Numerical integration algorithms. More...

Data Structures
struct	DoubleExpConfig
	Configuration for double exponential integration. More...

struct	KronrodConfig
	Configuration for Gauss-Kronrod integration. More...

struct	MultipleConfig
	Configuration for multi-dimensional Gauss integration. More...

struct	SetResult
	Result for vector function integration. More...

Functions
template<typename Function >
IntegResult	Gauss (Function &theFunc, double theLower, double theUpper, int theNbPoints=15)
	Gauss-Legendre quadrature for definite integrals. Computes integral of f(x) from theLower to theUpper using n-point Gauss-Legendre rule.

template<typename Function >
IntegResult	GaussAdaptive (Function &theFunc, double theLower, double theUpper, const IntegConfig &theConfig=IntegConfig())
	Adaptive Gauss-Legendre integration. Recursively subdivides interval until error estimate is below tolerance.

template<typename Function >
IntegResult	GaussComposite (Function &theFunc, double theLower, double theUpper, int theNbIntervals, int theNbPoints=7)
	Composite Gauss-Legendre integration. Divides interval into subintervals and applies Gauss-Legendre to each. Simple alternative to adaptive integration.

template<typename Function >
IntegResult	KronrodRule (Function &theFunc, double theLower, double theUpper, int theNbGauss=7)
	Apply Gauss-Kronrod rule to a single interval.

template<typename Function >
IntegResult	Kronrod (Function &theFunc, double theLower, double theUpper, const KronrodConfig &theConfig=KronrodConfig())
	Gauss-Kronrod adaptive integration.

template<typename Function >
IntegResult	KronrodAuto (Function &theFunc, double theLower, double theUpper, double theTolerance=1.0e-10, int theMaxOrder=30)
	Gauss-Kronrod integration with automatic order selection.

template<typename Function >
IntegResult	KronrodSemiInfinite (Function &theFunc, double theLower, const KronrodConfig &theConfig=KronrodConfig())
	Gauss-Kronrod integration over semi-infinite interval [a, +infinity).

template<typename Function >
IntegResult	KronrodInfinite (Function &theFunc, const KronrodConfig &theConfig=KronrodConfig())
	Gauss-Kronrod integration over infinite interval (-infinity, +infinity).

template<typename Function >
IntegResult	TanhSinh (Function &theFunc, double theLower, double theUpper, const DoubleExpConfig &theConfig=DoubleExpConfig())
	Tanh-Sinh (Double Exponential) quadrature for finite interval [a,b].

template<typename Function >
IntegResult	ExpSinh (Function &theFunc, double theLower, const DoubleExpConfig &theConfig=DoubleExpConfig())
	Exp-Sinh quadrature for semi-infinite interval [a, +infinity).

template<typename Function >
IntegResult	SinhSinh (Function &theFunc, const DoubleExpConfig &theConfig=DoubleExpConfig())
	Sinh-Sinh quadrature for infinite interval (-infinity, +infinity).

template<typename Function >
IntegResult	DoubleExponential (Function &theFunc, double theLower, double theUpper, const DoubleExpConfig &theConfig=DoubleExpConfig())
	Double exponential integration with automatic interval detection.

template<typename Function >
IntegResult	TanhSinhSingular (Function &theFunc, double theLower, double theUpper, double theTolerance=1.0e-10)
	Tanh-Sinh quadrature optimized for endpoint singularities.

template<typename Function >
IntegResult	TanhSinhWithSingularity (Function &theFunc, double theLower, double theUpper, double theSingularity, const DoubleExpConfig &theConfig=DoubleExpConfig())
	Integrates a function with a known singularity at an interior point.

template<typename Func >
IntegResult	GaussMultiple (Func &theFunc, int theNVars, const math_Vector &theLower, const math_Vector &theUpper, const math_IntegerVector &theOrder, const MultipleConfig &theConfig=MultipleConfig())
	Gauss-Legendre integration of a multi-variable function.

template<typename Func >
IntegResult	GaussMultipleUniform (Func &theFunc, int theNVars, const math_Vector &theLower, const math_Vector &theUpper, int theOrder)
	Gauss-Legendre integration with uniform order for all variables.

template<typename Func >
SetResult	GaussSet (Func &theFunc, double theLower, double theUpper, int theOrder)
	Gauss-Legendre integration of a vector-valued function.

template<typename Func >
SetResult	GaussSet (Func &theFunc, const math_Vector &theLower, const math_Vector &theUpper, int theOrder)
	Gauss-Legendre integration of vector function using math_Vector bounds.

