184587583370712e-8 + (0.73203433049229821618e-11 + 0.36087889048888888890e-13 * t) * t) * t) * t) * t) * t; + } + case 53: + { + double t = 2 * y100 - 107; + return 0.15626501395774612325e0 + (0.29226079376196624949e-2 + (0.25664553693768450545e-4 + (0.19808568415654461964e-6 + (0.13351257759815557897e-8 + (0.77658124891046760667e-11 + 0.38173420035555555555e-13 * t) * t) * t) * t) * t) * t; + } + case 54: + { + double t = 2 * y100 - 109; + return 0.16221449434620737567e0 + (0.30276865332726475672e-2 + (0.26885741326534564336e-4 + (0.20908350604346384143e-6 + (0.14151148144240728728e-8 + (0.82369170665974313027e-11 + 0.40360957457777777779e-13 * t) * t) * t) * t) * t) * t; + } + case 55: + { + double t = 2 * y100 - 111; + return 0.16837910595412130659e0 + (0.31377844510793082301e-2 + (0.28174873844911175026e-4 + (0.22074043807045782387e-6 + (0.14999481055996090039e-8 + (0.87348993661930809254e-11 + 0.42653528977777777779e-13 * t) * t) * t) * t) * t) * t; + } + case 56: + { + double t = 2 * y100 - 113; + return 0.17476916455659369953e0 + (0.32531815370903068316e-2 + (0.29536024347344364074e-4 + (0.23309632627767074202e-6 + (0.15899007843582444846e-8 + (0.92610375235427359475e-11 + 0.45054073102222222221e-13 * t) * t) * t) * t) * t) * t; + } + case 57: + { + double t = 2 * y100 - 115; + return 0.18139556223643701364e0 + (0.33741744168096996041e-2 + (0.30973511714709500836e-4 + (0.24619326937592290996e-6 + (0.16852609412267750744e-8 + (0.98166442942854895573e-11 + 0.47565418097777777779e-13 * t) * t) * t) * t) * t) * t; + } + case 58: + { + double t = 2 * y100 - 117; + return 0.18826980194443664549e0 + (0.35010775057740317997e-2 + (0.32491914440014267480e-4 + (0.26007572375886319028e-6 + (0.17863299617388376116e-8 + (0.10403065638343878679e-10 + 0.50190265831111111110e-13 * t) * t) * t) * t) * t) * t; + } + case 59: + { + double t = 2 * y100 - 119; + return 0.19540403413693967350e0 + (0.36342240767211326315e-2 + (0.34096085096200907289e-4 + (0.27479061117017637474e-6 + (0.18934228504790032826e-8 + (0.11021679075323598664e-10 + 0.52931171733333333334e-13 * t) * t) * t) * t) * t) * t; + } + case 60: + { + double t = 2 * y100 - 121; + return 0.20281109560651886959e0 + (0.37739673859323597060e-2 + (0.35791165457592409054e-4 + (0.29038742889416172404e-6 + (0.20068685374849001770e-8 + (0.11673891799578381999e-10 + 0.55790523093333333334e-13 * t) * t) * t) * t) * t) * t; + } + case 61: + { + double t = 2 * y100 - 123; + return 0.21050455062669334978e0 + (0.39206818613925652425e-2 + (0.37582602289680101704e-4 + (0.30691836231886877385e-6 + (0.21270101645763677824e-8 + (0.12361138551062899455e-10 + 0.58770520160000000000e-13 * t) * t) * t) * t) * t) * t; + } + case 62: + { + double t = 2 * y100 - 125; + return 0.21849873453703332479e0 + (0.40747643554689586041e-2 + (0.39476163820986711501e-4 + (0.32443839970139918836e-6 + (0.22542053491518680200e-8 + (0.13084879235290858490e-10 + 0.61873153262222222221e-13 * t) * t) * t) * t) * t) * t; + } + case 63: + { + double t = 2 * y100 - 127; + return 0.22680879990043229327e0 + (0.42366354648628516935e-2 + (0.41477956909656896779e-4 + (0.34300544894502810002e-6 + (0.23888264229264067658e-8 + (0.13846596292818514601e-10 + 0.65100183751111111110e-13 * t) * t) * t) * t) * t) * t; + } + case 64: + { + double t = 2 * y100 - 129; + return 0.23545076536988703937e0 + (0.44067409206365170888e-2 + (0.43594444916224700881e-4 + (0.36268045617760415178e-6 + (0.25312606430853202748e-8 + (0.14647791812837903061e-10 + 0.68453122631111111110e-13 * t) * t) * t) * t) * t) * t; + } + case 65: + { + double t = 2 * y100 - 131; + return 0.24444156740777432838e0 + (0.45855530511605787178e-2 + (0.45832466292683085475e-4 + (0.38352752590033030472e-6 + (0.26819103733055603460e-8 + (0.15489984390884756993e-10 + 0.71933206364444444445e-13 * t) * t) * t) * t) * t) * t; + } + case 66: + { + double t = 2 * y100 - 133; + return 0.25379911500634264643e0 + (0.47735723208650032167e-2 + (0.48199253896534185372e-4 + (0.40561404245564732314e-6 + (0.28411932320871165585e-8 + (0.16374705736458320149e-10 + 0.75541379822222222221e-13 * t) * t) * t) * t) * t) * t; + } + case 67: + { + double t = 2 * y100 - 135; + return 0.26354234756393613032e0 + (0.49713289477083781266e-2 + (0.50702455036930367504e-4 + (0.42901079254268185722e-6 + (0.30095422058900481753e-8 + (0.17303497025347342498e-10 + 0.79278273368888888890e-13 * t) * t) * t) * t) * t) * t; + } + case 68: + { + double t = 2 * y100 - 137; + return 0.27369129607732343398e0 + (0.51793846023052643767e-2 + (0.53350152258326602629e-4 + (0.45379208848865015485e-6 + (0.31874057245814381257e-8 + (0.18277905010245111046e-10 + 0.83144182364444444445e-13 * t) * t) * t) * t) * t) * t; + } + case 69: + { + double t = 2 * y100 - 139; + return 0.28426714781640316172e0 + (0.53983341916695141966e-2 + (0.56150884865255810638e-4 + (0.48003589196494734238e-6 + (0.33752476967570796349e-8 + (0.19299477888083469086e-10 + 0.87139049137777777779e-13 * t) * t) * t) * t) * t) * t; + } + case 70: + { + double t = 2 * y100 - 141; + return 0.29529231465348519920e0 + (0.56288077305420795663e-2 + (0.59113671189913307427e-4 + (0.50782393781744840482e-6 + (0.35735475025851713168e-8 + (0.20369760937017070382e-10 + 0.91262442613333333334e-13 * t) * t) * t) * t) * t) * t; + } + case 71: + { + double t = 2 * y100 - 143; + return 0.30679050522528838613e0 + (0.58714723032745403331e-2 + (0.62248031602197686791e-4 + (0.53724185766200945789e-6 + (0.37827999418960232678e-8 + (0.21490291930444538307e-10 + 0.95513539182222222221e-13 * t) * t) * t) * t) * t) * t; + } + case 72: + { + double t = 2 * y100 - 145; + return 0.31878680111173319425e0 + (0.61270341192339103514e-2 + (0.65564012259707640976e-4 + (0.56837930287837738996e-6 + (0.40035151353392378882e-8 + (0.22662596341239294792e-10 + 0.99891109760000000000e-13 * t) * t) * t) * t) * t) * t; + } + case 73: + { + double t = 2 * y100 - 147; + return 0.33130773722152622027e0 + (0.63962406646798080903e-2 + (0.69072209592942396666e-4 + (0.60133006661885941812e-6 + (0.42362183765883466691e-8 + (0.23888182347073698382e-10 + 0.10439349811555555556e-12 * t) * t) * t) * t) * t) * t; + } + case 74: + { + double t = 2 * y100 - 149; + return 0.34438138658041336523e0 + (0.66798829540414007258e-2 + (0.72783795518603561144e-4 + (0.63619220443228800680e-6 + (0.44814499336514453364e-8 + (0.25168535651285475274e-10 + 0.10901861383111111111e-12 * t) * t) * t) * t) * t) * t; + } + case 75: + { + double t = 2 * y100 - 151; + return 0.35803744972380175583e0 + (0.69787978834882685031e-2 + (0.76710543371454822497e-4 + (0.67306815308917386747e-6 + (0.47397647975845228205e-8 + (0.26505114141143050509e-10 + 0.11376390933333333333e-12 * t) * t) * t) * t) * t) * t; + } + case 76: + { + double t = 2 * y100 - 153; + return 0.37230734890119724188e0 + (0.72938706896461381003e-2 + (0.80864854542670714092e-4 + (0.71206484718062688779e-6 + (0.50117323769745883805e-8 + (0.27899342394100074165e-10 + 0.11862637614222222222e-12 * t) * t) * t) * t) * t) * t; + } + case 77: + { + double t = 2 * y100 - 155; + return 0.38722432730555448223e0 + (0.76260375162549802745e-2 + (0.85259785810004603848e-4 + (0.75329383305171327677e-6 + (0.52979361368388119355e-8 + (0.29352606054164086709e-10 + 0.12360253370666666667e-12 * t) * t) * t) * t) * t) * t; + } + case 78: + { + double t = 2 * y100 - 157; + return 0.40282355354616940667e0 + (0.79762880915029728079e-2 + (0.89909077342438246452e-4 + (0.79687137961956194579e-6 + (0.55989731807360403195e-8 + (0.30866246101464869050e-10 + 0.12868841946666666667e-12 * t) * t) * t) * t) * t) * t; + } + case 79: + { + double t = 2 * y100 - 159; + return 0.41914223158913787649e0 + (0.83456685186950463538e-2 + (0.94827181359250161335e-4 + (0.84291858561783141014e-6 + (0.59154537751083485684e-8 + (0.32441553034347469291e-10 + 0.13387957943111111111e-12 * t) * t) * t) * t) * t) * t; + } + case 80: + { + double t = 2 * y100 - 161; + return 0.43621971639463786896e0 + (0.87352841828289495773e-2 + (0.10002929142066799966e-3 + (0.89156148280219880024e-6 + (0.62480008150788597147e-8 + (0.34079760983458878910e-10 + 0.13917107176888888889e-12 * t) * t) * t) * t) * t) * t; + } + case 81: + { + double t = 2 * y100 - 163; + return 0.45409763548534330981e0 + (0.91463027755548240654e-2 + (0.10553137232446167258e-3 + (0.94293113464638623798e-6 + (0.65972492312219959885e-8 + (0.35782041795476563662e-10 + 0.14455745872000000000e-12 * t) * t) * t) * t) * t) * t; + } + case 82: + { + double t = 2 * y100 - 165; + return 0.47282001668512331468e0 + (0.95799574408860463394e-2 + (0.11135019058000067469e-3 + (0.99716373005509038080e-6 + (0.69638453369956970347e-8 + (0.37549499088161345850e-10 + 0.15003280712888888889e-12 * t) * t) * t) * t) * t) * t; + } + case 83: + { + double t = 2 * y100 - 167; + return 0.49243342227179841649e0 + (0.10037550043909497071e-1 + (0.11750334542845234952e-3 + (0.10544006716188967172e-5 + (0.73484461168242224872e-8 + (0.39383162326435752965e-10 + 0.15559069118222222222e-12 * t) * t) * t) * t) * t) * t; + } + case 84: + { + double t = 2 * y100 - 169; + return 0.51298708979209258326e0 + (0.10520454564612427224e-1 + (0.12400930037494996655e-3 + (0.11147886579371265246e-5 + (0.77517184550568711454e-8 + (0.41283980931872622611e-10 + 0.16122419680000000000e-12 * t) * t) * t) * t) * t) * t; + } + case 85: + { + double t = 2 * y100 - 171; + return 0.53453307979101369843e0 + (0.11030120618800726938e-1 + (0.13088741519572269581e-3 + (0.11784797595374515432e-5 + (0.81743383063044825400e-8 + (0.43252818449517081051e-10 + 0.16692592640000000000e-12 * t) * t) * t) * t) * t) * t; + } + case 86: + { + double t = 2 * y100 - 173; + return 0.55712643071169299478e0 + (0.11568077107929735233e-1 + (0.13815797838036651289e-3 + (0.12456314879260904558e-5 + (0.86169898078969313597e-8 + (0.45290446811539652525e-10 + 0.17268801084444444444e-12 * t) * t) * t) * t) * t) * t; + } + case 87: + { + double t = 2 * y100 - 175; + return 0.58082532122519320968e0 + (0.12135935999503877077e-1 + (0.14584223996665838559e-3 + (0.13164068573095710742e-5 + (0.90803643355106020163e-8 + (0.47397540713124619155e-10 + 0.17850211608888888889e-12 * t) * t) * t) * t) * t) * t; + } + case 88: + { + double t = 2 * y100 - 177; + return 0.60569124025293375554e0 + (0.12735396239525550361e-1 + (0.15396244472258863344e-3 + (0.13909744385382818253e-5 + (0.95651595032306228245e-8 + (0.49574672127669041550e-10 + 0.18435945564444444444e-12 * t) * t) * t) * t) * t) * t; + } + case 89: + { + double t = 2 * y100 - 179; + return 0.63178916494715716894e0 + (0.13368247798287030927e-1 + (0.16254186562762076141e-3 + (0.14695084048334056083e-5 + (0.10072078109604152350e-7 + (0.51822304995680707483e-10 + 0.19025081422222222222e-12 * t) * t) * t) * t) * t) * t; + } + case 90: + { + double t = 2 * y100 - 181; + return 0.65918774689725319200e0 + (0.14036375850601992063e-1 + (0.17160483760259706354e-3 + (0.15521885688723188371e-5 + (0.10601827031535280590e-7 + (0.54140790105837520499e-10 + 0.19616655146666666667e-12 * t) * t) * t) * t) * t) * t; + } + case 91: + { + double t = 2 * y100 - 183; + return 0.68795950683174433822e0 + (0.14741765091365869084e-1 + (0.18117679143520433835e-3 + (0.16392004108230585213e-5 + (0.11155116068018043001e-7 + (0.56530360194925690374e-10 + 0.20209663662222222222e-12 * t) * t) * t) * t) * t) * t; + } + case 92: + { + double t = 2 * y100 - 185; + return 0.71818103808729967036e0 + (0.15486504187117112279e-1 + (0.19128428784550923217e-3 + (0.17307350969359975848e-5 + (0.11732656736113607751e-7 + (0.58991125287563833603e-10 + 0.20803065333333333333e-12 * t) * t) * t) * t) * t) * t; + } + case 93: + { + double t = 2 * y100 - 187; + return 0.74993321911726254661e0 + (0.16272790364044783382e-1 + (0.20195505163377912645e-3 + (0.18269894883203346953e-5 + (0.12335161021630225535e-7 + (0.61523068312169087227e-10 + 0.21395783431111111111e-12 * t) * t) * t) * t) * t) * t; + } + case 94: + { + double t = 2 * y100 - 189; + return 0.78330143531283492729e0 + (0.17102934132652429240e-1 + (0.21321800585063327041e-3 + (0.19281661395543913713e-5 + (0.12963340087354341574e-7 + (0.64126040998066348872e-10 + 0.21986708942222222222e-12 * t) * t) * t) * t) * t) * t; + } + case 95: + { + double t = 2 * y100 - 191; + return 0.81837581041023811832e0 + (0.17979364149044223802e-1 + (0.22510330592753129006e-3 + (0.20344732868018175389e-5 + (0.13617902941839949718e-7 + (0.66799760083972474642e-10 + 0.22574701262222222222e-12 * t) * t) * t) * t) * t) * t; + } + case 96: + { + double t = 2 * y100 - 193; + return 0.85525144775685126237e0 + (0.18904632212547561026e-1 + (0.23764237370371255638e-3 + (0.21461248251306387979e-5 + (0.14299555071870523786e-7 + (0.69543803864694171934e-10 + 0.23158593688888888889e-12 * t) * t) * t) * t) * t) * t; + } + case 97: + { + double t = 2 * y100 - 195; + return 0.89402868170849933734e0 + (0.19881418399127202569e-1 + (0.25086793128395995798e-3 + (0.22633402747585233180e-5 + (0.15008997042116532283e-7 + (0.72357609075043941261e-10 + 0.23737194737777777778e-12 * t) * t) * t) * t) * t) * t; + } + case 98: + { + double t = 2 * y100 - 197; + return 0.93481333942870796363e0 + (0.20912536329780368893e-1 + (0.26481403465998477969e-3 + (0.23863447359754921676e-5 + (0.15746923065472184451e-7 + (0.75240468141720143653e-10 + 0.24309291271111111111e-12 * t) * t) * t) * t) * t) * t; + } + case 99: + { + double t = 2 * y100 - 199; + return 0.97771701335885035464e0 + (0.22000938572830479551e-1 + (0.27951610702682383001e-3 + (0.25153688325245314530e-5 + (0.16514019547822821453e-7 + (0.78191526829368231251e-10 + 0.24873652355555555556e-12 * t) * t) * t) * t) * t) * t; + } + } + // we only get here if y = 1, i.e. |x| < 4*eps, in which case + // erfcx is within 1e-15 of 1.. + return 1.0; +} + +double FADDEEVA_RE(erfcx)(double x) +{ + if (x >= 0) + { + if (x > 50) // continued-fraction expansion is faster + { + const double ispi = 0.56418958354775628694807945156; // 1 / sqrt(pi) + if (x > 5e7) // 1-term expansion, important to avoid overflow + { + return ispi / x; + } + /* 5-term expansion (rely on compiler for CSE), simplified from: + ispi / (x+0.5/(x+1/(x+1.5/(x+2/x)))) */ + return ispi * ((x * x) * (x * x + 4.5) + 2) / (x * ((x * x) * (x * x + 5) + 3.75)); + } + return erfcx_y100(400 / (4 + x)); + } + else + return x < -26.7 ? HUGE_VAL : (x < -6.1 ? 