Measures and Orthogonal Polynomials

Beta01Measure(a, b)

Beta probability measure on (0, 1).

Arguments

• a: first positive beta shape parameter.
• b: second positive beta shape parameter.

Fields

• w, dom, symmetric: measure data.
• ashapeParameter, bshapeParameter: validated beta parameters.

Beta01OrthoPoly(deg, shape_a, shape_b; Nrec = deg + 1,
                addQuadrature = true)

Construct the basis orthogonal to a beta probability measure on (0, 1).

Arguments

• deg: nonnegative maximum degree.
• shape_a, shape_b: positive beta shape parameters.

Keywords

• Nrec: recurrence coefficient count.
• addQuadrature: whether to attach Gaussian quadrature.

Fields

• deg: maximum represented degree.
• α, β: monic recurrence coefficients.
• measure: beta measure on (0, 1).
• quad: attached quadrature rule or EmptyQuad.

EmptyQuad()

Represent the absence of an attached quadrature rule.

EmptyQuad is returned when an orthogonal basis is constructed with addQuadrature = false. nw returns an empty 0 x 2 matrix for it; quadrature-dependent operations should reject it.

GammaMeasure(shape, rate = 1)

Gamma probability measure on (0, Inf) with unit rate.

Arguments

• shape: positive gamma shape.
• rate: currently required to equal 1.

Fields

• w, dom, symmetric: measure data.
• shapeParameter, rateParameter: validated gamma parameters.

GammaOrthoPoly(deg, shape, rate; Nrec = deg + 1, addQuadrature = true)

Construct the basis orthogonal to a unit-rate gamma probability measure.

Arguments

• deg: nonnegative maximum degree.
• shape: positive gamma shape.
• rate: currently required to equal 1.

Keywords

• Nrec: recurrence coefficient count.
• addQuadrature: whether to attach Gaussian quadrature.

Fields

• deg: maximum represented degree.
• α, β: monic recurrence coefficients.
• measure: unit-rate gamma measure.
• quad: attached quadrature rule or EmptyQuad.

GaussMeasure()

Standard Gaussian probability measure on the real line.

Fields

• w: standard-normal density.
• dom: (-Inf, Inf).
• symmetric: true.

GaussOrthoPoly(deg; Nrec = deg + 1, addQuadrature = true)

Construct the basis orthogonal to the standard Gaussian probability measure.

Arguments

• deg: nonnegative maximum degree.

Keywords

• Nrec: recurrence coefficient count.
• addQuadrature: whether to attach Gaussian quadrature.

Fields

• deg: maximum represented degree.
• α, β: monic recurrence coefficients.
• measure: standard Gaussian measure.
• quad: attached quadrature rule or EmptyQuad.

HermiteMeasure()

Canonical Hermite measure with weight exp(-t^2) on the real line.

Fields

• w: Hermite weight.
• dom: (-Inf, Inf).
• symmetric: true.

HermiteOrthoPoly(deg; Nrec = deg + 1, addQuadrature = true)

Construct the canonical Hermite basis.

Arguments

• deg: nonnegative maximum degree.

Keywords

• Nrec: recurrence coefficient count.
• addQuadrature: whether to attach Gaussian quadrature.

Fields

• deg: maximum represented degree.
• α, β: monic recurrence coefficients.
• measure: canonical Hermite measure.
• quad: attached quadrature rule or EmptyQuad.

JacobiMeasure(shape_a, shape_b)

Canonical Jacobi measure on (-1, 1).

Arguments

• shape_a: exponent of (1 - t); must exceed -1.
• shape_b: exponent of (1 + t); must exceed -1.

Fields

• w, dom, symmetric: measure data.
• ashapeParameter, bshapeParameter: validated shape parameters.

JacobiOrthoPoly(deg, shape_a, shape_b; Nrec = deg + 1, addQuadrature = true)

Construct the canonical Jacobi basis.

Arguments

• deg: nonnegative maximum degree.
• shape_a, shape_b: Jacobi parameters, each greater than -1.

Keywords

• Nrec: recurrence coefficient count.
• addQuadrature: whether to attach Gaussian quadrature.

Fields

The basis follows the AbstractOrthoPoly contract and stores a JacobiMeasure.

LaguerreMeasure()

Canonical Laguerre measure with weight exp(-t) on (0, Inf).

