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. More...

	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. More...

virtual	~AdaptiveIntegratorMultiDim ()
	destructor (no operations) More...

double	Error () const
	return integration error More...

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

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

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

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

double	RelError () const
	return relative error More...

double	Result () const
	return result of integration More...

void	SetAbsTolerance (double absTol)
	set absolute tolerance More...

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

void	SetMaxPts (unsigned int n)
	set max points More...

void	SetMinPts (unsigned int n)
	set min points More...

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

void	SetRelTolerance (double relTol)
	set relative tolerance More...

void	SetSize (unsigned int size)
	set workspace size More...

int	Status () const
	return status of integration More...

Public Member Functions inherited from ROOT::Math::VirtualIntegratorMultiDim
virtual	~VirtualIntegratorMultiDim ()
	destructor: no operation More...

virtual double	Integral (const double , const double )=0
	evaluate multi-dim integral More...

virtual ROOT::Math::IntegratorMultiDimOptions	Options () const =0
	get the option used for the integration impelement by derived class otherwise return default ones More...

virtual void	SetFunction (const IMultiGenFunction &)=0
	setting a multi-dim function More...

virtual void	SetOptions (const ROOT::Math::IntegratorMultiDimOptions &opt)
	set the options (if needed must be re-implemented by derived classes) More...

virtual ROOT::Math::IntegrationMultiDim::Type	Type () const

Public Member Functions inherited from ROOT::Math::VirtualIntegrator
virtual	~VirtualIntegrator ()

virtual double	Error () const =0
	return the estimate of the absolute Error of the last Integral calculation More...

virtual int	NEval () const
	return number of function evaluations in calculating the integral (if integrator do not implement this function returns -1) More...

virtual double	Result () const =0
	return the Result of the last Integral calculation More...

virtual void	SetAbsTolerance (double)=0
	set the desired absolute Error More...

virtual void	SetRelTolerance (double)=0
	set the desired relative Error More...

virtual int	Status () const =0
	return the Error Status of the last Integral calculation More...

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

Private Attributes
double	fAbsTol

unsigned int	fDim

double	fError

const IMultiGenFunction *	fFun

unsigned int	fMaxPts

unsigned int	fMinPts

int	fNEval

double	fRelError

double	fRelTol

double	fResult

unsigned int	fSize

int	fStatus

#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()

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

inlinevirtual

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

inlinevirtual

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
	)

inlinevirtual

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

inlinevirtual

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

virtual

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

inlinevirtual

return result of integration

Implements ROOT::Math::VirtualIntegrator.

Definition at line 131 of file AdaptiveIntegratorMultiDim.h.

◆ SetAbsTolerance()

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

virtual

set absolute tolerance

Implements ROOT::Math::VirtualIntegrator.

Definition at line 73 of file AdaptiveIntegratorMultiDim.cxx.

◆ SetFunction()

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

virtual

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 )

virtual

set the options

Reimplemented from ROOT::Math::VirtualIntegratorMultiDim.

Definition at line 405 of file AdaptiveIntegratorMultiDim.cxx.

◆ SetRelTolerance()

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

virtual

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

inlinevirtual

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.