2 * exp(x * x) + : 2 * exp(x * x) - erfcx_y100(400 / (4 - x))); +} + +///////////////////////////////////////////////////////////////////////// +/* Compute a scaled Dawson integral + FADDEEVA(w_im)(x) = 2*Dawson(x)/sqrt(pi) + equivalent to the imaginary part w(x) for real x. + + Uses methods similar to the erfcx calculation above: continued fractions + for large |x|, a lookup table of Chebyshev polynomials for smaller |x|, + and finally a Taylor expansion for |x|<0.01. + + Steven G. Johnson, October 2012. */ + +/* Given y100=100*y, where y = 1/(1+x) for x >= 0, compute w_im(x). + + Uses a look-up table of 100 different Chebyshev polynomials + for y intervals [0,0.01], [0.01,0.02], ...., [0.99,1], generated + with the help of Maple and a little shell script. This allows + the Chebyshev polynomials to be of significantly lower degree (about 1/30) + compared to fitting the whole [0,1] interval with a single polynomial. */ +static double w_im_y100(double y100, double x) +{ + switch ((int) y100) + { + case 0: + { + double t = 2 * y100 - 1; + return 0.28351593328822191546e-2 + (0.28494783221378400759e-2 + (0.14427470563276734183e-4 + (0.10939723080231588129e-6 + (0.92474307943275042045e-9 + (0.89128907666450075245e-11 + 0.92974121935111111110e-13 * t) * t) * t) * t) * t) * t; + } + case 1: + { + double t = 2 * y100 - 3; + return 0.85927161243940350562e-2 + (0.29085312941641339862e-2 + (0.15106783707725582090e-4 + (0.11716709978531327367e-6 + (0.10197387816021040024e-8 + (0.10122678863073360769e-10 + 0.10917479678400000000e-12 * t) * t) * t) * t) * t) * t; + } + case 2: + { + double t = 2 * y100 - 5; + return 0.14471159831187703054e-1 + (0.29703978970263836210e-2 + (0.15835096760173030976e-4 + (0.12574803383199211596e-6 + (0.11278672159518415848e-8 + (0.11547462300333495797e-10 + 0.12894535335111111111e-12 * t) * t) * t) * t) * t) * t; + } + case 3: + { + double t = 2 * y100 - 7; + return 0.20476320420324610618e-1 + (0.30352843012898665856e-2 + (0.16617609387003727409e-4 + (0.13525429711163116103e-6 + (0.12515095552507169013e-8 + (0.13235687543603382345e-10 + 0.15326595042666666667e-12 * t) * t) * t) * t) * t) * t; + } + case 4: + { + double t = 2 * y100 - 9; + return 0.26614461952489004566e-1 + (0.31034189276234947088e-2 + (0.17460268109986214274e-4 + (0.14582130824485709573e-6 + (0.13935959083809746345e-8 + (0.15249438072998932900e-10 + 0.18344741882133333333e-12 * t) * t) * t) * t) * t) * t; + } + case 5: + { + double t = 2 * y100 - 11; + return 0.32892330248093586215e-1 + (0.31750557067975068584e-2 + (0.18369907582308672632e-4 + (0.15761063702089457882e-6 + (0.15577638230480894382e-8 + (0.17663868462699097951e-10 + (0.22126732680711111111e-12 + 0.30273474177737853668e-14 * t) * t) * t) * t) * t) * t) * t; + } + case 6: + { + double t = 2 * y100 - 13; + return 0.39317207681134336024e-1 + (0.32504779701937539333e-2 + (0.19354426046513400534e-4 + (0.17081646971321290539e-6 + (0.17485733959327106250e-8 + (0.20593687304921961410e-10 + (0.26917401949155555556e-12 + 0.38562123837725712270e-14 * t) * t) * t) * t) * t) * t) * t; + } + case 7: + { + double t = 2 * y100 - 15; + return 0.45896976511367738235e-1 + (0.33300031273110976165e-2 + (0.20423005398039037313e-4 + (0.18567412470376467303e-6 + (0.19718038363586588213e-8 + (0.24175006536781219807e-10 + (0.33059982791466666666e-12 + 0.49756574284439426165e-14 * t) * t) * t) * t) * t) * t) * t; + } + case 8: + { + double t = 2 * y100 - 17; + return 0.52640192524848962855e-1 + (0.34139883358846720806e-2 + (0.21586390240603337337e-4 + (0.20247136501568904646e-6 + (0.22348696948197102935e-8 + (0.28597516301950162548e-10 + (0.41045502119111111110e-12 + 0.65151614515238361946e-14 * t) * t) * t) * t) * t) * t) * t; + } + case 9: + { + double t = 2 * y100 - 19; + return 0.59556171228656770456e-1 + (0.35028374386648914444e-2 + (0.22857246150998562824e-4 + (0.22156372146525190679e-6 + (0.25474171590893813583e-8 + (0.34122390890697400584e-10 + (0.51593189879111111110e-12 + 0.86775076853908006938e-14 * t) * t) * t) * t) * t) * t) * t; + } + case 10: + { + double t = 2 * y100 - 21; + return 0.66655089485108212551e-1 + (0.35970095381271285568e-2 + (0.24250626164318672928e-4 + (0.24339561521785040536e-6 + (0.29221990406518411415e-8 + (0.41117013527967776467e-10 + (0.65786450716444444445e-12 + 0.11791885745450623331e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 11: + { + double t = 2 * y100 - 23; + return 0.73948106345519174661e-1 + (0.36970297216569341748e-2 + (0.25784588137312868792e-4 + (0.26853012002366752770e-6 + (0.33763958861206729592e-8 + (0.50111549981376976397e-10 + (0.85313857496888888890e-12 + 0.16417079927706899860e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 12: + { + double t = 2 * y100 - 25; + return 0.81447508065002963203e-1 + (0.38035026606492705117e-2 + (0.27481027572231851896e-4 + (0.29769200731832331364e-6 + (0.39336816287457655076e-8 + (0.61895471132038157624e-10 + (0.11292303213511111111e-11 + 0.23558532213703884304e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 13: + { + double t = 2 * y100 - 27; + return 0.89166884027582716628e-1 + (0.39171301322438946014e-2 + (0.29366827260422311668e-4 + (0.33183204390350724895e-6 + (0.46276006281647330524e-8 + (0.77692631378169813324e-10 + (0.15335153258844444444e-11 + 0.35183103415916026911e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 14: + { + double t = 2 * y100 - 29; + return 0.97121342888032322019e-1 + (0.40387340353207909514e-2 + (0.31475490395950776930e-4 + (0.37222714227125135042e-6 + (0.55074373178613809996e-8 + (0.99509175283990337944e-10 + (0.21552645758222222222e-11 + 0.55728651431872687605e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 15: + { + double t = 2 * y100 - 31; + return 0.10532778218603311137e0 + (0.41692873614065380607e-2 + (0.33849549774889456984e-4 + (0.42064596193692630143e-6 + (0.66494579697622432987e-8 + (0.13094103581931802337e-9 + (0.31896187409777777778e-11 + 0.97271974184476560742e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 16: + { + double t = 2 * y100 - 33; + return 0.11380523107427108222e0 + (0.43099572287871821013e-2 + (0.36544324341565929930e-4 + (0.47965044028581857764e-6 + (0.81819034238463698796e-8 + (0.17934133239549647357e-9 + (0.50956666166186293627e-11 + (0.18850487318190638010e-12 + 0.79697813173519853340e-14 * t) * t) * t) * t) * t) * t) * t) * t; + } + case 17: + { + double t = 2 * y100 - 35; + return 0.12257529703447467345e0 + (0.44621675710026986366e-2 + (0.39634304721292440285e-4 + (0.55321553769873381819e-6 + (0.10343619428848520870e-7 + (0.26033830170470368088e-9 + (0.87743837749108025357e-11 + (0.34427092430230063401e-12 + 0.10205506615709843189e-13 * t) * t) * t) * t) * t) * t) * t) * t; + } + case 18: + { + double t = 2 * y100 - 37; + return 0.13166276955656699478e0 + (0.46276970481783001803e-2 + (0.43225026380496399310e-4 + (0.64799164020016902656e-6 + (0.13580082794704641782e-7 + (0.39839800853954313927e-9 + (0.14431142411840000000e-10 + 0.42193457308830027541e-12 * t) * t) * t) * t) * t) * t) * t; + } + case 19: + { + double t = 2 * y100 - 39; + return 0.14109647869803356475e0 + (0.48088424418545347758e-2 + (0.47474504753352150205e-4 + (0.77509866468724360352e-6 + (0.18536851570794291724e-7 + (0.60146623257887570439e-9 + (0.18533978397305276318e-10 + (0.41033845938901048380e-13 - 0.46160680279304825485e-13 * t) * t) * t) * t) * t) * t) * t) * t; + } + case 20: + { + double t = 2 * y100 - 41; + return 0.15091057940548936603e0 + (0.50086864672004685703e-2 + (0.52622482832192230762e-4 + (0.95034664722040355212e-6 + (0.25614261331144718769e-7 + (0.80183196716888606252e-9 + (0.12282524750534352272e-10 + (-0.10531774117332273617e-11 - 0.86157181395039646412e-13 * t) * t) * t) * t) * t) * t) * t) * t; + } + case 21: + { + double t = 2 * y100 - 43; + return 0.16114648116017010770e0 + (0.52314661581655369795e-2 + (0.59005534545908331315e-4 + (0.11885518333915387760e-5 + (0.33975801443239949256e-7 + (0.82111547144080388610e-9 + (-0.12357674017312854138e-10 + (-0.24355112256914479176e-11 - 0.75155506863572930844e-13 * t) * t) * t) * t) * t) * t) * t) * t; + } + case 22: + { + double t = 2 * y100 - 45; + return 0.17185551279680451144e0 + (0.54829002967599420860e-2 + (0.67013226658738082118e-4 + (0.14897400671425088807e-5 + (0.40690283917126153701e-7 + (0.44060872913473778318e-9 + (-0.52641873433280000000e-10 - 0.30940587864543343124e-11 * t) * t) * t) * t) * t) * t) * t; + } + case 23: + { + double t = 2 * y100 - 47; + return 0.18310194559815257381e0 + (0.57701559375966953174e-2 + (0.76948789401735193483e-4 + (0.18227569842290822512e-5 + (0.41092208344387212276e-7 + (-0.44009499965694442143e-9 + (-0.92195414685628803451e-10 + (-0.22657389705721753299e-11 + 0.10004784908106839254e-12 * t) * t) * t) * t) * t) * t) * t) * t; + } + case 24: + { + double t = 2 * y100 - 49; + return 0.19496527191546630345e0 + (0.61010853144364724856e-2 + (0.88812881056342004864e-4 + (0.21180686746360261031e-5 + (0.30652145555130049203e-7 + (-0.16841328574105890409e-8 + (-0.11008129460612823934e-9 + (-0.12180794204544515779e-12 + 0.15703325634590334097e-12 * t) * t) * t) * t) * t) * t) * t) * t; + } + case 25: + { + double t = 2 * y100 - 51; + return 0.20754006813966575720e0 + (0.64825787724922073908e-2 + (0.10209599627522311893e-3 + (0.22785233392557600468e-5 + (0.73495224449907568402e-8 + (-0.29442705974150112783e-8 + (-0.94082603434315016546e-10 + (0.23609990400179321267e-11 + 0.14141908654269023788e-12 * t) * t) * t) * t) * t) * t) * t) * t; + } + case 26: + { + double t = 2 * y100 - 53; + return 0.22093185554845172146e0 + (0.69182878150187964499e-2 + (0.11568723331156335712e-3 + (0.22060577946323627739e-5 + (-0.26929730679360840096e-7 + (-0.38176506152362058013e-8 + (-0.47399503861054459243e-10 + (0.40953700187172127264e-11 + 0.69157730376118511127e-13 * t) * t) * t) * t) * t) * t) * t) * t; + } + case 27: + { + double t = 2 * y100 - 55; + return 0.23524827304057813918e0 + (0.74063350762008734520e-2 + (0.12796333874615790348e-3 + (0.18327267316171054273e-5 + (-0.66742910737957100098e-7 + (-0.40204740975496797870e-8 + (0.14515984139495745330e-10 + (0.44921608954536047975e-11 - 0.18583341338983776219e-13 * t) * t) * t) * t) * t) * t) * t) * t; + } + case 28: + { + double t = 2 * y100 - 57; + return 0.25058626331812744775e0 + (0.79377285151602061328e-2 + (0.13704268650417478346e-3 + (0.11427511739544695861e-5 + (-0.10485442447768377485e-6 + (-0.34850364756499369763e-8 + (0.72656453829502179208e-10 + (0.36195460197779299406e-11 - 0.84882136022200714710e-13 * t) * t) * t) * t) * t) * t) * t) * t; + } + case 29: + { + double t = 2 * y100 - 59; + return 0.26701724900280689785e0 + (0.84959936119625864274e-2 + (0.14112359443938883232e-3 + (0.17800427288596909634e-6 + (-0.13443492107643109071e-6 + (-0.23512456315677680293e-8 + (0.11245846264695936769e-9 + (0.19850501334649565404e-11 - 0.11284666134635050832e-12 * t) * t) * t) * t) * t) * t) * t) * t; + } + case 30: + { + double t = 2 * y100 - 61; + return 0.28457293586253654144e0 + (0.90581563892650431899e-2 + (0.13880520331140646738e-3 + (-0.97262302362522896157e-6 + (-0.15077100040254187366e-6 + (-0.88574317464577116689e-9 + (0.12760311125637474581e-9 + (0.20155151018282695055e-12 - 0.10514169375181734921e-12 * t) * t) * t) * t) * t) * t) * t) * t; + } + case 31: + { + double t = 2 * y100 - 63; + return 0.30323425595617385705e0 + (0.95968346790597422934e-2 + (0.12931067776725883939e-3 + (-0.21938741702795543986e-5 + (-0.15202888584907373963e-6 + (0.61788350541116331411e-9 + (0.11957835742791248256e-9 + (-0.12598179834007710908e-11 - 0.75151817129574614194e-13 * t) * t) * t) * t) * t) * t) * t) * t; + } + case 32: + { + double t = 2 * y100 - 65; + return 0.32292521181517384379e0 + (0.10082957727001199408e-1 + (0.11257589426154962226e-3 + (-0.33670890319327881129e-5 + (-0.13910529040004008158e-6 + (0.19170714373047512945e-8 + (0.94840222377720494290e-10 + (-0.21650018351795353201e-11 - 0.37875211678024922689e-13 * t) * t) * t) * t) * t) * t) * t) * t; + } + case 33: + { + double t = 2 * y100 - 67; + return 0.34351233557911753862e0 + (0.10488575435572745309e-1 + (0.89209444197248726614e-4 + (-0.43893459576483345364e-5 + (-0.11488595830450424419e-6 + (0.28599494117122464806e-8 + (0.61537542799857777779e-10 - 0.24935749227658002212e-11 * t) * t) * t) * t) * t) * t) * t; + } + case 34: + { + double t = 2 * y100 - 69; + return 0.36480946642143669093e0 + (0.10789304203431861366e-1 + (0.60357993745283076834e-4 + (-0.51855862174130669389e-5 + (-0.83291664087289801313e-7 + (0.33898011178582671546e-8 + (0.27082948188277716482e-10 + (-0.23603379397408694974e-11 + 0.19328087692252869842e-13 * t) * t) * t) * t) * t) * t) * t) * t; + } + case 35: + { + double t = 2 * y100 - 71; + return 0.38658679935694939199e0 + (0.10966119158288804999e-1 + (0.27521612041849561426e-4 + (-0.57132774537670953638e-5 + (-0.48404772799207914899e-7 + (0.35268354132474570493e-8 + (-0.32383477652514618094e-11 + (-0.19334202915190442501e-11 + 0.32333189861286460270e-13 * t) * t) * t) * t) * t) * t) * t) * t; + } + case 36: + { + double t = 2 * y100 - 73; + return 0.40858275583808707870e0 + (0.11006378016848466550e-1 + (-0.76396376685213286033e-5 + (-0.59609835484245791439e-5 + (-0.13834610033859313213e-7 + (0.33406952974861448790e-8 + (-0.26474915974296612559e-10 + (-0.13750229270354351983e-11 + 0.36169366979417390637e-13 * t) * t) * t) * t) * t) * t) * t) * t; + } + case 37: + { + double t = 2 * y100 - 75; + return 0.43051714914006682977e0 + (0.10904106549500816155e-1 + (-0.43477527256787216909e-4 + (-0.59429739547798343948e-5 + (0.17639200194091885949e-7 + (0.29235991689639918688e-8 + (-0.41718791216277812879e-10 + (-0.81023337739508049606e-12 + 0.33618915934461994428e-13 * t) * t) * t) * t) * t) * t) * t) * t; + } + case 38: + { + double t = 2 * y100 - 77; + return 0.45210428135559607406e0 + (0.10659670756384400554e-1 + (-0.78488639913256978087e-4 + (-0.56919860886214735936e-5 + (0.44181850467477733407e-7 + (0.23694306174312688151e-8 + (-0.49492621596685443247e-10 + (-0.31827275712126287222e-12 + 0.27494438742721623654e-13 * t) * t) * t) * t) * t) * t) * t) * t; + } + case 39: + { + double t = 2 * y100 - 79; + return 0.47306491195005224077e0 + (0.10279006119745977570e-1 + (-0.11140268171830478306e-3 + (-0.52518035247451432069e-5 + (0.64846898158889479518e-7 + (0.17603624837787337662e-8 + (-0.51129481592926104316e-10 + (0.62674584974141049511e-13 + 0.20055478560829935356e-13 * t) * t) * t) * t) * t) * t) * t) * t; + } + case 40: + { + double t = 2 * y100 - 81; + return 0.49313638965719857647e0 + (0.97725799114772017662e-2 + (-0.14122854267291533334e-3 + (-0.46707252568834951907e-5 + (0.79421347979319449524e-7 + (0.11603027184324708643e-8 + (-0.48269605844397175946e-10 + (0.32477251431748571219e-12 + 0.12831052634143527985e-13 * t) * t) * t) * t) * t) * t) * t) * t; + } + case 41: + { + double t = 2 * y100 - 83; + return 0.51208057433416004042e0 + (0.91542422354009224951e-2 + (-0.16726530230228647275e-3 + (-0.39964621752527649409e-5 + (0.88232252903213171454e-7 + (0.61343113364949928501e-9 + (-0.42516755603130443051e-10 + (0.47910437172240209262e-12 + 0.66784341874437478953e-14 * t) * t) * t) * t) * t) * t) * t) * t; + } + case 42: + { + double t = 2 * y100 - 85; + return 0.52968945458607484524e0 + (0.84400880445116786088e-2 + (-0.18908729783854258774e-3 + (-0.32725905467782951931e-5 + (0.91956190588652090659e-7 + (0.14593989152420122909e-9 + (-0.35239490687644444445e-10 + 0.54613829888448694898e-12 * t) * t) * t) * t) * t) * t) * t; + } + case 43: + { + double t = 2 * y100 - 87; + return 0.54578857454330070965e0 + (0.76474155195880295311e-2 + (-0.20651230590808213884e-3 + (-0.25364339140543131706e-5 + (0.91455367999510681979e-7 + (-0.23061359005297528898e-9 + (-0.27512928625244444444e-10 + 0.54895806008493285579e-12 * t) * t) * t) * t) * t) * t) * t; + } + case 44: + { + double t = 2 * y100 - 89; + return 0.56023851910298493910e0 + (0.67938321739997196804e-2 + (-0.21956066613331411760e-3 + (-0.18181127670443266395e-5 + (0.87650335075416845987e-7 + (-0.51548062050366615977e-9 + (-0.20068462174044444444e-10 + 0.50912654909758187264e-12 * t) * t) * t) * t) * t) * t) * t; + } + case 45: + { + double t = 2 * y100 - 91; + return 0.57293478057455721150e0 + (0.58965321010394044087e-2 + (-0.22841145229276575597e-3 + (-0.11404605562013443659e-5 + (0.81430290992322326296e-7 + (-0.71512447242755357629e-9 + (-0.13372664928000000000e-10 + 0.44461498336689298148e-12 * t) * t) * t) * t) * t) * t) * t; + } + case 46: + { + double t = 2 * y100 - 93; + return 0.58380635448407827360e0 + (0.49717469530842831182e-2 + (-0.23336001540009645365e-3 + (-0.51952064448608850822e-6 + (0.73596577815411080511e-7 + (-0.84020916763091566035e-9 + (-0.76700972702222222221e-11 + 0.36914462807972467044e-12 * t) * t) * t) * t) * t) * t) * t; + } + case 47: + { + double t = 2 * y100 - 95; + return 0.59281340237769489597e0 + (0.40343592069379730568e-2 + (-0.23477963738658326185e-3 + (0.34615944987790224234e-7 + (0.64832803248395814574e-7 + (-0.90329163587627007971e-9 + (-0.30421940400000000000e-11 + 0.29237386653743536669e-12 * t) * t) * t) * t) * t) * t) * t; + } + case 48: + { + double t = 2 * y100 - 97; + return 0.59994428743114271918e0 + (0.30976579788271744329e-2 + (-0.23308875765700082835e-3 + (0.51681681023846925160e-6 + (0.55694594264948268169e-7 + (-0.91719117313243464652e-9 + (0.53982743680000000000e-12 + 0.22050829296187771142e-12 * t) * t) * t) * t) * t) * t) * t; + } + case 49: + { + double t = 2 * y100 - 99; + return 0.60521224471819875444e0 + (0.21732138012345456060e-2 + (-0.22872428969625997456e-3 + (0.92588959922653404233e-6 + (0.46612665806531930684e-7 + (-0.89393722514414153351e-9 + (0.31718550353777777778e-11 + 0.15705458816080549117e-12 * t) * t) * t) * t) * t) * t) * t; + } + case 50: + { + double t = 2 * y100 - 101; + return 0.60865189969791123620e0 + (0.12708480848877451719e-2 + (-0.22212090111534847166e-3 + (0.12636236031532793467e-5 + (0.37904037100232937574e-7 + (-0.84417089968101223519e-9 + (0.49843180828444444445e-11 + 0.10355439441049048273e-12 * t) * t) * t) * t) * t) * t) * t; + } + case 51: + { + double t = 2 * y100 - 103; + return 0.61031580103499200191e0 + (0.39867436055861038223e-3 + (-0.21369573439579869291e-3 + (0.15339402129026183670e-5 + (0.29787479206646594442e-7 + (-0.77687792914228632974e-9 + (0.61192452741333333334e-11 + 0.60216691829459295780e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 52: + { + double t = 2 * y100 - 105; + return 0.61027109047879835868e0 + (-0.43680904508059878254e-3 + (-0.20383783788303894442e-3 + (0.17421743090883439959e-5 + (0.22400425572175715576e-7 + (-0.69934719320045128997e-9 + (0.67152759655111111110e-11 + 0.26419960042578359995e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 53: + { + double t = 2 * y100 - 107; + return 0.60859639489217430521e0 + (-0.12305921390962936873e-2 + (-0.19290150253894682629e-3 + (0.18944904654478310128e-5 + (0.15815530398618149110e-7 + (-0.61726850580964876070e-9 + 0.68987888999111111110e-11 * t) * t) * t) * t) * t) * t; + } + case 54: + { + double t = 2 * y100 - 109; + return 0.60537899426486075181e0 + (-0.19790062241395705751e-2 + (-0.18120271393047062253e-3 + (0.19974264162313241405e-5 + (0.10055795094298172492e-7 + (-0.53491997919318263593e-9 + (0.67794550295111111110e-11 - 0.17059208095741511603e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 55: + { + double t = 2 * y100 - 111; + return 0.60071229457904110537e0 + (-0.26795676776166354354e-2 + (-0.16901799553627508781e-3 + (0.20575498324332621581e-5 + (0.51077165074461745053e-8 + (-0.45536079828057221858e-9 + (0.64488005516444444445e-11 - 0.29311677573152766338e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 56: + { + double t = 2 * y100 - 113; + return 0.59469361520112714738e0 + (-0.33308208190600993470e-2 + (-0.15658501295912405679e-3 + (0.20812116912895417272e-5 + (0.93227468760614182021e-9 + (-0.38066673740116080415e-9 + (0.59806790359111111110e-11 - 0.36887077278950440597e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 57: + { + double t = 2 * y100 - 115; + return 0.58742228631775388268e0 + (-0.39321858196059227251e-2 + (-0.14410441141450122535e-3 + (0.20743790018404020716e-5 + (-0.25261903811221913762e-8 + (-0.31212416519526924318e-9 + (0.54328422462222222221e-11 - 0.40864152484979815972e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 58: + { + double t = 2 * y100 - 117; + return 0.57899804200033018447e0 + (-0.44838157005618913447e-2 + (-0.13174245966501437965e-3 + (0.20425306888294362674e-5 + (-0.53330296023875447782e-8 + (-0.25041289435539821014e-9 + (0.48490437205333333334e-11 - 0.42162206939169045177e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 59: + { + double t = 2 * y100 - 119; + return 0.56951968796931245974e0 + (-0.49864649488074868952e-2 + (-0.11963416583477567125e-3 + (0.19906021780991036425e-5 + (-0.75580140299436494248e-8 + (-0.19576060961919820491e-9 + (0.42613011928888888890e-11 - 0.41539443304115604377e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 60: + { + double t = 2 * y100 - 121; + return 0.55908401930063918964e0 + (-0.54413711036826877753e-2 + (-0.10788661102511914628e-3 + (0.19229663322982839331e-5 + (-0.92714731195118129616e-8 + (-0.14807038677197394186e-9 + (0.36920870298666666666e-11 - 0.39603726688419162617e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 61: + { + double t = 2 * y100 - 123; + return 0.54778496152925675315e0 + (-0.58501497933213396670e-2 + (-0.96582314317855227421e-4 + (0.18434405235069270228e-5 + (-0.10541580254317078711e-7 + (-0.10702303407788943498e-9 + (0.31563175582222222222e-11 - 0.36829748079110481422e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 62: + { + double t = 2 * y100 - 125; + return 0.53571290831682823999e0 + (-0.62147030670760791791e-2 + (-0.85782497917111760790e-4 + (0.17553116363443470478e-5 + (-0.11432547349815541084e-7 + (-0.72157091369041330520e-10 + (0.26630811607111111111e-11 - 0.33578660425893164084e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 63: + { + double t = 2 * y100 - 127; + return 0.52295422962048434978e0 + (-0.65371404367776320720e-2 + (-0.75530164941473343780e-4 + (0.16613725797181276790e-5 + (-0.12003521296598910761e-7 + (-0.42929753689181106171e-10 + (0.22170894940444444444e-11 - 0.30117697501065110505e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 64: + { + double t = 2 * y100 - 129; + return 0.50959092577577886140e0 + (-0.68197117603118591766e-2 + (-0.65852936198953623307e-4 + (0.15639654113906716939e-5 + (-0.12308007991056524902e-7 + (-0.18761997536910939570e-10 + (0.18198628922666666667e-11 - 0.26638355362285200932e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 65: + { + double t = 2 * y100 - 131; + return 0.49570040481823167970e0 + (-0.70647509397614398066e-2 + (-0.56765617728962588218e-4 + (0.14650274449141448497e-5 + (-0.12393681471984051132e-7 + (0.92904351801168955424e-12 + (0.14706755960177777778e-11 - 0.23272455351266325318e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 66: + { + double t = 2 * y100 - 133; + return 0.48135536250935238066e0 + (-0.72746293327402359783e-2 + (-0.48272489495730030780e-4 + (0.13661377309113939689e-5 + (-0.12302464447599382189e-7 + (0.16707760028737074907e-10 + (0.11672928324444444444e-11 - 0.20105801424709924499e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 67: + { + double t = 2 * y100 - 135; + return 0.46662374675511439448e0 + (-0.74517177649528487002e-2 + (-0.40369318744279128718e-4 + (0.12685621118898535407e-5 + (-0.12070791463315156250e-7 + (0.29105507892605823871e-10 + (0.90653314645333333334e-12 - 0.17189503312102982646e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 68: + { + double t = 2 * y100 - 137; + return 0.45156879030168268778e0 + (-0.75983560650033817497e-2 + (-0.33045110380705139759e-4 + (0.11732956732035040896e-5 + (-0.11729986947158201869e-7 + (0.38611905704166441308e-10 + (0.68468768305777777779e-12 - 0.14549134330396754575e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 69: + { + double t = 2 * y100 - 139; + return 0.43624909769330896904e0 + (-0.77168291040309554679e-2 + (-0.26283612321339907756e-4 + (0.10811018836893550820e-5 + (-0.11306707563739851552e-7 + (0.45670446788529607380e-10 + (0.49782492549333333334e-12 - 0.12191983967561779442e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 70: + { + double t = 2 * y100 - 141; + return 0.42071877443548481181e0 + (-0.78093484015052730097e-2 + (-0.20064596897224934705e-4 + (0.99254806680671890766e-6 + (-0.10823412088884741451e-7 + (0.50677203326904716247e-10 + (0.34200547594666666666e-12 - 0.10112698698356194618e-13 * t) * t) * t) * t) * t) * t) * t; + } + case 71: + { + double t = 2 * y100 - 143; + return 0.40502758809710844280e0 + (-0.78780384460872937555e-2 + (-0.14364940764532853112e-4 + (0.90803709228265217384e-6 + (-0.10298832847014466907e-7 + (0.53981671221969478551e-10 + (0.21342751381333333333e-12 - 0.82975901848387729274e-14 * t) * t) * t) * t) * t) * t) * t; + } + case 72: + { + double t = 2 * y100 - 145; + return 0.38922115269731446690e0 + (-0.79249269708242064120e-2 + (-0.91595258799106970453e-5 + (0.82783535102217576495e-6 + (-0.97484311059617744437e-8 + (0.55889029041660225629e-10 + (0.10851981336888888889e-12 - 0.67278553237853459757e-14 * t) * t) * t) * t) * t) * t) * t; + } + case 73: + { + double t = 2 * y100 - 147; + return 0.37334112915460307335e0 + (-0.79519385109223148791e-2 + (-0.44219833548840469752e-5 + (0.75209719038240314732e-6 + (-0.91848251458553190451e-8 + (0.56663266668051433844e-10 + (0.23995894257777777778e-13 - 0.53819475285389344313e-14 * t) * t) * t) * t) * t) * t) * t; + } + case 74: + { + double t = 2 * y100 - 149; + return 0.35742543583374223085e0 + (-0.79608906571527956177e-2 + (-0.12530071050975781198e-6 + (0.68088605744900552505e-6 + (-0.86181844090844164075e-8 + (0.56530784203816176153e-10 + (-0.43120012248888888890e-13 - 0.42372603392496813810e-14 * t) * t) * t) * t) * t) * t) * t; + } + case 75: + { + double t = 2 * y100 - 151; + return 0.34150846431979618536e0 + (-0.79534924968773806029e-2 + (0.37576885610891515813e-5 + (0.61419263633090524326e-6 + (-0.80565865409945960125e-8 + (0.55684175248749269411e-10 + (-0.95486860764444444445e-13 - 0.32712946432984510595e-14 * t) * t) * t) * t) * t) * t) * t; + } + case 76: + { + double t = 2 * y100 - 153; + return 0.32562129649136346824e0 + (-0.79313448067948884309e-2 + (0.72539159933545300034e-5 + (0.55195028297415503083e-6 + (-0.75063365335570475258e-8 + (0.54281686749699595941e-10 - 0.13545424295111111111e-12 * t) * t) * t) * t) * t) * t; + } + case 77: + { + double t = 2 * y100 - 155; + return 0.30979191977078391864e0 + (-0.78959416264207333695e-2 + (0.10389774377677210794e-4 + (0.49404804463196316464e-6 + (-0.69722488229411164685e-8 + (0.52469254655951393842e-10 - 0.16507860650666666667e-12 * t) * t) * t) * t) * t) * t; + } + case 78: + { + double t = 2 * y100 - 157; + return 0.29404543811214459904e0 + (-0.78486728990364155356e-2 + (0.13190885683106990459e-4 + (0.44034158861387909694e-6 + (-0.64578942561562616481e-8 + (0.50354306498006928984e-10 - 0.18614473550222222222e-12 * t) * t) * t) * t) * t) * t; + } + case 79: + { + double t = 2 * y100 - 159; + return 0.27840427686253660515e0 + (-0.77908279176252742013e-2 + (0.15681928798708548349e-4 + (0.39066226205099807573e-6 + (-0.59658144820660420814e-8 + (0.48030086420373141763e-10 - 0.20018995173333333333e-12 * t) * t) * t) * t) * t) * t; + } + case 80: + { + double t = 2 * y100 - 161; + return 0.26288838011163800908e0 + (-0.77235993576119469018e-2 + (0.17886516796198660969e-4 + (0.34482457073472497720e-6 + (-0.54977066551955420066e-8 + (0.45572749379147269213e-10 - 0.20852924954666666667e-12 * t) * t) * t) * t) * t) * t; + } + case 81: + { + double t = 2 * y100 - 163; + return 0.24751539954181029717e0 + (-0.76480877165290370975e-2 + (0.19827114835033977049e-4 + (0.30263228619976332110e-6 + (-0.50545814570120129947e-8 + (0.43043879374212005966e-10 - 0.21228012028444444444e-12 * t) * t) * t) * t) * t) * t; + } + case 82: + { + double t = 2 * y100 - 165; + return 0.23230087411688914593e0 + (-0.75653060136384041587e-2 + (0.21524991113020016415e-4 + (0.26388338542539382413e-6 + (-0.46368974069671446622e-8 + (0.40492715758206515307e-10 - 0.21238627815111111111e-12 * t) * t) * t) * t) * t) * t; + } + case 83: + { + double t = 2 * y100 - 167; + return 0.21725840021297341931e0 + (-0.74761846305979730439e-2 + (0.23000194404129495243e-4 + (0.22837400135642906796e-6 + (-0.42446743058417541277e-8 + (0.37958104071765923728e-10 - 0.20963978568888888889e-12 * t) * t) * t) * t) * t) * t; + } + case 84: + { + double t = 2 * y100 - 169; + return 0.20239979200788191491e0 + (-0.73815761980493466516e-2 + (0.24271552727631854013e-4 + (0.19590154043390012843e-6 + (-0.38775884642456551753e-8 + (0.35470192372162901168e-10 - 0.20470131678222222222e-12 * t) * t) * t) * t) * t) * t; + } + case 85: + { + double t = 2 * y100 - 171; + return 0.18773523211558098962e0 + (-0.72822604530339834448e-2 + (0.25356688567841293697e-4 + (0.16626710297744290016e-6 + (-0.35350521468015310830e-8 + (0.33051896213898864306e-10 - 0.19811844544000000000e-12 * t) * t) * t) * t) * t) * t; + } + case 86: + { + double t = 2 * y100 - 173; + return 0.17327341258479649442e0 + (-0.71789490089142761950e-2 + (0.26272046822383820476e-4 + (0.13927732375657362345e-6 + (-0.32162794266956859603e-8 + (0.30720156036105652035e-10 - 0.19034196304000000000e-12 * t) * t) * t) * t) * t) * t; + } + case 87: + { + double t = 2 * y100 - 175; + return 0.15902166648328672043e0 + (-0.70722899934245504034e-2 + (0.27032932310132226025e-4 + (0.11474573347816568279e-6 + (-0.29203404091754665063e-8 + (0.28487010262547971859e-10 - 0.18174029063111111111e-12 * t) * t) * t) * t) * t) * t; + } + case 88: + { + double t = 2 * y100 - 177; + return 0.14498609036610283865e0 + (-0.69628725220045029273e-2 + (0.27653554229160596221e-4 + (0.92493727167393036470e-7 + (-0.26462055548683583849e-8 + (0.26360506250989943739e-10 - 0.17261211260444444444e-12 * t) * t) * t) * t) * t) * t; + } + case 89: + { + double t = 2 * y100 - 179; + return 0.13117165798208050667e0 + (-0.68512309830281084723e-2 + (0.28147075431133863774e-4 + (0.72351212437979583441e-7 + (-0.23927816200314358570e-8 + (0.24345469651209833155e-10 - 0.16319736960000000000e-12 * t) * t) * t) * t) * t) * t; + } + case 90: + { + double t = 2 * y100 - 181; + return 0.11758232561160626306e0 + (-0.67378491192463392927e-2 + (0.28525664781722907847e-4 + (0.54156999310046790024e-7 + (-0.21589405340123827823e-8 + (0.22444150951727334619e-10 - 0.15368675584000000000e-12 * t) * t) * t) * t) * t) * t; + } + case 91: + { + double t = 2 * y100 - 183; + return 0.10422112945361673560e0 + (-0.66231638959845581564e-2 + (0.28800551216363918088e-4 + (0.37758983397952149613e-7 + (-0.19435423557038933431e-8 + (0.20656766125421362458e-10 - 0.14422990012444444444e-12 * t) * t) * t) * t) * t) * t; + } + case 92: + { + double t = 2 * y100 - 185; + return 0.91090275493541084785e-1 + (-0.65075691516115160062e-2 + (0.28982078385527224867e-4 + (0.23014165807643012781e-7 + (-0.17454532910249875958e-8 + (0.18981946442680092373e-10 - 0.13494234691555555556e-12 * t) * t) * t) * t) * t) * t; + } + case 93: + { + double t = 2 * y100 - 187; + return 0.78191222288771379358e-1 + (-0.63914190297303976434e-2 + (0.29079759021299682675e-4 + (0.97885458059415717014e-8 + (-0.15635596116134296819e-8 + (0.17417110744051331974e-10 - 0.12591151763555555556e-12 * t) * t) * t) * t) * t) * t; + } + case 94: + { + double t = 2 * y100 - 189; + return 0.65524757106147402224e-1 + (-0.62750311956082444159e-2 + (0.29102328354323449795e-4 + (-0.20430838882727954582e-8 + (-0.13967781903855367270e-8 + (0.15958771833747057569e-10 - 0.11720175765333333333e-12 * t) * t) * t) * t) * t) * t; + } + case 95: + { + double t = 2 * y100 - 191; + return 0.53091065838453612773e-1 + (-0.61586898417077043662e-2 + (0.29057796072960100710e-4 + (-0.12597414620517987536e-7 + (-0.12440642607426861943e-8 + (0.14602787128447932137e-10 - 0.10885859114666666667e-12 * t) * t) * t) * t) * t) * t; + } + case 96: + { + double t = 2 * y100 - 193; + return 0.40889797115352738582e-1 + (-0.60426484889413678200e-2 + (0.28953496450191694606e-4 + (-0.21982952021823718400e-7 + (-0.11044169117553026211e-8 + (0.13344562332430552171e-10 - 0.10091231402844444444e-12 * t) * t) * t) * t) * t) * t; + } + case 97: + case 98: + case 99: + case 100: // use Taylor expansion for small x (|x| <= 0.0309...) + { + // (2/sqrt(pi)) * (x - 2/3 x^3 + 4/15 x^5 - 8/105 x^7 + 16/945 x^9) + double x2 = x * x; + return x * (1.1283791670955125739 + - x2 * (0.75225277806367504925 + - x2 * (0.30090111122547001970 + - x2 * (0.085971746064420005629 + - x2 * 0.016931216931216931217)))); + } + } + /* Since 0 <= y100 < 101, this is only reached if x is NaN, + in which case we should return NaN. */ + return NaN; +} + +double FADDEEVA(w_im)(double x) +{ + if (x >= 0) + { + if (x > 45) // continued-fraction expansion is faster + { + const double ispi = 0.56418958354775628694807945156; // 1 / sqrt(pi) + if (x > 5e7) // 1-term expansion, important to avoid overflow + { + return ispi / x; + } + /* 5-term expansion (rely on compiler for CSE), simplified from: + ispi / (x-0.5/(x-1/(x-1.5/(x-2/x)))) */ + return ispi * ((x * x) * (x * x - 4.5) + 2) / (x * ((x * x) * (x * x - 5) + 3.75)); + } + return w_im_y100(100 / (1 + x), x); + } + else // = -FADDEEVA(w_im)(-x) + { + if (x < -45) // continued-fraction expansion is faster + { + const double ispi = 0.56418958354775628694807945156; // 1 / sqrt(pi) + if (x < -5e7) // 1-term expansion, important to avoid overflow + { + return ispi / x; + } + /* 5-term expansion (rely on compiler for CSE), simplified from: + ispi / (x-0.5/(x-1/(x-1.5/(x-2/x)))) */ + return ispi * ((x * x) * (x * x - 4.5) + 2) / (x * ((x * x) * (x * x - 5) + 3.75)); + } + return -w_im_y100(100 / (1 - x), -x); + } +} + +///////////////////////////////////////////////////////////////////////// + +// Compile with -DTEST_FADDEEVA to compile a little test program +#ifdef TEST_FADDEEVA + +#ifdef __cplusplus +# include <cstdio> +#else +# include <stdio.h> +#endif + +// compute relative error |b-a|/|a|, handling case of NaN and Inf, +static double relerr(double a, double b) +{ + if (isnan(a) || isnan(b) || isinf(a) || isinf(b)) + { + if ((isnan(a) && !isnan(b)) || (!isnan(a) && isnan(b)) || + (isinf(a) && !isinf(b)) || (!isinf(a) && isinf(b)) || + (isinf(a) && isinf(b) && a * b < 0)) + { + return Inf; // "infinite" error + } + return 0; // matching infinity/nan results counted as zero error + } + if (a == 0) + { + return b == 0 ? 