User class for performing function integration.

It will use the Gauss Method for function integration in a given interval. This class is implemented from TF1::Integral().

Definition at line 43 of file GaussIntegrator.h.

Public Member Functions
virtual	~GaussIntegrator ()
	Destructor. More...

	GaussIntegrator (double absTol=-1, double relTol=-1)
	Default Constructor. More...

void	AbsValue (bool flag)
	Static function: set the fgAbsValue flag. More...

virtual void	SetRelTolerance (double eps)
	Set the desired relative Error. More...

virtual void	SetAbsTolerance (double eps)
	This method is not implemented. More...

double	Result () const
	Returns the result of the last Integral calculation. More...

double	Error () const
	Return the estimate of the absolute Error of the last Integral calculation. More...

int	Status () const
	return the status of the last integration - 0 in case of success More...

double	Integral (double a, double b)
	Returns Integral of function between a and b. More...

double	Integral ()
	Returns Integral of function on an infinite interval. More...

double	IntegralUp (double a)
	Returns Integral of function on an upper semi-infinite interval. More...

double	IntegralLow (double b)
	Returns Integral of function on a lower semi-infinite interval. More...

void	SetFunction (const IGenFunction &)
	Set integration function (flag control if function must be copied inside). More...

double	Integral (const std::vector< double > &pts)
	This method is not implemented. More...

double	IntegralCauchy (double a, double b, double c)
	This method is not implemented. More...

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

virtual void	SetOptions (const ROOT::Math::IntegratorOneDimOptions &opt)
	set the options (should be re-implemented by derived classes -if more options than tolerance exist More...

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

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

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

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

Protected Attributes
double	fEpsRel

double	fEpsAbs

bool	fUsedOnce

double	fLastResult

double	fLastError

const IGenFunction *	fFunction

Static Protected Attributes
static bool	fgAbsValue = false

Private Member Functions
virtual double	DoIntegral (double a, double b, const IGenFunction *func)
	Integration surrugate method. More...

#include <Math/GaussIntegrator.h>

Inheritance diagram for ROOT::Math::GaussIntegrator:

Collaboration diagram for ROOT::Math::GaussIntegrator:

Constructor & Destructor Documentation

§ ~GaussIntegrator()

ROOT::Math::GaussIntegrator::~GaussIntegrator ( )

virtual

Destructor.

Definition at line 40 of file GaussIntegrator.cxx.

§ GaussIntegrator()

ROOT::Math::GaussIntegrator::GaussIntegrator	(	double	absTol = `-1`,
		double	relTol = `-1`
	)

Default Constructor.

If the tolerance are not given, use default values specified in ROOT::Math::IntegratorOneDimOptions

Definition at line 21 of file GaussIntegrator.cxx.

Member Function Documentation

§ AbsValue()

void ROOT::Math::GaussIntegrator::AbsValue ( bool flag )

Static function: set the fgAbsValue flag.

By default TF1::Integral uses the original function value to compute the integral However, TF1::Moment, CentralMoment require to compute the integral using the absolute value of the function.

Definition at line 45 of file GaussIntegrator.cxx.

§ DoIntegral()

double ROOT::Math::GaussIntegrator::DoIntegral	(	double	a,
		double	b,
		const IGenFunction *	func
	)

privatevirtual

Integration surrugate method.

Return integral of passed function in interval [a,b] Derived class (like GaussLegendreIntegrator) can re-implement this method to modify to use an improved algorithm

Reimplemented in ROOT::Math::GaussLegendreIntegrator.

Definition at line 67 of file GaussIntegrator.cxx.

Referenced by Integral(), IntegralLow(), IntegralUp(), and SetAbsTolerance().

§ Error()

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

virtual

Return the estimate of the absolute Error of the last Integral calculation.

Implements ROOT::Math::VirtualIntegrator.

Definition at line 172 of file GaussIntegrator.cxx.

Referenced by TF1::IntegralOneDim(), and SetAbsTolerance().

§ Integral() [1/3]

double ROOT::Math::GaussIntegrator::Integral	(	double	a,
		double	b
	)

virtual

Returns Integral of function between a and b.

Based on original CERNLIB routine DGAUSS by Sigfried Kolbig converted to C++ by Rene Brun

This function computes, to an attempted specified accuracy, the value of the integral.

Method: For any interval [a,b] we define g8(a,b) and g16(a,b) to be the 8-point and 16-point Gaussian quadrature approximations to

\[ I = \int^{b}_{a} f(x)dx \]

and define

\[ r(a,b) = \frac{\left|g_{16}(a,b)-g_{8}(a,b)\right|}{1+\left|g_{16}(a,b)\right|} \]

Then,

\[ G = \sum_{i=1}^{k}g_{16}(x_{i-1},x_{i}) \]

where, starting with \(x_{0} = A\) and finishing with \(x_{k} = B\), the subdivision points \(x_{i}(i=1,2,...)\) are given by

\[ x_{i} = x_{i-1} + \lambda(B-x_{i-1}) \]

\(\lambda\) is equal to the first member of the sequence 1,1/2,1/4,... for which \(r(x_{i-1}, x_{i}) < EPS\). If, at any stage in the process of subdivision, the ratio

\[ q = \left|\frac{x_{i}-x_{i-1}}{B-A}\right| \]

is so small that 1+0.005q is indistinguishable from 1 to machine accuracy, an error exit occurs with the function value set equal to zero.

