Integral is a functional that represents integration of a function.

Supertype of Jacobi weights.

Supertype of quadratic forms that derive from a two-form.

If F is a TwoForm, then we call F(f,f) the associated quadratic form.

An algebraic singularity at a specific point.

Use a quadrature rule adapted to the measure.

A bilinear form is linear in its two arguments.

A canonical domain for the purposes of evaluating integrals.

Apply a quadrature rule defined on [-1,1].

The Chebyshev or ChebyshevT weight is the measure on [-1,1] with the Chebyshev weight function w(x) = 1/√(1-x^2).

The ChebyshevU weight is the measure on [-1,1] with the Chebyshev weight function of the second kind w(x) = √(1-x^2).

Supertype of singularities along a curve.

A continuous Dirac measure at a point

A DiscreteWeight is a measure defined in terms of a discrete set of points and an associated set of weights.

The measure implements the points and weights functions.

A quadrature rule associated with a domain.

The Duffy transform maps the unit square to the unit simplex.

It does so by collapsing the rightmost edge onto the x-axis.

A dummy domain to be associated with a discrete measure, for internal purposes.

Representation of an integrand function.

The Gaussian measure with weight exp(-|x|^2/2).

A generic discrete weight that stores points and weights.

A generic inner product that stores a positive definite Hermitian sesquilinear form.

A generic continuous weight function.

Apply a fixed quadrature rule defined on the halfline [0,∞).

The Hermite measure with weight exp(-x^2) on the real line.

A bilinear form induced by a measure.

It is the case that F(f,g) equals integral(x -> f(x)*g(x), measure(F)).

A sesquilinear form induced by a measure.

It is the case that F(f,g) equals integral(x -> f(x)*conj(g(x)), measure(F)).

An inner-product is a positive definite sesquilinear form.

Supertype of an integrand which evaluates to T.

The Jacobi weight on the interval [-1,1].

The generalised Laguerre measure on the halfline [0,∞).

The Lebesgue measure on the space FullSpace{T}.

Lebesgue measure supported on a general domain.

Supertype of Lebesgue measures

The Lebesgue measure on the unit interval [0,1].

The Legendre weight is a Lebesgue measure on the interval [-1,1].

A logarithmic singularity at a specific point.

Wrap a function and log any point at which it is evaluated.

The triangle below the diagonal of a square [a,b] × [a,b].

Representation of a weight mapped to a new weight.

Given a weight w(x)dx and a map y=m(x), the mapped weight is defined as w(m(x))/J(m(x))dx, where J is the jacobian determinant of m.

The definition is such that we have the following equality after a change of variables:

integral(t->f(t), domain, weight) = integral(t -> f(inverse(m,t))), m.(domain), MappedWeight(m, weight)

Supertype of measures.

A bilinear form induced by a generic measure.

A sesquilinear form induced by a generic measure.

No property of the integrand is known.

No singularity of the integrand is known.

A one-form is a map V → K, i.e., it is a generic function of one argument.

A one-form is applied to an argument using the F(f) syntax, which itself invokes form(F::OneForm, f). The form function can be specialized for concrete types of F and f.

The main use cases in DomainIntegrals.jl are a lazy Integral object and a quadratic form that derives from a bilinear form.

Supertype of all kinds of point singularities.

A product weight.

A Property type describes a property of the integrand that is independent from the other aspects of the integral (domain, measure).

The main examples are singularities of the integrand function, i.e., smoothness properties.

Adaptive quadrature using hcubature

Adaptive quadrature using quadgk

An adaptive quadrature strategy

A quadratic form derived from a stored two-form.

Apply a fixed quadrature rule defined on the real line.

A sesquilinear form is linear in the first and conjugate linear in the second argument.

An unspecified singularity occurring at a corner point.

An unspecified singularity occurring at a specific point.

For 2D integrands, a singularity along the line x=y.

The supertype of all kinds of singularities of an integrand.

An integrand that logs any transformations which are applied to it.

A triangle defined by three points in a plane.

A two-form is a map V x W → K, i.e., it is a generic function of two arguments.

The two-form is applied to two arguments using the F(f,g) syntax, which itself invokes form(F::TwoForm, f, g). The form function can be specialized for concrete types of F, f and g.

Apply a fixed quadrature rule defined on [0,1].

The triangle above the diagonal of a square [a,b] × [a,b].

A list of breakpoints of the integrand, for example because the integrand is piecewise smooth.

A Weight is a continuous measure that is defined in terms of a weight function: dμ = w(x) dx.

cauchy_integral(f [; a = 0, n  = 1, radius = 1])

Evaluate the integral of n!/(2πi) f(z)/(z-a)^n along a circular contour with radius radius around the point a.

If the integrand is analytic inside the contour except at the pole around z=a then its value should correspond to the derivative f^(n)(a).

Turn the interval rule into a composite quadrature rule with L segments on the same interval.

Split the integration domain into subdomains.

Does the integration domain naturally split into subdomains?

Apply a two-form to the given arguments.

form(F::TwoForm, f, g)

By default, form invokes form1 which in turn invokes form2. The form function itself can be specialized on the type of F without ambiguity. Similarly, form1 and form2 can be specialized without ambiguity on the types of f and g respectively.

Each of these functions can be specialized on all three types involved, as long as the main type is at least as specific as any other specialization.

Helper function for form(F, f, g) that can be specialized on the type of f.

Helper function for form(F, f, g) that can be specialized on the type of g.

Construct a graded quadrature rule towards the left of the unit interval.

The rule is a composite quadrature rule on intervals of the form σ^(k+1):σ^k. The number of points on interval i is ⌈n*i*μ⌉, where μ is an optional parameter.

Construct a graded quadrature rule towards the right of the unit interval.

See: graded_rule_left.

Is the domain associated with a simpler domain for numerical integration?

Compute an integral of the given integrand on the given domain, or with the given measure.

Example:

integral(cos, 0.0..1.0)

Like integral, but also returns an error estimate (if applicable).

Return the integration domain associated with d.

Is the measure continuous?

Is the measure discrete?

Is the given measure the map of another measure?

Is the measure normalized?

Does the discrete measure have equal weights?

The Lebesgue measure associated with the given domain

Return the map from the integration domain of d to d.

Map the weight using the given map.

Convert a union of domains into a vector of domains without overlap.

Transform the integral problem based on the supplied properties.

For example, if the integrand is known to be singular, then the domain may be split into several components such that there are no singularities in the interior.

Construct the product of the given measures.

Split the domain according to a singularity at the point x.

Split the domain into a list of domains, such that any singularity lies on the boundary of one or more of the constituting parts.

Care is taken to avoid a singularity in the interior due to roundoff errors.

The suggested quadrature strategy based on the integral arguments.

By default, adaptive quadrature is chosen.

Return the support of the measure.

Evaluate the weight function associated with the measure.

What is the codomain type of the measure?