Fields

• w: Laguerre weight.
• dom: (0, Inf).
• symmetric: false.

LaguerreOrthoPoly(deg; Nrec = deg + 1, addQuadrature = true)

Construct the canonical Laguerre basis.

Arguments

• deg: nonnegative maximum degree.

Keywords

• Nrec: recurrence coefficient count.
• addQuadrature: whether to attach Gaussian quadrature.

Fields

• deg: maximum represented degree.
• α, β: monic recurrence coefficients.
• measure: canonical Laguerre measure.
• quad: attached quadrature rule or EmptyQuad.

LegendreMeasure()

Canonical Legendre measure with unit weight on (-1, 1).

Fields

• w: the Legendre weight function.
• dom: (-1.0, 1.0).
• symmetric: true.

Examples

julia> using PolyChaos

julia> LegendreMeasure().dom
(-1.0, 1.0)

LegendreOrthoPoly(deg; Nrec = deg + 1, addQuadrature = true)

Construct the canonical Legendre basis through degree deg.

Arguments

• deg: nonnegative maximum polynomial degree.

Keywords

• Nrec: number of recurrence coefficients; must be at least deg + 1.
• addQuadrature: attach the associated Gaussian quadrature rule.

Fields

deg, alpha, beta, measure, and quad satisfy the AbstractOrthoPoly contract.

LogisticMeasure()

Standard logistic probability measure on the real line.

Fields

• w: logistic density.
• dom: (-Inf, Inf).
• symmetric: true.

LogisticOrthoPoly(deg; Nrec = deg + 1, addQuadrature = true)

Construct the basis orthogonal to the standard logistic measure.

Arguments

• deg: nonnegative maximum degree.

Keywords

• Nrec: recurrence coefficient count.
• addQuadrature: whether to attach Gaussian quadrature.

Fields

• deg: maximum represented degree.
• α, β: monic recurrence coefficients.
• measure: standard logistic measure.
• quad: attached quadrature rule or EmptyQuad.

Measure(name, w, dom, symmetric, pars = Dict())

Construct a user-defined univariate measure.

Arguments

• name: descriptive measure name; it is stored in lowercase.
• w: nonnegative weight function on dom.
• dom: ordered finite or infinite support bounds.
• symmetric: whether the measure is symmetric about zero.
• pars: optional parameter dictionary retained with the measure.

Fields

• name: lowercase measure name.
• w: weight function.
• dom: support bounds.
• symmetric: symmetry flag used by scalar-product algorithms.
• pars: user-supplied metadata.

Examples

julia> using PolyChaos

julia> Measure("uniform", x -> 1.0, (0.0, 1.0), false).name
"uniform"

MeixnerPollaczekMeasure(lambda, phi)

Meixner-Pollaczek measure on the real line.

Arguments

• lambda: positive shape parameter.
• phi: angle parameter in (0, pi).

Fields

• w, dom, symmetric: measure data.
• λParameter, ϕParameter: validated family parameters.

MeixnerPollaczekOrthoPoly(deg, lambda, phi; Nrec = deg + 1,
                          addQuadrature = true)

Construct a Meixner-Pollaczek basis.

Arguments

• deg: nonnegative maximum degree.
• lambda: positive family parameter.
• phi: angle parameter in (0, pi).

Keywords

• Nrec: recurrence coefficient count.
• addQuadrature: whether to attach Gaussian quadrature.

Fields

• deg: maximum represented degree.
• α, β: monic recurrence coefficients.
• measure: Meixner-Pollaczek measure.
• quad: attached quadrature rule or EmptyQuad.

MultiOrthoPoly(uniOrthoPolys, deg)

Construct a total-degree multivariate orthogonal-polynomial basis from univariate bases.

Arguments

• uniOrthoPolys: one basis per coordinate; each must represent at least deg.
• deg: requested nonnegative total degree.

Fields

• name: coordinate basis names.
• deg, dim: total degree and number of multivariate polynomials.
• ind: total-degree multi-index matrix.
• measure: product measure.
• uni: coordinate bases.

