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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.

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.

destructor (no operations)

double Error () const
return integration error

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

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

double RelError () const
return relative error

double Result () const
return result of integration

void SetAbsTolerance (double absTol)
set absolute tolerance

void SetFunction (const IMultiGenFunction &f)
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)
set the options

void SetRelTolerance (double relTol)
set relative tolerance

void SetSize (unsigned int size)
set workspace size

int Status () const
return status of integration Public Member Functions inherited from ROOT::Math::VirtualIntegratorMultiDim
virtual ~VirtualIntegratorMultiDim ()
destructor: no operation

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

## Protected Member Functions

double DoIntegral (const double *xmin, const double *xmax, bool absVal=false)

## Private Attributes

double fAbsTol

unsigned int fDim

double fError

const IMultiGenFunctionfFun

unsigned int fMaxPts

unsigned int fMinPts

int fNEval

double fRelError

double fRelTol

double fResult

unsigned int fSize

int fStatus

#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.

inlinevirtual

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()

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()

inlinevirtual

return number of function evaluations in calculating the integral

Reimplemented from ROOT::Math::VirtualIntegrator.

Definition at line 152 of file AdaptiveIntegratorMultiDim.h.

## ◆ Options()

virtual

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()

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()

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.

## ◆ fAbsTol

private

Definition at line 186 of file AdaptiveIntegratorMultiDim.h.

## ◆ fDim

private

Definition at line 182 of file AdaptiveIntegratorMultiDim.h.

## ◆ fError

private

Definition at line 190 of file AdaptiveIntegratorMultiDim.h.

## ◆ fFun

private

Definition at line 195 of file AdaptiveIntegratorMultiDim.h.

## ◆ fMaxPts

private

Definition at line 184 of file AdaptiveIntegratorMultiDim.h.

## ◆ fMinPts

private

Definition at line 183 of file AdaptiveIntegratorMultiDim.h.

## ◆ fNEval

private

Definition at line 192 of file AdaptiveIntegratorMultiDim.h.

## ◆ fRelError

private

Definition at line 191 of file AdaptiveIntegratorMultiDim.h.

## ◆ fRelTol

private

Definition at line 187 of file AdaptiveIntegratorMultiDim.h.

## ◆ fResult

private

Definition at line 189 of file AdaptiveIntegratorMultiDim.h.

## ◆ fSize

private

Definition at line 185 of file AdaptiveIntegratorMultiDim.h.

## ◆ fStatus 