Class Quadrature

Quadrature nodes and weights

Inheritance

Object

Quadrature

Inherited Members

Object.Equals(Object)

Object.Equals(Object, Object)

Object.GetHashCode()

Object.GetType()

Object.MemberwiseClone()

Object.ReferenceEquals(Object, Object)

Object.ToString()

Namespace: Microsoft.ML.Probabilistic.Math

Assembly: Microsoft.ML.Probabilistic.dll

Syntax

public static class Quadrature

Fields

HermiteNodesAndWeights

Hermite nodes and weights

Declaration

public static double[, ][] HermiteNodesAndWeights

Field Value

Type	Description
Double[,][]

LaguerreNodesAndWeights

Laguerre nodes and weights

Declaration

public static double[, ][] LaguerreNodesAndWeights

Field Value

Type	Description
Double[,][]

LegendreNodesAndWeights

Legendre nodes and weights

Declaration

public static double[, ][] LegendreNodesAndWeights

Field Value

Type	Description
Double[,][]

Methods

AdaptiveClenshawCurtis(Converter<Double, Double>, Double, Int32, Double)

Integrate the function f from -Infinity to Infinity

Declaration

public static double AdaptiveClenshawCurtis(Converter<double, double> f, double scale, int nodeCount, double relTol)

Parameters

Type	Name	Description
Converter<Double, Double>	f	The function to integrate
Double	scale	A positive tuning parameter. f is assumed to be negligible outside of [-scale,scale]
Int32	nodeCount	The initial number of nodes. Should be at least 2 and a power of 2.
Double	relTol	A threshold to stop subdividing

Returns

Type	Description
Double

AdaptiveExpSinh(Converter<Double, Double>, Double, Double)

Integrate the function f from 0 to +Infinity using exp-sinh quadrature.

Declaration

public static double AdaptiveExpSinh(Converter<double, double> f, double scale, double relTol)

Parameters

Type	Name	Description
Converter<Double, Double>	f	The function to integrate
Double	scale	A positive tuning parameter. f is assumed to be continuous and at least one of f(0) and cosh(arsinh(2 * ln(scale)/pi)) * scale * f(scale) to be non-zero.
Double	relTol	A threshold to stop subdividing

Returns

Type	Description
Double

AdaptivePositiveHalfAxisTrapeziumRule(Converter<Double, Double>, Double, Double, Int32)

Integrate the function f from 0 to +Infinity

Declaration

public static double AdaptivePositiveHalfAxisTrapeziumRule(Converter<double, double> f, double scale, double relTol, int maxNodes)

Parameters

Type	Name	Description
Converter<Double, Double>	f	The function to integrate. Must have at most one extremum.
Double	scale	A positive tuning parameter. f is assumed to be continuous on (0, + Infinity) and non-negligible somewhere on (0, scale]
Double	relTol	A threshold to stop subdividing
Int32	maxNodes	Another threshold to stop subdividing

Returns

Type	Description
Double

GammaNodesAndWeights(Double, Double, IList<Double>, IList<Double>)

Quadrature nodes for Gamma expectations.

Declaration

public static void GammaNodesAndWeights(double a, double b, IList<double> nodes, IList<double> logWeights)

Parameters

Type	Name	Description
Double	a
Double	b
IList<Double>	nodes	A list in which to store the nodes.
IList<Double>	logWeights	A list in which to store the weights.

Remarks

The nodes and weights lists are modified to have the property that for any function f with a fast-converging Taylor series, sum_i weights[i] f(nodes[i]) =approx int_{0..inf} f(x) Ga(x; a, b) dx where Ga(x; a, b) = x^aexp(-xb)b^(a+1)/Gamma(a+1). For example, to approximate E[xx] where x ~ Ga(2,3):

Vector nodes = new Vector(3);
Vector logWeights = new Vector(3);
Quadrature.GammaNodesAndWeights(2,3,nodes,logWeights);
double result = (exp(logWeights)*nodes*nodes).Sum();

The result is mean^2 + variance = ((a+1)^2 + (a+1))/b^2 = 4/3.

GaussianNodesAndWeights(Double, Double, IList<Double>, IList<Double>)

Quadrature nodes for Gaussian expectations.

Declaration

public static void GaussianNodesAndWeights(double mean, double variance, IList<double> nodes, IList<double> weights)

Parameters

Type	Name	Description
Double	mean
Double	variance
IList<Double>	nodes	A list in which to store the nodes.
IList<Double>	weights	A list in which to store the weights.

Remarks

The nodes and weights lists are modified to have the property that for any function f with a fast-converging Taylor series, sum_i weights[i] f(nodes[i]) =approx int_{-inf..inf} f(x) N(x; m, v) dx. If f is a polynomial of order 2n-1, then the result is exact. For example, to compute E[xx] where x ~ N(2,3):

Vector nodes = new Vector(2);
Vector weights = new Vector(2);
Quadrature.GaussianNodesAndWeights(2,3,nodes,weights);
double result = (weights*nodes*nodes).Sum();

The result is mean^2 + variance = 7.

LaguerreGammaNodesAndWeights(Double, Double, IList<Double>, IList<Double>)

Declaration

public static void LaguerreGammaNodesAndWeights(double a, double b, IList<double> nodes, IList<double> weights)

Parameters

Type	Name	Description
Double	a
Double	b
IList<Double>	nodes
IList<Double>	weights

UniformNodesAndWeights(Double, Double, IList<Double>, IList<Double>)

Quadrature nodes for integrals on [low,high].

Declaration

public static void UniformNodesAndWeights(double low, double high, IList<double> nodes, IList<double> weights)

Parameters

Type	Name	Description
Double	low	Lower limit of integration. Must be finite.
Double	high	Upper limit of integration. Must be finite.
IList<Double>	nodes	A list in which to store the nodes.
IList<Double>	weights	A list in which to store the weights.

Remarks

The nodes and weights lists are modified to have the property that for any function f with a fast-converging Taylor series, sum_i weights[i] f(nodes[i]) =approx int_{low..high} f(x) dx. If f is a polynomial of order 2*n-1, then the result is exact. For example, to compute int_{0..1} x^3 dx:

Vector nodes = new Vector(2);
Vector weights = new Vector(2);
Quadrature.UniformNodesAndWeights(0,1,nodes,weights);
double result = (weights*nodes*nodes*nodes).Sum();

The result is 1/4.