mitsuba::GaussLobattoIntegrator Class Reference

Computes the integral of a one-dimensional function using adaptive Gauss-Lobatto quadrature. More...

#include <mitsuba/core/quad.h>

Public Types
typedef boost::function< Float(Float)>	Integrand

Public Member Functions
	GaussLobattoIntegrator (size_t maxEvals, Float absError=0, Float relError=0, bool useConvergenceEstimate=true, bool warn=true)
Float	integrate (const Integrand &f, Float a, Float b, size_t *evals=NULL) const
	Integrate the function f from a to b. More...

Protected Member Functions
Float	adaptiveGaussLobattoStep (const boost::function< Float(Float)> &f, Float a, Float b, Float fa, Float fb, Float is, size_t &evals) const
	Perform one step of the 4-point Gauss-Lobatto rule, then compute the same integral using a 7-point Kronrod extension and compare. If the accuracy is deemed too low, recurse. More...
Float	calculateAbsTolerance (const boost::function< Float(Float)> &f, Float a, Float b, size_t &evals) const

Protected Attributes
Float	m_absError
Float	m_relError
size_t	m_maxEvals
bool	m_useConvergenceEstimate
bool	m_warn

Static Protected Attributes
static const Float	m_alpha
static const Float	m_beta
static const Float	m_x1
static const Float	m_x2
static const Float	m_x3

Detailed Description

Computes the integral of a one-dimensional function using adaptive Gauss-Lobatto quadrature.

Given a target error \( \epsilon \), the integral of a function \( f \) between \( a \) and \( b \) is calculated by means of the Gauss-Lobatto formula.

References: This algorithm is a C++ implementation of the algorithm outlined in

W. Gander and W. Gautschi, Adaptive Quadrature - Revisited. BIT, 40(1):84-101, March 2000. CS technical report: ftp.inf.ethz.ch/pub/publications/tech-reports/3xx/306.ps.gz

The original MATLAB version can be downloaded here http://www.inf.ethz.ch/personal/gander/adaptlob.m

This particular implementation is based on code in QuantLib, a free-software/open-source library for financial quantitative analysts and developers - http://quantlib.org/

Member Typedef Documentation

typedef boost::function<Float (Float)> mitsuba::GaussLobattoIntegrator::Integrand

Constructor & Destructor Documentation

mitsuba::GaussLobattoIntegrator::GaussLobattoIntegrator	(	size_t	maxEvals,
		Float	absError = 0,
		Float	relError = 0,
		bool	useConvergenceEstimate = true,
		bool	warn = true
	)

Initialize a Gauss-Lobatto integration scheme

Parameters

maxEvals	Maximum number of function evaluations. The integrator will print a warning when this limit is exceeded. It will then stop the recursion, but a few further evaluations may still take place. Hence the limit is not a strict one.
absError	Absolute error requirement (0 to disable)
relError	Relative error requirement (0 to disable)
useConvergenceEstimate	Estimate the convergence behavior of the GL-quadrature by comparing the 4, 7 and 13-point variants and increase the absolute tolerance accordingly.
warn	Should the integrator warn when the number of function evaluations is exceeded?

Member Function Documentation

Float mitsuba::GaussLobattoIntegrator::adaptiveGaussLobattoStep ( const boost::function< Float(Float)> &  f, Float  a, Float  b, Float  fa, Float  fb, Float  is, size_t &  evals  ) const

protected

Perform one step of the 4-point Gauss-Lobatto rule, then compute the same integral using a 7-point Kronrod extension and compare. If the accuracy is deemed too low, recurse.

Parameters

f	Function to integrate
a	Lower integration limit
b	Upper integration limit
fa	Function evaluated at the lower limit
fb	Function evaluated at the upper limit
is	Absolute tolerance in epsilons

Float mitsuba::GaussLobattoIntegrator::calculateAbsTolerance ( const boost::function< Float(Float)> &  f, Float  a, Float  b, size_t &  evals  ) const

protected

Compute the absolute error tolerance using a 13-point Gauss-Lobatto rule.

Float mitsuba::GaussLobattoIntegrator::integrate	(	const Integrand &	f,
		Float	a,
		Float	b,
		size_t *	evals = NULL
	)		const

Integrate the function f from a to b.

Also returns the total number of evaluations if requested

Member Data Documentation

Float mitsuba::GaussLobattoIntegrator::m_absError

protected

const Float mitsuba::GaussLobattoIntegrator::m_alpha

staticprotected

const Float mitsuba::GaussLobattoIntegrator::m_beta

staticprotected

size_t mitsuba::GaussLobattoIntegrator::m_maxEvals

protected

Float mitsuba::GaussLobattoIntegrator::m_relError

protected

bool mitsuba::GaussLobattoIntegrator::m_useConvergenceEstimate

protected

bool mitsuba::GaussLobattoIntegrator::m_warn

protected

const Float mitsuba::GaussLobattoIntegrator::m_x1

staticprotected

const Float mitsuba::GaussLobattoIntegrator::m_x2

staticprotected

const Float mitsuba::GaussLobattoIntegrator::m_x3

staticprotected

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The documentation for this class was generated from the following file:

• include/mitsuba/core/quad.h