ROOT   Reference Guide

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:

1. 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.
2. 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...

destructor (no operations) More...

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

double Integral (const double *xmin, const double *xmax) override
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 override
return number of function evaluations in calculating the integral More...

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

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

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

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

void SetFunction (const IMultiGenFunction &f) override
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) override
set the options More...

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

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

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

Public Member Functions inherited from ROOT::Math::VirtualIntegratorMultiDim
~VirtualIntegratorMultiDim () override
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 implement 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
return type of integrator More...

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

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
absolute tolerance More...

unsigned int fDim
dimensionality of integrand More...

double fError
integration error More...

const IMultiGenFunctionfFun

unsigned int fMaxPts
maximum number of function evaluation requested More...

unsigned int fMinPts
minimum number of function evaluation requested More...

int fNEval
number of function evaluation More...

double fRelError
Relative error. More...

double fRelTol
relative tolerance More...

double fResult
last integration result More...

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

int fStatus
status of algorithm (error if not zero) More...

#include <Math/AdaptiveIntegratorMultiDim.h>

[legend]

## Constructor & Destructor Documentation

 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.

 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.

inlineoverride

destructor (no operations)

Definition at line 113 of file AdaptiveIntegratorMultiDim.h.

## ◆ 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()

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()

inlineoverridevirtual

return number of function evaluations in calculating the integral

Reimplemented from ROOT::Math::VirtualIntegrator.

Definition at line 152 of file AdaptiveIntegratorMultiDim.h.

## ◆ Options()

overridevirtual

get the option used for the integration

Implements ROOT::Math::VirtualIntegratorMultiDim.

Definition at line 394 of file AdaptiveIntegratorMultiDim.cxx.

## ◆ RelError()

inline

return relative error

Definition at line 137 of file AdaptiveIntegratorMultiDim.h.

## ◆ Result()

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()

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.

## ◆ fAbsTol

private

absolute tolerance

Definition at line 186 of file AdaptiveIntegratorMultiDim.h.

## ◆ fDim

private

dimensionality of integrand

Definition at line 182 of file AdaptiveIntegratorMultiDim.h.

## ◆ fError

private

integration error

Definition at line 190 of file AdaptiveIntegratorMultiDim.h.

## ◆ fFun

private

Definition at line 195 of file AdaptiveIntegratorMultiDim.h.

## ◆ fMaxPts

private

maximum number of function evaluation requested

Definition at line 184 of file AdaptiveIntegratorMultiDim.h.

## ◆ fMinPts

private

minimum number of function evaluation requested

Definition at line 183 of file AdaptiveIntegratorMultiDim.h.

## ◆ fNEval

private

number of function evaluation

Definition at line 192 of file AdaptiveIntegratorMultiDim.h.

## ◆ fRelError

private

Relative error.

Definition at line 191 of file AdaptiveIntegratorMultiDim.h.

## ◆ fRelTol

private

relative tolerance

Definition at line 187 of file AdaptiveIntegratorMultiDim.h.

## ◆ fResult

private

last integration result

Definition at line 189 of file AdaptiveIntegratorMultiDim.h.

## ◆ fSize

private

max size of working array (explode with dimension)

Definition at line 185 of file AdaptiveIntegratorMultiDim.h.