# Gauss quadrature in 3D with change of variables

**URL:** https://discourse.julialang.org/t/gauss-quadrature-in-3d-with-change-of-variables/113486
**Category:** Numerics
**Created:** [April 25, 2024, 7:52am UTC](https://discourse.julialang.org/t/gauss-quadrature-in-3d-with-change-of-variables/113486 "2024-04-25T07:52:08Z")
**Posts on this page:** 1
**Showing post:** 2

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### Author: ![stevengj](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/stevengj/32/71_2.png) [@stevengj](https://discourse.julialang.org/u/stevengj)
#### Post date: [April 25, 2024, 12:45pm UTC](https://discourse.julialang.org/t/gauss-quadrature-in-3d-with-change-of-variables/113486/2 "2024-04-25T12:45:43Z")

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Your basic idea is correct, but you have a bug/typo in your code somewhere, and you also didn’t supply `gauss_quadrature_nodes_2d` so it is not runnable. I found your code a bit hard to read so I re-implemented it, and mine seems to work fine:

```julia
using QuadGK
function custom_gaussquad(f, n)
    ξ, w = QuadGK.gauss(n, -1, 1) # 1d gauss rule of order n
    integral = 0.0 # not 0, for type stability
    for (ξ₁,w₁) in zip(ξ, w), (ξ₂,w₂) in zip(ξ, w) # tensor product of 1d rules
        x = (ξ₁+1)/2
        y = ((1-x)*ξ₂ + (1+x))/2
        integral += (w₁ * w₂) * f(x, y) * (1-x)/4
    end
    return integral
end

```

which gives the correct answer:

```julia
julia> custom_gaussquad((x,y) -> 1, 11) # area of triangle
0.49999999999999983

julia> custom_gaussquad((x,y) -> x+y, 11)
0.5000000000000002

julia> quadgk(x -> quadgk(y -> x+y, x, 1)[1], 0,1)[1]
0.49999999999999994

julia> custom_gaussquad((x,y) -> exp(x^2 + sin(x*y)), 11)
0.8215954352465259

julia> quadgk(x -> quadgk(y -> exp(x^2 + sin(x*y)), x, 1)[1], 0,1)[1]
0.8215954352465216

```

PS. Note that `integral = 0` is [type unstable](https://docs.julialang.org/en/v1/manual/performance-tips/#Avoid-changing-the-type-of-a-variable). If you know you want a `Float64` result, you can use `integral = 0.0` to initialize it.

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