Add some special math function
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@ -114,6 +114,9 @@ if (Boost_FOUND)
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add_dependencies(graficToDensity ${all_deps})
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endif()
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endif()
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add_executable(test_special test_special.cpp)
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target_link_libraries(test_special ${tolink})
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endif (Boost_FOUND)
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IF (ENABLE_OPENMP AND YORICK_SUPPORT)
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12
sample/test_special.cpp
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12
sample/test_special.cpp
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#include <iostream>
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#include "special_math.hpp"
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int main()
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{
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std::cout << CosmoTool::log1p_exp(2.0) << std::endl;
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std::cout << CosmoTool::log_modified_bessel_first_kind(100.0, 0.1) << std::endl;
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return 0;
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}
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231
src/special_math.hpp
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231
src/special_math.hpp
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// Original code derived from Boost and is distributed here
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// under the Boost license (licenses/boost-license.txt)
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// Copyright (c) 2006 Xiaogang Zhang
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// Copyright (c) 2007, 2017 John Maddock
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// Secondary code copyright by its author and is distributed here
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// under the BSD-3 license (LICENSE.md). Derived from
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// stan/math/prim/fun/log_modified_bessel_first_kind.hpp
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#ifndef __COSMOTOOL_SPECIAL_MATH_HPP
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#define __COSMOTOOL_SPECIAL_MATH_HPP
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#include <CosmoTool/algo.hpp>
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#include <boost/math/constants/constants.hpp>
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#include <boost/math/tools/rational.hpp>
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#include <cmath>
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#include <limits>
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// Taken and adapted from
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// https://github.com/stan-dev/math/blob/develop/stan/math/prim/fun/log_modified_bessel_first_kind.hpp
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namespace CosmoTool {
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template <typename T> T log1p_exp(T x) {
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if (x > T(0)) {
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return x + std::log1p(std::exp(-x));
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}
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return std::log1p(std::exp(x));
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}
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template <typename T> T multiply_log(T a, T b) {
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if (a == 0 && b == 0)
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return 0;
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return a * std::log(b);
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}
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template <typename T> T inf() { return std::numeric_limits<T>::infinity(); }
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template <typename T> T log_sum_exp(T const a, T const b) {
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if (a == -inf<T>()) {
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return b;
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}
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if (a == inf<T>() && b == inf<T>()) {
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return inf<T>();
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}
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if (a > b) {
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return a + log1p_exp(b - a);
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}
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return b + log1p_exp(a - b);
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}
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/* Log of the modified Bessel function of the first kind,
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* which is better known as the Bessel I function. See
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* modified_bessel_first_kind.hpp for the function definition.
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* The derivatives are known to be incorrect for v = 0 because a
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* simple constant 0 is returned.
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*
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* @tparam T common type for calculation
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* @param v Order, can be a non-integer but must be at least -1
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* @param z Real non-negative number
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* @throws std::domain_error if either v or z is NaN, z is
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* negative, or v is less than -1
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* @return log of Bessel I function
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*/
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template <typename T> T log_modified_bessel_first_kind(T const v, T const z) {
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using boost::math::tools::evaluate_polynomial;
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using std::log;
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using std::pow;
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using std::sqrt;
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static const double LOG_TWO = std::log(2.0);
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static const double EPSILON = std::numeric_limits<double>::epsilon();
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static const double TWO_PI = 2.0 * boost::math::constants::pi<double>();
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if (z == 0) {
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if (v == 0) {
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return 0.0;
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}
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if (v > 0) {
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return -std::numeric_limits<T>::infinity();
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}
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return std::numeric_limits<T>::infinity();
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}
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if (std::isinf(z)) {
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return z;
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}
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if (v == 0) {
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// Modified Bessel function of the first kind of order zero
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// we use the approximating forms derived in:
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// "Rational Approximations for the Modified Bessel Function of the
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// First Kind -- I0(x) for Computations with Double Precision"
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// by Pavel Holoborodko, see
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// http://www.advanpix.com/2015/11/11/rational-approximations-for-the-modified-bessel-function-of-the-first-kind-i0-computations-double-precision
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// The actual coefficients used are [Boost's] own, and extend
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// Pavel's work to precisions other than double.