0 : Inf; + } + else + { + return fabs((b - a) / a); + } +} + +int main(void) +{ + double errmax_all = 0; + { + printf("############# w(z) tests #############\n"); +#define NTST 57 // define instead of const for C compatibility + cmplx z[NTST] = + { + C(624.2, -0.26123), + C(-0.4, 3.), + C(0.6, 2.), + C(-1., 1.), + C(-1., -9.), + C(-1., 9.), + C(-0.0000000234545, 1.1234), + C(-3., 5.1), + C(-53, 30.1), + C(0.0, 0.12345), + C(11, 1), + C(-22, -2), + C(9, -28), + C(21, -33), + C(1e5, 1e5), + C(1e14, 1e14), + C(-3001, -1000), + C(1e160, -1e159), + C(-6.01, 0.01), + C(-0.7, -0.7), + C(2.611780000000000e+01, 4.540909610972489e+03), + C(0.8e7, 0.3e7), + C(-20, -19.8081), + C(1e-16, -1.1e-16), + C(2.3e-8, 1.3e-8), + C(6.3, -1e-13), + C(6.3, 1e-20), + C(1e-20, 6.3), + C(1e-20, 16.3), + C(9, 1e-300), + C(6.01, 0.11), + C(8.01, 1.01e-10), + C(28.01, 1e-300), + C(10.01, 1e-200), + C(10.01, -1e-200), + C(10.01, 0.99e-10), + C(10.01, -0.99e-10), + C(1e-20, 7.01), + C(-1, 7.01), + C(5.99, 7.01), + C(1, 0), + C(55, 0), + C(-0.1, 0), + C(1e-20, 0), + C(0, 5e-14), + C(0, 51), + C(Inf, 0), + C(-Inf, 0), + C(0, Inf), + C(0, -Inf), + C(Inf, Inf), + C(Inf, -Inf), + C(NaN, NaN), + C(NaN, 0), + C(0, NaN), + C(NaN, Inf), + C(Inf, NaN) + }; + cmplx w[NTST] = { /* w(z), computed with WolframAlpha + ... note that WolframAlpha is problematic + some of the above inputs, so I had to + use the continued-fraction expansion + in WolframAlpha in some cases, or switch + to Maple */ + C(-3.78270245518980507452677445620103199303131110e-7, + 0.000903861276433172057331093754199933411710053155), + C(0.1764906227004816847297495349730234591778719532788, + -0.02146550539468457616788719893991501311573031095617), + C(0.2410250715772692146133539023007113781272362309451, + 0.06087579663428089745895459735240964093522265589350), + C(0.30474420525691259245713884106959496013413834051768, + -0.20821893820283162728743734725471561394145872072738), + C(7.317131068972378096865595229600561710140617977e34, + 8.321873499714402777186848353320412813066170427e34), + C(0.0615698507236323685519612934241429530190806818395, + -0.00676005783716575013073036218018565206070072304635), + C(0.3960793007699874918961319170187598400134746631, + -5.593152259116644920546186222529802777409274656e-9), + C(0.08217199226739447943295069917990417630675021771804, + -0.04701291087643609891018366143118110965272615832184), + C(0.00457246000350281640952328010227885008541748668738, + -0.00804900791411691821818731763401840373998654987934), + C(0.8746342859608052666092782112565360755791467973338452, + 0.), + C(0.00468190164965444174367477874864366058339647648741, + 0.0510735563901306197993676329845149741675029197050), + C(-0.0023193175200187620902125853834909543869428763219, + -0.025460054739731556004902057663500272721780776336), + C(9.11463368405637174660562096516414499772662584e304, + 3.97101807145263333769664875189354358563218932e305), + C(-4.4927207857715598976165541011143706155432296e281, + -2.8019591213423077494444700357168707775769028e281), + C(2.820947917809305132678577516325951485807107151e-6, + 2.820947917668257736791638444590253942253354058e-6), + C(2.82094791773878143474039725787438662716372268e-15, + 2.82094791773878143474039725773333923127678361e-15), + C(-0.0000563851289696244350147899376081488003110150498, + -0.000169211755126812174631861529808288295454992688), + C(-5.586035480670854326218608431294778077663867e-162, + 5.586035480670854326218608431294778077663867e-161), + C(0.00016318325137140451888255634399123461580248456, + -0.095232456573009287370728788146686162555021209999), + C(0.69504753678406939989115375989939096800793577783885, + -1.8916411171103639136680830887017670616339912024317), + C(0.0001242418269653279656612334210746733213167234822, + 7.145975826320186888508563111992099992116786763e-7), + C(2.318587329648353318615800865959225429377529825e-8, + 6.182899545728857485721417893323317843200933380e-8), + C(-0.0133426877243506022053521927604277115767311800303, + -0.0148087097143220769493341484176979826888871576145), + C(1.00000000000000012412170838050638522857747934, + 1.12837916709551279389615890312156495593616433e-16), + C(0.9999999853310704677583504063775310832036830015, + 2.595272024519678881897196435157270184030360773e-8), + C(-1.4731421795638279504242963027196663601154624e-15, + 0.090727659684127365236479098488823462473074709), + C(5.79246077884410284575834156425396800754409308e-18, + 0.0907276596841273652364790985059772809093822374), + C(0.0884658993528521953466533278764830881245144368, + 1.37088352495749125283269718778582613192166760e-22), + C(0.0345480845419190424370085249304184266813447878, + 2.11161102895179044968099038990446187626075258e-23), + C(6.63967719958073440070225527042829242391918213e-36, + 0.0630820900592582863713653132559743161572639353), + C(0.00179435233208702644891092397579091030658500743634, + 0.0951983814805270647939647438459699953990788064762), + C(9.09760377102097999924241322094863528771095448e-13, + 0.0709979210725138550986782242355007611074966717), + C(7.2049510279742166460047102593255688682910274423e-304, + 0.0201552956479526953866611812593266285000876784321), + C(3.04543604652250734193622967873276113872279682e-44, + 0.0566481651760675042930042117726713294607499165), + C(3.04543604652250734193622967873276113872279682e-44, + 0.0566481651760675042930042117726713294607499165), + C(0.5659928732065273429286988428080855057102069081e-12, + 0.056648165176067504292998527162143030538756683302), + C(-0.56599287320652734292869884280802459698927645e-12, + 0.0566481651760675042929985271621430305387566833029), + C(0.0796884251721652215687859778119964009569455462, + 1.11474461817561675017794941973556302717225126e-22), + C(0.07817195821247357458545539935996687005781943386550, + -0.01093913670103576690766705513142246633056714279654), + C(0.04670032980990449912809326141164730850466208439937, + 0.03944038961933534137558064191650437353429669886545), + C(0.36787944117144232159552377016146086744581113103176, + 0.60715770584139372911503823580074492116122092866515), + C(0, + 0.010259688805536830986089913987516716056946786526145), + C(0.99004983374916805357390597718003655777207908125383, + -0.11208866436449538036721343053869621153527769495574), + C(0.99999999999999999999999999999999999999990000, + 1.12837916709551257389615890312154517168802603e-20), + C(0.999999999999943581041645226871305192054749891144158, + 0), + C(0.0110604154853277201542582159216317923453996211744250, + 0), + C(0, 0), + C(0, 0), + C(0, 0), + C(Inf, 0), + C(0, 0), + C(NaN, NaN), + C(NaN, NaN), + C(NaN, NaN), + C(NaN, 0), + C(NaN, NaN), + C(NaN, NaN) + }; + double errmax = 0; + for (int i = 0; i < NTST; ++i) + { + cmplx fw = FADDEEVA(w)(z[i], 0.); + double re_err = relerr(creal(w[i]), creal(fw)); + double im_err = relerr(cimag(w[i]), cimag(fw)); + printf("w(%g%+gi) = %g%+gi (vs. %g%+gi), re/im rel. err. = %0.2g/%0.2g)\n", + creal(z[i]), cimag(z[i]), creal(fw), cimag(fw), creal(w[i]), cimag(w[i]), + re_err, im_err); + if (re_err > errmax) + { + errmax = re_err; + } + if (im_err > errmax) + { + errmax = im_err; + } + } + if (errmax > 1e-13) + { + printf("FAILURE -- relative error %g too large!\n", errmax); + return 1; + } + printf("SUCCESS (max relative error = %g)\n", errmax); + if (errmax > errmax_all) + { + errmax_all = errmax; + } + } + { +#undef NTST +#define NTST 41 // define instead of const for C compatibility + cmplx z[NTST] = + { + C(1, 2), + C(-1, 2), + C(1, -2), + C(-1, -2), + C(9, -28), + C(21, -33), + C(1e3, 1e3), + C(-3001, -1000), + C(1e160, -1e159), + C(5.1e-3, 1e-8), + C(-4.9e-3, 4.95e-3), + C(4.9e-3, 0.5), + C(4.9e-4, -0.5e1), + C(-4.9e-5, -0.5e2), + C(5.1e-3, 0.5), + C(5.1e-4, -0.5e1), + C(-5.1e-5, -0.5e2), + C(1e-6, 2e-6), + C(0, 2e-6), + C(0, 2), + C(0, 20), + C(0, 200), + C(Inf, 0), + C(-Inf, 0), + C(0, Inf), + C(0, -Inf), + C(Inf, Inf), + C(Inf, -Inf), + C(NaN, NaN), + C(NaN, 0), + C(0, NaN), + C(NaN, Inf), + C(Inf, NaN), + C(1e-3, NaN), + C(7e-2, 7e-2), + C(7e-2, -7e-4), + C(-9e-2, 7e-4), + C(-9e-2, 9e-2), + C(-7e-4, 9e-2), + C(7e-2, 0.9e-2), + C(7e-2, 1.1e-2) + }; + cmplx w[NTST] = // erf(z[i]), evaluated with Maple + { + C(-0.5366435657785650339917955593141927494421, + -5.049143703447034669543036958614140565553), + C(0.5366435657785650339917955593141927494421, + -5.049143703447034669543036958614140565553), + C(-0.5366435657785650339917955593141927494421, + 5.049143703447034669543036958614140565553), + C(0.5366435657785650339917955593141927494421, + 5.049143703447034669543036958614140565553), + C(0.3359473673830576996788000505817956637777e304, + -0.1999896139679880888755589794455069208455e304), + C(0.3584459971462946066523939204836760283645e278, + 0.3818954885257184373734213077678011282505e280), + C(0.9996020422657148639102150147542224526887, + 0.00002801044116908227889681753993542916894856), + C(-1, 0), + C(1, 0), + C(0.005754683859034800134412990541076554934877, + 0.1128349818335058741511924929801267822634e-7), + C(-0.005529149142341821193633460286828381876955, + 0.005585388387864706679609092447916333443570), + C(0.007099365669981359632319829148438283865814, + 0.6149347012854211635026981277569074001219), + C(0.3981176338702323417718189922039863062440e8, + -0.8298176341665249121085423917575122140650e10), + C(-Inf, + -Inf), + C(0.007389128308257135427153919483147229573895, + 0.6149332524601658796226417164791221815139), + C(0.4143671923267934479245651547534414976991e8, + -0.8298168216818314211557046346850921446950e10), + C(-Inf, + -Inf), + C(0.1128379167099649964175513742247082845155e-5, + 0.2256758334191777400570377193451519478895e-5), + C(0, + 0.2256758334194034158904576117253481476197e-5), + C(0, + 18.56480241457555259870429191324101719886), + C(0, + 0.1474797539628786202447733153131835124599e173), + C(0, + Inf), + C(1, 0), + C(-1, 0), + C(0, Inf), + C(0, -Inf), + C(NaN, NaN), + C(NaN, NaN), + C(NaN, NaN), + C(NaN, 0), + C(0, NaN), + C(NaN, NaN), + C(NaN, NaN), + C(NaN, NaN), + C(0.07924380404615782687930591956705225541145, + 0.07872776218046681145537914954027729115247), + C(0.07885775828512276968931773651224684454495, + -0.0007860046704118224342390725280161272277506), + C(-0.1012806432747198859687963080684978759881, + 0.0007834934747022035607566216654982820299469), + C(-0.1020998418798097910247132140051062512527, + 0.1010030778892310851309082083238896270340), + C(-0.0007962891763147907785684591823889484764272, + 0.1018289385936278171741809237435404896152), + C(0.07886408666470478681566329888615410479530, + 0.01010604288780868961492224347707949372245), + C(0.07886723099940260286824654364807981336591, + 0.01235199327873258197931147306290916629654) + }; +#define TST(f,isc) \ + printf("############# " #f "(z) tests #############\n"); \ + double errmax = 0; \ + for (int i = 0; i < NTST; ++i) { \ + cmplx fw = FADDEEVA(f)(z[i],0.); \ + double re_err = relerr(creal(w[i]), creal(fw)); \ + double im_err = relerr(cimag(w[i]), cimag(fw)); \ + printf(#f "(%g%+gi) = %g%+gi (vs. %g%+gi), re/im rel. err. = %0.2g/%0.2g)\n", \ + creal(z[i]),cimag(z[i]), creal(fw),cimag(fw), creal(w[i]),cimag(w[i]), \ + re_err, im_err); \ + if (re_err > errmax) errmax = re_err; \ + if (im_err > errmax) errmax = im_err; \ + } \ + if (errmax > 1e-13) { \ + printf("FAILURE -- relative error %g too large!\n", errmax); \ + return 1; \ + } \ + printf("Checking " #f "(x) special case...\n"); \ + for (int i = 0; i < 10000; ++i) { \ + double x = pow(10., -300. + i * 600. / (10000 - 1)); \ + double re_err = relerr(FADDEEVA_RE(f)(x), \ + creal(FADDEEVA(f)(C(x,x*isc),0.))); \ + if (re_err > errmax) errmax = re_err; \ + re_err = relerr(FADDEEVA_RE(f)(-x), \ + creal(FADDEEVA(f)(C(-x,x*isc),0.))); \ + if (re_err > errmax) errmax = re_err; \ + } \ + { \ + double re_err = relerr(FADDEEVA_RE(f)(Inf), \ + creal(FADDEEVA(f)(C(Inf,0.),0.))); \ + if (re_err > errmax) errmax = re_err; \ + re_err = relerr(FADDEEVA_RE(f)(-Inf), \ + creal(FADDEEVA(f)(C(-Inf,0.),0.))); \ + if (re_err > errmax) errmax = re_err; \ + re_err = relerr(FADDEEVA_RE(f)(NaN), \ + creal(FADDEEVA(f)(C(NaN,0.),0.))); \ + if (re_err > errmax) errmax = re_err; \ + } \ + if (errmax > 1e-13) { \ + printf("FAILURE -- relative error %g too large!\n", errmax); \ + return 1; \ + } \ + printf("SUCCESS (max relative error = %g)\n", errmax); \ + if (errmax > errmax_all) errmax_all = errmax + + TST(erf, 1e-20); + } + { + // since erfi just calls through to erf, just one test should + // be sufficient to make sure I didn't screw up the signs or something +#undef NTST +#define NTST 1 // define instead of const for C compatibility + cmplx z[NTST] = { C(1.234, 0.5678) }; + cmplx w[NTST] = // erfi(z[i]), computed with Maple + { + C(1.081032284405373149432716643834106923212, + 1.926775520840916645838949402886591180834) + }; + TST(erfi, 0); + } + { + // since erfcx just calls through to w, just one test should + // be sufficient to make sure I didn't screw up the signs or something +#undef NTST +#define NTST 1 // define instead of const for C compatibility + cmplx z[NTST] = { C(1.234, 0.5678) }; + cmplx w[NTST] = // erfcx(z[i]), computed with Maple + { + C(0.3382187479799972294747793561190487832579, + -0.1116077470811648467464927471872945833154) + }; + TST(erfcx, 0); + } + { +#undef NTST +#define NTST 30 // define instead of const for C compatibility + cmplx z[NTST] = + { + C(1, 2), + C(-1, 2), + C(1, -2), + C(-1, -2), + C(9, -28), + C(21, -33), + C(1e3, 1e3), + C(-3001, -1000), + C(1e160, -1e159), + C(5.1e-3, 1e-8), + C(0, 2e-6), + C(0, 2), + C(0, 20), + C(0, 200), + C(2e-6, 0), + C(2, 0), + C(20, 0), + C(200, 0), + C(Inf, 0), + C(-Inf, 0), + C(0, Inf), + C(0, -Inf), + C(Inf, Inf), + C(Inf, -Inf), + C(NaN, NaN), + C(NaN, 0), + C(0, NaN), + C(NaN, Inf), + C(Inf, NaN), + C(88, 0) + }; + cmplx w[NTST] = // erfc(z[i]), evaluated with Maple + { + C(1.536643565778565033991795559314192749442, + 5.049143703447034669543036958614140565553), + C(0.4633564342214349660082044406858072505579, + 5.049143703447034669543036958614140565553), + C(1.536643565778565033991795559314192749442, + -5.049143703447034669543036958614140565553), + C(0.4633564342214349660082044406858072505579, + -5.049143703447034669543036958614140565553), + C(-0.3359473673830576996788000505817956637777e304, + 0.1999896139679880888755589794455069208455e304), + C(-0.3584459971462946066523939204836760283645e278, + -0.3818954885257184373734213077678011282505e280), + C(0.0003979577342851360897849852457775473112748, + -0.00002801044116908227889681753993542916894856), + C(2, 0), + C(0, 0), + C(0.9942453161409651998655870094589234450651, + -0.1128349818335058741511924929801267822634e-7), + C(1, + -0.2256758334194034158904576117253481476197e-5), + C(1, + -18.56480241457555259870429191324101719886), + C(1, + -0.1474797539628786202447733153131835124599e173), + C(1, -Inf), + C(0.9999977432416658119838633199332831406314, + 0), + C(0.004677734981047265837930743632747071389108, + 0), + C(0.5395865611607900928934999167905345604088e-175, + 0), + C(0, 0), + C(0, 0), + C(2, 0), + C(1, -Inf), + C(1, Inf), + C(NaN, NaN), + C(NaN, NaN), + C(NaN, NaN), + C(NaN, 0), + C(1, NaN), + C(NaN, NaN), + C(NaN, NaN), + C(0, 0) + }; + TST(erfc, 1e-20); + } + { +#undef NTST +#define NTST 48 // define instead of const for C compatibility + cmplx z[NTST] = + { + C(2, 1), + C(-2, 1), + C(2, -1), + C(-2, -1), + C(-28, 9), + C(33, -21), + C(1e3, 1e3), + C(-1000, -3001), + C(1e-8, 5.1e-3), + C(4.95e-3, -4.9e-3), + C(5.1e-3, 5.1e-3), + C(0.5, 4.9e-3), + C(-0.5e1, 4.9e-4), + C(-0.5e2, -4.9e-5), + C(0.5e3, 4.9e-6), + C(0.5, 5.1e-3), + C(-0.5e1, 5.1e-4), + C(-0.5e2, -5.1e-5), + C(1e-6, 2e-6), + C(2e-6, 0), + C(2, 0), + C(20, 0), + C(200, 0), + C(0, 4.9e-3), + C(0, -5.1e-3), + C(0, 2e-6), + C(0, -2), + C(0, 20), + C(0, -200), + C(Inf, 0), + C(-Inf, 0), + C(0, Inf), + C(0, -Inf), + C(Inf, Inf), + C(Inf, -Inf), + C(NaN, NaN), + C(NaN, 0), + C(0, NaN), + C(NaN, Inf), + C(Inf, NaN), + C(39, 6.4e-5), + C(41, 6.09e-5), + C(4.9e7, 5e-11), + C(5.1e7, 4.8e-11), + C(1e9, 2.4e-12), + C(1e11, 2.4e-14), + C(1e13, 2.4e-16), + C(1e300, 2.4e-303) + }; + cmplx w[NTST] = // dawson(z[i]), evaluated with Maple + { + C(0.1635394094345355614904345232875688576839, + -0.1531245755371229803585918112683241066853), + C(-0.1635394094345355614904345232875688576839, + -0.1531245755371229803585918112683241066853), + C(0.1635394094345355614904345232875688576839, + 