Class for adaptive quadrature integration in multi-dimensions using rectangular regions.

Algorithm from A.C. Genz, A.A. Malik, An adaptive algorithm for numerical integration over an N-dimensional rectangular region, J. Comput. Appl. Math. 6 (1980) 295-302.

Converted/adapted by R.Brun to C++ from Fortran CERNLIB routine RADMUL (D120) The new code features many changes compared to the Fortran version.

Control parameters are:

\( minpts \): Minimum number of function evaluations requested. Must not exceed maxpts. if minpts < 1 minpts is set to \( 2^n +2n(n+1) +1 \) where n is the function dimension
\( maxpts \): Maximum number of function evaluations to be allowed. \( maxpts >= 2^n +2n(n+1) +1 \) if \( maxpts<minpts \), \( maxpts \) is set to \( 10minpts \)
\( epstol \), \( epsrel \) : Specified relative and absolute accuracy.

The integral will stop if the relative error is less than relative tolerance OR the absolute error is less than the absolute tolerance

The class computes in addition to the integral of the function in the desired interval:

an estimation of the relative accuracy of the result.
number of function evaluations performed.
status code: 0. Normal exit. . At least minpts and at most maxpts calls to the function were performed.
1. maxpts is too small for the specified accuracy eps. The result and relerr contain the values obtainable for the specified value of maxpts.
2. size is too small for the specified number MAXPTS of function evaluations.
3. n<2 or n>15

Method:

An integration rule of degree seven is used together with a certain strategy of subdivision. For a more detailed description of the method see References.

Notes:

1..Multi-dimensional integration is time-consuming. For each rectangular subregion, the routine requires function evaluations. Careful programming of the integrand might result in substantial saving of time. 2..Numerical integration usually works best for smooth functions. Some analysis or suitable transformations of the integral prior to numerical work may contribute to numerical efficiency.

References:

A.C. Genz and A.A. Malik, Remarks on algorithm 006: An adaptive algorithm for numerical integration over an N-dimensional rectangular region, J. Comput. Appl. Math. 6 (1980) 295-302.
A. van Doren and L. de Ridder, An adaptive algorithm for numerical integration over an n-dimensional cube, J.Comput. Appl. Math. 2 (1976) 207-217.

Definition at line 84 of file AdaptiveIntegratorMultiDim.h.

Public Member Functions
	AdaptiveIntegratorMultiDim (const IMultiGenFunction &f, double absTol=0.0, double relTol=1E-9, unsigned int maxcall=100000, unsigned int size=0)
	Construct with a reference to the integrand function and given optionally tolerance (absolute and relative), maximum number of function evaluation (maxpts) and size of the working array.

	AdaptiveIntegratorMultiDim (double absTol=0.0, double relTol=1E-9, unsigned int maxpts=100000, unsigned int size=0)
	Construct given optionally tolerance (absolute and relative), maximum number of function evaluation (maxpts) and size of the working array.

	~AdaptiveIntegratorMultiDim () override
	destructor (no operations)

double	Error () const override
	return integration error

double	Integral (const double xmin, const double xmax) override
	evaluate the integral with the previously given function between xmin[] and xmax[]

double	Integral (const IMultiGenFunction &f, const double xmin, const double xmax)
	evaluate the integral passing a new function

int	NEval () const override
	return number of function evaluations in calculating the integral

ROOT::Math::IntegratorMultiDimOptions	Options () const override
	get the option used for the integration

double	RelError () const
	return relative error

double	Result () const override
	return result of integration

void	SetAbsTolerance (double absTol) override
	set absolute tolerance

void	SetFunction (const IMultiGenFunction &f) override
	set the integration function (must implement multi-dim function interface: IBaseFunctionMultiDim)

void	SetMaxPts (unsigned int n)
	set max points

void	SetMinPts (unsigned int n)
	set min points

void	SetOptions (const ROOT::Math::IntegratorMultiDimOptions &opt) override
	set the options

void	SetRelTolerance (double relTol) override
	set relative tolerance

void	SetSize (unsigned int size)
	set workspace size

int	Status () const override
	return status of integration

Public Member Functions inherited from ROOT::Math::VirtualIntegratorMultiDim
	~VirtualIntegratorMultiDim () override
	destructor: no operation

virtual ROOT::Math::IntegrationMultiDim::Type	Type () const
	return type of integrator

