diff --git a/CMakeLists.txt b/CMakeLists.txt index 7df8eaedb0..05f03626e7 100644 --- a/CMakeLists.txt +++ b/CMakeLists.txt @@ -1,5 +1,5 @@ cmake_minimum_required(VERSION 2.8 FATAL_ERROR) -project(openmc Fortran) +project(openmc Fortran C) # Setup output directories set(CMAKE_ARCHIVE_OUTPUT_DIRECTORY ${CMAKE_BINARY_DIR}/lib) @@ -107,49 +107,62 @@ if(CMAKE_Fortran_COMPILER_ID STREQUAL GNU) message(FATAL_ERROR "gfortran version must be 4.6 or higher") endif() - # GNU Fortran compiler options + # GCC compiler options list(APPEND f90flags -cpp -std=f2008 -fbacktrace) + list(APPEND cflags -cpp -std=c99) if(debug) if(NOT (GCC_VERSION VERSION_LESS 4.7)) list(APPEND f90flags -Wall) + list(APPEND cflags -Wall) endif() list(APPEND f90flags -g -pedantic -fbounds-check -ffpe-trap=invalid,overflow,underflow) + list(APPEND cflags -g -pedantic -fbounds-check) list(APPEND ldflags -g) endif() if(profile) list(APPEND f90flags -pg) + list(APPEND cflags -pg) list(APPEND ldflags -pg) endif() if(optimize) list(APPEND f90flags -O3 -flto -fuse-linker-plugin) + list(APPEND cflags -O3 -flto -fuse-linker-plugin) endif() if(openmp) list(APPEND f90flags -fopenmp) + list(APPEND cflags -fopenmp) list(APPEND ldflags -fopenmp) endif() if(coverage) list(APPEND f90flags -coverage) + list(APPEND cflags -coverage) list(APPEND ldflags -coverage) endif() elseif(CMAKE_Fortran_COMPILER_ID STREQUAL Intel) - # Intel Fortran compiler options + # Intel compiler options list(APPEND f90flags -fpp -std08 -assume byterecl -traceback) + list(APPEND cflags -std=c99) if(debug) list(APPEND f90flags -g -warn -ftrapuv -fp-stack-check "-check all" -fpe0) + list(APPEND cflags -g -warn -ftrapuv -fp-stack-check + "-check all" -fpe0) list(APPEND ldflags -g) endif() if(profile) list(APPEND f90flags -pg) + list(APPEND cflags -pg) list(APPEND ldflags -pg) endif() if(optimize) list(APPEND f90flags -O3) + list(APPEND cflags -O3) endif() if(openmp) list(APPEND f90flags -openmp) + list(APPEND cflags -openmp) list(APPEND ldflags -openmp) endif() @@ -243,6 +256,12 @@ add_subdirectory(src/xml/fox) # which point to directories outside the build tree to the install RPATH set(CMAKE_INSTALL_RPATH_USE_LINK_PATH TRUE) +#=============================================================================== +# Build faddeeva library +#=============================================================================== + +add_library(faddeeva STATIC src/Faddeeva.c) + #=============================================================================== # Build OpenMC executable #=============================================================================== @@ -263,11 +282,14 @@ endif() # set compile flags. Note that this sets the COMPILE_OPTIONS property (also # available only in 2.8.12+) rather than the COMPILE_FLAGS property, which is # deprecated. The former can handle lists whereas the latter cannot. -if(CMAKE_VERSION VERSION_LESS 2.8.12) +if (CMAKE_VERSION VERSION_LESS 2.8.12) string(REPLACE ";" " " f90flags "${f90flags}") + string(REPLACE ";" " " cflags "${cflags}") set_property(TARGET ${program} PROPERTY COMPILE_FLAGS "${f90flags}") + set_property(TARGET faddeeva PROPERTY COMPILE_FLAGS "${cflags}") else() target_compile_options(${program} PUBLIC ${f90flags}) + target_compile_options(faddeeva PRIVATE ${cflags}) endif() # Add HDF5 library directories to link line with -L @@ -277,7 +299,7 @@ endforeach() # target_link_libraries treats any arguments starting with - but not -l as # linker flags. Thus, we can pass both linker flags and libraries together. -target_link_libraries(${program} ${ldflags} ${HDF5_LIBRARIES} fox_dom) +target_link_libraries(${program} ${ldflags} ${HDF5_LIBRARIES} fox_dom faddeeva) #=============================================================================== # Install executable, scripts, manpage, license diff --git a/docs/source/conf.py b/docs/source/conf.py index 65db07b25e..6ca551a431 100644 --- a/docs/source/conf.py +++ b/docs/source/conf.py @@ -154,7 +154,8 @@ html_title = "OpenMC Documentation" # so a file named "default.css" will overwrite the builtin "default.css". html_static_path = ['_static'] -html_context = {'css_files': ['_static/theme_overrides.css']} +def setup(app): + app.add_stylesheet('theme_overrides.css') # If not '', a 'Last updated on:' timestamp is inserted at every page bottom, # using the given strftime format. diff --git a/docs/source/usersguide/input.rst b/docs/source/usersguide/input.rst index d1dac8caff..87cab5ebaf 100644 --- a/docs/source/usersguide/input.rst +++ b/docs/source/usersguide/input.rst @@ -987,6 +987,15 @@ Each ```` element can have the following attributes or sub-elements: *Default*: A region filling all space. + :temperature: + The temperature of the cell in Kelvin. If windowed-multipole data is + avalable, this temperature will be used to Doppler broaden some cross + sections in the resolved resonance region. A list of temperatures can be + specified for the "distributed temperature" feature. This will give each + unique instance of the cell its own temperature. + + *Default*: The temperature of the coldest nuclide in the cell's material(s) + :rotation: If the cell is filled with a universe, this element specifies the angles in degrees about the x, y, and z axes that the filled universe should be diff --git a/docs/source/usersguide/output/summary.rst b/docs/source/usersguide/output/summary.rst index 83602e5065..3be0cfdfe9 100644 --- a/docs/source/usersguide/output/summary.rst +++ b/docs/source/usersguide/output/summary.rst @@ -98,6 +98,10 @@ The current revision of the summary file format is 1. material. The data is an array if the cell uses distributed materials, otherwise it is a scalar. +**/geometry/cells/cell /temperature** (*double[]*) + + Temperature of the cell in Kelvin. + **/geometry/cells/cell /offset** (*int[]*) Offsets used for distribcell tally filter. This dataset is present only if diff --git a/openmc/arithmetic.py b/openmc/arithmetic.py index 65e12c5d27..8574c4873e 100644 --- a/openmc/arithmetic.py +++ b/openmc/arithmetic.py @@ -823,7 +823,8 @@ class AggregateFilter(object): """ - if filter_bin not in self.bins: + if filter_bin not in self.bins and \ + filter_bin != self._aggregate_filter.bins: msg = 'Unable to get the bin index for AggregateFilter since ' \ '"{0}" is not one of the bins'.format(filter_bin) raise ValueError(msg) diff --git a/openmc/filter.py b/openmc/filter.py index a0c777fa07..54814a6b6f 100644 --- a/openmc/filter.py +++ b/openmc/filter.py @@ -673,9 +673,11 @@ class Filter(object): # Assign entry to Lattice Multi-index column else: + # Reverse y index per lattice ordering in OpenCG level_dict[lat_id_key][offset] = coords._lattice._id level_dict[lat_x_key][offset] = coords._lat_x - level_dict[lat_y_key][offset] = coords._lat_y + level_dict[lat_y_key][offset] = \ + coords._lattice.dimension[1] - coords._lat_y - 1 level_dict[lat_z_key][offset] = coords._lat_z # Move to next node in LocalCoords linked list diff --git a/openmc/mgxs/library.py b/openmc/mgxs/library.py index 4b8d8a9149..e87b4bdaef 100644 --- a/openmc/mgxs/library.py +++ b/openmc/mgxs/library.py @@ -568,8 +568,9 @@ class Library(object): for domain in self.domains: for mgxs_type in self.mgxs_types: mgxs = subdomain_avg_library.get_mgxs(domain, mgxs_type) - avg_mgxs = mgxs.get_subdomain_avg_xs() - subdomain_avg_library.all_mgxs[domain.id][mgxs_type] = avg_mgxs + if mgxs.domain_type == 'distribcell': + avg_mgxs = mgxs.get_subdomain_avg_xs() + subdomain_avg_library.all_mgxs[domain.id][mgxs_type] = avg_mgxs return subdomain_avg_library diff --git a/openmc/summary.py b/openmc/summary.py index eb14d3bb81..614a022068 100644 --- a/openmc/summary.py +++ b/openmc/summary.py @@ -264,6 +264,9 @@ class Summary(object): self._f['geometry/cells'][key]['rotation'][...] rotation = np.asarray(rotation, dtype=np.int) cell.rotation = rotation + elif fill_type == 'normal': + cell.temperature = \ + self._f['geometry/cells'][key]['temperature'][...] # Store Cell fill information for after Universe/Lattice creation self._cell_fills[index] = (fill_type, fill) @@ -367,9 +370,9 @@ class Summary(object): # Set the universes for the lattice lattice.universes = universes + # Set the distribcell offsets for the lattice if offsets is not None: - offsets = np.swapaxes(offsets, 0, 2) - lattice.offsets = offsets + lattice.offsets = offsets[:, ::-1, :] # Add the Lattice to the global dictionary of all Lattices self.lattices[index] = lattice diff --git a/openmc/tallies.py b/openmc/tallies.py index da209f527a..ae9493b16a 100644 --- a/openmc/tallies.py +++ b/openmc/tallies.py @@ -2785,7 +2785,7 @@ class Tally(object): bin_indices.append(bin_index) num_bins += 1 - find_filter.bins = set(find_filter.bins[bin_indices]) + find_filter.bins = np.unique(find_filter.bins[bin_indices]) find_filter.num_bins = num_bins # Update the new tally's filter strides diff --git a/openmc/universe.py b/openmc/universe.py index 237ddce1f7..edc4619dd2 100644 --- a/openmc/universe.py +++ b/openmc/universe.py @@ -51,9 +51,13 @@ class Cell(object): name : str Name of the cell fill : Material or Universe or Lattice or 'void' or iterable of Material - Indicates what the region of space is filled with + Indicates what the region of space is filled with. Multiple materials + can be given to give each distributed cell instance a unique material. region : openmc.region.Region Region of space that is assigned to the cell. + temperature : float or iterable of float + Temperature of the cell in Kelvin. Multiple temperatures can be given + to give each distributed cell instance a unique temperature. rotation : ndarray If the cell is filled with a universe, this array specifies the angles in degrees about the x, y, and z axes that the filled universe should be @@ -75,6 +79,7 @@ class Cell(object): self._fill = None self._type = None self._region = None + self._temperature = None self._rotation = None self._translation = None self._offsets = None @@ -91,6 +96,8 @@ class Cell(object): return False elif self.region != other.region: return False + elif self.temperature != other.temperature: + return False elif self.rotation != other.rotation: return False elif self.translation != other.translation: @@ -126,6 +133,10 @@ class Cell(object): string += '{0: <16}{1}{2}\n'.format('\tRegion', '=\t', self._region) + if self.fill_type == 'material': + string += '\t{0: <15}=\t{1}\n'.format('Temperature', + self.temperature) + string += '{0: <16}{1}{2}\n'.format('\tRotation', '=\t', self._rotation) string += '{0: <16}{1}{2}\n'.format('\tTranslation', '=\t', @@ -150,7 +161,7 @@ class Cell(object): @property def fill_type(self): - if isinstance(self.fill, openmc.Material): + if isinstance(self.fill, (openmc.Material, Iterable)): return 'material' elif isinstance(self.fill, openmc.Universe): return 'universe' @@ -163,6 +174,10 @@ class Cell(object): def region(self): return self._region + @property + def temperature(self): + return self._temperature + @property def rotation(self): return self._rotation @@ -251,6 +266,17 @@ class Cell(object): cv.check_type('cell region', region, Region) self._region = region + @temperature.setter + def temperature(self, temperature): + cv.check_type('cell temperature', temperature, (Iterable, Real)) + if isinstance(temperature, Iterable): + cv.check_type('cell temperature', temperature, Iterable, Real) + for T in temperature: + cv.check_greater_than('cell temperature', T, 0.0, True) + else: + cv.check_greater_than('cell temperature', temperature, 0.0, True) + self._temperature = temperature + @distribcell_index.setter def distribcell_index(self, ind): cv.check_type('distribcell index', ind, Integral) @@ -445,6 +471,13 @@ class Cell(object): # Call the recursive function from the top node create_surface_elements(self.region, xml_element) + if self.temperature is not None: + if isinstance(self.temperature, Iterable): + element.set("temperature", ' '.join( + [str(t) for t in self.temperature])) + else: + element.set("temperature", str(self.temperature)) + if self.translation is not None: element.set("translation", ' '.join(map(str, self.translation))) @@ -1095,14 +1128,14 @@ class RectLattice(Lattice): # For 2D Lattices if len(self._dimension) == 2: - offset = self._offsets[i[1]-1, i[2]-1, 0, distribcell_index-1] + offset = self._offsets[i[3]-1, i[2]-1, i[1]-1, distribcell_index-1] offset += self._universes[i[1]-1][i[2]-1].get_cell_instance(path, distribcell_index) # For 3D Lattices else: - offset = self._offsets[i[1]-1, i[2]-1, i[3]-1, distribcell_index-1] - offset += self._universes[i[1]-1][i[2]-1][i[3]-1].get_cell_instance( + offset = self._offsets[i[3]-1, i[2]-1, i[1]-1, distribcell_index-1] + offset += self._universes[i[3]-1][i[2]-1][i[1]-1].get_cell_instance( path, distribcell_index) return offset diff --git a/setup.py b/setup.py index 0bf4549c03..87fdff68cf 100644 --- a/setup.py +++ b/setup.py @@ -32,7 +32,7 @@ kwargs = {'name': 'openmc', if have_setuptools: kwargs.update({ # Required dependencies - 'install_requires': ['numpy', 'h5py', 'matplotlib'], + 'install_requires': ['numpy>=1.9', 'h5py', 'matplotlib'], # Optional dependencies 'extras_require': { diff --git a/src/Faddeeva.c b/src/Faddeeva.c new file mode 100644 index 0000000000..78f5714efa --- /dev/null +++ b/src/Faddeeva.c @@ -0,0 +1,3 @@ +/* The Faddeeva.cc file contains macros to let it compile as C code + (assuming C99 complex-number support), so just #include it. */ +#include "Faddeeva.cc" diff --git a/src/Faddeeva.cc b/src/Faddeeva.cc new file mode 100644 index 0000000000..80e42228c1 --- /dev/null +++ b/src/Faddeeva.cc @@ -0,0 +1,2516 @@ +// -*- 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, 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 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 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 "config.h" +#endif + +///////////////////////////////////////////////////////////////////////// +// macros to allow us to use either C++ or C (with C99 features) + +#ifdef __cplusplus + +# include "Faddeeva.hh" + +# include +# include +# include +using namespace std; + +// use std::numeric_limits, since 1./0. and 0./0. fail with some compilers (MS) +# define Inf numeric_limits::infinity() +# define NaN numeric_limits::quiet_NaN() + +typedef complex 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 +# 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 (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 +# include + +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 +#else +# include +#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/src/Faddeeva.h b/src/Faddeeva.h new file mode 100644 index 0000000000..4293861907 --- /dev/null +++ b/src/Faddeeva.h @@ -0,0 +1,68 @@ +/* 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.c; see Faddeeva.cc for more information. */ + +#ifndef FADDEEVA_H +#define FADDEEVA_H 1 + +// Require C99 complex-number support +#include + +#ifdef __cplusplus +extern "C" +{ +#endif /* __cplusplus */ + +// compute w(z) = exp(-z^2) erfc(-iz) [ Faddeeva / scaled complex error func ] +extern double complex Faddeeva_w(double complex z,double relerr); +extern double Faddeeva_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) +extern double complex Faddeeva_erfcx(double complex z, double relerr); +extern double Faddeeva_erfcx_re(double x); // special case for real x + +// compute erf(z), the error function of complex arguments +extern double complex Faddeeva_erf(double complex z, double relerr); +extern double Faddeeva_erf_re(double x); // special case for real x + +// compute erfi(z) = -i erf(iz), the imaginary error function +extern double complex Faddeeva_erfi(double complex z, double relerr); +extern double Faddeeva_erfi_re(double x); // special case for real x + +// compute erfc(z) = 1 - erf(z), the complementary error function +extern double complex Faddeeva_erfc(double complex z, double relerr); +extern double Faddeeva_erfc_re(double x); // special case for real x + +// compute Dawson(z) = sqrt(pi)/2 * exp(-z^2) * erfi(z) +extern double complex Faddeeva_Dawson(double complex z, double relerr); +extern double Faddeeva_Dawson_re(double x); // special case for real x + +#ifdef __cplusplus +} +#endif /* __cplusplus */ + +#endif // FADDEEVA_H diff --git a/src/ace.F90 b/src/ace.F90 index 8d73f8bd49..813e06ec2a 100644 --- a/src/ace.F90 +++ b/src/ace.F90 @@ -11,6 +11,8 @@ module ace use global use list_header, only: ListInt use material_header, only: Material + use multipole, only: multipole_read + use multipole_header, only: max_L, max_poles, max_poly use output, only: write_message use set_header, only: SetChar use secondary_header, only: AngleEnergy @@ -111,6 +113,9 @@ contains end do end if + ! Read multipole file into the appropriate entry on the nuclides array + call read_multipole_data(i_nuclide) + ! Add name and alias to dictionary call already_read % add(name) call already_read % add(alias) @@ -414,6 +419,69 @@ contains end subroutine read_ace_table +!=============================================================================== +! READ_MULTIPOLE_DATA checks for the existence of a multipole library in the +! directory and loads it using multipole_read +!=============================================================================== + + subroutine read_multipole_data(i_table) + + integer, intent(in) :: i_table ! index in nuclides/sab_tables + + logical :: file_exists ! does multipole library exist? + character(7) :: readable ! is multipole library readable? + character(6) :: zaid_string ! String of the ZAID + character(9) :: filename ! path to multipole cross section library + type(Nuclide), pointer :: nuc => null() + + ! For the time being, and I know this is a bit hacky, we just assume + ! that the file will be zaid.h5. + associate (nuc => nuclides(i_table)) + + write(zaid_string,'(I6.6)') nuc % zaid + filename = zaid_string // ".h5" + + ! Check if Multipole library exists and is readable + inquire(FILE=filename, EXIST=file_exists, READ=readable) + if (.not. file_exists) then + nuc % mp_present = .false. + return + elseif (readable(1:3) == 'NO') then + call fatal_error("Multipole library '" // trim(filename) // "' is not readable! & + &Change file permissions with chmod command.") + end if + + ! display message + call write_message("Loading Multipole XS table: " // filename, 6) + + allocate(nuc % multipole) + + ! Call the read routine + call multipole_read(filename, nuc % multipole, i_table) + nuc % mp_present = .true. + + ! Update the maximum number of poles, l indices, and polynomial order + if (nuc % multipole % max_w > max_poles) then + max_poles = nuc % multipole % max_w + end if + + if (nuc % multipole % num_l > max_L) then + max_L = nuc % multipole % num_l + end if + + if (nuc % multipole % fit_order + 1 > max_poly) then + max_poly = nuc % multipole % fit_order + 1 + end if + + ! Recreate nu-fission tables + if (nuc % fissionable) then + call generate_nu_fission(nuc) + end if + + end associate + + end subroutine read_multipole_data + !=============================================================================== ! READ_ESZ - reads through the ESZ block. This block contains the energy grid, ! total xs, absorption xs, elastic scattering xs, and heating numbers. diff --git a/src/ace_header.F90 b/src/ace_header.F90 index 2331f9117c..2d1ca94751 100644 --- a/src/ace_header.F90 +++ b/src/ace_header.F90 @@ -3,6 +3,7 @@ module ace_header use constants, only: MAX_FILE_LEN, ZERO use dict_header, only: DictIntInt use endf_header, only: Tab1 + use multipole_header, only: MultipoleArray use secondary_header, only: SecondaryDistribution, AngleEnergyContainer use stl_vector, only: VectorInt @@ -110,6 +111,10 @@ module ace_header integer :: urr_inelastic type(UrrData), pointer :: urr_data => null() + ! Multipole data + logical :: mp_present + type(MultipoleArray), allocatable :: multipole + ! Reactions integer :: n_reaction ! # of reactions type(Reaction), allocatable :: reactions(:) @@ -233,6 +238,9 @@ module ace_header ! Information for URR probability table use logical :: use_ptable ! in URR range with probability tables? real(8) :: last_prn + + ! Information for Doppler broadening + real(8) :: last_sqrtkT = ZERO ! last temperature in sqrt(Boltzmann constant * temperature (MeV)) end type NuclideMicroXS !