0.1531245755371229803585918112683241066853), + C(-0.1635394094345355614904345232875688576839, + 0.1531245755371229803585918112683241066853), + C(-0.01619082256681596362895875232699626384420, + -0.005210224203359059109181555401330902819419), + C(0.01078377080978103125464543240346760257008, + 0.006866888783433775382193630944275682670599), + C(-0.5808616819196736225612296471081337245459, + 0.6688593905505562263387760667171706325749), + C(Inf, + -Inf), + C(0.1000052020902036118082966385855563526705e-7, + 0.005100088434920073153418834680320146441685), + C(0.004950156837581592745389973960217444687524, + -0.004899838305155226382584756154100963570500), + C(0.005100176864319675957314822982399286703798, + 0.005099823128319785355949825238269336481254), + C(0.4244534840871830045021143490355372016428, + 0.002820278933186814021399602648373095266538), + C(-0.1021340733271046543881236523269967674156, + -0.00001045696456072005761498961861088944159916), + C(-0.01000200120119206748855061636187197886859, + 0.9805885888237419500266621041508714123763e-8), + C(0.001000002000012000023960527532953151819595, + -0.9800058800588007290937355024646722133204e-11), + C(0.4244549085628511778373438768121222815752, + 0.002935393851311701428647152230552122898291), + C(-0.1021340732357117208743299813648493928105, + -0.00001088377943049851799938998805451564893540), + C(-0.01000200120119126652710792390331206563616, + 0.1020612612857282306892368985525393707486e-7), + C(0.1000000000007333333333344266666666664457e-5, + 0.2000000000001333333333323199999999978819e-5), + C(0.1999999999994666666666675199999999990248e-5, + 0), + C(0.3013403889237919660346644392864226952119, + 0), + C(0.02503136792640367194699495234782353186858, + 0), + C(0.002500031251171948248596912483183760683918, + 0), + C(0, 0.004900078433419939164774792850907128053308), + C(0, -0.005100088434920074173454208832365950009419), + C(0, 0.2000000000005333333333341866666666676419e-5), + C(0, -48.16001211429122974789822893525016528191), + C(0, 0.4627407029504443513654142715903005954668e174), + C(0, -Inf), + C(0, 0), + C(-0, 0), + C(0, Inf), + C(0, -Inf), + C(NaN, NaN), + C(NaN, NaN), + C(NaN, NaN), + C(NaN, 0), + C(0, NaN), + C(NaN, NaN), + C(NaN, NaN), + C(0.01282473148489433743567240624939698290584, + -0.2105957276516618621447832572909153498104e-7), + C(0.01219875253423634378984109995893708152885, + -0.1813040560401824664088425926165834355953e-7), + C(0.1020408163265306334945473399689037886997e-7, + -0.1041232819658476285651490827866174985330e-25), + C(0.9803921568627452865036825956835185367356e-8, + -0.9227220299884665067601095648451913375754e-26), + C(0.5000000000000000002500000000000000003750e-9, + -0.1200000000000000001800000188712838420241e-29), + C(5.00000000000000000000025000000000000000000003e-12, + -1.20000000000000000000018000000000000000000004e-36), + C(5.00000000000000000000000002500000000000000000e-14, + -1.20000000000000000000000001800000000000000000e-42), + C(5e-301, 0) + }; + TST(Dawson, 1e-20); + } + printf("#####################################\n"); + printf("SUCCESS (max relative error = %g)\n", errmax_all); +} + +#endif diff --git a/modules/special_functions/src/cpp/faddeeva.h b/modules/special_functions/src/cpp/faddeeva.h new file mode 100755 index 000000000..854e41710 --- /dev/null +++ b/modules/special_functions/src/cpp/faddeeva.h @@ -0,0 +1,64 @@ +/* Copyright (c) 2012 Massachusetts Institute of Technology + * + * Permission is hereby granted, free of charge, to any person obtaining + * a copy of this software and associated documentation files (the + * "Software"), to deal in the Software without restriction, including + * without limitation the rights to use, copy, modify, merge, publish, + * distribute, sublicense, and/or sell copies of the Software, and to + * permit persons to whom the Software is furnished to do so, subject to + * the following conditions: + * + * The above copyright notice and this permission notice shall be + * included in all copies or substantial portions of the Software. + * + * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, + * EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF + * MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND + * NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE + * LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION + * OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION + * WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. + */ + +/* Available at: http://ab-initio.mit.edu/Faddeeva + + Header file for Faddeeva.cc; see that file for more information. */ + +#ifndef FADDEEVA_HH +#define FADDEEVA_HH 1 + +#include <complex> +#include "dynlib_elementary_functions.h" + +namespace Faddeeva +{ + +// compute w(z) = exp(-z^2) erfc(-iz) [ Faddeeva / scaled complex error func ] +std::complex<double> w(std::complex<double> z, double relerr = 0); +double w_im(double x); // special-case code for Im[w(x)] of real x + +// Various functions that we can compute with the help of w(z) + +// compute erfcx(z) = exp(z^2) erfc(z) +std::complex<double> erfcx(std::complex<double> z, double relerr = 0); +double erfcx(double x); // special case for real x + +// compute erf(z), the error function of complex arguments +std::complex<double> erf(std::complex<double> z, double relerr = 0); +double erf(double x); // special case for real x + +// compute erfi(z) = -i erf(iz), the imaginary error function +std::complex<double> erfi(std::complex<double> z, double relerr = 0); +double erfi(double x); // special case for real x + +// compute erfc(z) = 1 - erf(z), the complementary error function +std::complex<double> erfc(std::complex<double> z, double relerr = 0); +double erfc(double x); // special case for real x + +// compute Dawson(z) = sqrt(pi)/2 * exp(-z^2) * erfi(z) +std::complex<double> Dawson(std::complex<double> z, double relerr = 0); +double Dawson(double x); // special case for real x + +} // namespace Faddeeva + +#endif // FADDEEVA_HH diff --git a/modules/special_functions/src/cpp/libscispecial_functions_algo_la-faddeeva.lo b/modules/special_functions/src/cpp/libscispecial_functions_algo_la-faddeeva.lo new file mode 100755 index 000000000..daa5e7d28 --- /dev/null +++ b/modules/special_functions/src/cpp/libscispecial_functions_algo_la-faddeeva.lo @@ -0,0 +1,12 @@ +# src/cpp/libscispecial_functions_algo_la-faddeeva.lo - a libtool object file +# Generated by libtool (GNU libtool) 2.4.2 +# +# Please DO NOT delete this file! +# It is necessary for linking the library. + +# Name of the PIC object. +pic_object='.libs/libscispecial_functions_algo_la-faddeeva.o' + +# Name of the non-PIC object +non_pic_object=none + diff --git a/modules/special_functions/src/fortran/.deps/.dirstamp b/modules/special_functions/src/fortran/.deps/.dirstamp new file mode 100755 index 000000000..e69de29bb --- /dev/null +++ b/modules/special_functions/src/fortran/.deps/.dirstamp diff --git a/modules/special_functions/src/fortran/.dirstamp b/modules/special_functions/src/fortran/.dirstamp new file mode 100755 index 000000000..e69de29bb --- /dev/null +++ b/modules/special_functions/src/fortran/.dirstamp diff --git a/modules/special_functions/src/fortran/.libs/dbesig.o b/modules/special_functions/src/fortran/.libs/dbesig.o Binary files differnew file mode 100755 index 000000000..245b57af2 --- /dev/null +++ b/modules/special_functions/src/fortran/.libs/dbesig.o diff --git a/modules/special_functions/src/fortran/.libs/dbesjg.o b/modules/special_functions/src/fortran/.libs/dbesjg.o Binary files differnew file mode 100755 index 000000000..f0ac5402a --- /dev/null +++ b/modules/special_functions/src/fortran/.libs/dbesjg.o diff --git a/modules/special_functions/src/fortran/.libs/dbeskg.o b/modules/special_functions/src/fortran/.libs/dbeskg.o Binary files differnew file mode 100755 index 000000000..70e7197aa --- /dev/null +++ b/modules/special_functions/src/fortran/.libs/dbeskg.o diff --git a/modules/special_functions/src/fortran/.libs/dbesyg.o b/modules/special_functions/src/fortran/.libs/dbesyg.o Binary files differnew file mode 100755 index 000000000..445bfcb78 --- /dev/null +++ b/modules/special_functions/src/fortran/.libs/dbesyg.o diff --git a/modules/special_functions/src/fortran/.libs/psi.o b/modules/special_functions/src/fortran/.libs/psi.o Binary files differnew file mode 100755 index 000000000..965877094 --- /dev/null +++ b/modules/special_functions/src/fortran/.libs/psi.o diff --git a/modules/special_functions/src/fortran/.libs/zbesig.o b/modules/special_functions/src/fortran/.libs/zbesig.o Binary files differnew file mode 100755 index 000000000..4121bb9a4 --- /dev/null +++ b/modules/special_functions/src/fortran/.libs/zbesig.o diff --git a/modules/special_functions/src/fortran/.libs/zbesjg.o b/modules/special_functions/src/fortran/.libs/zbesjg.o Binary files differnew file mode 100755 index 000000000..931477c58 --- /dev/null +++ b/modules/special_functions/src/fortran/.libs/zbesjg.o diff --git a/modules/special_functions/src/fortran/.libs/zbeskg.o b/modules/special_functions/src/fortran/.libs/zbeskg.o Binary files differnew file mode 100755 index 000000000..cd5723252 --- /dev/null +++ b/modules/special_functions/src/fortran/.libs/zbeskg.o diff --git a/modules/special_functions/src/fortran/.libs/zbesyg.o b/modules/special_functions/src/fortran/.libs/zbesyg.o Binary files differnew file mode 100755 index 000000000..e4e0cf962 --- /dev/null +++ b/modules/special_functions/src/fortran/.libs/zbesyg.o diff --git a/modules/special_functions/src/fortran/core_Import.def b/modules/special_functions/src/fortran/core_Import.def new file mode 100755 index 000000000..aeb5e0b3a --- /dev/null +++ b/modules/special_functions/src/fortran/core_Import.def @@ -0,0 +1,21 @@ + LIBRARY core.dll + + +EXPORTS +; +;core +; +com_ +checkrhs_ +checklhs_ +vstk_ +iop_ +stack_ +cremat_ +getmat_ +getscalar_ +recu_ +errgst_ +cha1_ +adre_ +intersci_
\ No newline at end of file diff --git a/modules/special_functions/src/fortran/dbesig.f b/modules/special_functions/src/fortran/dbesig.f new file mode 100755 index 000000000..452a96209 --- /dev/null +++ b/modules/special_functions/src/fortran/dbesig.f @@ -0,0 +1,156 @@ +c Scilab ( http://www.scilab.org/ ) - This file is part of Scilab +c Copyright (C) 2005 - INRIA - Serge STEER +c +c This file must be used under the terms of the CeCILL. +c This source file is licensed as described in the file COPYING, which +c you should have received as part of this distribution. The terms +c are also available at +c http://www.cecill.info/licences/Licence_CeCILL_V2.1-en.txt +c + subroutine dbesig (x1, alpha, kode, n, y, nz,w,ierr) +c Author Serge Steer, 2005 +c extends dbesi for the case where alpha is negative + double precision x1,alpha,y(n),w(n) + integer kode,n,nz,ierr +c + double precision a,pi,inf,x,a1 + integer ier2 + double precision dlamch + data pi /3.14159265358979324D0/ + + inf=dlamch('o')*2.0d0 + x=x1 + ier2=0 + if (x.ne.x.or.alpha.ne.alpha) then +c . NaN case + call dset(n,inf-inf,y,1) + ierr=4 + elseif (alpha .ge. 0.0d0) then + call dbesi(abs(x),alpha,kode,n,y,nz,ierr) + if (ierr.eq.2) call dset(n,inf,y,1) + if(x.lt.0.0d0) then + i0=mod(int(abs(alpha))+1,2) + call dscal((n-i0+1)/2,-1.0d0,y(1+i0),2) + endif + else if (alpha .eq. dint(alpha)) then +c . alpha <0 and integer, +c . transform to positive value of alpha + if(alpha-1+n.ge.0) then +c . 0 is between alpha and alpha+n + a1=0.0d0 + nn=min(n,int(-alpha)) + else + a1=-(alpha-1+n) + nn=n + endif + call dbesi(abs(x),a1,kode,n,w,nz,ierr) + if (ierr.eq.2) then + call dset(n,inf,y,1) + else + if(n.gt.nn) then +c . 0 is between alpha and alpha+n + call dcopy(n-nn,w,1,y(nn+1),1) + call dcopy(nn,w(2),-1,y,1) + else +c . alpha and alpha+n are negative + call dcopy(nn,w,-1,y,1) + endif + endif + if(x.lt.0.0d0) then + i0=mod(int(abs(alpha))+1,2) + call dscal((n-i0+1)/2,-1.0d0,y(1+i0),2) + endif + else if (x .eq. 0.0d0) then +c . alpha <0 and x==0 + if(alpha-1.0d0+n.ge.0.0d0) then +c . 0 is between alpha and alpha+n + nn=int(-alpha)+1 + else + nn=n + endif + ierr=2 + call dset(nn,-inf,y,1) + if (n.gt.nn) call dset(n-nn,0.0d0,y(nn+1),1) + else +c . first alpha is negative non integer, x should be positive (with +C . x negative the result is complex. CHECKED +c . transform to positive value of alpha + if(alpha-1.0d0+n.ge.0.0d0) then +c . 0 is between alpha and alpha+n + nn=int(-alpha)+1 + else + nn=n + endif +c . compute for negative value of alpha+k, transform problem for +c . a1:a1+(nn-1) with a1 positive a1+k =abs(alpha+nn-k) + a1=-(alpha-1.0d0+nn) + call dbesi(x,a1,kode,nn,w,nz1,ierr) + call dbesk(x,a1,1,nn,y,nz2,ier) + ierr=max(ierr,ier) + nz=max(nz1,nz2) + if (ierr.eq.0) then + a=(2.0d0/pi)*dsin(a1*pi) + if (kode.eq.2) a=a*dexp(-x) +c . change sign to take into account that sin((a1+k)*pi) +C . changes sign with k + if (nn.ge.2) call dscal(nn/2,-1.0d0,y(2),2) + + call daxpy(nn,a,y,1,w,1) + elseif (ierr.eq.2) then + call dset(nn,inf,w,1) + elseif (ierr.eq.4) then + call dset(nn,inf-inf,w,1) + endif +c . store the result in the correct order + call dcopy(nn,w,-1,y,1) +c . compute for positive value of alpha+k is any (note that x>0) + if (n.gt.nn) then + call dbesi(x,1.0d0-a1,kode,n-nn,y(nn+1),nz,ier) + if (ier.eq.2) call dset(n-nn,inf,y(nn+1),1) + ierr=max(ierr,ier) + endif + endif + end + + subroutine dbesiv (x,nx,alpha,na, kode,y,w,ierr) +c Author Serge Steer, Copyright INRIA, 2005 +c compute besseli function for x and alpha given by vectors +c w : working array of size 2*na (used only if nz>0 and alpha contains negative +C values + double precision x(nx),alpha(na),y(*),w(*) + integer kode,nx,na,ier + double precision dlamch,w1,eps + eps=dlamch('p') + ierr=0 + if (na.lt.0) then +c . element wise case x and alpha are supposed to have the same size + do i=1,nx + call dbesig (x(i), alpha(i),kode,1,y(i), nz, w1,ier) + ierr=max(ierr,ier) + enddo + elseif (na.eq.1) then +c . element wise case x and alpha are supposed to have the same size + do i=1,nx + call dbesig (x(i), alpha(1),kode,1,y(i), nz, w1,ier) + ierr=max(ierr,ier) + enddo + else +c . compute besseli(x(i),y(j)), i=1,nx,j=1,na + j0=1 + 05 n=0 + 10 n=n+1 + j=j0+n + if (j.le.na.and.abs((1+alpha(j-1))-alpha(j)).le.eps) then + goto 10 + endif + do i=1,nx + call dbesig(x(i),alpha(j0),kode,n, w, nz, w(na+1),ier) + ierr=max(ierr,ier) + call dcopy(n,w,1,y(i+(j0-1)*nx),nx) + enddo + j0=j + if (j0.le.na) goto 05 + endif + end + + diff --git a/modules/special_functions/src/fortran/dbesig.lo b/modules/special_functions/src/fortran/dbesig.lo new file mode 100755 index 000000000..ceaf6c9a1 --- /dev/null +++ b/modules/special_functions/src/fortran/dbesig.lo @@ -0,0 +1,12 @@ +# src/fortran/dbesig.lo - a libtool object file +# Generated by libtool (GNU libtool) 2.4.2 +# +# Please DO NOT delete this file! +# It is necessary for linking the library. + +# Name of the PIC object. +pic_object='.libs/dbesig.o' + +# Name of the non-PIC object +non_pic_object=none + diff --git a/modules/special_functions/src/fortran/dbesjg.f b/modules/special_functions/src/fortran/dbesjg.f new file mode 100755 index 000000000..a341d82a9 --- /dev/null +++ b/modules/special_functions/src/fortran/dbesjg.f @@ -0,0 +1,169 @@ +c Scilab ( http://www.scilab.org/ ) - This file is part of Scilab +c Copyright (C) 2005 - INRIA - Serge STEER +c +c This file must be used under the terms of the CeCILL. +c This source file is licensed as described in the file COPYING, which +c you should have received as part of this distribution. The terms +c are also available at +c http://www.cecill.info/licences/Licence_CeCILL_V2.1-en.txt +c + subroutine dbesjg (x1, alpha, n, y, nz,w,ierr) +c Author Serge Steer, 2005 +c extends dbesj for the case where alpha is negative +c if alpha is negative and non integer and x1 is negative:error + double precision x1,alpha,y(n),w(n) + integer n,nz,ierr +c + double precision a,b,pi,inf,x,a1 + integer ier2 + double precision dlamch + data pi /3.14159265358979324D0/ + + inf=dlamch('o')*2.0d0 + x=x1 + ierr=0 + ier2=0 + if (x.ne.x.or.alpha.ne.alpha) then +c . NaN case + call dset(n,inf-inf,y,1) + ierr=4 + elseif (alpha .ge. 0.0d0) then +c . alpha <0 and x>=0 or x<0 + call dbesj(abs(x),alpha,n,y,nz,ierr) + if (ierr.eq.2) call dset(n,inf,y,1) + if(x.lt.0.0d0) then + i0=mod(int(alpha)+1,2) + call dscal((n-i0+1)/2,-1.0d0,y(1+i0),2) + endif + else if (alpha .eq. dint(alpha)) then +c . alpha <0 and integer, CHECKED +c . transform to positive value of alpha + if(alpha-1+n.ge.0) then +c . 