Public Member Functions inherited from ROOT::Math::VirtualIntegrator
virtual	~VirtualIntegrator ()
	destructor: no operation

Protected Member Functions
double	DoIntegral (const double xmin, const double xmax, bool absVal=false)

Private Attributes
double	fAbsTol
	absolute tolerance

unsigned int	fDim
	dimensionality of integrand

double	fError
	integration error

const IMultiGenFunction *	fFun

unsigned int	fMaxPts
	maximum number of function evaluation requested

unsigned int	fMinPts
	minimum number of function evaluation requested

int	fNEval
	number of function evaluation

double	fRelError
	Relative error.

double	fRelTol
	relative tolerance

double	fResult
	last integration result

unsigned int	fSize
	max size of working array (explode with dimension)

int	fStatus
	status of algorithm (error if not zero)

#include <Math/AdaptiveIntegratorMultiDim.h>

Inheritance diagram for ROOT::Math::AdaptiveIntegratorMultiDim:

[legend]

Constructor & Destructor Documentation

◆ AdaptiveIntegratorMultiDim() [1/2]

ROOT::Math::AdaptiveIntegratorMultiDim::AdaptiveIntegratorMultiDim	(	double	absTol = `0.0`,
		double	relTol = `1E-9`,
		unsigned int	maxpts = `100000`,
		unsigned int	size = `0`
	)

explicit

Construct given optionally tolerance (absolute and relative), maximum number of function evaluation (maxpts) and size of the working array.

The integration will stop when the absolute error is less than the absolute tolerance OR when the relative error is less than the relative tolerance. The absolute tolerance by default is not used (it is equal to zero). The size of working array represents the number of sub-division used for calculating the integral. Higher the dimension, larger sizes are required for getting the same accuracy. The size must be larger than \( (2n + 3) (1 + maxpts/(2^n + 2n(n + 1) + 1))/2) \). For smaller value passed, the minimum allowed will be used

Definition at line 17 of file AdaptiveIntegratorMultiDim.cxx.

◆ AdaptiveIntegratorMultiDim() [2/2]

ROOT::Math::AdaptiveIntegratorMultiDim::AdaptiveIntegratorMultiDim	(	const IMultiGenFunction &	f,
		double	absTol = `0.0`,
		double	relTol = `1E-9`,
		unsigned int	maxcall = `100000`,
		unsigned int	size = `0`
	)

explicit

Construct with a reference to the integrand function and given optionally tolerance (absolute and relative), maximum number of function evaluation (maxpts) and size of the working array.

Definition at line 37 of file AdaptiveIntegratorMultiDim.cxx.

◆ ~AdaptiveIntegratorMultiDim()

ROOT::Math::AdaptiveIntegratorMultiDim::~AdaptiveIntegratorMultiDim ( )

inlineoverride

destructor (no operations)

Definition at line 113 of file AdaptiveIntegratorMultiDim.h.

Member Function Documentation

◆ DoIntegral()

double ROOT::Math::AdaptiveIntegratorMultiDim::DoIntegral	(	const double *	xmin,
		const double *	xmax,
		bool	absVal = `false`
	)

protected

Definition at line 76 of file AdaptiveIntegratorMultiDim.cxx.

◆ Error()

double ROOT::Math::AdaptiveIntegratorMultiDim::Error ( ) const

inlineoverridevirtual

return integration error

Implements ROOT::Math::VirtualIntegrator.

Definition at line 134 of file AdaptiveIntegratorMultiDim.h.

◆ Integral() [1/2]

double ROOT::Math::AdaptiveIntegratorMultiDim::Integral	(	const double *	xmin,
		const double *	xmax
	)

inlineoverridevirtual

evaluate the integral with the previously given function between xmin[] and xmax[]

Implements ROOT::Math::VirtualIntegratorMultiDim.

Definition at line 119 of file AdaptiveIntegratorMultiDim.h.