=============================================================================== @@ -271,6 +279,10 @@ module ace_header if (associated(this % urr_data)) deallocate(this % urr_data) + if (this % mp_present) then + deallocate(this % multipole) + end if + if (allocated(this % reactions)) then do i = 1, size(this % reactions) call this % reactions(i) % clear() diff --git a/src/constants.F90 b/src/constants.F90 index 72df3c216d..37f85b8303 100644 --- a/src/constants.F90 +++ b/src/constants.F90 @@ -63,6 +63,7 @@ module constants real(8), parameter :: & PI = 3.1415926535898_8, & ! pi + SQRT_PI = 1.7724538509055_8, & ! square root of pi MASS_NEUTRON = 1.008664916_8, & ! mass of a neutron in amu MASS_NEUTRON_MEV = 939.565379_8, & ! mass of a neutron in MeV/c^2 MASS_PROTON = 1.007276466812_8, & ! mass of a proton in amu @@ -77,6 +78,7 @@ module constants TWO = 2.0_8, & THREE = 3.0_8, & FOUR = 4.0_8 + complex(8), parameter :: ONEI = (ZERO, ONE) ! ============================================================================ ! GEOMETRY-RELATED CONSTANTS diff --git a/src/cross_section.F90 b/src/cross_section.F90 index f874eb2a74..033e80b174 100644 --- a/src/cross_section.F90 +++ b/src/cross_section.F90 @@ -1,19 +1,31 @@ module cross_section - use ace_header, only: Nuclide, SAlphaBeta, Reaction, UrrData + use ace_header, only: Nuclide, SAlphaBeta, Reaction, UrrData use constants - use energy_grid, only: grid_method, log_spacing - use error, only: fatal_error - use fission, only: nu_total + use energy_grid, only: grid_method, log_spacing + use error, only: fatal_error + use fission, only: nu_total use global - use list_header, only: ListElemInt - use material_header, only: Material - use particle_header, only: Particle - use random_lcg, only: prn - use search, only: binary_search + use list_header, only: ListElemInt + use material_header, only: Material + use math, only: w, broaden_n_polynomials + use multipole_header, only: FORM_RM, FORM_MLBW, MP_EA, RM_RT, RM_RA, RM_RF, & + MLBW_RT, MLBW_RX, MLBW_RA, MLBW_RF, FIT_T, FIT_A, FIT_F, & + MultipoleArray, max_poly, max_L, max_poles + use particle_header, only: Particle + use random_lcg, only: prn + use search, only: binary_search implicit none + ! Allocatable arrays for multipole that are allocated once for speed purposes + complex(8), allocatable :: sigT_factor(:) + real(8), allocatable :: twophi(:) + real(8), allocatable :: broadened_polynomials(:) + logical :: mp_already_alloc = .false. + +!$omp threadprivate(sigT_factor, twophi, broadened_polynomials, mp_already_alloc) + contains !=============================================================================== @@ -93,10 +105,13 @@ contains i_nuclide = mat % nuclide(i) ! Calculate microscopic cross section for this nuclide - if (p % E /= micro_xs(i_nuclide) % last_E) then - call calculate_nuclide_xs(i_nuclide, i_sab, p % E, p % material, i, i_grid) + if (p % E /= micro_xs(i_nuclide) % last_E & + .or. p % sqrtkT /= micro_xs(i_nuclide) % last_sqrtkT) then + call calculate_nuclide_xs(i_nuclide, i_sab, p % E, p % material, i, & + i_grid, p % sqrtkT) else if (i_sab /= micro_xs(i_nuclide) % last_index_sab) then - call calculate_nuclide_xs(i_nuclide, i_sab, p % E, p % material, i, i_grid) + call calculate_nuclide_xs(i_nuclide, i_sab, p % E, p % material, i, & + i_grid, p % sqrtkT) end if ! ======================================================================== @@ -133,7 +148,8 @@ contains ! given index in the nuclides array at the energy of the given particle !=============================================================================== - subroutine calculate_nuclide_xs(i_nuclide, i_sab, E, i_mat, i_nuc_mat, i_log_union) + subroutine calculate_nuclide_xs(i_nuclide, i_sab, E, i_mat, i_nuc_mat, & + i_log_union, sqrtkT) integer, intent(in) :: i_nuclide ! index into nuclides array integer, intent(in) :: i_sab ! index into sab_tables array real(8), intent(in) :: E ! energy @@ -141,11 +157,13 @@ contains integer, intent(in) :: i_nuc_mat ! index into nuclides array for a material integer, intent(in) :: i_log_union ! index into logarithmic mapping array or ! material union energy grid + real(8), intent(in) :: sqrtkT ! Square root of kT, material dependent integer :: i_grid ! index on nuclide energy grid integer :: i_low ! lower logarithmic mapping index integer :: i_high ! upper logarithmic mapping index real(8) :: f ! interp factor on nuclide energy grid + real(8) :: sigT, sigA, sigF ! Intermediate multipole variables type(Nuclide), pointer :: nuc type(Material), pointer :: mat @@ -153,53 +171,111 @@ contains nuc => nuclides(i_nuclide) mat => materials(i_mat) - ! Determine index on nuclide energy grid - select case (grid_method) - case (GRID_MAT_UNION) + ! If MP, don't interpolate, it's all already baked in. + if (nuc % mp_present .and. & + (E >= nuc % multipole % start_E/1.0e6_8 .and.& + E <= nuc % multipole % end_E/1.0e6_8)) then - i_grid = mat % nuclide_grid_index(i_nuc_mat, i_log_union) + ! Call multipole kernel + call multipole_eval(nuc % multipole, E, sqrtkT, sigT, sigA, sigF) - case (GRID_LOGARITHM) - ! Determine the energy grid index using a logarithmic mapping to reduce - ! the energy range over which a binary search needs to be performed + micro_xs(i_nuclide) % total = sigT + micro_xs(i_nuclide) % absorption = sigA + micro_xs(i_nuclide) % elastic = sigT - sigA - if (E < nuc % energy(1)) then - i_grid = 1 - elseif (E > nuc % energy(nuc % n_grid)) then - i_grid = nuc % n_grid - 1 + if (nuc % fissionable) then + micro_xs(i_nuclide) % fission = sigF + micro_xs(i_nuclide) % nu_fission = sigF * nu_total(nuc, E) else - ! Determine bounding indices based on which equal log-spaced interval - ! the energy is in - i_low = nuc % grid_index(i_log_union) - i_high = nuc % grid_index(i_log_union + 1) + 1 - - ! Perform binary search over reduced range - i_grid = binary_search(nuc % energy(i_low:i_high), & - i_high - i_low + 1, E) + i_low - 1 + micro_xs(i_nuclide) % fission = ZERO + micro_xs(i_nuclide) % nu_fission = ZERO end if - case (GRID_NUCLIDE) - ! Perform binary search on the nuclide energy grid in order to determine - ! which points to interpolate between + ! Ensure these values are set + ! Note, the only time either is used is in one of 4 places: + ! 1. physics.F90 - scatter - For inelastic scatter. + ! 2. physics.F90 - sample_fission - For partial fissions. + ! 3. tally.F90 - score_general - For tallying on MTxxx reactions. + ! 4. cross_section.F90 - calculate_urr_xs - For unresolved purposes. + ! It is worth noting that none of these occur in the resolved + ! resonance range, so the value here does not matter. + micro_xs(i_nuclide) % index_grid = 0 + micro_xs(i_nuclide) % interp_factor = ZERO + else + ! Determine index on nuclide energy grid + select case (grid_method) + case (GRID_MAT_UNION) - if (E <= nuc % energy(1)) then - i_grid = 1 - elseif (E > nuc % energy(nuc % n_grid)) then - i_grid = nuc % n_grid - 1 - else - i_grid = binary_search(nuc % energy, nuc % n_grid, E) + i_grid = mat % nuclide_grid_index(i_nuc_mat, i_log_union) + + case (GRID_LOGARITHM) + ! Determine the energy grid index using a logarithmic mapping to reduce + ! the energy range over which a binary search needs to be performed + + if (E < nuc % energy(1)) then + i_grid = 1 + elseif (E > nuc % energy(nuc % n_grid)) then + i_grid = nuc % n_grid - 1 + else + ! Determine bounding indices based on which equal log-spaced interval + ! the energy is in + i_low = nuc % grid_index(i_log_union) + i_high = nuc % grid_index(i_log_union + 1) + 1 + + ! Perform binary search over reduced range + i_grid = binary_search(nuc % energy(i_low:i_high), & + i_high - i_low + 1, E) + i_low - 1 + end if + + case (GRID_NUCLIDE) + ! Perform binary search on the nuclide energy grid in order to determine + ! which points to interpolate between + + if (E <= nuc % energy(1)) then + i_grid = 1 + elseif (E > nuc % energy(nuc % n_grid)) then + i_grid = nuc % n_grid - 1 + else + i_grid = binary_search(nuc % energy, nuc % n_grid, E) + end if + + end select + + ! check for rare case where two energy points are the same + if (nuc % energy(i_grid) == nuc % energy(i_grid+1)) i_grid = i_grid + 1 + + ! calculate interpolation factor + f = (E - nuc%energy(i_grid))/(nuc%energy(i_grid+1) - nuc%energy(i_grid)) + + micro_xs(i_nuclide) % index_grid = i_grid + micro_xs(i_nuclide) % interp_factor = f + + ! Initialize nuclide cross-sections to zero + micro_xs(i_nuclide) % fission = ZERO + micro_xs(i_nuclide) % nu_fission = ZERO + + ! Calculate microscopic nuclide total cross section + micro_xs(i_nuclide) % total = (ONE - f) * nuc % total(i_grid) & + + f * nuc % total(i_grid+1) + + ! Calculate microscopic nuclide elastic cross section + micro_xs(i_nuclide) % elastic = (ONE - f) * nuc % elastic(i_grid) & + + f * nuc % elastic(i_grid+1) + + ! Calculate microscopic nuclide absorption cross section + micro_xs(i_nuclide) % absorption = (ONE - f) * nuc % absorption( & + i_grid) + f * nuc % absorption(i_grid+1) + + if (nuc % fissionable) then + ! Calculate microscopic nuclide total cross section + micro_xs(i_nuclide) % fission = (ONE - f) * nuc % fission(i_grid) & + + f * nuc % fission(i_grid+1) + + ! Calculate microscopic nuclide nu-fission cross section + micro_xs(i_nuclide) % nu_fission = (ONE - f) * nuc % nu_fission( & + i_grid) + f * nuc % nu_fission(i_grid+1) end if - - end select - - ! check for rare case where two energy points are the same - if (nuc % energy(i_grid) == nuc % energy(i_grid+1)) i_grid = i_grid + 1 - - ! calculate interpolation factor - f = (E - nuc%energy(i_grid))/(nuc%energy(i_grid+1) - nuc%energy(i_grid)) - - micro_xs(i_nuclide) % index_grid = i_grid - micro_xs(i_nuclide) % interp_factor = f + end if ! Initialize sab treatment to false micro_xs(i_nuclide) % index_sab = NONE @@ -208,32 +284,6 @@ contains ! Initialize URR probability table treatment to false micro_xs(i_nuclide) % use_ptable = .false. - ! Initialize nuclide cross-sections to zero - micro_xs(i_nuclide) % fission = ZERO - micro_xs(i_nuclide) % nu_fission = ZERO - - ! Calculate microscopic nuclide total cross section - micro_xs(i_nuclide) % total = (ONE - f) * nuc % total(i_grid) & - + f * nuc % total(i_grid+1) - - ! Calculate microscopic nuclide elastic cross section - micro_xs(i_nuclide) % elastic = (ONE - f) * nuc % elastic(i_grid) & - + f * nuc % elastic(i_grid+1) - - ! Calculate microscopic nuclide absorption cross section - micro_xs(i_nuclide) % absorption = (ONE - f) * nuc % absorption( & - i_grid) + f * nuc % absorption(i_grid+1) - - if (nuc % fissionable) then - ! Calculate microscopic nuclide total cross section - micro_xs(i_nuclide) % fission = (ONE - f) * nuc % fission(i_grid) & - + f * nuc % fission(i_grid+1) - - ! Calculate microscopic nuclide nu-fission cross section - micro_xs(i_nuclide) % nu_fission = (ONE - f) * nuc % nu_fission( & - i_grid) + f * nuc % nu_fission(i_grid+1) - end if - ! If there is S(a,b) data for this nuclide, we need to do a few ! things. Since the total cross section was based on non-S(a,b) data, we ! need to correct it by subtracting the non-S(a,b) elastic cross section and @@ -253,6 +303,7 @@ contains micro_xs(i_nuclide) % last_E = E micro_xs(i_nuclide) % last_index_sab = i_sab + micro_xs(i_nuclide) % last_sqrtkT = sqrtkT end subroutine calculate_nuclide_xs @@ -525,6 +576,192 @@ contains end function find_energy_index +!