Accuracy: The user provides absolute and relative error bounds (epsrel and epsabs) and the algorithm will stop when the estimated error is less than the epsabs OR is less than |I| * epsrel. Unless there is severe cancellation of positive and negative values of f(x) over the interval [A,B], the relative error may be considered as specifying a bound on the relative error of I in the case |I|>1, and a bound on the absolute error in the case |I|<1. More precisely, if k is the number of sub-intervals contributing to the approximation (see Method), and if

\[ I_{abs} = \int^{B}_{A} \left|f(x)\right|dx \]

then the relation

\[ \frac{\left|G-I\right|}{I_{abs}+k} < EPS \]

will nearly always be true, provided the routine terminates without printing an error message. For functions f having no singularities in the closed interval [A,B] the accuracy will usually be much higher than this.

Error handling: The requested accuracy cannot be obtained (see Method). The function value is set equal to zero.

Note 1: Values of the function f(x) at the interval end-points A and B are not required. The subprogram may therefore be used when these values are undefined

Implements ROOT::Math::VirtualIntegratorOneDim.

Definition at line 48 of file GaussIntegrator.cxx.

Referenced by TF1::CentralMoment(), TF1::IntegralFast(), TF1::IntegralOneDim(), and TF1::Moment().

§ Integral() [2/3]

double ROOT::Math::GaussIntegrator::Integral ( )

virtual

Returns Integral of function on an infinite interval.

This function computes, to an attempted specified accuracy, the value of the integral:

\[ I = \int^{\infty}_{-\infty} f(x)dx \]

Usage: In any arithmetic expression, this function has the approximate value of the integral I.

The integral is mapped onto [0,1] using a transformation then integral computation is surrogated to DoIntegral.

Implements ROOT::Math::VirtualIntegratorOneDim.

Definition at line 52 of file GaussIntegrator.cxx.

Referenced by SetAbsTolerance().

§ Integral() [3/3]

double ROOT::Math::GaussIntegrator::Integral ( const std::vector< double > & pts )

virtual

This method is not implemented.

Implements ROOT::Math::VirtualIntegratorOneDim.

Definition at line 186 of file GaussIntegrator.cxx.

§ IntegralCauchy()

double ROOT::Math::GaussIntegrator::IntegralCauchy	(	double	a,
		double	b,
		double	c
	)

virtual

This method is not implemented.

Implements ROOT::Math::VirtualIntegratorOneDim.

Definition at line 193 of file GaussIntegrator.cxx.

Referenced by SetAbsTolerance().

§ IntegralLow()

double ROOT::Math::GaussIntegrator::IntegralLow ( double b )

virtual

Returns Integral of function on a lower semi-infinite interval.

This function computes, to an attempted specified accuracy, the value of the integral:

\[ I = \int^{B}_{-\infty} f(x)dx \]

Usage: In any arithmetic expression, this function has the approximate value of the integral I.

B: upper end-point of integration interval.

The integral is mapped onto [0,1] using a transformation then integral computation is surrogated to DoIntegral.

Implements ROOT::Math::VirtualIntegratorOneDim.

Definition at line 62 of file GaussIntegrator.cxx.

Referenced by TF1::IntegralOneDim(), and SetAbsTolerance().

§ IntegralUp()

double ROOT::Math::GaussIntegrator::IntegralUp ( double a )

virtual

Returns Integral of function on an upper semi-infinite interval.

This function computes, to an attempted specified accuracy, the value of the integral:

\[ I = \int^{\infty}_{A} f(x)dx \]

Usage: In any arithmetic expression, this function has the approximate value of the integral I.

A: lower end-point of integration interval.

The integral is mapped onto [0,1] using a transformation then integral computation is surrogated to DoIntegral.

Implements ROOT::Math::VirtualIntegratorOneDim.

Definition at line 57 of file GaussIntegrator.cxx.

Referenced by TF1::IntegralOneDim(), and SetAbsTolerance().

§ Options()

ROOT::Math::IntegratorOneDimOptions ROOT::Math::GaussIntegrator::Options ( ) const

virtual

get the option used for the integration

Implements ROOT::Math::VirtualIntegratorOneDim.

Reimplemented in ROOT::Math::GaussLegendreIntegrator.

Definition at line 206 of file GaussIntegrator.cxx.

Referenced by SetAbsTolerance().

§ Result()

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

virtual

Returns the result of the last Integral calculation.

Implements ROOT::Math::VirtualIntegrator.

Definition at line 162 of file GaussIntegrator.cxx.

Referenced by SetAbsTolerance().

§ SetAbsTolerance()

virtual void ROOT::Math::GaussIntegrator::SetAbsTolerance ( double eps )

inlinevirtual

This method is not implemented.