Detailed Description

Numerical integration algorithms.

Function Documentation

◆ DoubleExponential()

template<typename Function >

IntegResult MathInteg::DoubleExponential	(	Function &	theFunc,
		double	theLower,
		double	theUpper,
		const DoubleExpConfig &	theConfig = DoubleExpConfig() )

Double exponential integration with automatic interval detection.

Selects the appropriate DE quadrature based on the interval:

Finite [a,b]: TanhSinh
Semi-infinite [a, +infinity): ExpSinh
Infinite (-infinity, +infinity): SinhSinh

Template Parameters

Function type with Value(double theX, double& theF) method

Parameters

theFunc	function to integrate
theLower	lower bound (use -HUGE_VAL for -infinity)
theUpper	upper bound (use HUGE_VAL for +infinity)
theConfig	integration configuration

Returns: integration result

◆ ExpSinh()

template<typename Function >

IntegResult MathInteg::ExpSinh	(	Function &	theFunc,
		double	theLower,
		const DoubleExpConfig &	theConfig = DoubleExpConfig() )

Exp-Sinh quadrature for semi-infinite interval [a, +infinity).

The transformation maps [0, +infinity) to (-infinity, +infinity): x = a + exp(pi/2 * sinh(t))

Template Parameters

Function type with Value(double theX, double& theF) method

Parameters

theFunc	function to integrate
theLower	lower bound a
theConfig	integration configuration

Returns: integration result

◆ Gauss()

template<typename Function >

IntegResult MathInteg::Gauss	(	Function &	theFunc,
		double	theLower,
		double	theUpper,
		int	theNbPoints = 15 )

Gauss-Legendre quadrature for definite integrals. Computes integral of f(x) from theLower to theUpper using n-point Gauss-Legendre rule.

Algorithm:

Transform interval [theLower, theUpper] to [-1, 1]
Evaluate f at Gauss-Legendre points
Sum weighted function values

Exact for polynomials of degree up to 2n-1.

Template Parameters

Function type with Value(double theX, double& theF) method

Parameters

theFunc	function to integrate
theLower	lower integration bound
theUpper	upper integration bound
theNbPoints	number of quadrature points (>= 1)

Returns: result containing integral value

◆ GaussAdaptive()

template<typename Function >

IntegResult MathInteg::GaussAdaptive	(	Function &	theFunc,
		double	theLower,
		double	theUpper,
		const IntegConfig &	theConfig = IntegConfig() )

Adaptive Gauss-Legendre integration. Recursively subdivides interval until error estimate is below tolerance.

Algorithm:

Compute integral using n and 2n points
Estimate error as difference between the two
If error > tolerance, subdivide and recurse

Template Parameters

Function type with Value(double theX, double& theF) method

Parameters

theFunc	function to integrate
theLower	lower integration bound
theUpper	upper integration bound
theConfig	integration configuration

Returns: result containing integral value and error estimate

◆ GaussComposite()

template<typename Function >

IntegResult MathInteg::GaussComposite	(	Function &	theFunc,
		double	theLower,
		double	theUpper,
		int	theNbIntervals,
		int	theNbPoints = 7 )

Composite Gauss-Legendre integration. Divides interval into subintervals and applies Gauss-Legendre to each. Simple alternative to adaptive integration.

Template Parameters

Function type with Value(double theX, double& theF) method

Parameters

theFunc	function to integrate
theLower	lower integration bound
theUpper	upper integration bound
theNbIntervals	number of subintervals
theNbPoints	Gauss points per interval (>= 1)

Returns: result containing integral value

◆ GaussMultiple()

template<typename Func >

IntegResult MathInteg::GaussMultiple	(	Func &	theFunc,
		int	theNVars,
		const math_Vector &	theLower,
		const math_Vector &	theUpper,
		const math_IntegerVector &	theOrder,
		const MultipleConfig &	theConfig = MultipleConfig() )

Gauss-Legendre integration of a multi-variable function.

Computes the N-dimensional integral using tensor product of 1D Gauss-Legendre quadrature: I = integral_{Lower}^{Upper} F(x1,...,xN) dx1...dxN

Uses recursive summation over all Gauss points in each dimension.