=============================================================================== +! MULTIPOLE_EVAL_ALLOCATE allocates fixed-length arrays that vary based on +! what nuclides are loaded into the problem +!=============================================================================== + + subroutine multipole_eval_allocate() + allocate(sigT_factor(max_L)) + allocate(twophi(max_L)) + allocate(broadened_polynomials(max_poly)) + + mp_already_alloc = .true. + end subroutine + +!=============================================================================== +! MULTIPOLE_EVAL evaluates the windowed multipole equations for cross +! sections in the resolved resonance regions +!=============================================================================== + + subroutine multipole_eval(multipole, Emev, sqrtkT_, sigT, sigA, sigF) + type(MultipoleArray), intent(in) :: multipole ! The windowed multipole + ! object to process. + real(8), intent(in) :: Emev ! The energy at which to + ! evaluate the cross section + ! in MeV + real(8), intent(in) :: sqrtkT_ ! The temperature in the form + ! sqrt(kT (in MeV)), at which + ! to evaluate the XS. + real(8), intent(out) :: sigT ! Total cross section + real(8), intent(out) :: sigA ! Absorption cross section + real(8), intent(out) :: sigF ! Fission cross section + complex(8) :: psi_ki ! The value of the psi-ki function for the asymptotic + ! form + complex(8) :: c_temp ! complex temporary variable + complex(8) :: w_val ! The faddeeva function evaluated at Z + complex(8) :: Z ! sqrt(atomic weight ratio / kT) * (sqrt(E) - pole) + real(8) :: sqrtE ! sqrt(E), eV + real(8) :: invE ! 1/E, eV + real(8) :: dopp ! sqrt(atomic weight ratio / kT) + real(8) :: dopp_ecoef ! sqrt(atomic weight ratio * pi / kT) / E + real(8) :: temp ! real temporary value + real(8) :: E ! energy, eV + real(8) :: sqrtkT ! sqrt(kT (in eV)) + integer :: iP ! index of pole + integer :: iC ! index of curvefit + integer :: iW ! index of window + integer :: startw ! window start pointer (for poles) + integer :: startw_1 ! window start pointer - 1 + integer :: startw_endw ! window start pointer - window end pointer + integer :: endw ! window end pointer + + ! Convert to eV + E = Emev * 1.0e6_8 + sqrtkT = sqrtkT_ * 1.0e3_8 + + sqrtE = sqrt(E) + invE = ONE/E + + if(.not. mp_already_alloc) then + call multipole_eval_allocate() + end if + + ! Locate us + iW = floor((sqrtE - sqrt(multipole % start_E))/multipole % spacing + ONE) + + startw = multipole % w_start(iW) + startw_1 = startw - 1 ! This is an index shift parameter. + endw = multipole % w_end(iW) + startw_endw = endw - startw + 1 + + ! Fill in factors + if (startw <= endw) then + call fill_factors(multipole, sqrtE, sigT_factor, twophi, multipole % num_l) + end if + + ! Generate some doppler broadening parameters + + ! dopp_ecoef is inverse of dopp, divided by E, multiplied by sqrt(pi). + dopp = multipole % sqrtAWR / sqrtKT + dopp_ecoef = dopp * invE * SQRT_PI + + sigT = ZERO + sigA = ZERO + sigF = ZERO + ! Evaluate linefit first + + if(sqrtkT /= 0 .and. multipole % broaden_poly(iW) == 1) then ! Broaden the curvefit. + call broaden_n_polynomials(E, dopp, multipole % fit_order + 1, broadened_polynomials) + + do iC = 1, multipole % fit_order+1 + sigT = sigT + multipole % curvefit(FIT_T, iC, iW)*broadened_polynomials(iC) + sigA = sigA + multipole % curvefit(FIT_A, iC, iW)*broadened_polynomials(iC) + if (multipole % fissionable) then + sigF = sigF + multipole % curvefit(FIT_F, iC, iW)*broadened_polynomials(iC) + end if + end do + else ! Evaluate as if it were a polynomial + temp = invE + do iC = 1, multipole % fit_order+1 + + sigT = sigT + multipole % curvefit(FIT_T, iC, iW)*temp + sigA = sigA + multipole % curvefit(FIT_A, iC, iW)*temp + if (multipole % fissionable) then + sigF = sigF + multipole % curvefit(FIT_F, iC, iW)*temp + end if + + temp = temp * sqrtE + end do + end if + + ! Then get the poles we want and broaden them. + + if (sqrtkT == ZERO) then + ! If at 0K, use asymptotic form. + do iP = startw, endw + psi_ki = -ONEI/(multipole % data(MP_EA, iP) - sqrtE) + c_temp = psi_ki/E + if (multipole % formalism == FORM_MLBW) then + sigT = sigT + real(multipole % data(MLBW_RT, iP) * c_temp * & + sigT_factor(multipole % l_value(iP))) & + + real(multipole % data(MLBW_RX, iP) * c_temp) + sigA = sigA + real(multipole % data(MLBW_RA, iP) * c_temp) + sigF = sigF + real(multipole % data(MLBW_RF, iP) * c_temp) + else if (multipole % formalism == FORM_RM) then + sigT = sigT + real(multipole % data(RM_RT, iP) * c_temp* & + sigT_factor(multipole % l_value(iP))) + sigA = sigA + real(multipole % data(RM_RA, iP) * c_temp) + sigF = sigF + real(multipole % data(RM_RF, iP) * c_temp) + end if + end do + else + ! At temperature, use Faddeeva function-based form. + if(endw >= startw) then + do iP = startw, endw + Z = (sqrtE - multipole % data(MP_EA, iP)) * dopp + w_val = w(Z) * dopp_ecoef + + if (multipole % formalism == FORM_MLBW) then + sigT = sigT + real((multipole % data(MLBW_RT, iP) * & + sigT_factor(multipole%l_value(iP)) + & + multipole % data(MLBW_RX, iP)) * w_val) + sigA = sigA + real(multipole % data(MLBW_RA, iP) * w_val) + sigF = sigF + real(multipole % data(MLBW_RF, iP) * w_val) + else if (multipole % formalism == FORM_RM) then + sigT = sigT + real(multipole % data(RM_RT, iP) * w_val * & + sigT_factor(multipole % l_value(iP))) + sigA = sigA + real(multipole % data(RM_RA, iP) * w_val) + sigF = sigF + real(multipole % data(RM_RF, iP) * w_val) + end if + end do + end if + end if + end subroutine + +!=============================================================================== +! FILL_FACTORS calculates the value of phi, the hardsphere phase shift factor, +! and sigT_factor, a factor inside of the sigT equation not present in the +! sigA and sigF equations. +!=============================================================================== + + subroutine fill_factors(multipole, sqrtE, sigT_factor, twophi, max_L) + type(MultipoleArray), intent(in) :: multipole + real(8), intent(in) :: sqrtE + integer, intent(in) :: max_L + complex(8), intent(out) :: sigT_factor(max_L) + real(8), intent(out) :: twophi(max_L) + + integer :: iL + real(8) :: arg + + do iL = 1, max_L + twophi(iL) = multipole % pseudo_k0RS(iL) * sqrtE + if (iL == 2) then + twophi(iL) = twophi(iL) - atan(twophi(iL)) + else if (iL == 3) then + arg = 3.0_8 * twophi(iL) / (3.0_8 - twophi(iL)**2) + twophi(iL) = twophi(iL) - atan(arg) + else if (iL == 4) then + arg = twophi(iL) * (15.0_8 - twophi(iL)**2) / (15.0_8 - 6.0_8 * twophi(iL)**2) + twophi(iL) = twophi(iL) - atan(arg) + end if + end do + + twophi = 2.0_8 * twophi + sigT_factor = cmplx(cos(twophi),-sin(twophi), KIND=8) + end subroutine + !=============================================================================== ! 0K_ELASTIC_XS determines the microscopic 0K elastic cross section ! for a given nuclide at the trial relative energy used in resonance scattering diff --git a/src/geometry.F90 b/src/geometry.F90 index 8a38f982b5..5f37f1047b 100644 --- a/src/geometry.F90 +++ b/src/geometry.F90 @@ -279,6 +279,36 @@ contains p % material = c % material(offset + 1) end if + ! Set the particle temperature + if (size(c % sqrtkT) == 1) then + ! Only one temperature for this cell; assign that one to the particle. + p % sqrtkT = c % sqrtkT(1) + else + ! Distributed instances of this cell have different temperatures. + ! Determine which instance this is and assign the matching temp. + distribcell_index = c % distribcell_index + offset = 0 + do k = 1, p % n_coord + if (cells(p % coord(k) % cell) % type == CELL_FILL) then + offset = offset + cells(p % coord(k) % cell) % & + offset(distribcell_index) + elseif (cells(p % coord(k) % cell) % type == CELL_LATTICE) then + if (lattices(p % coord(k + 1) % lattice) % obj & + % are_valid_indices([& + p % coord(k + 1) % lattice_x, & + p % coord(k + 1) % lattice_y, & + p % coord(k + 1) % lattice_z])) then + offset = offset + lattices(p % coord(k + 1) % lattice) % obj % & + offset(distribcell_index, & + p % coord(k + 1) % lattice_x, & + p % coord(k + 1) % lattice_y, & + p % coord(k + 1) % lattice_z) + end if + end if + end do + p % sqrtkT = c % sqrtkT(offset + 1) + end if + elseif (c % type == CELL_FILL) then CELL_TYPE ! ====================================================================== ! CELL CONTAINS LOWER UNIVERSE, RECURSIVELY FIND CELL diff --git a/src/geometry_header.F90 b/src/geometry_header.F90 index 519b15bff4..0756e23dc4 100644 --- a/src/geometry_header.F90 +++ b/src/geometry_header.F90 @@ -139,6 +139,9 @@ module geometry_header ! only) integer :: distribcell_index ! Index corresponding to this cell in ! distribcell arrays + real(8), allocatable :: sqrtkT(:) ! Square root of k_Boltzmann * + ! temperature in MeV. Multiple for + ! distribcell ! Rotation matrix and translation vector real(8), allocatable :: translation(:) diff --git a/src/hdf5_interface.F90 b/src/hdf5_interface.F90 index 01e50d983a..14c9b9eb19 100644 --- a/src/hdf5_interface.F90 +++ b/src/hdf5_interface.F90 @@ -62,6 +62,7 @@ module hdf5_interface module procedure read_string_1D module procedure read_tally_result_1D module procedure read_tally_result_2D + module procedure read_complex_2D end interface read_dataset public :: write_dataset @@ -1921,4 +1922,93 @@ contains mpio = (driver == H5FD_MPIO_F) end function using_mpio_device +!=============================================================================== +! READ_COMPLEX_2D reads double precision complex 2-D array data as output by +! the h5py HDF5 python module. +!