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if (z < 7.75) {
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// Bessel I0 over[10 ^ -16, 7.75]
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// Max error in interpolated form : 3.042e-18
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// Max Error found at double precision = Poly : 5.106609e-16
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// Cheb : 5.239199e-16
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static const double P[] = {
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1.00000000000000000e+00, 2.49999999999999909e-01,
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2.77777777777782257e-02, 1.73611111111023792e-03,
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6.94444444453352521e-05, 1.92901234513219920e-06,
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3.93675991102510739e-08, 6.15118672704439289e-10,
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7.59407002058973446e-12, 7.59389793369836367e-14,
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6.27767773636292611e-16, 4.34709704153272287e-18,
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2.63417742690109154e-20, 1.13943037744822825e-22,
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9.07926920085624812e-25};
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return log1p_exp(multiply_log(2.0, z) - log(4.0) +
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log(evaluate_polynomial(P, 0.25 * square(z))));
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}
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if (z < 500) {
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// Max error in interpolated form : 1.685e-16
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// Max Error found at double precision = Poly : 2.575063e-16
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// Cheb : 2.247615e+00
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static const double P[] = {
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3.98942280401425088e-01, 4.98677850604961985e-02,
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2.80506233928312623e-02, 2.92211225166047873e-02,
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4.44207299493659561e-02, 1.30970574605856719e-01,
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-3.35052280231727022e+00, 2.33025711583514727e+02,
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-1.13366350697172355e+04, 4.24057674317867331e+05,
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-1.23157028595698731e+07, 2.80231938155267516e+08,
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-5.01883999713777929e+09, 7.08029243015109113e+10,
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-7.84261082124811106e+11, 6.76825737854096565e+12,
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-4.49034849696138065e+13, 2.24155239966958995e+14,
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-8.13426467865659318e+14, 2.02391097391687777e+15,
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-3.08675715295370878e+15, 2.17587543863819074e+15};
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return z + log(evaluate_polynomial(P, 1 / z)) - multiply_log(0.5, z);
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}
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// Max error in interpolated form : 2.437e-18
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// Max Error found at double precision = Poly : 1.216719e-16
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static const double P[] = {3.98942280401432905e-01, 4.98677850491434560e-02,
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2.80506308916506102e-02, 2.92179096853915176e-02,
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4.53371208762579442e-02};
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return z + log(evaluate_polynomial(P, 1 / z)) - multiply_log(0.5, z);
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}
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if (v == 1) { // WARNING: will not autodiff for v = 1 correctly
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// modified from Boost's bessel_i1_imp in the double precision case
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// see credits above in the v == 0 case
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if (z < 7.75) {
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// Bessel I0 over[10 ^ -16, 7.75]
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// Max error in interpolated form: 5.639e-17
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// Max Error found at double precision = Poly: 1.795559e-16
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static const double P[] = {
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8.333333333333333803e-02, 6.944444444444341983e-03,
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3.472222222225921045e-04, 1.157407407354987232e-05,
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2.755731926254790268e-07, 4.920949692800671435e-09,
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6.834657311305621830e-11, 7.593969849687574339e-13,
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6.904822652741917551e-15, 5.220157095351373194e-17,
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3.410720494727771276e-19, 1.625212890947171108e-21,
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1.332898928162290861e-23};
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T a = square(z) * 0.25;
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T Q[3] = {1, 0.5, evaluate_polynomial(P, a)};
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return log(z) + log(evaluate_polynomial(Q, a)) - LOG_TWO;
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}
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if (z < 500) {
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// Max error in interpolated form: 1.796e-16
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// Max Error found at double precision = Poly: 2.898731e-16
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static const double P[] = {
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3.989422804014406054e-01, -1.496033551613111533e-01,
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-4.675104253598537322e-02, -4.090895951581637791e-02,
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-5.719036414430205390e-02, -1.528189554374492735e-01,
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3.458284470977172076e+00, -2.426181371595021021e+02,
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1.178785865993440669e+04, -4.404655582443487334e+05,
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1.277677779341446497e+07, -2.903390398236656519e+08,
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5.192386898222206474e+09, -7.313784438967834057e+10,
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8.087824484994859552e+11, -6.967602516005787001e+12,
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4.614040809616582764e+13, -2.298849639457172489e+14,
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8.325554073334618015e+14, -2.067285045778906105e+15,
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3.146401654361325073e+15, -2.213318202179221945e+15};
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return z + log(evaluate_polynomial(P, 1 / z)) - multiply_log(0.5, z);
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}
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// Max error in interpolated form: 1.320e-19
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// Max Error found at double precision = Poly: 7.065357e-17
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static const double P[] = {
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3.989422804014314820e-01, -1.496033551467584157e-01,
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-4.675105322571775911e-02, -4.090421597376992892e-02,
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-5.843630344778927582e-02};
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return z + log(evaluate_polynomial(P, 1 / z)) - multiply_log(0.5, z);
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}
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if (z > 100) {
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// Boost does something like this in asymptotic_bessel_i_large_x
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T lim = pow((square(v) + 2.5) / (2 * z), 3) / 24;
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if (lim < (EPSILON * 10)) {
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T s = 1;
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T mu = 4 * square(v);
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T ex = 8 * z;
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T num = mu - 1;
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T denom = ex;
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s -= num / denom;
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num *= mu - 9;
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denom *= ex * 2;
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s += num / denom;
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num *= mu - 25;
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denom *= ex * 3;
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s -= num / denom;
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s = z - log(sqrt(z * TWO_PI)) + log(s);
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return s;
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}
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}
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T log_half_z = log(0.5 * z);
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T lgam = (v > -1) ? lgamma(v + 1.0) : 0;
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T lcons = (2.0 + v) * log_half_z;
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T out;
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if (v > -1) {
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out = log_sum_exp(v * log_half_z - lgam, lcons - lgamma(v + 2));
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lgam += log1p(v);
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} else {
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out = lcons;
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}
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double m = 2;
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double lfac = 0;
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T old_out;
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do {
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lfac += log(m);
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lgam += log(v + m);
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lcons += 2 * log_half_z;
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old_out = out;
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out = log_sum_exp(out, lcons - lfac - lgam); // underflows eventually
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m++;
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} while (out > old_out || out < old_out);
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return out;
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}
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} // namespace CosmoTool
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#endif
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