Implements ROOT::Math::VirtualIntegrator.

Reimplemented in ROOT::Math::GaussLegendreIntegrator.

Definition at line 71 of file GaussIntegrator.h.

Referenced by SetOptions().

§ SetFunction()

void ROOT::Math::GaussIntegrator::SetFunction ( const IGenFunction & function )

virtual

Set integration function (flag control if function must be copied inside).

@param f Function to be used in the calculations.

Implements ROOT::Math::VirtualIntegratorOneDim.

Definition at line 178 of file GaussIntegrator.cxx.

Referenced by TF1::CentralMoment(), TF1::IntegralFast(), TF1::IntegralOneDim(), TF1::Moment(), and SetAbsTolerance().

§ SetOptions()

void ROOT::Math::GaussIntegrator::SetOptions ( const ROOT::Math::IntegratorOneDimOptions & opt )

virtual

set the options (should be re-implemented by derived classes -if more options than tolerance exist

Reimplemented from ROOT::Math::VirtualIntegratorOneDim.

Reimplemented in ROOT::Math::GaussLegendreIntegrator.

Definition at line 200 of file GaussIntegrator.cxx.

Referenced by SetAbsTolerance().

§ SetRelTolerance()

virtual void ROOT::Math::GaussIntegrator::SetRelTolerance ( double eps )

inlinevirtual

Set the desired relative Error.

Implements ROOT::Math::VirtualIntegrator.

Reimplemented in ROOT::Math::GaussLegendreIntegrator.

Definition at line 68 of file GaussIntegrator.h.

Referenced by TF1::CentralMoment(), TF1::Moment(), and SetOptions().

§ Status()

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

virtual

return the status of the last integration - 0 in case of success

Implements ROOT::Math::VirtualIntegrator.

Definition at line 175 of file GaussIntegrator.cxx.

Referenced by TF1::IntegralOneDim(), and SetAbsTolerance().

Member Data Documentation

§ fEpsAbs

double ROOT::Math::GaussIntegrator::fEpsAbs

protected

Definition at line 226 of file GaussIntegrator.h.

Referenced by DoIntegral(), GaussIntegrator(), Options(), and SetAbsTolerance().

§ fEpsRel

double ROOT::Math::GaussIntegrator::fEpsRel

protected

Definition at line 225 of file GaussIntegrator.h.

Referenced by ROOT::Math::GaussLegendreIntegrator::CalcGaussLegendreSamplingPoints(), DoIntegral(), GaussIntegrator(), ROOT::Math::GaussLegendreIntegrator::Options(), Options(), ROOT::Math::GaussLegendreIntegrator::SetOptions(), ROOT::Math::GaussLegendreIntegrator::SetRelTolerance(), and SetRelTolerance().

§ fFunction

const IGenFunction* ROOT::Math::GaussIntegrator::fFunction

protected

Definition at line 230 of file GaussIntegrator.h.

Referenced by DoIntegral(), GaussIntegrator(), Integral(), IntegralLow(), IntegralUp(), and SetFunction().

§ fgAbsValue

bool ROOT::Math::GaussIntegrator::fgAbsValue = false

staticprotected

Definition at line 224 of file GaussIntegrator.h.

Referenced by AbsValue(), and DoIntegral().

§ fLastError

double ROOT::Math::GaussIntegrator::fLastError

protected

Definition at line 229 of file GaussIntegrator.h.

Referenced by DoIntegral(), Error(), and GaussIntegrator().

§ fLastResult

double ROOT::Math::GaussIntegrator::fLastResult

protected

Definition at line 228 of file GaussIntegrator.h.

Referenced by ROOT::Math::GaussLegendreIntegrator::DoIntegral(), DoIntegral(), GaussIntegrator(), and Result().

§ fUsedOnce

bool ROOT::Math::GaussIntegrator::fUsedOnce

protected

Definition at line 227 of file GaussIntegrator.h.

Referenced by ROOT::Math::GaussLegendreIntegrator::DoIntegral(), DoIntegral(), GaussIntegrator(), Result(), SetFunction(), and Status().

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

math/mathcore/inc/Math/GaussIntegrator.h
math/mathcore/src/GaussIntegrator.cxx

Public Member Functions

Protected Attributes

Static Protected Attributes

Private Member Functions

Constructor & Destructor Documentation

§ ~GaussIntegrator()

§ GaussIntegrator()

Member Function Documentation

§ AbsValue()

§ DoIntegral()

§ Error()

§ Integral() [1/3]

§ Integral() [2/3]

§ Integral() [3/3]

§ IntegralCauchy()

§ IntegralLow()

§ IntegralUp()

§ Options()

§ Result()

§ SetAbsTolerance()

§ SetFunction()

§ SetOptions()

§ SetRelTolerance()

§ Status()

Member Data Documentation

§ fEpsAbs

§ fEpsRel

§ fFunction

§ fgAbsValue

§ fLastError

§ fLastResult

§ fUsedOnce