=============================================================================== + + subroutine read_complex_2D(group_id, name, buffer, indep) + integer(HID_T), intent(in) :: group_id + character(*), intent(in) :: name ! name of data + complex(8), intent(inout), target :: buffer(:,:) ! data to write + logical, intent(in), optional :: indep ! independent I/O + + integer(HSIZE_T) :: dims(2) + + dims(:) = shape(buffer) + if (present(indep)) then + call read_complex_2D_explicit(group_id, dims, name, buffer, indep) + else + call read_complex_2D_explicit(group_id, dims, name, buffer) + end if + end subroutine read_complex_2D + + subroutine read_complex_2D_explicit(group_id, dims, name, buffer, indep) + integer(HID_T), intent(in) :: group_id + integer(HSIZE_T), intent(in) :: dims(2) + character(*), intent(in) :: name ! name of data + complex(8), intent(inout), target :: buffer(dims(1),dims(2)) + logical, intent(in), optional :: indep ! independent I/O + + real(8), target :: buffer_r(dims(1), dims(2)) + real(8), target :: buffer_i(dims(1), dims(2)) + + integer(HSIZE_T) :: i, j + + integer :: hdf5_err + integer :: data_xfer_mode +#ifdef PHDF5 + integer(HID_T) :: plist ! property list +#endif + integer(HID_T) :: dset ! data set handle + type(c_ptr) :: f_ptr_r, f_ptr_i + + ! Components needed for complex type support + integer(HID_T) :: dtype_real + integer(HID_T) :: dtype_imag + integer(SIZE_T) :: size_double + integer :: error + + ! Create the complex type + call h5tget_size_f(H5T_NATIVE_DOUBLE, size_double, error) + + ! Insert the 'r' and 'i' identifiers + call h5tcreate_f(H5T_COMPOUND_F, size_double, dtype_real, error) + call h5tcreate_f(H5T_COMPOUND_F, size_double, dtype_imag, error) + call h5tinsert_f(dtype_real, "r", 0_8, H5T_NATIVE_DOUBLE, error) + call h5tinsert_f(dtype_imag, "i", 0_8, H5T_NATIVE_DOUBLE, error) + + ! Set up collective vs. independent I/O + data_xfer_mode = H5FD_MPIO_COLLECTIVE_F + if (present(indep)) then + if (indep) data_xfer_mode = H5FD_MPIO_INDEPENDENT_F + end if + + call h5dopen_f(group_id, trim(name), dset, hdf5_err) + f_ptr_r = c_loc(buffer_r) + f_ptr_i = c_loc(buffer_i) + + if (using_mpio_device(group_id)) then +#ifdef PHDF5 + call h5pcreate_f(H5P_DATASET_XFER_F, plist, hdf5_err) + call h5pset_dxpl_mpio_f(plist, data_xfer_mode, hdf5_err) + call h5dread_f(dset, dtype_real, f_ptr_r, hdf5_err, xfer_prp=plist) + call h5dread_f(dset, dtype_imag, f_ptr_i, hdf5_err, xfer_prp=plist) + call h5pclose_f(plist, hdf5_err) +#endif + else + call h5dread_f(dset, dtype_real, f_ptr_r, hdf5_err) + call h5dread_f(dset, dtype_imag, f_ptr_i, hdf5_err) + end if + + ! Reconstitute the complex numbers + do i = 1,dims(1) + do j = 1,dims(2) + buffer(i,j) = cmplx(buffer_r(i,j), buffer_i(i,j), kind=8) + end do + end do + + call h5dclose_f(dset, hdf5_err) + end subroutine read_complex_2D_explicit + end module hdf5_interface diff --git a/src/initialize.F90 b/src/initialize.F90 index ae62838741..4b412fb29c 100644 --- a/src/initialize.F90 +++ b/src/initialize.F90 @@ -120,6 +120,9 @@ contains ! Create linked lists for multiple instances of the same nuclide call same_nuclide_list() + ! Set undefined cell temperatures to match the material data. + call lookup_material_temperatures() + ! Construct unionized or log energy grid for cross-sections select case (grid_method) case (GRID_NUCLIDE) @@ -976,10 +979,11 @@ contains end do end do - ! We also need distribcell if any distributed materials are present. + ! We also need distribcell if any distributed materials or distributed + ! temperatues are present. if (.not. distribcell_active) then do i = 1, n_cells - if (size(cells(i) % material) > 1) then + if (size(cells(i) % material) > 1 .or. size(cells(i) % sqrtkT) > 1) then distribcell_active = .true. exit end if @@ -1004,8 +1008,8 @@ contains end do end do - ! Make sure the number of materials matches the number of cell instances for - ! distributed materials. + ! Make sure the number of materials and temperatures matches the number of + ! cell instances. do i = 1, n_cells associate (c => cells(i)) if (size(c % material) > 1) then @@ -1017,6 +1021,15 @@ contains &equal one or the number of instances.") end if end if + if (size(c % sqrtkT) > 1) then + if (size(c % sqrtkT) /= c % instances) then + call fatal_error("Cell " // trim(to_str(c % id)) // " was & + &specified with " // trim(to_str(size(c % sqrtkT))) & + // " temperatures but has " // trim(to_str(c % instances)) & + // " distributed instances. The number of temperatures must & + &equal one or the number of instances.") + end if + end if end associate end do @@ -1058,9 +1071,9 @@ contains end do end do - ! List all cells with multiple (distributed) materials. + ! List all cells with multiple (distributed) materials or temperatures. do i = 1, n_cells - if (size(cells(i) % material) > 1) then + if (size(cells(i) % material) > 1 .or. size(cells(i) % sqrtkT) > 1) then call cell_list % add(i) end if end do @@ -1129,4 +1142,57 @@ contains end subroutine allocate_offsets +!=============================================================================== +! LOOKUP_MATERIAL_TEMPERATURES If any cells have undefined temperatures, try to +! find their temperatures from material data. +!=============================================================================== + + subroutine lookup_material_temperatures() + integer :: i, j, k + real(8) :: min_temp + logical :: warning_given + + warning_given = .false. + do i = 1, n_cells + ! Ignore non-normal cells and cells with defined temperature. + if (cells(i) % type /= CELL_NORMAL) cycle + if (cells(i) % sqrtkT(1) /= ERROR_REAL) cycle + + ! Set the number of temperatures equal to the number of materials. + deallocate(cells(i) % sqrtkT) + allocate(cells(i) % sqrtkT(size(cells(i) % material))) + + ! Check each of the cell materials for temperature data. + do j = 1, size(cells(i) % material) + ! Arbitrarily set void regions to 0K. + if (cells(i) % material(j) == MATERIAL_VOID) then + cells(i) % sqrtkT(j) = ZERO + cycle + end if + + associate (mat => materials(cells(i) % material(j))) + ! Find the temperature of the coldest nuclide. + min_temp = nuclides(mat % nuclide(1)) % kT + do k = 2, mat % n_nuclides + ! Warn the user if the nuclides don't have identical temperatues. + if (nuclides(mat % nuclide(k)) % kT /= min_temp & + .and. .not. warning_given) then + call warning("OpenMC cannot & + &identify the temperature of at least one cell. For the & + &purposes of multipole cross section evaluations, all cells & + &with unknown temperature will be set to the coldest & + &temperature found in the nuclear data for that cell's & + &material") + warning_given = .true. + end if + min_temp = min(min_temp, nuclides(mat % nuclide(k)) % kT) + end do + + ! Set the temperature for this cell instance. + cells(i) % sqrtkT(j) = sqrt(min_temp) + end associate + end do + end do + end subroutine lookup_material_temperatures + end module initialize diff --git a/src/input_xml.F90 b/src/input_xml.F90 index f29661352b..2d29734b72 100644 --- a/src/input_xml.F90 +++ b/src/input_xml.F90 @@ -1298,6 +1298,40 @@ contains call get_node_array(node_cell, "translation", c % translation) end if + ! Read cell temperatures. If the temperature is not specified, set it to + ! ERROR_REAL for now. During initialization we'll replace ERROR_REAL with + ! the temperature from the material data. + if (check_for_node(node_cell, "temperature")) then + n = get_arraysize_double(node_cell, "temperature") + if (n > 0) then + ! Make sure this is a "normal" cell. + if (c % material(1) == NONE) call fatal_error("Cell " & + // trim(to_str(c % id)) // " was specified with a temperature & + &but no material. Temperature specification is only valid for & + &cells filled with a material.") + + ! Copy in temperatures + allocate(c % sqrtkT(n)) + call get_node_array(node_cell, "temperature", c % sqrtkT) + + ! Make sure all temperatues are positive + do j = 1, size(c % sqrtkT) + if (c % sqrtkT(j) < ZERO) call fatal_error("Cell " & + // trim(to_str(c % id)) // " was specified with a negative & + &temperature. All cell temperatures must be non-negative.") + end do + + ! Convert to sqrt(kT) + c % sqrtkT(:) = sqrt(K_BOLTZMANN * c % sqrtkT(:)) + else + allocate(c % sqrtkT(1)) + c % sqrtkT(1) = ERROR_REAL + end if + else + allocate(c % sqrtkT(1)) + c % sqrtkT = ERROR_REAL + end if + ! Add cell to dictionary call cell_dict % add_key(c % id, i) @@ -1867,6 +1901,7 @@ contains type(Node), pointer :: doc => null() type(Node), pointer :: node_mat => null() type(Node), pointer :: node_dens => null() + type(Node), pointer :: node_temp => null() type(Node), pointer :: node_nuc => null() type(Node), pointer :: node_ele => null() type(Node), pointer :: node_sab => null() diff --git a/src/math.F90 b/src/math.F90 index 9a64c88d7d..ae3ed1a714 100644 --- a/src/math.F90 +++ b/src/math.F90 @@ -2,9 +2,25 @@ module math use constants use random_lcg, only: prn + use ISO_C_BINDING implicit none +!=============================================================================== +! FADDEEVA_W evaluates the scaled complementary error function. This +! interfaces with the MIT C library +!=============================================================================== + + interface + function faddeeva_w(z, relerr) bind(C, name='Faddeeva_w') result(w) + use ISO_C_BINDING + implicit none + complex(C_DOUBLE_COMPLEX), value :: z + real(C_DOUBLE), value :: relerr + complex(C_DOUBLE_COMPLEX) :: w + end function faddeeva_w + end interface + contains !=============================================================================== @@ -659,4 +675,91 @@ contains end function watt_spectrum +!=============================================================================== +! W acts as a front end to the MIT Faddeeva function, Faddeeva_w. +!=============================================================================== + + function w(z) result(wv) + complex(C_DOUBLE_COMPLEX), intent(in) :: z ! The point to evaluate Z at + complex(8) :: wv ! The resulting w(z) value + real(C_DOUBLE) :: relerr ! Target relative error in inner loop of MIT Faddeeva + + ! Technically, the value we want is given by the equation: + ! w(z) = I/Pi * Integrate[Exp[-t^2]/(z-t), {t, -Infinity, Infinity}] + ! as shown in Equation 63 from Hwang, R. N. "A rigorous pole + ! representation of multilevel cross sections and its practical + ! applications." Nuclear Science and Engineering 96.3 (1987): 192-209. + ! + ! The MIT Faddeeva function evaluates w(z) = exp(-z^2)erfc(-iz). These + ! two forms of the Faddeeva function are related by a transformation. + ! + ! If we call the integral form w_int, and the function form w_fun: + ! For imag(z) > 0, w_int(z) = w_fun(z) + ! For imag(z) < 0, w_int(z) = -conjg(w_fun(conjg(z))) + + relerr = ZERO + if (aimag(z) > ZERO) then + wv = faddeeva_w(z, relerr) + else + wv = -conjg(faddeeva_w(conjg(z), relerr)) + end if + + end function w + +!=============================================================================== +! BROADEN_N_POLYNOMIALS doppler broadens polynomials of the form +! a/En + b/sqrt(En) + c + d sqrt(En) ... +! exactly and quickly. +!=============================================================================== + + subroutine broaden_n_polynomials(En, dopp, n, factors) + real(8), intent(in) :: En ! Energy to evaluate at + real(8), intent(in) :: dopp ! sqrt(atomic weight ratio / kT), kT given in eV. + integer, intent(in) :: n ! number of components to polynomial + real(8), intent(out):: factors(n) ! output leading coefficient + + integer :: i + + real(8) :: sqrtE ! Sqrt(energy) + real(8) :: beta ! sqrt(atomic weight ratio * E / kT) + real(8) :: half_inv_dopp2 ! 