0 is between alpha and alpha+n + a1=0.0d0 + nn=min(n,int(-alpha)) + else + a1=-(alpha-1+n) + nn=n + endif + call dbesj(abs(x),a1,n,w,nz,ierr) + if (ierr.eq.2) then + call dset(n,inf,y,1) + else + if(n.gt.nn) then +c . 0 is between alpha and alpha+n + call dcopy(n-nn,w,1,y(nn+1),1) + call dcopy(nn,w(2),-1,y,1) + else +c . alpha and alpha+n are negative + call dcopy(nn,w,-1,y,1) + endif + endif +c . apply parity + i0=mod(int(abs(alpha))+1,2) + if(x.gt.0.0d0) then + call dscal((nn-i0+1)/2,-1.0d0,y(1+i0),2) + else + call dscal((n-nn)/2,-1.0d0,y(nn+2),2) + endif + else if (x .eq. 0.0d0) then +c . alpha <0 and x==0 CHECKED + if(alpha-1.0d0+n.ge.0.0d0) then +c . 0 is between alpha and alpha+n + nn=int(-alpha)+1 + else + nn=n + endif + ierr=2 + call dset(nn,-inf,y,1) + if (n.gt.nn) call dset(n-nn,0.0d0,y(nn+1),1) + else +c . first alpha is negative non integer, x should be positive (with +C . x negative the result is complex. CHECKED +c . transform to positive value of alpha + if(alpha-1.0d0+n.ge.0.0d0) then +c . 0 is between alpha and alpha+n + nn=int(-alpha)+1 + else + nn=n + endif +c . compute for negative value of alpha+k, transform problem for +c . a1:a1+(nn-1) with a1 positive a1+k =abs(alpha+nn-k) + a1=-(alpha-1.0d0+nn) + call dbesj(x,a1,nn,y,nz1,ierr) + call dbesy(x,a1,nn,w,ier) + ierr=max(ierr,ier) + if (ierr.eq.0) then + a=cos(a1*pi) + b=-sin(a1*pi) +c . to avoid numerical errors if a is near 0 (or b near 0) +c . and y is big (or w is big) + if(abs(abs(a)-1.0d0).lt.eps) b=0.0d0 + if(abs(abs(b)-1.0d0).lt.eps) a=0.0d0 + call dscal(nn,b,w,1) + call daxpy(nn,a,y,1,w,1) + elseif (ierr.eq.2) then + call dset(nn,inf,w,1) + elseif (ierr.eq.4) then + call dset(nn,inf-inf,w,1) + endif +c . change sign to take into account that cos((a1+k)*pi) and +C . sin((a1+k)*pi) changes sign with k + if (nn.ge.2) call dscal(nn/2,-1.0d0,w(2),2) + +c . store the result in the correct order + call dcopy(nn,w,-1,y,1) + +c . compute for positive value of alpha+k is any (note that x>0) + if (n.gt.nn) then +c . this code is taken from the alpha>0 case above + call dbesj(abs(x),1.0d0-a1,n-nn,y(nn+1),nz1,ier) + if (ier.eq.2) call dset(n-nn,inf,y(nn+1),1) + ierr=max(ierr,ier) + endif + endif + end + + + subroutine dbesjv (x,nx,alpha,na,kode,y,w,ierr) +c Author Serge Steer, Copyright INRIA, 2005 +c compute besselj function for x and alpha given by vectors +c w : working array of size 2*na (used only if nz>0 and alpha contains negative +C values + double precision x(nx),alpha(na),y(*),w(*) + integer kode,nx,na,ier + double precision dlamch,w1,eps + eps=dlamch('p') + ierr=0 + if (na.lt.0) then +c . element wise case x and alpha are supposed to have the same size + do i=1,nx + call dbesjg (x(i), alpha(i),1,y(i), nz, w1,ier) + ierr=max(ierr,ier) + enddo + elseif (na.eq.1) then +c . element wise case x and alpha are supposed to have the same size + do i=1,nx + call dbesjg (x(i), alpha(1),1,y(i), nz, w1,ier) + ierr=max(ierr,ier) + enddo + else +c . compute besselj(x(i),y(j)), i=1,nx,j=1,na + j0=1 + 05 n=0 + 10 n=n+1 + j=j0+n + if (j.le.na.and.abs((1+alpha(j-1))-alpha(j)).le.eps) then + goto 10 + endif + do i=1,nx + call dbesjg(x(i),alpha(j0),n, w, nz, w(na+1),ier) + ierr=max(ierr,ier) + call dcopy(n,w,1,y(i+(j0-1)*nx),nx) + enddo + j0=j + if (j0.le.na) goto 05 + endif + end + + diff --git a/modules/special_functions/src/fortran/dbesjg.lo b/modules/special_functions/src/fortran/dbesjg.lo new file mode 100755 index 000000000..2baac211d --- /dev/null +++ b/modules/special_functions/src/fortran/dbesjg.lo @@ -0,0 +1,12 @@ +# src/fortran/dbesjg.lo - a libtool object file +# Generated by libtool (GNU libtool) 2.4.2 +# +# Please DO NOT delete this file! +# It is necessary for linking the library. + +# Name of the PIC object. +pic_object='.libs/dbesjg.o' + +# Name of the non-PIC object +non_pic_object=none + diff --git a/modules/special_functions/src/fortran/dbeskg.f b/modules/special_functions/src/fortran/dbeskg.f new file mode 100755 index 000000000..507c24cab --- /dev/null +++ b/modules/special_functions/src/fortran/dbeskg.f @@ -0,0 +1,98 @@ +c Scilab ( http://www.scilab.org/ ) - This file is part of Scilab +c Copyright (C) 2005 - INRIA - Serge STEER +c +c This file must be used under the terms of the CeCILL. +c This source file is licensed as described in the file COPYING, which +c you should have received as part of this distribution. The terms +c are also available at +c http://www.cecill.info/licences/Licence_CeCILL_V2.1-en.txt +c + subroutine dbeskg (x1, alpha, kode, n, y, nz,ierr) +c Author Serge Steer, 2005 +c extends dbesk for the case where alpha is negative +c x is supposed to be positive (besselk,with x<0 is complex) + double precision x1,alpha,y(n) + integer kode,n,nz,ierr +c + double precision inf,x,dlamch,a1,temp + + inf=dlamch('o')*2.0d0 + x=x1 + ierr=0 + if (x.ne.x.or.alpha.ne.alpha) then +c . NaN case + call dset(n,inf-inf,y,1) + ierr=4 + elseif (x .eq. 0.0d0) then + call dset(n,-inf,y,1) + ierr=2 + elseif (alpha.ge.0.0d0) then + call dbesk(x,alpha,kode,n,y,nz,ierr) + if (ierr.eq.2) call dset(n,inf,y,1) + else + if(alpha-1.0d0+n.ge.0.0d0) then +c . 0 is between alpha and alpha+n + nn=int(-alpha)+1 + else + nn=n + endif + a1=-(alpha-1.0d0+nn) + call dbesk(x,a1,kode,nn,y,nz,ierr) + if (ierr.eq.2) call dset(nn,inf,y,1) +c . swap the result to have it in correct order + if (nn.ge.2) then + do i=1,nn/2 + temp=y(i) + y(i)=y(nn+1-i) + y(nn+1-i)=temp + enddo + endif + if (n.gt.nn) then + call dbesk(x,1.0d0-a1,kode,n-nn,y(nn+1),nz,ier) + if (ier.eq.2) call dset(n-nn,inf,y(nn+1),1) + ierr=max(ierr,ier) + endif + endif + end + + subroutine dbeskv (x,nx,alpha,na, kode,y,w,ierr) +c Author Serge Steer, Copyright INRIA, 2005 +c compute besseli function for x and alpha given by vectors +c w : working array of size 2*na (used only if nz>0 and alpha contains negative +C values + double precision x(nx),alpha(na),y(*),w(*) + integer kode,nx,na,ier + double precision dlamch,eps + eps=dlamch('p') + ierr=0 + if (na.lt.0) then +c . element wise case x and alpha are supposed to have the same size + do i=1,nx + call dbeskg (abs(x(i)), alpha(i),kode,1,y(i), nz,ier) + ierr=max(ierr,ier) + enddo + elseif (na.eq.1) then + do i=1,nx + call dbeskg (abs(x(i)), alpha(1),kode,1,y(i), nz,ier) + ierr=max(ierr,ier) + enddo + else +c . compute besseli(x(i),y(j)), i=1,nx,j=1,na + j0=1 + 05 n=0 + 10 n=n+1 + j=j0+n + if (j.le.na.and.abs((1+alpha(j-1))-alpha(j)).le.eps) then + goto 10 + endif + do i=1,nx + call dbeskg(abs(x(i)),alpha(j0),kode,n, w, nz,ier) + ierr=max(ierr,ier) + call dcopy(n,w,1,y(i+(j0-1)*nx),nx) + enddo + j0=j + if (j0.le.na) goto 05 + endif + end + + diff --git a/modules/special_functions/src/fortran/dbeskg.lo b/modules/special_functions/src/fortran/dbeskg.lo new file mode 100755 index 000000000..19a8ef965 --- /dev/null +++ b/modules/special_functions/src/fortran/dbeskg.lo @@ -0,0 +1,12 @@ +# src/fortran/dbeskg.lo - a libtool object file +# Generated by libtool (GNU libtool) 2.4.2 +# +# Please DO NOT delete this file! +# It is necessary for linking the library. + +# Name of the PIC object. +pic_object='.libs/dbeskg.o' + +# Name of the non-PIC object +non_pic_object=none + diff --git a/modules/special_functions/src/fortran/dbesyg.f b/modules/special_functions/src/fortran/dbesyg.f new file mode 100755 index 000000000..106508033 --- /dev/null +++ b/modules/special_functions/src/fortran/dbesyg.f @@ -0,0 +1,150 @@ +c Scilab ( http://www.scilab.org/ ) - This file is part of Scilab +c Copyright (C) 2005 - INRIA - Serge STEER +c +c This file must be used under the terms of the CeCILL. +c This source file is licensed as described in the file COPYING, which +c you should have received as part of this distribution. The terms +c are also available at +c http://www.cecill.info/licences/Licence_CeCILL_V2.1-en.txt +c + subroutine dbesyg (x1, alpha, n, y, nz,w,ierr) +c Author Serge Steer, 2005 +c extends dbesy for the case where alpha is negative +c x is assumed to be >0 (if negative bessely(alpha,x) is complex) + double precision x1,alpha,y(n),w(n) + integer n,nz,ierr +c + double precision a,b,pi,inf,x,a1,eps + integer ier2 + double precision dlamch + data pi /3.14159265358979324D0/ + + inf=dlamch('o')*2.0d0 + eps=dlamch('p') + x=x1 + ier2=0 + if (x.ne.x.or.alpha.ne.alpha) then +c . NaN case + call dset(n,inf-inf,y,1) + ierr=4 + else if (x .eq. 0.0d0) then + ierr=2 + call dset(n,-inf,y,1) + elseif (alpha .ge. 0.0d0) then + call dbesy(x,alpha,n,y,ierr) + if (ierr.eq.2) call dset(n,inf,y,1) + else if (alpha .eq. dint(alpha)) then +c . alpha <0 and integer, +c . transform to positive value of alpha + if(alpha-1+n.ge.0) then +c . 0 is between alpha and alpha+n + a1=0.0d0 + nn=min(n,int(-alpha)) + else + a1=-(alpha-1+n) + nn=n + endif + call dbesy(x,a1,n,w,ierr) + if (ierr.eq.2) then + call dset(n,inf,y,1) + else + if(n.gt.nn) then +c . 0 is between alpha and alpha+n + call dcopy(n-nn,w,1,y(nn+1),1) + call dcopy(nn,w(2),-1,y,1) + else +c . alpha and alpha+n are negative + call dcopy(nn,w,-1,y,1) + endif + endif + i0=mod(int(abs(alpha))+1,2) + call dscal((nn-i0+1)/2,-1.0d0,y(1+i0),2) + else +c . first alpha is negative non integer, x should be positive (with +C . x negative the result is complex. CHECKED +c . transform to positive value of alpha + if(alpha-1.0d0+n.ge.0.0d0) then +c . 0 is between alpha and alpha+n + nn=int(-alpha)+1 + else + nn=n + endif +c . compute for negative value of alpha+k, transform problem for +c . a1:a1+(nn-1) with a1 positive a1+k =abs(alpha+nn-k) + a1=-(alpha-1.0d0+nn) + + call dbesj(x,a1,nn,y,nz1,ierr) + call dbesy(x,a1,nn,w,ier) + ierr=max(ierr,ier) + if (ierr.eq.0) then + a=sin(a1*pi) + b=cos(a1*pi) +c . to avoid numerical errors if a is near 0 (or b near 0) +c . and y is big (or w is big) + if(abs(abs(a)-1.0d0).lt.eps) b=0.0d0 + if(abs(abs(b)-1.0d0).lt.eps) a=0.0d0 + call dscal(nn,b,w,1) + call daxpy(nn,a,y,1,w,1) + elseif (ierr.eq.2) then + call dset(nn,inf,w,1) + elseif (ierr.eq.4) then + call dset(nn,inf-inf,w,1) + endif + +c . change sign to take into account that cos((a1+k)*pi) and +C . sin((a1+k)*pi) changes sign with k + if (nn.ge.2) call dscal(nn/2,-1.0d0,w(2),2) +c . store the result in the correct order + call dcopy(nn,w,-1,y,1) +c . compute for positive value of alpha+k is any + if (n.gt.nn) then + call dbesy(x,1.0d0-a1,n-nn,y(nn+1),ier) + if (ierr.eq.2) call dset(n-nn,inf,y(nn+1),1) + ierr=max(ierr,ier) + endif + endif + + end + + subroutine dbesyv (x,nx,alpha,na,kode,y,w,ierr) +c Author Serge Steer, Copyright INRIA, 2005 +c compute bessely function for x and alpha given by vectors +c w : working array of size 2*na (used only if nz>0 and alpha contains negative +C values + double precision x(nx),alpha(na),y(*),w(*) + integer kode, nx,na,ier + double precision dlamch,w1,eps + eps=dlamch('p') + ierr=0 + if (na.lt.0) then +c . element wise case x and alpha are supposed to have the same size + do i=1,nx + call dbesyg (abs(x(i)), alpha(i),1,y(i), nz, w1,ier) + ierr=max(ierr,ier) + enddo + elseif (na.eq.1) then +c . element wise case x and alpha are supposed to have the same size + do i=1,nx + call dbesyg (abs(x(i)), alpha(1),1,y(i), nz, w1,ier) + ierr=max(ierr,ier) + enddo + else +c . compute bessely(x(i),y(j)), i=1,nx,j=1,na + j0=1 + 05 n=0 + 10 n=n+1 + j=j0+n + if (j.le.na.and.abs((1+alpha(j-1))-alpha(j)).le.eps) then + goto 10 + endif + do i=1,nx + call dbesyg(abs(x(i)),alpha(j0),n, w, nz, w(na+1),ier) + ierr=max(ierr,ier) + call dcopy(n,w,1,y(i+(j0-1)*nx),nx) + enddo + j0=j + if (j0.le.na) goto 05 + endif + end + + diff --git a/modules/special_functions/src/fortran/dbesyg.lo b/modules/special_functions/src/fortran/dbesyg.lo new file mode 100755 index 000000000..20a02a9f3 --- /dev/null +++ b/modules/special_functions/src/fortran/dbesyg.lo @@ -0,0 +1,12 @@ +# src/fortran/dbesyg.lo - a libtool object file +# Generated by libtool (GNU libtool) 2.4.2 +# +# Please DO NOT delete this file! +# It is necessary for linking the library. + +# Name of the PIC object. +pic_object='.libs/dbesyg.o' + +# Name of the non-PIC object +non_pic_object=none + diff --git a/modules/special_functions/src/fortran/elementary_functions_f_Import.def b/modules/special_functions/src/fortran/elementary_functions_f_Import.def new file mode 100755 index 000000000..e88940fe3 --- /dev/null +++ b/modules/special_functions/src/fortran/elementary_functions_f_Import.def @@ -0,0 +1,9 @@ + LIBRARY elementary_functions_f.dll + + +EXPORTS +; +;elementary_functions_f +dset_ + + diff --git a/modules/special_functions/src/fortran/output_stream_Import.def b/modules/special_functions/src/fortran/output_stream_Import.def new file mode 100755 index 000000000..4e3a0c527 --- /dev/null +++ b/modules/special_functions/src/fortran/output_stream_Import.def @@ -0,0 +1,7 @@ + LIBRARY output_stream.dll + + +EXPORTS +; +;output_stream +error_ diff --git a/modules/special_functions/src/fortran/psi.f b/modules/special_functions/src/fortran/psi.f new file mode 100755 index 000000000..29b86fb49 --- /dev/null +++ b/modules/special_functions/src/fortran/psi.f @@ -0,0 +1,261 @@ + FUNCTION PSI(XX) +C---------------------------------------------------------------------- +C +C This function program evaluates the logarithmic derivative of the +C gamma function, +C +C psi(x) = d/dx (gamma(x)) / gamma(x) = d/dx (ln gamma(x)) +C +C for real x, where either +C +C -xmax1 < x < -xmin (x not a negative integer), or +C xmin < x. +C +C The calling sequence for this function is +C +C Y = PSI(X) +C +C The main computation uses rational Chebyshev approximations +C published in Math. Comp. 27, 123-127 (1973) by Cody, Strecok and +C Thacher. This transportable program is patterned after the +C machine-dependent FUNPACK program PSI(X), but cannot match that +C version for efficiency or accuracy. This version uses rational +C approximations that are theoretically accurate to 20 significant +C decimal digits. The accuracy achieved depends on the arithmetic +C system, the compiler, the intrinsic functions, and proper selection +C of the machine-dependent constants. +C +C******************************************************************* +C******************************************************************* +C +C Explanation of machine-dependent constants +C +C XINF = largest positive machine number +C XMAX1 = beta ** (p-1), where beta is the radix for the +C floating-point system, and p is the number of base-beta +C digits in the floating-point significand. This is an +C upper bound on non-integral floating-point numbers, and +C the negative of the lower bound on acceptable negative +C arguments for PSI. If rounding is necessary, round this +C value down. +C XMIN1 = the smallest in magnitude acceptable argument. We +C recommend XMIN1 = MAX(1/XINF,xmin) rounded up, where +C xmin is the smallest positive floating-point number. +C XSMALL = absolute argument below which PI*COTAN(PI*X) may be +C represented by 1/X. We recommend XSMALL < sqrt(3 eps)/pi, +C where eps is the smallest positive number such that +C 1+eps > 1. +C XLARGE = argument beyond which PSI(X) may be represented by +C LOG(X). The solution to the equation +C x*ln(x) = beta ** p +C is a safe value. +C +C Approximate values for some important machines are +C +C beta p eps xmin XINF +C +C CDC 7600 (S.P.) 2 48 7.11E-15 3.13E-294 1.26E+322 +C CRAY-1 (S.P.) 2 48 7.11E-15 4.58E-2467 5.45E+2465 +C IEEE (IBM/XT, +C SUN, etc.) (S.P.) 2 24 1.19E-07 1.18E-38 3.40E+38 +C IEEE (IBM/XT, +C SUN, etc.) (D.P.) 2 53 1.11D-16 2.23E-308 1.79D+308 +C IBM 3033 (D.P.) 16 14 1.11D-16 5.40D-79 7.23D+75 +C SUN 3/160 (D.P.) 2 53 1.11D-16 2.23D-308 1.79D+308 +C VAX 11/780 (S.P.) 2 24 5.96E-08 2.94E-39 1.70E+38 +C (D.P.) 2 56 1.39D-17 2.94D-39 1.70D+38 +C (G Format) (D.P.) 2 53 1.11D-16 5.57D-309 8.98D+307 +C +C XMIN1 XMAX1 XSMALL XLARGE +C +C CDC 7600 (S.P.) 3.13E-294 1.40E+14 4.64E-08 9.42E+12 +C CRAY-1 (S.P.) 1.84E-2466 1.40E+14 4.64E-08 9.42E+12 +C IEEE (IBM/XT, +C SUN, etc.) (S.P.) 1.18E-38 8.38E+06 1.90E-04 1.20E+06 +C IEEE (IBM/XT, +C SUN, etc.) (D.P.) 2.23D-308 4.50D+15 5.80D-09 2.71D+14 +C IBM 3033 (D.P.) 