CODEWITHSUNDEEP
julianumerical-integration
Student Debil
Student Debil

Reputation: 21

Julia HCubature.jl limits dependinding on other variables

I want to calculate double integral with HCubature, but one of the variables' limit depends on the other. For example: enter image description here

And I've tried primitively:

using HCubature
F(x, y) = x+y
hcubature(r->F(r[1], r[2]), [0,0], [r[2], 1])

and of course it doesn't work and gives an error:

ERROR: LoadError: UndefVarError: r not defined

How can I calculate such integral? Maybe using another package? If so, which package and how to do this?

Upvotes: 1

Views: 212

Answers (2)

SGJ
SGJ

Reputation: 1058

Use a change of variables: see the links here

I'm the author of the HCubature.jl package, and this is the way to handle non-rectangular domains in such methods. In this example, you make the explicit change of variables x=uy (dx = y du) then integrate u from 0 to 1:

julia> using HCubature

julia> hcubature(r -> F(r[1]*r[2], r[2]) * r[2], (0,0), (1,1))
(0.5, 1.6653345369377348e-16)

Upvotes: -1

Dan Getz
Dan Getz

Reputation: 18227

This is an iterated integral. hcubature can be used to calculate the integrals from the inside out, like so:

julia> F(x, y) = x+y
F (generic function with 1 method)

julia> hcubature(r->hcubature(s->F(s[1],r[1]), [0], [r[1]])[1], [0], [1])
(0.4999999999999999, 1.1102230246251565e-16)

So the answer is 0.5. In case the semantics is unclear, the integration is over a triangle (0,0)-(1,0)-(1,1). The function ranges from 0 to 2 on this triangle and so the answer is reasonable.

Upvotes: 1

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