https://github.com/juliaapproximation/annuliorthogonalpolynomials.jl

https://github.com/juliaapproximation/annuliorthogonalpolynomials.jl

Last synced: 5 months ago
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• Host: GitHub
• URL: https://github.com/juliaapproximation/annuliorthogonalpolynomials.jl
• Owner: JuliaApproximation
• License: mit
• Created: 2023-07-18T19:58:01.000Z (almost 3 years ago)
• Default Branch: main
• Last Pushed: 2025-03-24T16:06:20.000Z (over 1 year ago)
• Last Synced: 2025-12-17T06:38:50.256Z (6 months ago)
• Language: Julia
• Size: 77.1 KB
• Stars: 1
• Watchers: 2
• Forks: 0
• Open Issues: 3
• Metadata Files:
  • Readme: README.md
  • License: LICENSE

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README

          
## AnnuliOrthogonalPolynomials.jl

[![Build Status](https://github.com/JuliaApproximation/AnnuliOrthogonalPolynomials.jl/workflows/CI/badge.svg)](https://github.com/JuliaApproximation/AnnuliOrthogonalPolynomials.jl/actions)

[![codecov.io](http://codecov.io/github/JuliaApproximation/AnnuliOrthogonalPolynomials.jl/coverage.svg?branch=main)](http://codecov.io/github/JuliaApproximation/AnnuliOrthogonalPolynomials.jl?branch=main)

A Julia package for orthogonal polynomials on an annulus.

This code used to be part of [AlgebraicCurveOrthogonalPolynomials.jl](https://github.com/JuliaApproximation/AlgebraicCurveOrthogonalPolynomials.jl).

This experimental package implements `ZernikeAnnulus` and `ComplexZernikeAnnulus`,

families of orthogonal polynomials on the annulus `{ρ² ≤ x² + y² ≤ 1}`

with respect to the weight `(r² - ρ²)ᵃ * (1-r²)ᵇ`. This builds on top of

[SemiclassicalOrthogonalPolynomials.jl](https://github.com/JuliaApproximation/SemiclassicalOrthogonalPolynomials.jl) and [MultivariateOrthogonalPolynomials.jl](https://github.com/JuliaApproximation/MultivariateOrthogonalPolynomials.jl).

```julia

julia> using AnnuliOrthogonalPolynomials, StaticArrays

julia> ρ, a, b = 0.5, 0.0, 0.0;

julia> Z = ZernikeAnnulus(ρ, a, b)

ZernikeAnnulus{Float64}(0.5, 0.0, 0.0)

julia> Z[SVector(0.5,0.2), Block.(1:3)] # Blocked by degree

3-blocked 6-element BlockVector{Float64}:

 1.0

 ───────────────────

 0.20000000000000004

 0.5000000000000001

 ───────────────────

 1.547298721428197

 0.20000000000000007

 0.21000000000000008

julia> x,y = first.(axes(Z,1)), last.(axes(Z,1));

julia> u = Z * (Z \ (exp.(x) .* cos.(y))) # Expand in annulus OPs

ZernikeAnnulus{Float64}(0.5, 0.0, 0.0) * [1.0, 9.352800883640776e-18, 0.9999999999999999, -5.2512913921638607e-17, -2.337754850841795e-18, 0.5, 2.153350906504162e-18, 1.1925107582660448e-17, -5.513529126039963e-18, 0.1666666666666666  …  ]

julia> u[SVector(0.5,0.2)] # Evaluate expansion

1.6158566135891377

```