1.39D-76 4.50D+15 5.80D-09 2.05D+15 +C SUN 3/160 (D.P.) 2.23D-308 4.50D+15 5.80D-09 2.71D+14 +C VAX 11/780 (S.P.) 5.89E-39 8.38E+06 1.35E-04 1.20E+06 +C (D.P.) 5.89D-39 3.60D+16 2.05D-09 2.05D+15 +C (G Format) (D.P.) 1.12D-308 4.50D+15 5.80D-09 2.71D+14 +C +C******************************************************************* +C******************************************************************* +C +C Error Returns +C +C The program returns XINF for X < -XMAX1, for X zero or a negative +C integer, or when X lies in (-XMIN1, 0), and returns -XINF +C when X lies in (0, XMIN1). +C +C Intrinsic functions required are: +C +C ABS, AINT, DBLE, INT, LOG, REAL, TAN +C +C +C Author: W. J. Cody +C Mathematics and Computer Science Division +C Argonne National Laboratory +C Argonne, IL 60439 +C +C Latest modification: June 8, 1988 +C +C---------------------------------------------------------------------- + INTEGER I,N,NQ +CS REAL + DOUBLE PRECISION + 1 AUG,CONV,DEN,PSI,FOUR,FOURTH,HALF,ONE,P1,P2,PIOV4,Q1,Q2, + 2 SGN,THREE,XLARGE,UPPER,W,X,XINF,XMAX1,XMIN1,XSMALL,X01, + 3 X01D,X02,XX,Z,ZERO + DIMENSION P1(9),P2(7),Q1(8),Q2(6) +C---------------------------------------------------------------------- +C Mathematical constants. PIOV4 = pi / 4 +C---------------------------------------------------------------------- +CS DATA ZERO,FOURTH,HALF,ONE/0.0E0,0.25E0,0.5E0,1.0E0/ +CS DATA THREE,FOUR/3.0E0,4.0E0/,PIOV4/7.8539816339744830962E-01/ + DATA ZERO,FOURTH,HALF,ONE/0.0D0,0.25D0,0.5D0,1.0D0/ + DATA THREE,FOUR/3.0D0,4.0D0/,PIOV4/7.8539816339744830962D-01/ +C---------------------------------------------------------------------- +C Machine-dependent constants +C---------------------------------------------------------------------- +CS DATA XINF/1.70E+38/, XMIN1/5.89E-39/, XMAX1/8.38E+06/, +CS 1 XSMALL/1.35E-04/, XLARGE/1.20E+06/ + DATA XINF/1.79D+308/, XMIN1/2.23D-308/, XMAX1/4.50D+15/, + 1 XSMALL/5.80D-09/, XLARGE/2.71D+14/ +C---------------------------------------------------------------------- +C Zero of psi(x) +C---------------------------------------------------------------------- +CS DATA X01/187.0E0/,X01D/128.0E0/,X02/6.9464496836234126266E-04/ + DATA X01/187.0D0/,X01D/128.0D0/,X02/6.9464496836234126266D-04/ +C---------------------------------------------------------------------- +C Coefficients for approximation to psi(x)/(x-x0) over [0.5, 3.0] +C---------------------------------------------------------------------- +CS DATA P1/4.5104681245762934160E-03,5.4932855833000385356E+00, +CS 1 3.7646693175929276856E+02,7.9525490849151998065E+03, +CS 2 7.1451595818951933210E+04,3.0655976301987365674E+05, +CS 3 6.3606997788964458797E+05,5.8041312783537569993E+05, +CS 4 1.6585695029761022321E+05/ +CS DATA Q1/9.6141654774222358525E+01,2.6287715790581193330E+03, +CS 1 2.9862497022250277920E+04,1.6206566091533671639E+05, +CS 2 4.3487880712768329037E+05,5.4256384537269993733E+05, +CS 3 2.4242185002017985252E+05,6.4155223783576225996E-08/ + DATA P1/4.5104681245762934160D-03,5.4932855833000385356D+00, + 1 3.7646693175929276856D+02,7.9525490849151998065D+03, + 2 7.1451595818951933210D+04,3.0655976301987365674D+05, + 3 6.3606997788964458797D+05,5.8041312783537569993D+05, + 4 1.6585695029761022321D+05/ + DATA Q1/9.6141654774222358525D+01,2.6287715790581193330D+03, + 1 2.9862497022250277920D+04,1.6206566091533671639D+05, + 2 4.3487880712768329037D+05,5.4256384537269993733D+05, + 3 2.4242185002017985252D+05,6.4155223783576225996D-08/ +C---------------------------------------------------------------------- +C Coefficients for approximation to psi(x) - ln(x) + 1/(2x) +C for x > 3.0 +C---------------------------------------------------------------------- +CS DATA P2/-2.7103228277757834192E+00,-1.5166271776896121383E+01, +CS 1 -1.9784554148719218667E+01,-8.8100958828312219821E+00, +CS 2 -1.4479614616899842986E+00,-7.3689600332394549911E-02, +CS 3 -6.5135387732718171306E-21/ +CS DATA Q2/ 4.4992760373789365846E+01, 2.0240955312679931159E+02, +CS 1 2.4736979003315290057E+02, 1.0742543875702278326E+02, +CS 2 1.7463965060678569906E+01, 8.8427520398873480342E-01/ + DATA P2/-2.7103228277757834192D+00,-1.5166271776896121383D+01, + 1 -1.9784554148719218667D+01,-8.8100958828312219821D+00, + 2 -1.4479614616899842986D+00,-7.3689600332394549911D-02, + 3 -6.5135387732718171306D-21/ + DATA Q2/ 4.4992760373789365846D+01, 2.0240955312679931159D+02, + 1 2.4736979003315290057D+02, 1.0742543875702278326D+02, + 2 1.7463965060678569906D+01, 8.8427520398873480342D-01/ +C---------------------------------------------------------------------- +CS CONV(I) = REAL(I) + CONV(I) = DBLE(I) + X = XX + W = ABS(X) + AUG = ZERO +C---------------------------------------------------------------------- +C Check for valid arguments, then branch to appropriate algorithm +C---------------------------------------------------------------------- + IF ((-X .GE. XMAX1) .OR. (W .LT. XMIN1)) THEN + GO TO 410 + ELSE IF (X .GE. HALF) THEN + GO TO 200 +C---------------------------------------------------------------------- +C X < 0.5, use reflection formula: psi(1-x) = psi(x) + pi * cot(pi*x) +C Use 1/X for PI*COTAN(PI*X) when XMIN1 < |X| <= XSMALL. +C---------------------------------------------------------------------- + ELSE IF (W .LE. XSMALL) THEN + AUG = -ONE / X + GO TO 150 + END IF +C---------------------------------------------------------------------- +C Argument reduction for cot +C---------------------------------------------------------------------- + 100 IF (X .LT. ZERO) THEN + SGN = PIOV4 + ELSE + SGN = -PIOV4 + END IF + W = W - AINT(W) + NQ = INT(W * FOUR) + W = FOUR * (W - CONV(NQ) * FOURTH) +C---------------------------------------------------------------------- +C W is now related to the fractional part of 4.0 * X. +C Adjust argument to correspond to values in the first +C quadrant and determine the sign. +C---------------------------------------------------------------------- + N = NQ / 2 + IF ((N+N) .NE. NQ) W = ONE - W + Z = PIOV4 * W + IF (MOD(N,2) .NE. 0) SGN = - SGN +C---------------------------------------------------------------------- +C determine the final value for -pi * cotan(pi*x) +C---------------------------------------------------------------------- + N = (NQ + 1) / 2 + IF (MOD(N,2) .EQ. 0) THEN +C---------------------------------------------------------------------- +C Check for singularity +C---------------------------------------------------------------------- + IF (Z .EQ. ZERO) GO TO 410 + AUG = SGN * (FOUR / TAN(Z)) + ELSE + AUG = SGN * (FOUR * TAN(Z)) + END IF + 150 X = ONE - X + 200 IF (X .GT. THREE) GO TO 300 +C---------------------------------------------------------------------- +C 0.5 <= X <= 3.0 +C---------------------------------------------------------------------- + DEN = X + UPPER = P1(1) * X + DO 210 I = 1, 7 + DEN = (DEN + Q1(I)) * X + UPPER = (UPPER + P1(I+1)) * X + 210 CONTINUE + DEN = (UPPER + P1(9)) / (DEN + Q1(8)) + X = (X-X01/X01D) - X02 + PSI = DEN * X + AUG + GO TO 500 +C---------------------------------------------------------------------- +C 3.0 < X +C---------------------------------------------------------------------- + 300 IF (X .LT. XLARGE) THEN + W = ONE / (X * X) + DEN = W + UPPER = P2(1) * W + DO 310 I = 1, 5 + DEN = (DEN + Q2(I)) * W + UPPER = (UPPER + P2(I+1)) * W + 310 CONTINUE + AUG = (UPPER + P2(7)) / (DEN + Q2(6)) - HALF / X + AUG + END IF + PSI = AUG + LOG(X) + GO TO 500 +C---------------------------------------------------------------------- +C Error return +C---------------------------------------------------------------------- + 410 PSI = XINF + IF (X .GT. ZERO) PSI = -XINF + 500 RETURN +C---------- Last card of PSI ---------- + END diff --git a/modules/special_functions/src/fortran/psi.lo b/modules/special_functions/src/fortran/psi.lo new file mode 100755 index 000000000..d8b83c811 --- /dev/null +++ b/modules/special_functions/src/fortran/psi.lo @@ -0,0 +1,12 @@ +# src/fortran/psi.lo - a libtool object file +# Generated by libtool (GNU libtool) 2.4.2 +# +# Please DO NOT delete this file! +# It is necessary for linking the library. + +# Name of the PIC object. +pic_object='.libs/psi.o' + +# Name of the non-PIC object +non_pic_object=none + diff --git a/modules/special_functions/src/fortran/slatec_f_Import.def b/modules/special_functions/src/fortran/slatec_f_Import.def new file mode 100755 index 000000000..6343ded8e --- /dev/null +++ b/modules/special_functions/src/fortran/slatec_f_Import.def @@ -0,0 +1,16 @@ + LIBRARY slatec_f.dll + + +EXPORTS +; +;slatec_f +dbesi_ +dbesj_ +dbesk_ +dbesy_ +dgammacody_ +zbesi_ +zbesj_ +zbesk_ +zbesy_ + diff --git a/modules/special_functions/src/fortran/special_functions_f.rc b/modules/special_functions/src/fortran/special_functions_f.rc new file mode 100755 index 000000000..fc25f7647 --- /dev/null +++ b/modules/special_functions/src/fortran/special_functions_f.rc @@ -0,0 +1,95 @@ +// Microsoft Visual C++ generated resource script. +// + + +#define APSTUDIO_READONLY_SYMBOLS +///////////////////////////////////////////////////////////////////////////// +// +// Generated from the TEXTINCLUDE 2 resource. +// +#define APSTUDIO_HIDDEN_SYMBOLS +#include "windows.h" +///////////////////////////////////////////////////////////////////////////// +#undef APSTUDIO_READONLY_SYMBOLS + +///////////////////////////////////////////////////////////////////////////// +// French (France) resources + +#if !defined(AFX_RESOURCE_DLL) || defined(AFX_TARG_FRA) +#ifdef _WIN32 +LANGUAGE LANG_FRENCH, SUBLANG_FRENCH +#pragma code_page(1252) +#endif //_WIN32 + +#ifdef APSTUDIO_INVOKED +///////////////////////////////////////////////////////////////////////////// +// +// TEXTINCLUDE +// + +1 TEXTINCLUDE +BEGIN + "resource.h\0" +END + +3 TEXTINCLUDE +BEGIN + "\r\n" + "\0" +END + +#endif // APSTUDIO_INVOKED + + +///////////////////////////////////////////////////////////////////////////// +// +// Version +// + +VS_VERSION_INFO VERSIONINFO + FILEVERSION 5,5,2,0 + PRODUCTVERSION 5,5,2,0 + FILEFLAGSMASK 0x17L +#ifdef _DEBUG + FILEFLAGS 0x1L +#else + FILEFLAGS 0x0L +#endif + FILEOS 0x4L + FILETYPE 0x2L + FILESUBTYPE 0x0L +BEGIN + BLOCK "StringFileInfo" + BEGIN + BLOCK "040c04b0" + BEGIN + VALUE "FileDescription", "elementary_functions_f module" + VALUE "FileVersion", "5, 5, 2, 0" + VALUE "InternalName", "elementary_functions_f module" + VALUE "LegalCopyright", "Copyright (C) 2017" + VALUE "OriginalFilename", "elementary_functions_f.dll" + VALUE "ProductName", "elementary_functions_f module" + VALUE "ProductVersion", "5, 5, 2, 0" + END + END + BLOCK "VarFileInfo" + BEGIN + VALUE "Translation", 0x40c, 1200 + END +END + +#endif // French (France) resources +///////////////////////////////////////////////////////////////////////////// + + + +#ifndef APSTUDIO_INVOKED +///////////////////////////////////////////////////////////////////////////// +// +// Generated from the TEXTINCLUDE 3 resource. +// + + +///////////////////////////////////////////////////////////////////////////// +#endif // not APSTUDIO_INVOKED + diff --git a/modules/special_functions/src/fortran/special_functions_f.vfproj b/modules/special_functions/src/fortran/special_functions_f.vfproj new file mode 100755 index 000000000..f25fadf18 --- /dev/null +++ b/modules/special_functions/src/fortran/special_functions_f.vfproj @@ -0,0 +1,108 @@ +<?xml version="1.0" encoding="UTF-8"?> +<VisualStudioProject ProjectType="typeDynamicLibrary" ProjectCreator="Intel Fortran" Keyword="Dll" Version="11.0" ProjectIdGuid="{E3A01BE7-815F-4A06-9F0A-C95E80873966}"> + <Platforms> + <Platform Name="Win32"/> + 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\ No newline at end of file diff --git a/modules/special_functions/src/fortran/zbesig.f b/modules/special_functions/src/fortran/zbesig.f new file mode 100755 index 000000000..b1adf589c --- /dev/null +++ b/modules/special_functions/src/fortran/zbesig.f @@ -0,0 +1,176 @@ +c Scilab ( http://www.scilab.org/ ) - This file is part of Scilab +c Copyright (C) 2005 - INRIA - Serge STEER +c +c This file must be used under the terms of the CeCILL. +c This source file is licensed as described in the file COPYING, which +c you should have received as part of this distribution. The terms +c are also available at +c http://www.cecill.info/licences/Licence_CeCILL_V2.1-en.txt +c + subroutine zbesig (x1r,x1i, alpha, kode, n, yr,yi, nz,wr,wi,ierr) +c Author Serge Steer, 2005 +c extends cbesi for the case where alpha is negative + double precision alpha + double precision x1r, x1i, yr(n), yi(n), wr(n),wi(n) + integer kode,n,nz,ierr +c + double precision a,pi,inf,xr, xi,a1 + integer ier2 + double precision dlamch + + data pi /3.14159265358979324D0/ + inf=dlamch('o')*2.0d0 + + xr=x1r + xi=x1i + ier2=0 + if (xr.ne.xr.or.xi.ne.xi.or.alpha.ne.alpha) then +c . NaN case + call dset(n,inf-inf,yr,1) + call dset(n,inf-inf,yi,1) + ierr=4 + elseif (alpha .ge. 0.0d0) then + call zbesi(xr,xi,alpha,kode,n,yr,yi,nz,ierr) + if (ierr.eq.2) then + call dset(n,inf,yr,1) + call dset(n,inf,yi,1) + elseif(ierr.ge.4) then + call dset(n,inf-inf,yr,1) + call dset(n,inf-inf,yi,1) + endif + else if (alpha .eq. dint(alpha)) then +c . alpha <0 and integer, +c . transform to positive value of alpha + if(alpha-1+n.ge.0) then +c . 0 is between alpha and alpha+n + a1=0.0d0 + nn=min(n,int(-alpha)) + else + a1=-(alpha-1+n) + nn=n + endif + call zbesi(xr,xi,a1,kode,n,wr,wi,nz,ierr) + if (ierr.eq.2) then + call dset(n,inf,yr,1) + call dset(n,inf,yi,1) + else + if(n.gt.nn) then +c . 0 is between alpha and alpha+n + call dcopy(n-nn,wr,1,yr(nn+1),1) + call dcopy(n-nn,wi,1,yi(nn+1),1) + call dcopy(nn,wr(2),-1,yr,1) + call dcopy(nn,wi(2),-1,yi,1) + else +c . alpha and alpha+n are negative + call dcopy(n,wr,-1,yr,1) + call dcopy(n,wi,-1,yi,1) + endif + endif + + else if (xr .eq. 0.0d0.and.xi .eq.0.0d0) then + call dset(n,-inf,yr,1) + call dset(n,0.0d0,yi,1) + ierr=2 + else +c . first alpha is negative non integer, +c . transform to positive value of alpha + if(alpha-1.0d0+n.ge.0.0d0) then +c . 