Examples

julia> using PolyChaos

julia> MultiOrthoPoly([LegendreOrthoPoly(2), HermiteOrthoPoly(2)], 2).dim
6

OrthoPoly(name, deg, alpha, beta, measure; addQuadrature = true)
OrthoPoly(name, deg, measure; Nrec = deg + 1, Nquad = 10Nrec,
          quadrature = clenshaw_curtis, discretization = stieltjes,
          addQuadrature = true)

Construct a univariate orthogonal-polynomial basis from recurrence coefficients or by discretizing a measure.

Arguments

• name: descriptive basis name.
• deg: maximum represented polynomial degree.
• alpha, beta: equal-length monic recurrence coefficient vectors.
• measure: source AbstractMeasure.

Keywords

• Nrec: number of recurrence coefficients; at least deg + 1.
• Nquad: number of nodes used during numerical discretization.
• quadrature: numerical quadrature generator.
• discretization: recurrence-recovery procedure.
• addQuadrature: attach a Golub-Welsch quadrature rule when true.

Fields

• name, deg: basis identity and maximum degree.
• alpha, beta: recurrence coefficients.
• measure, quad: source measure and attached quadrature or EmptyQuad.

Examples

julia> using PolyChaos

julia> op = OrthoPoly("legendre", 2, LegendreMeasure()); deg(op)
2

PolyChaos

Orthogonal-polynomial, quadrature, and polynomial-chaos tools for uncertainty quantification. Construct a canonical basis such as LegendreOrthoPoly, evaluate it with evaluate, and use its quadrature rule through integrate.

ProductMeasure(w, measures)

Represent a product measure assembled from univariate component measures.

Arguments

• w: product weight function accepting one point per component.
• measures: component AbstractMeasures in coordinate order.

Fields

• w: product weight function.
• measures: univariate component measures.

Quad(name, N, nodes, weights)
Quad(N, alpha, beta)
Quad(N, measure; quadrature = clenshaw_curtis)

Construct a quadrature rule.

Arguments

• name: descriptive rule name for the direct constructor.
• N: positive number of quadrature nodes.
• nodes, weights: equal-length vectors of nodes and weights.
• alpha, beta: recurrence coefficients for the Golub-Welsch constructor.
• measure: measure used by numerical quadrature.

Keywords

• quadrature: rule generator used for numerical measure discretization.

Fields

• name: lowercase rule name.
• Nquad: number of nodes.
• nodes, weights: paired quadrature vectors.

Examples

julia> using PolyChaos

julia> Quad("rule", 2, [-1.0, 1.0], [1.0, 1.0]).Nquad
2

Tensor(dim, basis)

Construct a sparse tensor of scalar products of order dim for an orthogonal-polynomial basis.

Arguments

• dim: positive scalar-product order.
• basis: a univariate AbstractOrthoPoly or multivariate MultiOrthoPoly with attached quadrature rules.

Fields

• dim: scalar-product order.
• T: sparse vector of tensor entries.
• get: index-to-entry accessor.
• op: source orthogonal-polynomial basis.

Examples

julia> using PolyChaos

julia> Tensor(2, LegendreOrthoPoly(2; Nrec = 4)).dim
2

Uniform01Measure()

Uniform probability measure on (0, 1).

Fields

• w: unit density on (0, 1).
• dom: (0, 1).
• symmetric: true for the centered polynomial family.

Uniform01OrthoPoly(deg; Nrec = deg + 1, addQuadrature = true)

Construct the basis orthogonal to the uniform probability measure on (0, 1).

Arguments

• deg: nonnegative maximum degree.

Keywords

• Nrec: recurrence coefficient count.
• addQuadrature: whether to attach Gaussian quadrature.

Fields

• deg: maximum represented degree.
• α, β: monic recurrence coefficients.
• measure: uniform measure on (0, 1).
• quad: attached quadrature rule or EmptyQuad.

Uniform_11Measure()

Uniform probability measure on (-1, 1).

Fields

• w: density 1 / 2.
• dom: (-1, 1).
• symmetric: true.

Uniform_11OrthoPoly(deg; Nrec = deg + 1, addQuadrature = true)

Construct the basis orthogonal to the uniform probability measure on (-1, 1).

Arguments

• deg: nonnegative maximum degree.

Keywords

• Nrec: recurrence coefficient count.
• addQuadrature: whether to attach Gaussian quadrature.

Fields

• deg: maximum represented degree.
• α, β: monic recurrence coefficients.
• measure: uniform measure on (-1, 1).
• quad: attached quadrature rule or EmptyQuad.