0.5 / dopp**2 + real(8) :: quarter_inv_dopp4 ! 0.25 / dopp**4 + real(8) :: erfbeta ! error function of beta + real(8) :: exp_m_beta2 ! exp(-beta**2) + + sqrtE = sqrt(En) + beta = sqrtE * dopp + half_inv_dopp2 = HALF / dopp**2 + quarter_inv_dopp4 = half_inv_dopp2**2 + + if (beta > 6.0_8) then + ! Save time, ERF(6) is 1 to machine precision. + ! beta/sqrtpi*exp(-beta**2) is also approximately 1 machine epsilon. + erfBeta = ONE + exp_m_beta2 = ZERO + else + erfBeta = erf(beta) + exp_m_beta2 = exp(-beta**2) + end if + + ! Assume that, for sure, we'll use a second order (1/E, 1/V, const) + ! fit, and no less. + + factors(1) = erfbeta / En + factors(2) = ONE / sqrtE + factors(3) = factors(1) * (half_inv_dopp2 + En) + exp_m_beta2 / (beta * SQRT_PI) + + ! Perform recursive broadening of high order components + do i = 1, n-3 + if (i /= 1) then + factors(i+3) = -factors(i-1) * (i - ONE) * i * quarter_inv_dopp4 + & + factors(i+1) * (En + (ONE + TWO * i) * half_inv_dopp2) + else + ! Although it's mathematically identical, factors(0) will contain + ! nothing, and we don't want to have to worry about memory. + factors(i+3) = factors(i+1)*(En + (ONE + TWO * i) * half_inv_dopp2) + end if + end do + end subroutine broaden_n_polynomials + end module math diff --git a/src/multipole.F90 b/src/multipole.F90 new file mode 100644 index 0000000000..4ed28a2dbf --- /dev/null +++ b/src/multipole.F90 @@ -0,0 +1,183 @@ +module multipole + + use constants + use global + use hdf5 + use hdf5_interface + use multipole_header, only: MultipoleArray, FIT_T, FIT_A, FIT_F, max_L, & + max_poles, max_poly, MP_FISS, FORM_MLBW, FORM_RM + + implicit none + +contains + +!=============================================================================== +! MULTIPOLE_READ Reads in a multipole HDF5 file with the original API +! specification. Subject to change as the library format matures. +!=============================================================================== + + subroutine multipole_read(filename, multipole, i_table) + character(len=*), intent(in) :: filename ! Filename of the multipole library to load + type(MultipoleArray), intent(out), target :: multipole ! The object to fill + integer, intent(in) :: i_table ! index in nuclides/sab_tables + + type(Nuclide), pointer :: nuc => null() + + integer(HID_T) :: file_id + integer(HID_T) :: group_id + + ! Intermediate loading components + integer :: NMT + integer :: i, j, k + integer, allocatable :: MT(:) + logical :: accumulated_fission + character(len=3) :: MT_string + character(len=24) :: MT_n ! Takes the form '/nuclide/reactions/MT???' + integer :: is_fissionable + + associate (nuc => nuclides(i_table)) + + ! Open file for reading and move into the /isotope group + file_id = file_open(filename, 'r', parallel=.true.) + group_id = open_group(file_id, "/nuclide") + + ! Load in all the array size scalars + call read_dataset(group_id, "length", multipole % length) + call read_dataset(group_id, "windows", multipole % windows) + call read_dataset(group_id, "num_l", multipole % num_l) + call read_dataset(group_id, "fit_order", multipole % fit_order) + call read_dataset(group_id, "max_w", multipole % max_w) + call read_dataset(group_id, "fissionable", is_fissionable) + if (is_fissionable == MP_FISS) then + multipole % fissionable = .true. + else + multipole % fissionable = .false. + end if + call read_dataset(group_id, "formalism", multipole % formalism) + + call read_dataset(group_id, "spacing", multipole % spacing) + call read_dataset(group_id, "sqrtAWR", multipole % sqrtAWR) + call read_dataset(group_id, "start_E", multipole % start_E) + call read_dataset(group_id, "end_E", multipole % end_E) + + ! Allocate the multipole array components + call multipole % allocate() + + ! Read in arrays + call read_dataset(group_id, "data", multipole % data) + call read_dataset(group_id, "pseudo_K0RS", multipole % pseudo_k0RS) + call read_dataset(group_id, "l_value", multipole % l_value) + call read_dataset(group_id, "w_start", multipole % w_start) + call read_dataset(group_id, "w_end", multipole % w_end) + call read_dataset(group_id, "broaden_poly", multipole % broaden_poly) + + call read_dataset(group_id, "curvefit", multipole % curvefit) + + ! Delete ACE pointwise data + call read_dataset(group_id, "n_grid", nuc % n_grid) + + deallocate(nuc % energy) + deallocate(nuc % total) + deallocate(nuc % elastic) + deallocate(nuc % fission) + deallocate(nuc % nu_fission) + deallocate(nuc % absorption) + + allocate(nuc % energy(nuc % n_grid)) + allocate(nuc % total(nuc % n_grid)) + allocate(nuc % elastic(nuc % n_grid)) + allocate(nuc % fission(nuc % n_grid)) + allocate(nuc % nu_fission(nuc % n_grid)) + allocate(nuc % absorption(nuc % n_grid)) + + nuc % total = ZERO + nuc % absorption = ZERO + nuc % fission = ZERO + + ! Read in new energy axis (converting eV to MeV) + call read_dataset(group_id, "energy_points", nuc % energy) + nuc % energy = nuc % energy / 1.0D6 + + ! Get count and list of MT tables + call read_dataset(group_id, "MT_count", NMT) + allocate(MT(NMT)) + + call read_dataset(group_id, "MT_list", MT) + + call close_group(group_id) + + accumulated_fission = .false. + + ! Loop over each MT entry and load it into a reaction. + do i = 1, NMT + write(MT_string, '(I3.3)') MT(i) + MT_n = "/nuclide/reactions/MT" // MT_string + + group_id = open_group(file_id, MT_n) + + ! Each MT needs to be treated slightly differently. + select case (MT(i)) + case(ELASTIC) + call read_dataset(group_id, "MT_sigma", nuc % elastic) + nuc % total = nuc % total + nuc % elastic + case(N_FISSION) + call read_dataset(group_id, "MT_sigma", nuc % fission) + nuc % total = nuc % total + nuc % fission + nuc % absorption = nuc % absorption + nuc % fission + accumulated_fission = .true. + case default + ! Search through all of our secondary reactions + do j = 1, nuc % n_reaction + if (nuc % reactions(j) % MT == MT(i)) then + ! Match found + + ! Individual Fission components exist, so remove the combined + ! fission cross section. + if ( (MT(i) == N_F .or. MT(i) == N_NF .or. MT(i) == N_2NF & + .or. MT(i) == N_3NF) .and. accumulated_fission) then + nuc % total = nuc % total - nuc % fission + nuc % absorption = nuc % absorption - nuc % fission + nuc % fission = 0.0_8 + accumulated_fission = .false. + end if + + deallocate(nuc % reactions(j) % sigma) + allocate(nuc % reactions(j) % sigma(nuc % n_grid)) + + call read_dataset(group_id, "MT_sigma", nuc % reactions(j) % sigma) + call read_dataset(group_id, "Q_value", nuc % reactions(j) % Q_value) + call read_dataset(group_id, "threshold", nuc % reactions(j) % threshold) + nuc % reactions(j) % threshold = 1 ! TODO: reconsider implications. + nuc % reactions(j) % Q_value = nuc % reactions(j) % Q_value / 1.0D6 + + ! Accumulate total + if (MT(i) /= N_LEVEL .and. MT(i) <= N_DA) then + nuc % total = nuc % total + nuc % reactions(j) % sigma + end if + + ! Accumulate absorption + if (MT(i) >= N_GAMMA .and. MT(i) <= N_DA) then + nuc % absorption = nuc % absorption + nuc % reactions(j) % sigma + end if + + ! Accumulate fission (if needed) + if ( (MT(i) == N_F .or. MT(i) == N_NF .or. MT(i) == N_2NF & + .or. MT(i) == N_3NF) ) then + nuc % fission = nuc % fission + nuc % reactions(j) % sigma + nuc % absorption = nuc % absorption + nuc % reactions(j) % sigma + end if + end if + end do + end select + + call close_group(group_id) + end do + + ! Close file + call file_close(file_id) + + end associate + + end subroutine multipole_read + +end module multipole diff --git a/src/multipole_header.F90 b/src/multipole_header.F90 new file mode 100644 index 0000000000..a3642b890b --- /dev/null +++ b/src/multipole_header.F90 @@ -0,0 +1,133 @@ +module multipole_header + + implicit none + + !======================================================================== + ! Multipole related constants + + ! Formalisms + integer, parameter :: FORM_MLBW = 2, & + FORM_RM = 3, & + FORM_RML = 7 + + ! Constants that determine which value to access + integer, parameter :: MP_EA = 1 ! Pole + + ! Reich-Moore indices + integer, parameter :: RM_RT = 2, & ! Residue total + RM_RA = 3, & ! Residue absorption + RM_RF = 4 ! Residue fission + + ! Multi-level Breit Wigner indices + integer, parameter :: MLBW_RT = 2, & ! Residue total + MLBW_RX = 3, & ! Residue compettitive + MLBW_RA = 4, & ! Residue absorption + MLBW_RF = 5 ! Residue fission + + ! Polynomial fit indices + integer, parameter :: FIT_T = 1, & ! Total + FIT_A = 2, & ! Absorption + FIT_F = 3 ! Fission + + ! Value of 'true' when checking if nuclide is fissionable + integer, parameter :: MP_FISS = 1 + + ! These variables store the maximum value from every nuclide in order + ! to preallocate some arrays to improve performance. + integer :: max_poly ! Maximum number of polynomials we expect + integer :: max_poles ! Maximum number of poles in the problem for allocation + integer :: max_L ! Maximum L value for allocation + +!=============================================================================== +! MULTIPOLE contains all the components needed for the windowed multipole +! temperature dependent cross section libraries for the resolved resonance +! region. +!=============================================================================== + + type MultipoleArray + + !========================================================================= + ! Isotope Properties + logical :: fissionable = .false. ! Is this isotope fissionable? + integer :: length ! Number of poles + integer, allocatable :: l_value(:) ! The l index of the pole + real(8), allocatable :: pseudo_k0RS(:) ! The value (sqrt(2*mass neutron)/reduced planck constant) * AWR/(AWR + 1) * scattering radius for each l + complex(8), allocatable :: data(:,:) ! Contains all of the pole-residue data + real(8) :: sqrtAWR ! Square root of the atomic weight ratio + + !========================================================================= + ! Windows + + integer :: windows ! Number of windows + integer :: fit_order ! Order of the fit. 1 linear, 2 quadratic, etc. + real(8) :: start_E ! Start energy for the windows + real(8) :: end_E ! End energy for the windows + real(8) :: spacing ! The actual spacing in sqrt(E) space. + ! spacing = sqrt(multipole_w%endE - multipole_w%startE)/multipole_w%windows + integer, allocatable :: w_start(:) ! Contains the index of the pole at the start of the window + integer, allocatable :: w_end(:) ! Contains the index of the pole at the end of the window + real(8), allocatable :: curvefit(:,:,:) ! Contains the fitting function. (reaction type, coeff index, window index) + + integer, allocatable :: broaden_poly(:) ! if 1, broaden, if 0, don't. + + !