◆ Integral() [2/2]

double ROOT::Math::AdaptiveIntegratorMultiDim::Integral	(	const IMultiGenFunction &	f,
		const double *	xmin,
		const double *	xmax
	)

evaluate the integral passing a new function

Definition at line 386 of file AdaptiveIntegratorMultiDim.cxx.

◆ NEval()

int ROOT::Math::AdaptiveIntegratorMultiDim::NEval ( ) const

inlineoverridevirtual

return number of function evaluations in calculating the integral

Reimplemented from ROOT::Math::VirtualIntegrator.

Definition at line 152 of file AdaptiveIntegratorMultiDim.h.

◆ Options()

ROOT::Math::IntegratorMultiDimOptions ROOT::Math::AdaptiveIntegratorMultiDim::Options ( ) const

overridevirtual

get the option used for the integration

Implements ROOT::Math::VirtualIntegratorMultiDim.

Definition at line 394 of file AdaptiveIntegratorMultiDim.cxx.

◆ RelError()

double ROOT::Math::AdaptiveIntegratorMultiDim::RelError ( ) const

inline

return relative error

Definition at line 137 of file AdaptiveIntegratorMultiDim.h.

◆ Result()

double ROOT::Math::AdaptiveIntegratorMultiDim::Result ( ) const

inlineoverridevirtual

return result of integration

Implements ROOT::Math::VirtualIntegrator.

Definition at line 131 of file AdaptiveIntegratorMultiDim.h.

◆ SetAbsTolerance()

void ROOT::Math::AdaptiveIntegratorMultiDim::SetAbsTolerance ( double absTol )

overridevirtual

set absolute tolerance

Implements ROOT::Math::VirtualIntegrator.

Definition at line 73 of file AdaptiveIntegratorMultiDim.cxx.

◆ SetFunction()

void ROOT::Math::AdaptiveIntegratorMultiDim::SetFunction ( const IMultiGenFunction & f )

overridevirtual

set the integration function (must implement multi-dim function interface: IBaseFunctionMultiDim)

Implements ROOT::Math::VirtualIntegratorMultiDim.

Definition at line 63 of file AdaptiveIntegratorMultiDim.cxx.

◆ SetMaxPts()

void ROOT::Math::AdaptiveIntegratorMultiDim::SetMaxPts ( unsigned int n )

inline

set max points

Definition at line 167 of file AdaptiveIntegratorMultiDim.h.

◆ SetMinPts()

void ROOT::Math::AdaptiveIntegratorMultiDim::SetMinPts ( unsigned int n )

inline

set min points

Definition at line 164 of file AdaptiveIntegratorMultiDim.h.

◆ SetOptions()

void ROOT::Math::AdaptiveIntegratorMultiDim::SetOptions ( const ROOT::Math::IntegratorMultiDimOptions & opt )

overridevirtual

set the options

Reimplemented from ROOT::Math::VirtualIntegratorMultiDim.

Definition at line 405 of file AdaptiveIntegratorMultiDim.cxx.

◆ SetRelTolerance()

void ROOT::Math::AdaptiveIntegratorMultiDim::SetRelTolerance ( double relTol )

overridevirtual

set relative tolerance

Implements ROOT::Math::VirtualIntegrator.

Definition at line 70 of file AdaptiveIntegratorMultiDim.cxx.

◆ SetSize()

void ROOT::Math::AdaptiveIntegratorMultiDim::SetSize ( unsigned int size )

inline

set workspace size

Definition at line 161 of file AdaptiveIntegratorMultiDim.h.

◆ Status()

int ROOT::Math::AdaptiveIntegratorMultiDim::Status ( ) const

inlineoverridevirtual

return status of integration

status = 0 successful integration
status = 1 MAXPTS is too small for the specified accuracy EPS. The result contain the values obtainable for the specified value of MAXPTS.
status = 2 size is too small for the specified number MAXPTS of function evaluations.
status = 3 wrong dimension , N<2 or N > 15. Returned result and error are zero

Implements ROOT::Math::VirtualIntegrator.

Definition at line 149 of file AdaptiveIntegratorMultiDim.h.