Template Parameters

Func	function type with bool Value(const math_Vector&, double&)

Parameters

theFunc	N-dimensional function to integrate
theNVars	number of variables
theLower	lower bounds for each variable
theUpper	upper bounds for each variable
theOrder	integration order for each variable (max 61)
theConfig	optional configuration

Returns: IntegResult containing the integral value

◆ GaussMultipleUniform()

template<typename Func >

IntegResult MathInteg::GaussMultipleUniform	(	Func &	theFunc,
		int	theNVars,
		const math_Vector &	theLower,
		const math_Vector &	theUpper,
		int	theOrder )

Gauss-Legendre integration with uniform order for all variables.

Template Parameters

Func	function type with bool Value(const math_Vector&, double&)

Parameters

theFunc	N-dimensional function to integrate
theNVars	number of variables
theLower	lower bounds for each variable
theUpper	upper bounds for each variable
theOrder	integration order for all variables

Returns: IntegResult containing the integral value

◆ GaussSet() [1/2]

template<typename Func >

SetResult MathInteg::GaussSet	(	Func &	theFunc,
		const math_Vector &	theLower,
		const math_Vector &	theUpper,
		int	theOrder )

Gauss-Legendre integration of vector function using math_Vector bounds.

Convenience overload that extracts scalar bounds from math_Vector.

Template Parameters

Func	function type

Parameters

theFunc	vector-valued function
theLower	lower bound vector (only first element used)
theUpper	upper bound vector (only first element used)
theOrder	integration order

Returns: SetResult containing vector of integrals

◆ GaussSet() [2/2]

template<typename Func >

SetResult MathInteg::GaussSet	(	Func &	theFunc,
		double	theLower,
		double	theUpper,
		int	theOrder )

Gauss-Legendre integration of a vector-valued function.

Integrates F: R -> R^N over interval [Lower, Upper]: Result[i] = integral_{Lower}^{Upper} F_i(x) dx

Uses Gauss-Legendre quadrature applied to each component.

Note: Only 1D input is supported (M=1). Multi-dimensional input is not implemented in the legacy API.

Template Parameters

Func	function type with: int NbEquations() - number of output components bool Value(const math_Vector& theX, math_Vector& theF)

Parameters

theFunc	vector-valued function to integrate
theLower	lower bound
theUpper	upper bound
theOrder	integration order (max 61)

Returns: SetResult containing vector of integrals

◆ Kronrod()

template<typename Function >

IntegResult MathInteg::Kronrod	(	Function &	theFunc,
		double	theLower,
		double	theUpper,
		const KronrodConfig &	theConfig = KronrodConfig() )

Gauss-Kronrod adaptive integration.

Uses adaptive bisection to achieve the requested tolerance. At each subdivision, the interval with the largest error estimate is bisected, and both halves are reintegrated.

Template Parameters

Function type with Value(double theX, double& theF) method

Parameters

theFunc	function to integrate
theLower	lower integration bound
theUpper	upper integration bound
theConfig	integration configuration

Returns: integration result with error estimate

◆ KronrodAuto()

template<typename Function >

IntegResult MathInteg::KronrodAuto	(	Function &	theFunc,
		double	theLower,
		double	theUpper,
		double	theTolerance = 1.0e-10,
		int	theMaxOrder = 30 )

Gauss-Kronrod integration with automatic order selection.

Starts with a low-order rule and increases the order until the tolerance is met or the maximum order is reached.

Template Parameters

Function type with Value(double theX, double& theF) method

Parameters

theFunc	function to integrate
theLower	lower integration bound
theUpper	upper integration bound
theTolerance	relative tolerance
theMaxOrder	maximum Gauss order to try

Returns: integration result

◆ KronrodInfinite()

template<typename Function >

IntegResult MathInteg::KronrodInfinite	(	Function &	theFunc,
		const KronrodConfig &	theConfig = KronrodConfig() )

Gauss-Kronrod integration over infinite interval (-infinity, +infinity).

Uses the substitution x = t / (1 - t^2) to map (-infinity, +infinity) to (-1, 1). The function must decay sufficiently fast at infinity.

Template Parameters

Function type with Value(double theX, double& theF) method

Parameters

theFunc	function to integrate
theConfig	integration configuration

Returns: integration result

◆ KronrodRule()

template<typename Function >

IntegResult MathInteg::KronrodRule	(	Function &	theFunc,
		double	theLower,
		double	theUpper,
		int	theNbGauss = 7 )

Apply Gauss-Kronrod rule to a single interval.