========================================================================= + ! Storage Helpers + integer :: num_l + integer :: max_w + + integer :: formalism + + contains + procedure :: allocate => multipole_allocate ! Allocates Multipole + end type MultipoleArray + +contains + +!=============================================================================== +! MULTIPOLE_ALLOCATE allocates necessary data for Multipole. +!=============================================================================== + + subroutine multipole_allocate(multipole) + class(MultipoleArray), intent(inout) :: multipole ! Multipole object to allocate. + + ! This function assumes length, numL, fissionable, windows, fitorder, + ! and formalism are known + + ! Allocate the pole-residue storage. + ! MLBW has one more pole than Reich-Moore, and fissionable nuclides + ! have further one more. + if (multipole % formalism == FORM_MLBW) then + if (multipole % fissionable) then + allocate(multipole % data(5, multipole % length)) + else + allocate(multipole % data(4, multipole % length)) + end if + else if (multipole % formalism == FORM_RM) then + if (multipole % fissionable) then + allocate(multipole % data(4, multipole % length)) + else + allocate(multipole % data(3, multipole % length)) + end if + end if + + ! Allocate the l value table for each pole-residue set. + allocate(multipole % l_value(multipole % length)) + + ! Allocate the table of pseudo_k0RS values at each l. + allocate(multipole % pseudo_k0RS(multipole % num_l)) + + ! Allocate window start, window end + allocate(multipole % w_start(multipole % windows)) + allocate(multipole % w_end(multipole % windows)) + + ! Allocate broaden_poly + allocate(multipole % broaden_poly(multipole % windows)) + + ! Allocate curvefit + if(multipole % fissionable) then + allocate(multipole % curvefit(FIT_F, multipole % fit_order+1, multipole % windows)) + else + allocate(multipole % curvefit(FIT_A, multipole % fit_order+1, multipole % windows)) + end if + end subroutine +end module multipole_header diff --git a/src/output.F90 b/src/output.F90 index f9b025f4e3..fb4375b548 100644 --- a/src/output.F90 +++ b/src/output.F90 @@ -1434,7 +1434,7 @@ contains do i = 1, n ! If the cell matches the goal and the offset matches final, write to the ! geometry stack - if (univ % cells(i) == goal .AND. offset == final) then + if (univ % cells(i) == goal .and. offset == final) then c => cells(univ % cells(i)) path = trim(path) // "->" // to_str(c % id) return diff --git a/src/particle_header.F90 b/src/particle_header.F90 index 0b9b251ee5..fb02bc6a56 100644 --- a/src/particle_header.F90 +++ b/src/particle_header.F90 @@ -2,7 +2,7 @@ module particle_header use bank_header, only: Bank use constants, only: NEUTRON, ONE, NONE, ZERO, MAX_SECONDARY, & - MAX_DELAYED_GROUPS + MAX_DELAYED_GROUPS, ERROR_REAL use error, only: fatal_error use geometry_header, only: BASE_UNIVERSE @@ -79,6 +79,9 @@ module particle_header integer :: material ! index for current material integer :: last_material ! index for last material + ! Temperature of the current cell + real(8) :: sqrtkT ! sqrt(k_Boltzmann * temperature) in MeV + ! Statistical data integer :: n_collision ! # of collisions @@ -86,7 +89,7 @@ module particle_header logical :: write_track = .false. ! Secondary particles created - integer :: n_secondary = 0 + integer(8) :: n_secondary = 0 type(Bank) :: secondary_bank(MAX_SECONDARY) contains @@ -124,6 +127,7 @@ contains this % absorb_wgt = ZERO this % n_bank = 0 this % wgt_bank = ZERO + this % sqrtkT = ERROR_REAL this % n_collision = 0 this % fission = .false. this % delayed_group = 0 diff --git a/src/physics.F90 b/src/physics.F90 index 31eb981abb..da2682ebf7 100644 --- a/src/physics.F90 +++ b/src/physics.F90 @@ -88,9 +88,14 @@ contains ! change when sampling fission sites. The following block handles all ! absorption (including fission) - if (nuc % fissionable .and. run_mode == MODE_EIGENVALUE) then + if (nuc % fissionable) then call sample_fission(i_nuclide, i_reaction) - call create_fission_sites(p, i_nuclide, i_reaction) + if (run_mode == MODE_EIGENVALUE) then + call create_fission_sites(p, i_nuclide, i_reaction, fission_bank, n_bank) + elseif (run_mode == MODE_FIXEDSOURCE) then + call create_fission_sites(p, i_nuclide, i_reaction, & + p % secondary_bank, p % n_secondary) + end if end if ! If survival biasing is being used, the following subroutine adjusts the @@ -1071,10 +1076,12 @@ contains ! neutrons produced from fission and creates appropriate bank sites. !=============================================================================== - subroutine create_fission_sites(p, i_nuclide, i_reaction) + subroutine create_fission_sites(p, i_nuclide, i_reaction, bank_array, size_bank) type(Particle), intent(inout) :: p integer, intent(in) :: i_nuclide integer, intent(in) :: i_reaction + type(Bank), intent(inout) :: bank_array(:) + integer(8), intent(inout) :: size_bank integer :: nu_d(MAX_DELAYED_GROUPS) ! number of delayed neutrons born integer :: i ! loop index @@ -1123,27 +1130,35 @@ contains nu = int(nu_t) + 1 end if - ! Check for fission bank size getting hit - if (n_bank + nu > size(fission_bank)) then - if (master) call warning("Maximum number of sites in fission bank & - &reached. This can result in irreproducible results using different & - &numbers of processes/threads.") + ! Check for bank size getting hit. For fixed source calculations, this is a + ! fatal error. For eigenvalue calculations, it just means that k-effective + ! was too high for a single batch. + if (size_bank + nu > size(bank_array)) then + if (run_mode == MODE_FIXEDSOURCE) then + call fatal_error("Secondary particle bank size limit reached. If you & + &are running a subcritical multiplication problem, k-effective & + &may be too close to one.") + else + if (master) call warning("Maximum number of sites in fission bank & + &reached. This can result in irreproducible results using different & + &numbers of processes/threads.") + end if end if ! Bank source neutrons - if (nu == 0 .or. n_bank == size(fission_bank)) return + if (nu == 0 .or. size_bank == size(bank_array)) return ! Initialize counter of delayed neutrons encountered for each delayed group ! to zero. nu_d(:) = 0 p % fission = .true. ! Fission neutrons will be banked - do i = int(n_bank,4) + 1, int(min(n_bank + nu, int(size(fission_bank),8)),4) + do i = int(size_bank,4) + 1, int(min(size_bank + nu, int(size(bank_array),8)),4) ! Bank source neutrons by copying particle data - fission_bank(i) % xyz = p % coord(1) % xyz + bank_array(i) % xyz = p % coord(1) % xyz ! Set weight of fission bank site - fission_bank(i) % wgt = ONE/weight + bank_array(i) % wgt = ONE/weight ! Sample cosine of angle -- fission neutrons are always emitted ! isotropically. Sometimes in ACE data, fission reactions actually have @@ -1153,17 +1168,17 @@ contains ! Sample azimuthal angle uniformly in [0,2*pi) phi = TWO*PI*prn() - fission_bank(i) % uvw(1) = mu - fission_bank(i) % uvw(2) = sqrt(ONE - mu*mu) * cos(phi) - fission_bank(i) % uvw(3) = sqrt(ONE - mu*mu) * sin(phi) + bank_array(i) % uvw(1) = mu + bank_array(i) % uvw(2) = sqrt(ONE - mu*mu) * cos(phi) + bank_array(i) % uvw(3) = sqrt(ONE - mu*mu) * sin(phi) ! Sample secondary energy distribution for fission reaction and set energy ! in fission bank - fission_bank(i) % E = sample_fission_energy(nuc, nuc%reactions(& - i_reaction), p) + bank_array(i) % E = sample_fission_energy(nuc, & + nuc % reactions(i_reaction), p) ! Set the delayed group of the neutron - fission_bank(i) % delayed_group = p % delayed_group + bank_array(i) % delayed_group = p % delayed_group ! Increment the number of neutrons born delayed if (p % delayed_group > 0) then @@ -1172,7 +1187,7 @@ contains end do ! increment number of bank sites - n_bank = min(n_bank + nu, int(size(fission_bank),8)) + size_bank = min(size_bank + nu, int(size(bank_array),8)) ! Store total and delayed weight banked for analog fission tallies p % n_bank = nu diff --git a/src/relaxng/geometry.rnc b/src/relaxng/geometry.rnc index 8d25789f5a..1bca009a61 100644 --- a/src/relaxng/geometry.rnc +++ b/src/relaxng/geometry.rnc @@ -9,6 +9,8 @@ element geometry { (element material { ( xsd:int | "void" )+ } | attribute material { ( xsd:int | "void" )+ }) ) & + (element temperature { list { xsd:double+ } } | + attribute temperature { list { xsd:double+ } } )? & (element region { xsd:string } | attribute region { xsd:string })? & (element rotation { list { xsd:double+ } } | attribute rotation { list { xsd:double+ } })? & (element translation { list { xsd:double+ } } | attribute translation { list { xsd:double+ } })? diff --git a/src/relaxng/geometry.rng b/src/relaxng/geometry.rng index d40401b281..ba8eab73d3 100644 --- a/src/relaxng/geometry.rng +++ b/src/relaxng/geometry.rng @@ -64,6 +64,24 @@ + + + + + + + + + + + + + + + + + + diff --git a/src/summary.F90 b/src/summary.F90 index c31068eb9f..4ebf5fdfd2 100644 --- a/src/summary.F90 +++ b/src/summary.F90 @@ -108,6 +108,7 @@ contains integer :: i, j, k, m integer, allocatable :: lattice_universes(:,:,:) integer, allocatable :: cell_materials(:) + real(8), allocatable :: cell_temperatures(:) integer(HID_T) :: geom_group integer(HID_T) :: cells_group, cell_group integer(HID_T) :: surfaces_group, surface_group @@ -151,6 +152,7 @@ contains select case (c%type) case (CELL_NORMAL) call write_dataset(cell_group, "fill_type", "normal") + if (size(c % material) == 1) then if (c % material(1) == MATERIAL_VOID) then call write_dataset(cell_group, "material", MATERIAL_VOID) @@ -171,6 +173,12 @@ contains deallocate(cell_materials) end if + allocate(cell_temperatures(size(c % sqrtkT))) + cell_temperatures(:) = c % sqrtkT(:) + cell_temperatures(:) = cell_temperatures(:)**2 / K_BOLTZMANN + call write_dataset(cell_group, "temperature", cell_temperatures) + deallocate(cell_temperatures) + case (CELL_FILL) call write_dataset(cell_group, "fill_type", "universe") call write_dataset(cell_group, "fill", universes(c%fill)%id) diff --git a/src/tracking.F90 b/src/tracking.F90 index f20e5e413b..de2c54a75a 100644 --- a/src/tracking.F90 +++ b/src/tracking.F90 @@ -213,6 +213,7 @@ contains if (p % n_secondary > 0) then call p % initialize_from_source(p % secondary_bank(p % n_secondary)) p % n_secondary = p % n_secondary - 1 + n_event = 0 ! Enter new particle in particle track file if (p % write_track) call add_particle_track() diff --git a/tests/test_asymmetric_lattice/inputs_true.dat b/tests/test_asymmetric_lattice/inputs_true.dat new file mode 100644 index 0000000000..e3b00b185c --- /dev/null +++ b/tests/test_asymmetric_lattice/inputs_true.dat @@ -0,0 +1 @@ +b9b4222c4beea80fe6083590f6b785303d174972d80671fb661bac8e030db6f4a61648240cfad6162799361fc0e08a23c61d31aff844d978528d6dad5b5fbc63 \ No newline at end of file diff --git a/tests/test_asymmetric_lattice/results_true.dat b/tests/test_asymmetric_lattice/results_true.dat new file mode 100644 index 0000000000..ec4b883886 --- /dev/null +++ b/tests/test_asymmetric_lattice/results_true.dat @@ -0,0 +1 @@ +b5f96919ca474cd1c9c9d0acde3b8aac4a1cf636443c72a38b6c5a4221a8ce3e90182aaef2f664e44b9175ca257a89db2328b63e19388ee0e5006de4b3d92ce6 \ No newline at end of file diff --git a/tests/test_asymmetric_lattice/test_asymmetric_lattice.py b/tests/test_asymmetric_lattice/test_asymmetric_lattice.py new file mode 100644 index 0000000000..5a1d47ef85 --- /dev/null +++ b/tests/test_asymmetric_lattice/test_asymmetric_lattice.py @@ -0,0 +1,129 @@ +#!