0 is between alpha and alpha+n + nn=int(-alpha)+1 + else + nn=n + endif +c . compute for negative value of alpha+k, transform problem for +c . a1:a1+(nn-1) with a1 positive a1+k =abs(alpha+nn-k) + a1=-(alpha-1.0d0+nn) + + call zbesi(xr,xi,a1,kode,n,wr,wi,nz1,ierr) + call zbesk(xr,xi,a1,1,n,yr,yi,nz2,ier) + ierr=max(ierr,ier) + nz=max(nz1,nz2) + if (ierr.eq.0) then + a=(2.0d0/pi)*dsin(a1*pi) + if (kode.eq.2) a=a*dexp(-abs(xr)) +c . change sign to take into account that sin((a1+k)*pi) +C . changes sign with k + if (nn.ge.2) then + call dscal(nn/2,-1.0d0,yr(2),2) + call dscal(nn/2,-1.0d0,yi(2),2) + endif + call daxpy(nn,a,yr,1,wr,1) + call daxpy(nn,a,yi,1,wi,1) + + elseif (ierr.eq.2) then + call dset(nn,inf,wr,1) + call dset(nn,inf,wi,1) + elseif(ierr.ge.4) then + call dset(nn,inf-inf,wr,1) + call dset(nn,inf-inf,wi,1) + endif +c . store the result in the correct order + call dcopy(nn,wr,-1,yr,1) + call dcopy(nn,wi,-1,yi,1) +c . compute for positive value of alpha+k is any (note that x>0) + if (n.gt.nn) then + a1=1.0d0-a1 + call zbesi(xr,xi,a1,kode,n-nn,yr(nn+1),yi(nn+1),nz,ier) + if (ier.eq.2) then + call dset(n-nn,inf,yr(nn+1),1) + call dset(n-nn,inf,yi(nn+1),1) + elseif(ier.ge.4) then + call dset(n-nn,inf-inf,yr(nn+1),1) + call dset(n-nn,inf-inf,yi(nn+1),1) + endif + ierr=max(ierr,ier) + endif + endif + end + + subroutine zbesiv (xr,xi,nx,alpha,na, kode,yr,yi,wr,wi,ierr) +c Author Serge Steer, Copyright INRIA, 2005 +c compute besseli function for x and alpha given by vectors +c w : working array of size 2*na (used only if nz>0 and alpha +C contains negative +C values + double precision xr(nx),xi(nx),alpha(na),yr(*),yi(*),wr(*),wi(*) + double precision dlamch,eps + integer kode,nx,na,ier + ierr=0 + eps=dlamch('p') + if (na.lt.0) then +c . element wise case x and alpha are supposed to have the same +C size + do i=1,nx + call zbesig (xr(i), xi(i), alpha(i), kode, 1, yr(i),yi(i), + $ nz,wr,wi,ier) + ierr=max(ierr,ier) + enddo + elseif (na.eq.1) then + do i=1,nx + call zbesig (xr(i), xi(i), alpha(1), kode, 1, yr(i),yi(i), + $ nz,wr,wi,ier) + ierr=max(ierr,ier) + enddo + else +c . compute besseli(x(i),y(j)), i=1,nx,j=1,na + j0=1 + 05 n=0 + 10 n=n+1 + j=j0+n + if (j.le.na.and.abs((1+alpha(j-1))-alpha(j)).le.eps) then + goto 10 + endif + + do i=1,nx + call zbesig (xr(i), xi(i), alpha(j0) ,kode, n, wr, wi, nz, + $ wr(na+1),wi(na+1),ier) + ierr=max(ierr,ier) + call dcopy(n,wr,1,yr(i+(j0-1)*nx),nx) + call dcopy(n,wi,1,yi(i+(j0-1)*nx),nx) + enddo + j0=j + if (j0.le.na) goto 05 + endif + end + + diff --git a/modules/special_functions/src/fortran/zbesig.lo b/modules/special_functions/src/fortran/zbesig.lo new file mode 100755 index 000000000..d046de2f2 --- /dev/null +++ b/modules/special_functions/src/fortran/zbesig.lo @@ -0,0 +1,12 @@ +# src/fortran/zbesig.lo - a libtool object file +# Generated by libtool (GNU libtool) 2.4.2 +# +# Please DO NOT delete this file! +# It is necessary for linking the library. + +# Name of the PIC object. +pic_object='.libs/zbesig.o' + +# Name of the non-PIC object +non_pic_object=none + diff --git a/modules/special_functions/src/fortran/zbesjg.f b/modules/special_functions/src/fortran/zbesjg.f new file mode 100755 index 000000000..80cd6b4ca --- /dev/null +++ b/modules/special_functions/src/fortran/zbesjg.f @@ -0,0 +1,182 @@ +c Scilab ( http://www.scilab.org/ ) - This file is part of Scilab +c Copyright (C) 2005 - INRIA - Serge STEER +c +c This file must be used under the terms of the CeCILL. +c This source file is licensed as described in the file COPYING, which +c you should have received as part of this distribution. The terms +c are also available at +c http://www.cecill.info/licences/Licence_CeCILL_V2.1-en.txt +c + subroutine zbesjg (x1r,x1i, alpha, kode, n, yr,yi, nz,wr,wi,ierr) +c Author Serge Steer, 2005 +c extends cbesj for the case where alpha is negative + double precision alpha + double precision x1r, x1i, yr(n), yi(n), wr(2*n),wi(2*n) + integer kode,n,nz,ierr +c + double precision a,b,pi,inf,eps,xr, xi,a1 + integer ier1,ier2 + double precision dlamch + + data pi /3.14159265358979324D0/ + inf=dlamch('o')*2.0d0 + eps=dlamch('p') + + xr=x1r + xi=x1i + ier2=0 + if (xr.ne.xr.or.xi.ne.xi.or.alpha.ne.alpha) then +c . NaN case + call dset(n,inf-inf,yr,1) + call dset(n,inf-inf,yi,1) + ier1=4 + elseif (alpha .ge. 0.0d0) then +c . alpha >= 0 the simple case CHECKED + call zbesj(xr,xi,alpha,kode,n,yr,yi,nz,ierr) + if (ierr.eq.2) then + call dset(n,inf,yr,1) + call dset(n,inf,yi,1) + elseif(ierr.ge.4) then + call dset(n,inf-inf,yr,1) + call dset(n,inf-inf,yi,1) + endif + else if (alpha .eq. dint(alpha)) then +c . alpha <0 and integer, CHECKED +c . transform to positive value of alpha + if(alpha-1+n.ge.0) then +c . 0 is between alpha and alpha+n + a1=0.0d0 + nn=min(n,int(-alpha)) + else + a1=-(alpha-1+n) + nn=n + endif + call zbesj(xr,xi,a1,kode,n,wr,wi,nz,ierr) + if (ierr.eq.2) then + call dset(n,inf,yr,1) + call dset(n,inf,yi,1) + else + if(n.gt.nn) then +c . 0 is between alpha and alpha+n + call dcopy(n-nn,wr,1,yr(nn+1),1) + call dcopy(n-nn,wi,1,yi(nn+1),1) + call dcopy(nn,wr(2),-1,yr,1) + call dcopy(nn,wi(2),-1,yi,1) + else +c . alpha and alpha+n are negative + call dcopy(nn,wr,-1,yr,1) + call dcopy(nn,wi,-1,yi,1) + endif + endif +c . apply parity + i0=mod(int(abs(alpha))+1,2) + call dscal((nn-i0+1)/2,-1.0d0,yr(1+i0),2) + call dscal((nn-i0+1)/2,-1.0d0,yi(1+i0),2) + else if (xr .eq. 0.0d0.and.xi .eq.0.0d0) then + call dset(n,-inf,yr,1) + call dset(n,0.0d0,yi,1) + ierr=2 + else +c . alpha is negative non integer CHECKED +c . transform to positive value of alpha + if(alpha-1.0d0+n.ge.0.0d0) then +c . 0 is between alpha and alpha+n + nn=int(-alpha)+1 + else + nn=n + endif + a1=-(alpha-1.0d0+nn) + call zbesj(xr,xi,a1,kode,nn,yr,yi,nz1,ierr) + call zbesy(xr,xi,a1,kode,nn,wr,wi,nz2,wr(n+1),wi(n+1),ier) + ierr=max(ierr,ier) + + if (ierr.eq.0) then + a=cos(a1*pi) + b=-sin(a1*pi) +c . to avoid numerical errors if a is near 0 (or b near 0) +c . and y is big (or w is big) + if(abs(abs(a)-1.0d0).lt.eps) b=0.0d0 + if(abs(abs(b)-1.0d0).lt.eps) a=0.0d0 + call dscal(nn,b,wr,1) + call dscal(nn,b,wi,1) + call daxpy(nn,a,yr,1,wr,1) + call daxpy(nn,a,yi,1,wi,1) + elseif (ierr.eq.2) then + call dset(nn,inf,wr,1) + call dset(nn,inf,wi,1) + elseif (ierr.eq.4) then + call dset(nn,inf-inf,wr,1) + call dset(nn,inf-inf,wi,1) + endif +c . change sign to take into account that cos((a1+k)*pi) and +C . sin((a1+k)*pi) change sign with k + if (nn.ge.2) then + call dscal(nn/2,-1.0d0,wr(2),2) + call dscal(nn/2,-1.0d0,wi(2),2) + endif +c . store the result in the correct order + call dcopy(nn,wr,-1,yr,1) + call dcopy(nn,wi,-1,yi,1) +c . compute for positive value of alpha+k is any + if (n.gt.nn) then +c . this code is taken from the alpha>0 case above + a1=1.0d0-a1 + call zbesj(xr,xi,a1,kode,n-nn,yr(nn+1),yi(nn+1),nz,ier) + if (ier.eq.2) then + call dset(n-nn,inf,yr(nn+1),1) + call dset(n-nn,inf,yi(nn+1),1) + elseif(ier.ge.4) then + call dset(n-nn,inf-inf,yr(nn+1),1) + call dset(n-nn,inf-inf,yi(nn+1),1) + endif + ierr=max(ierr,ier) + endif + + endif + end + + subroutine zbesjv (xr,xi,nx,alpha,na, kode,yr,yi,wr,wi,ierr) +c Author Serge Steer, Copyright INRIA, 2005 +c compute besseli function for x and alpha given by vectors +c w : working array of size 3*na (used only if nz>0 and alpha contains negative +C values + double precision xr(nx),xi(nx),alpha(na),yr(*),yi(*),wr(*),wi(*) + double precision dlamch,eps + integer kode,nx,na,ier + ierr=0 + eps=dlamch('p') + if (na.lt.0) then +c . element wise case x and alpha are supposed to have the same size + do i=1,nx + call zbesjg (xr(i), xi(i), alpha(i), kode, 1, yr(i),yi(i), + $ nz,wr,wi,ier) + enddo + elseif (na.eq.1) then + do i=1,nx + call zbesjg (xr(i), xi(i), alpha(1), kode, 1, yr(i),yi(i), + $ nz,wr,wi,ier) + ierr=max(ierr,ier) + enddo + else +c . compute besseli(x(i),y(j)), i=1,nx,j=1,na + j0=1 + 05 n=0 + 10 n=n+1 + j=j0+n + if (j.le.na.and.abs((1+alpha(j-1))-alpha(j)).le.eps) then + goto 10 + endif + + do i=1,nx + call zbesjg (xr(i), xi(i), alpha(j0) ,kode, n, wr, wi, nz, + $ wr(na+1),wi(na+1),ier) + ierr=max(ierr,ier) + call dcopy(n,wr,1,yr(i+(j0-1)*nx),nx) + call dcopy(n,wi,1,yi(i+(j0-1)*nx),nx) + enddo + j0=j + if (j0.le.na) goto 05 + endif + end + + diff --git a/modules/special_functions/src/fortran/zbesjg.lo b/modules/special_functions/src/fortran/zbesjg.lo new file mode 100755 index 000000000..e03f66dc4 --- /dev/null +++ b/modules/special_functions/src/fortran/zbesjg.lo @@ -0,0 +1,12 @@ +# src/fortran/zbesjg.lo - a libtool object file +# Generated by libtool (GNU libtool) 2.4.2 +# +# Please DO NOT delete this file! +# It is necessary for linking the library. + +# Name of the PIC object. +pic_object='.libs/zbesjg.o' + +# Name of the non-PIC object +non_pic_object=none + diff --git a/modules/special_functions/src/fortran/zbeskg.f b/modules/special_functions/src/fortran/zbeskg.f new file mode 100755 index 000000000..34dc30e59 --- /dev/null +++ b/modules/special_functions/src/fortran/zbeskg.f @@ -0,0 +1,126 @@ +c Scilab ( http://www.scilab.org/ ) - This file is part of Scilab +c Copyright (C) 2005 - INRIA - Serge STEER +c +c This file must be used under the terms of the CeCILL. +c This source file is licensed as described in the file COPYING, which +c you should have received as part of this distribution. The terms +c are also available at +c http://www.cecill.info/licences/Licence_CeCILL_V2.1-en.txt +c + subroutine zbeskg (x1r,x1i, alpha, kode, n, yr,yi, nz,ierr) +c Author Serge Steer, 2005 +c extends cbesk for the case where alpha is negative + double precision alpha + double precision x1r, x1i, yr(n), yi(n) + integer kode,n,nz,ierr +c + double precision inf,xr, xi,a1,temp + double precision dlamch + inf=dlamch('o')*2.0d0 + + xr=x1r + xi=x1i + ierr=0 + if (xr.ne.xr.or.xi.ne.xi.or.alpha.ne.alpha) then +c . NaN case + call dset(n,inf-inf,yr,1) + call dset(n,inf-inf,yi,1) + ierr=4 + elseif (alpha.ge.0.0d0) then + call zbesk(xr,xi,abs(alpha),kode,n,yr,yi,nz,ierr) + if (ierr.eq.2) then + call dset(n,inf,yr,1) + call dset(n,inf,yi,1) + elseif(ierr.ge.4) then + call dset(n,inf-inf,yr,1) + call dset(n,inf-inf,yi,1) + endif + else + if(alpha-1.0d0+n.ge.0.0d0) then +c . 0 is between alpha and alpha+n + nn=int(-alpha)+1 + else + nn=n + endif + a1=-(alpha-1.0d0+nn) + call zbesk(xr,xi,a1,kode,nn,yr,yi,nz,ierr) + if (ierr.eq.0) then +c . swap the result to have it in correct order + if (nn.ge.2) then + do i=1,nn/2 + temp=yr(i) + yr(i)=yr(nn+1-i) + yr(nn+1-i)=temp + temp=yi(i) + yi(i)=yi(nn+1-i) + yi(nn+1-i)=temp + enddo + endif + elseif(ierr.eq.2) then + call dset(n,inf,yr,1) + call dset(n,inf,yi,1) + elseif(ierr.ge.4) then + call dset(n,inf-inf,yr,1) + call dset(n,inf-inf,yi,1) + endif + if (n.gt.nn) then + a1=1.0d0-a1 + call zbesk(xr,xi,a1,kode,n-nn,yr(nn+1),yi(nn+1),nz,ier) + if (ier.eq.2) then + call dset(n-nn,inf,yr(nn+1),1) + call dset(n-nn,inf,yi(nn+1),1) + elseif(ierr.ge.4) then + call dset(n-nn,inf-inf,yr(nn+1),1) + call dset(n-nn,inf-inf,yi(nn+1),1) + endif + ierr=max(ierr,ier) + endif + endif + + end + + subroutine zbeskv (xr,xi,nx,alpha,na, kode,yr,yi,wr,wi,ierr) +c Author Serge Steer, Copyright INRIA, 2005 +c compute besseli function for x and alpha given by vectors +c w : working array of size 2*na (used only if nz>0 and alpha contains negative +C values + double precision xr(nx),xi(nx),alpha(na),yr(*),yi(*),wr(*),wi(*) + double precision dlamch,eps + integer kode,nx,na,ier + eps=dlamch('p') + ierr=0 + if (na.lt.0) then +c . element wise case x and alpha are supposed to have the same size + do i=1,nx + call zbeskg (xr(i), xi(i), alpha(i), kode, 1, yr(i),yi(i), + $ nz,ier) + ierr=max(ierr,ier) + enddo + elseif (na.eq.1) then + do i=1,nx + call zbeskg (xr(i), xi(i), alpha(1), kode, 1, yr(i),yi(i), + $ nz,ier) + ierr=max(ierr,ier) + enddo + else +c . compute besseli(x(i),y(j)), i=1,nx,j=1,na + j0=1 + 05 n=0 + 10 n=n+1 + j=j0+n + if (j.le.na.and.abs((1.0d0+alpha(j-1))-alpha(j)).le.eps) then + goto 10 + endif + + do i=1,nx + call zbeskg (xr(i),xi(i),alpha(j0),kode, n, wr, wi, nz,ier) + ierr=max(ierr,ier) + call dcopy(n,wr,1,yr(i+(j0-1)*nx),nx) + call dcopy(n,wi,1,yi(i+(j0-1)*nx),nx) + enddo + j0=j + if (j0.le.na) goto 05 + endif + end + + diff --git a/modules/special_functions/src/fortran/zbeskg.lo b/modules/special_functions/src/fortran/zbeskg.lo new file mode 100755 index 000000000..48184e977 --- /dev/null +++ b/modules/special_functions/src/fortran/zbeskg.lo @@ -0,0 +1,12 @@ +# src/fortran/zbeskg.lo - a libtool object file +# Generated by libtool (GNU libtool) 2.4.2 +# +# Please DO NOT delete this file! +# It is necessary for linking the library. + +# Name of the PIC object. +pic_object='.libs/zbeskg.o' + +# Name of the non-PIC object +non_pic_object=none + diff --git a/modules/special_functions/src/fortran/zbesyg.f b/modules/special_functions/src/fortran/zbesyg.f new file mode 100755 index 000000000..e8f37e80d --- /dev/null +++ b/modules/special_functions/src/fortran/zbesyg.f @@ -0,0 +1,182 @@ +c Scilab ( http://www.scilab.org/ ) - This file is part of Scilab +c Copyright (C) 2005 - INRIA - Serge STEER +c +c This file must be used under the terms of the CeCILL. +c This source file is licensed as described in the file COPYING, which +c you should have received as part of this distribution. The terms +c are also available at +c http://www.cecill.info/licences/Licence_CeCILL_V2.1-en.txt +c + subroutine zbesyg (x1r,x1i, alpha, kode, n, yr,yi, nz,wr,wi,ierr) +c Author Serge Steer, 2005 +c extends cbesy for the case where alpha is negative + double precision alpha + double precision x1r, x1i, yr(n), yi(n), wr(2*n),wi(2*n) + integer kode,n,nz,ierr +c + double precision a,b,pi,inf,eps,xr, xi,a1 + integer ier2 + double precision dlamch + + data pi /3.14159265358979324D0/ + inf=dlamch('o')*2.0d0 + eps=dlamch('p') + + xr=x1r + xi=x1i + ier2=0 + if (xr.ne.xr.or.xi.ne.xi.or.alpha.ne.alpha) then +c . NaN case + call dset(n,inf-inf,yr,1) + call dset(n,inf-inf,yi,1) + ierr=4 + else if (xr .eq. 0.0d0.and.xi .eq.0.0d0) then + call dset(n,-inf,yr,1) + call dset(n,0.0d0,yi,1) + ierr=2 + elseif (alpha .ge. 0.0d0) then + call zbesy(xr,xi,alpha,kode,n,yr,yi,nz,wr,wi,ierr) + if (ierr.eq.2) then + call dset(n,inf,yr,1) + call dset(n,inf,yi,1) + elseif(ierr.ge.4) then + call dset(n,inf-inf,yr,1) + call dset(n,inf-inf,yi,1) + endif + else if (alpha .eq. dint(alpha)) then +c . alpha <0 and integer, +c . transform to positive value of alpha + if(alpha-1+n.ge.0) then +c . 0 is between alpha and alpha+n + a1=0.0d0 + nn=min(n,int(-alpha)) + else + a1=-(alpha-1+n) + nn=n + endif + call zbesy(xr,xi,a1,kode,n,wr,wi,nz,yr,yi,ierr) + if (ierr.eq.2) then + call dset(n,inf,yr,1) + call dset(n,inf,yi,1) + else + if(n.gt.nn) then +c . 0 is between alpha and alpha+n + call dcopy(n-nn,wr,1,yr(nn+1),1) + call dcopy(n-nn,wi,1,yi(nn+1),1) + call dcopy(nn,wr(2),-1,yr,1) + call dcopy(nn,wi(2),-1,yi,1) + else +c . alpha and alpha+n are negative + call dcopy(nn,wr,-1,yr,1) + call dcopy(nn,wi,-1,yi,1) + endif + endif + i0=mod(int(abs(alpha))+1,2) + call dscal((nn-i0+1)/2,-1.0d0,yr(1+i0),2) + call dscal((nn-i0+1)/2,-1.0d0,yi(1+i0),2) + + else +c . first alpha is negative non integer, x should be positive (with +C . x negative the result is complex. CHECKED +c . transform to positive value of alpha + if(alpha-1.0d0+n.ge.0.0d0) then +c . 0 is between alpha and alpha+n + nn=int(-alpha)+1 + else + nn=n + endif +c . compute for negative value of alpha+k, transform problem for +c . a1:a1+(nn-1) with a1 positive a1+k =abs(alpha+nn-k) + a1=-(alpha-1.0d0+nn) + call zbesj(xr,xi,a1,kode,nn,yr,yi,nz1,ierr) + call zbesy(xr,xi,a1,kode,nn,wr,wi,nz2,wr(n+1),wi(n+1),ier) + nz=max(nz1,nz2) + ierr=max(ierr,ier) + if (ierr.eq.0) then + a=sin(a1*pi) + b=cos(a1*pi) +c . to avoid numerical errors if a is near 0 (or b near 0) +c . and y is big (or w is big) + if(abs(abs(a)-1.0d0).lt.eps) b=0.0d0 + if(abs(abs(b)-1.0d0).lt.eps) a=0.0d0 + call dscal(nn,b,wr,1) + call dscal(nn,b,wi,1) + call daxpy(nn,a,yr,1,wr,1) + call daxpy(nn,a,yi,1,wi,1) + elseif (ierr.eq.2) then + call dset(nn,inf,wr,1) + call dset(nn,0.0d0,wi,1) + elseif (ierr.eq.4) then + call dset(nn,inf-inf,wr,1) + call dset(nn,inf-inf,wi,1) + endif +c . change sign to take into account that cos((a1+k)*pi) and +C . sin((a1+k)*pi) changes sign with k + if (nn.ge.2) then + call dscal(nn/2,-1.0d0,wr(2),2) + call dscal(nn/2,-1.0d0,wi(2),2) + endif +c . store the result in the correct order + call dcopy(nn,wr,-1,yr,1) + call dcopy(nn,wi,-1,yi,1) +c . compute for positive value of alpha+k is any + if (n.gt.nn) then + call zbesy(xr,xi,1.0d0-a1,kode,n-nn,yr(nn+1),yi(nn+1),nz, + $ wr,wi,ier) + if (ier.eq.2) then + call dset(n-nn,inf,yr(nn+1),1) + call dset(n-nn,inf,yi(nn+1),1) + elseif(ier.ge.4) then + call dset(n-nn,inf-inf,yr(nn+1),1) + call dset(n-nn,inf-inf,yi(nn+1),1) + endif + ierr=max(ierr,ier) + endif + endif + end + + subroutine zbesyv (xr,xi,nx,alpha,na, kode,yr,yi,wr,wi,ierr) +c Author Serge Steer, Copyright INRIA, 2005 +c compute besseli function for x and alpha given by vectors +c w : working array of size 3*na (used only if nz>0 and alpha contains negative +C values + double precision xr(nx),xi(nx),alpha(na),yr(*),yi(*),wr(*),wi(*) + double precision dlamch,eps + integer kode,nx,na,ier + ierr=0 + eps=dlamch('p') + if (na.lt.0) then +c . element wise case x and alpha are supposed to have the same size + do i=1,nx + call zbesyg (xr(i), xi(i), alpha(i), kode, 1, yr(i),yi(i), + $ nz,wr,wi,ier) + enddo + elseif (na.eq.1) then + do i=1,nx + call zbesyg (xr(i), xi(i), alpha(1), kode, 1, yr(i),yi(i), + $ nz,wr,wi,ier) + ierr=max(ierr,ier) + enddo + else +c . compute besseli(x(i),y(j)), i=1,nx,j=1,na + j0=1 + 05 n=0 + 10 n=n+1 + j=j0+n + if (j.le.na.and.abs((1+alpha(j-1))-alpha(j)).le.eps) then + goto 10 + endif + + do i=1,nx + call zbesyg (xr(i), xi(i), alpha(j0) ,kode, n, wr, wi, nz, + $ wr(na+1),wi(na+1),ier) + ierr=max(ierr,ier) + call dcopy(n,wr,1,yr(i+(j0-1)*nx),nx) + call dcopy(n,wi,1,yi(i+(j0-1)*nx),nx) + enddo + j0=j + if (j0.le.na) goto 05 + endif + end + + diff --git a/modules/special_functions/src/fortran/zbesyg.lo b/modules/special_functions/src/fortran/zbesyg.lo new file mode 100755 index 000000000..5fa81cc84 --- /dev/null +++ b/modules/special_functions/src/fortran/zbesyg.lo @@ -0,0 +1,12 @@ +# src/fortran/zbesyg.lo - a libtool object file +# Generated by libtool (GNU libtool) 2.4.2 +# +# Please DO NOT delete this file! +# It is necessary for linking the library. + +# Name of the PIC object. +pic_object='.libs/zbesyg.o' + +# Name of the non-PIC object +non_pic_object=none + |