The Gauss-Kronrod rule uses n Gauss points embedded in 2n+1 Kronrod points. The difference between the Gauss and Kronrod estimates provides an error estimate without additional function evaluations.

Template Parameters

Function type with Value(double theX, double& theF) method

Parameters

theFunc	function to integrate
theLower	lower integration bound
theUpper	upper integration bound
theNbGauss	number of Gauss points (determines rule order)

Returns: integration result with error estimate

◆ KronrodSemiInfinite()

template<typename Function >

IntegResult MathInteg::KronrodSemiInfinite	(	Function &	theFunc,
		double	theLower,
		const KronrodConfig &	theConfig = KronrodConfig() )

Gauss-Kronrod integration over semi-infinite interval [a, +infinity).

Uses the substitution x = a + t / (1 - t) to map [a, +infinity) to [0, 1).

Template Parameters

Function type with Value(double theX, double& theF) method

Parameters

theFunc	function to integrate
theLower	lower bound a
theConfig	integration configuration

Returns: integration result

◆ SinhSinh()

template<typename Function >

IntegResult MathInteg::SinhSinh	(	Function &	theFunc,
		const DoubleExpConfig &	theConfig = DoubleExpConfig() )

Sinh-Sinh quadrature for infinite interval (-infinity, +infinity).

The transformation maps (-infinity, +infinity) to (-infinity, +infinity): x = sinh(pi/2 * sinh(t))

Template Parameters

Function type with Value(double theX, double& theF) method

Parameters

theFunc	function to integrate
theConfig	integration configuration

Returns: integration result

◆ TanhSinh()

template<typename Function >

IntegResult MathInteg::TanhSinh	(	Function &	theFunc,
		double	theLower,
		double	theUpper,
		const DoubleExpConfig &	theConfig = DoubleExpConfig() )

Tanh-Sinh (Double Exponential) quadrature for finite interval [a,b].

The double exponential transformation maps [a,b] to (-inf, +inf): x = (b+a)/2 + (b-a)/2 * tanh(pi/2 * sinh(t))

The integrand is multiplied by the Jacobian: dx/dt = (b-a)/2 * (pi/2) * cosh(t) / cosh^2(pi/2 * sinh(t))

Properties:

Excellent for functions with endpoint singularities
Exponential convergence for analytic functions
Self-adaptive through level refinement
Handles algebraic and logarithmic singularities

Template Parameters

Function type with Value(double theX, double& theF) method

Parameters

theFunc	function to integrate
theLower	lower integration bound
theUpper	upper integration bound
theConfig	integration configuration

Returns: integration result with error estimate

◆ TanhSinhSingular()

template<typename Function >

IntegResult MathInteg::TanhSinhSingular	(	Function &	theFunc,
		double	theLower,
		double	theUpper,
		double	theTolerance = 1.0e-10 )

Tanh-Sinh quadrature optimized for endpoint singularities.

This variant uses precomputed nodes and weights for better performance when integrating functions with algebraic singularities at the endpoints.

Template Parameters

Function type with Value(double theX, double& theF) method

Parameters

theFunc	function to integrate
theLower	lower bound
theUpper	upper bound
theTolerance	relative tolerance

Returns: integration result

◆ TanhSinhWithSingularity()

template<typename Function >

IntegResult MathInteg::TanhSinhWithSingularity	(	Function &	theFunc,
		double	theLower,
		double	theUpper,
		double	theSingularity,
		const DoubleExpConfig &	theConfig = DoubleExpConfig() )

Integrates a function with a known singularity at an interior point.

Splits the interval at the singularity and integrates each part.

Template Parameters

Function type with Value(double theX, double& theF) method

Parameters

theFunc	function to integrate
theLower	lower bound
theUpper	upper bound
theSingularity	location of interior singularity
theConfig	integration configuration

Returns: combined integration result

Data Structures

Functions

Detailed Description

Function Documentation

◆ DoubleExponential()

◆ ExpSinh()

◆ Gauss()

◆ GaussAdaptive()

◆ GaussComposite()

◆ GaussMultiple()

◆ GaussMultipleUniform()

◆ GaussSet() [1/2]

◆ GaussSet() [2/2]

◆ Kronrod()

◆ KronrodAuto()

◆ KronrodInfinite()

◆ KronrodRule()

◆ KronrodSemiInfinite()

◆ SinhSinh()

◆ TanhSinh()

◆ TanhSinhSingular()

◆ TanhSinhWithSingularity()