/usr/bin/env python + +import os +import sys +import glob +import hashlib +sys.path.insert(0, os.pardir) +from testing_harness import PyAPITestHarness +import openmc +from openmc.source import Source +from openmc.stats import Box + + +class AsymmetricLatticeTestHarness(PyAPITestHarness): + + def _build_inputs(self): + """Build an axis-asymmetric lattice of fuel assemblies""" + + # Build full core geometry from underlying input set + self._input_set.build_default_materials_and_geometry() + + # Extract all universes from the full core geometry + geometry = self._input_set.geometry.geometry + all_univs = geometry.get_all_universes() + + # Extract universes encapsulating fuel and water assemblies + water = all_univs[7] + fuel = all_univs[8] + + # Construct a 3x3 lattice of fuel assemblies + core_lat = openmc.RectLattice(name='3x3 Core Lattice', lattice_id=202) + core_lat.dimension = (3, 3) + core_lat.lower_left = (-32.13, -32.13) + core_lat.pitch = (21.42, 21.42) + core_lat.universes = [[fuel, water, water], + [fuel, fuel, fuel], + [water, water, water]] + + # Create bounding surfaces + min_x = openmc.XPlane(x0=-32.13, boundary_type='reflective') + max_x = openmc.XPlane(x0=+32.13, boundary_type='reflective') + min_y = openmc.YPlane(y0=-32.13, boundary_type='reflective') + max_y = openmc.YPlane(y0=+32.13, boundary_type='reflective') + min_z = openmc.ZPlane(z0=0, boundary_type='reflective') + max_z = openmc.ZPlane(z0=+32.13, boundary_type='reflective') + + # Define root universe + root_univ = openmc.Universe(universe_id=0, name='root universe') + root_cell = openmc.Cell(cell_id=1) + root_cell.region = +min_x & -max_x & +min_y & -max_y & +min_z & -max_z + root_cell.fill = core_lat + root_univ.add_cell(root_cell) + + # Over-ride geometry in the input set with this 3x3 lattice + self._input_set.geometry.geometry.root_universe = root_univ + + # Initialize a "distribcell" filter for the fuel pin cell + distrib_filter = openmc.Filter(type='distribcell', bins=[27]) + + # Initialize the tallies + tally = openmc.Tally(name='distribcell tally', tally_id=27) + tally.add_filter(distrib_filter) + tally.add_score('nu-fission') + + # Initialize the tallies file + tallies_file = openmc.TalliesFile() + tallies_file.add_tally(tally) + + # Assign the tallies file to the input set + self._input_set.tallies = tallies_file + + # Build default settings + self._input_set.build_default_settings() + + # Specify summary output and correct source sampling box + source = Source(space=Box([-32, -32, 0], [32, 32, 32])) + source.space.only_fissionable = True + self._input_set.settings.source = source + self._input_set.settings.output = {'summary': True} + + # Write input XML files + self._input_set.export() + + def _get_results(self, hash_output=True): + """Digest info in statepoint and summary and return as a string.""" + + # Read the statepoint file + statepoint = glob.glob(os.path.join(os.getcwd(), self._sp_name))[0] + sp = openmc.StatePoint(statepoint) + + # Read the summary file + summary = glob.glob(os.path.join(os.getcwd(), 'summary.h5'))[0] + su = openmc.Summary(summary) + sp.link_with_summary(su) + + # Extract the tally of interest + tally = sp.get_tally(name='distribcell tally') + + # Create a string of all mean, std. dev. values for both tallies + outstr = '' + outstr += ', '.join(map(str, tally.mean.flatten())) + '\n' + outstr += ', '.join(map(str, tally.std_dev.flatten())) + '\n' + + # Extract fuel assembly lattices from the summary + all_cells = su.openmc_geometry.get_all_cells() + fuel = all_cells[80].fill + core = all_cells[1].fill + + # Append a string of lattice distribcell offsets to the string + outstr += ', '.join(map(str, fuel.offsets.flatten())) + '\n' + outstr += ', '.join(map(str, core.offsets.flatten())) + '\n' + + # Hash the results if necessary + if hash_output: + sha512 = hashlib.sha512() + sha512.update(outstr.encode('utf-8')) + outstr = sha512.hexdigest() + + return outstr + + def _cleanup(self): + super(AsymmetricLatticeTestHarness, self)._cleanup() + f = os.path.join(os.getcwd(), 'tallies.xml') + if os.path.exists(f): os.remove(f) + + +if __name__ == '__main__': + harness = AsymmetricLatticeTestHarness('statepoint.10.h5', True) + harness.main() diff --git a/tests/test_distribmat/results_true.dat b/tests/test_distribmat/results_true.dat index 70464fbc6c..73a48907cb 100644 --- a/tests/test_distribmat/results_true.dat +++ b/tests/test_distribmat/results_true.dat @@ -5,6 +5,7 @@ Cell Name = Material = [2, 3, void, 2] Region = -10000 + Temperature = [ 293.60594237 293.60594237 0. 293.60594237] Rotation = None Translation = None Offset = None diff --git a/tests/test_fixed_source/materials.xml b/tests/test_fixed_source/materials.xml index 5242021d34..ab7a76bc00 100644 --- a/tests/test_fixed_source/materials.xml +++ b/tests/test_fixed_source/materials.xml @@ -4,6 +4,7 @@ + diff --git a/tests/test_fixed_source/results_true.dat b/tests/test_fixed_source/results_true.dat index 0940301886..b3def050e8 100644 --- a/tests/test_fixed_source/results_true.dat +++ b/tests/test_fixed_source/results_true.dat @@ -1,6 +1,6 @@ tally 1: -4.538791E+02 -2.073271E+04 +4.563929E+02 +2.091711E+04 leakage: -9.830000E+00 -9.663900E+00 +9.780000E+00 +9.566400E+00 diff --git a/tests/test_multipole/inputs_true.dat b/tests/test_multipole/inputs_true.dat new file mode 100644 index 0000000000..a429c9b8d0 --- /dev/null +++ b/tests/test_multipole/inputs_true.dat @@ -0,0 +1 @@ +5c1cec635da5c4c869bdf58f62924a4cb1648e4ddecf0ce71b823cf45178767ba7f2e089b76d36e8616b1b21ffa43b32ab1b17d20bb74120f900b9e3e9ab9bcc \ No newline at end of file diff --git a/tests/test_multipole/results_true.dat b/tests/test_multipole/results_true.dat new file mode 100644 index 0000000000..19dd4d26b7 --- /dev/null +++ b/tests/test_multipole/results_true.dat @@ -0,0 +1,12 @@ +k-combined: +1.445285E+00 9.521660E-03 +Cell + ID = 11 + Name = + Material = 2 + Region = -10000 + Temperature = [ 500. 0. 700. 800.] + Rotation = None + Translation = None + Offset = None + Distribcell index= 1 diff --git a/tests/test_multipole/test_multipole.py b/tests/test_multipole/test_multipole.py new file mode 100644 index 0000000000..a5437e433a --- /dev/null +++ b/tests/test_multipole/test_multipole.py @@ -0,0 +1,133 @@ +#!/usr/bin/env python + +import os +import sys +sys.path.insert(0, os.pardir) +from testing_harness import TestHarness, PyAPITestHarness +import openmc +from openmc.stats import Box +from openmc.source import Source + + +class DistribmatTestHarness(PyAPITestHarness): + def _build_inputs(self): + #################### + # Materials + #################### + + moderator = openmc.Material(material_id=1) + moderator.set_density('g/cc', 1.0) + moderator.add_nuclide('H-1', 2.0) + moderator.add_nuclide('O-16', 1.0) + + dense_fuel = openmc.Material(material_id=2) + dense_fuel.set_density('g/cc', 4.5) + dense_fuel.add_nuclide('U-235', 1.0) + + mats_file = openmc.MaterialsFile() + mats_file.default_xs = '71c' + mats_file.add_materials([moderator, dense_fuel]) + mats_file.export_to_xml() + + + #################### + # Geometry + #################### + + c1 = openmc.Cell(cell_id=1) + c1.fill = moderator + mod_univ = openmc.Universe(universe_id=1) + mod_univ.add_cell(c1) + + r0 = openmc.ZCylinder(R=0.3) + c11 = openmc.Cell(cell_id=11) + c11.region = -r0 + c11.fill = dense_fuel + c11.temperature = [500, 0, 700, 800] + c12 = openmc.Cell(cell_id=12) + c12.region = +r0 + c12.fill = moderator + fuel_univ = openmc.Universe(universe_id=11) + fuel_univ.add_cells((c11, c12)) + + lat = openmc.RectLattice(lattice_id=101) + lat.dimension = [2, 2] + lat.lower_left = [-2.0, -2.0] + lat.pitch = [2.0, 2.0] + lat.universes = [[fuel_univ]*2]*2 + lat.outer = mod_univ + + x0 = openmc.XPlane(x0=-3.0) + x1 = openmc.XPlane(x0=3.0) + y0 = openmc.YPlane(y0=-3.0) + y1 = openmc.YPlane(y0=3.0) + for s in [x0, x1, y0, y1]: + s.boundary_type = 'reflective' + c101 = openmc.Cell(cell_id=101) + c101.region = +x0 & -x1 & +y0 & -y1 + c101.fill = lat + root_univ = openmc.Universe(universe_id=0) + root_univ.add_cell(c101) + + geometry = openmc.Geometry() + geometry.root_universe = root_univ + geo_file = openmc.GeometryFile() + geo_file.geometry = geometry + geo_file.export_to_xml() + + + #################### + # Settings + #################### + + sets_file = openmc.SettingsFile() + sets_file.batches = 5 + sets_file.inactive = 0 + sets_file.particles = 1000 + sets_file.source = Source(space=Box([-1, -1, -1], [1, 1, 1])) + sets_file.output = {'summary': True} + sets_file.export_to_xml() + + + #################### + # Plots + #################### + + plots_file = openmc.PlotsFile() + + plot = openmc.Plot(plot_id=1) + plot.basis = 'xy' + plot.color = 'cell' + plot.filename = 'cellplot' + plot.origin = (0, 0, 0) + plot.width = (7, 7) + plot.pixels = (400, 400) + plots_file.add_plot(plot) + + plot = openmc.Plot(plot_id=2) + plot.basis = 'xy' + plot.color = 'mat' + plot.filename = 'matplot' + plot.origin = (0, 0, 0) + plot.width = (7, 7) + plot.pixels = (400, 400) + plots_file.add_plot(plot) + + plots_file.export_to_xml() + + def _get_results(self): + outstr = super(DistribmatTestHarness, self)._get_results() + su = openmc.Summary('summary.h5') + outstr += str(su.get_cell_by_id(11)) + return outstr + + def _cleanup(self): + f = os.path.join(os.getcwd(), 'plots.xml') + if os.path.exists(f): + os.remove(f) + super(DistribmatTestHarness, self)._cleanup() + + +if __name__ == '__main__': + harness = DistribmatTestHarness('statepoint.5.*') + harness.main()