/usr/include/shogun/mathematics/Integration.h is in libshogun-dev 3.2.0-7.5.
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* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation; either version 3 of the License, or
* (at your option) any later version.
*
* Written (W) 2013 Roman Votyakov
*
* The abscissae and weights for Gauss-Kronrod rules are taken form
* QUADPACK, which is in public domain.
* http://www.netlib.org/quadpack/
*
* See method comments which functions are adapted from GNU Octave,
* file quadgk.m: Copyright (C) 2008-2012 David Bateman under GPLv3
* http://www.gnu.org/software/octave/
*/
#ifndef _INTEGRATION_H_
#define _INTEGRATION_H_
#include <shogun/lib/config.h>
#ifdef HAVE_EIGEN3
#include <shogun/base/SGObject.h>
#include <shogun/lib/DynamicArray.h>
#include <shogun/mathematics/Math.h>
#include <shogun/mathematics/Function.h>
namespace shogun
{
/** @brief Class that contains certain methods related to numerical
* integration
*/
class CIntegration : public CSGObject
{
public:
/** numerically evaluate definite integral \f$\int_a^b f(x) dx\f$,
* where \f$f(x)\f$ - function of one variable, using adaptive
* Gauss-Kronrod quadrature formula
*
* \f[
* \int_a^b f(x)\dx \approx \sum_{i=1}^n w_i f(x_i)
* \f]
*
* where x_i and w_i - Gauss-Kronrod nodes and weights
* respectively.
*
* This function applies the Gauss-Kronrod 21-point integration
* rule for finite bounds \f$[a, b]\f$ and 15-point rule for
* infinite ones.
*
* Based on ideas form GNU Octave (file quadgk.m) under GPLv3.
*
* @param f integrable function of one variable
* @param a lower bound of the domain of integration
* @param b upper bound of the domain of integration
* @param abs_tol absolute tolerance of the quadrature
* @param rel_tol relative tolerance of the quadrature
* @param max_iter maximum number of iterations of the method
* @param sn initial number of subintervals
*
* @return approximate value of definite integral of the function
* on given domain
*/
static float64_t integrate_quadgk(CFunction* f, float64_t a,
float64_t b, float64_t abs_tol=1e-10, float64_t rel_tol=1e-5,
uint32_t max_iter=1000, index_t sn=10);
/** numerically evaluate integral of the following kind
*
* \f[
* \int_{-\infty}^{\infty}e^{-x^2}f(x)dx
* \f]
*
* using 64-point Gauss-Hermite rule
*
* \f[
* \int_{-\infty}^{\infty}e^{-x^2}f(x)dx \approx
* \sum_{i=1}^{64} w_if(x_i)
* \f]
*
* where x_i and w_i - ith node and weight for the 64-point
* Gauss-Hermite formula respectively.
*
* @param f integrable function of one variable
*
* @return approximate value of the
* integral \f$\int_{-\infty}^{\infty}e^{-x^2}f(x)dx\f$
*/
static float64_t integrate_quadgh(CFunction* f);
/** get object name
*
* @return name Integration
*/
virtual const char* get_name() const { return "Integration"; }
private:
/** evaluate definite integral of a function and error on each
* subinterval using Gauss-Kronrod quadrature formula of order n
*
* Adapted form GNU Octave (file quadgk.m) under GPLv3.
*
* @param f integrable function of one variable
* @param subs subintervals of integration
* @param q approximate value of definite integral of the function
* on each subinterval
* @param err error on each subinterval
* @param n order of the Gauss-Kronrod rule
* @param xgk Gauss-Kronrod nodes
* @param wg Gauss weights
* @param wgk Gauss-Kronrod weights
*/
static void evaluate_quadgk(CFunction* f, CDynamicArray<float64_t>* subs,
CDynamicArray<float64_t>* q, CDynamicArray<float64_t>* err, index_t n,
float64_t* xgk, float64_t* wg, float64_t* wgk);
/** evaluate definite integral of a function and error on each
* subinterval using Gauss-Kronrod quadrature formula of order 15.
*
* Gauss-Kronrod nodes, Gauss weights and Gauss-Kronrod weights
* are precomputed.
*
* The abscissae and weights for 15-point rule are taken from from
* QUADPACK (file dqk15.f).
*
* @param f integrable function of one variable
* @param subs subintervals of integration
* @param q approximate value of definite integral of the function
* on each subinterval
* @param err error on each subinterval
*/
static void evaluate_quadgk15(CFunction* f, CDynamicArray<float64_t>* subs,
CDynamicArray<float64_t>* q, CDynamicArray<float64_t>* err);
/** evaluate definite integral of a function and error on each
* subinterval using Gauss-Kronrod quadrature formula of order 21.
*
* Gauss-Kronrod nodes, Gauss weights and Gauss-Kronrod weights
* are precomputed.
*
* The abscissae and weights for 21-point rule are taken from
* QUADPACK (file dqk21.f).
*
* @param f integrable function of one variable
* @param subs subintervals of integration
* @param q approximate value of definite integral of the function
* on each subinterval
* @param err error on each subinterval
*/
static void evaluate_quadgk21(CFunction* f, CDynamicArray<float64_t>* subs,
CDynamicArray<float64_t>* q, CDynamicArray<float64_t>* err);
/** evaluate integral \f$\int_{-\infty}^{\infty}e^{-x^2}f(x)dx\f$
* using Gauss-Hermite quadrature formula of order n
*
* @param f integrable function of one variable
* @param n order of the Gauss-Hermite rule
* @param xh Gauss-Hermite nodes
* @param wh Gauss-Hermite weights
*
* @return approximate value of the integral
* \f$\int_{-\infty}^{\infty}e^{-x^2}f(x)dx\f$
*/
static float64_t evaluate_quadgh(CFunction* f, index_t n, float64_t* xh,
float64_t* wh);
/** evaluate integral \f$\int_{-\infty}^{\infty}e^{-x^2}f(x)dx\f$
* using Gauss-Hermite quadrature formula of order 64.
*
* Gauss-Hermite nodes \f$x_i\f$ and weights \f$w_i\f$ are
* precomputed: \f$x_i\f$ - the i-th zero of \f$H_n(x)\f$,
* \f$w_i=\frac{2^{n-1}n!\sqrt{\pi}}{n^2[H_{n-1}(x_i)]^2}\f$,
* where \f$H_n(x)\f$ is physicists' Hermite polynomials.
*
* @param f integrable function of one variable
*
* @return approximate value of the integral
* \f$\int_{-\infty}^{\infty}e^{-x^2}f(x)dx\f$
*/
static float64_t evaluate_quadgh64(CFunction* f);
};
}
#endif /* HAVE_EIGEN3 */
#endif /* _INTEGRATION_H_ */
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