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

A Julia package for working with spherical harmonic expansions
https://github.com/juliaapproximation/harmonicorthogonalpolynomials.jl

Last synced: over 1 year ago
JSON representation

A Julia package for working with spherical harmonic expansions

• Host: GitHub
• URL: https://github.com/juliaapproximation/harmonicorthogonalpolynomials.jl
• Owner: JuliaApproximation
• License: mit
• Created: 2020-03-16T12:14:17.000Z (over 6 years ago)
• Default Branch: master
• Last Pushed: 2025-01-30T11:02:19.000Z (over 1 year ago)
• Last Synced: 2025-02-28T06:27:43.182Z (over 1 year ago)
• Language: Julia
• Size: 106 KB
• Stars: 23
• Watchers: 5
• Forks: 2
• Open Issues: 13
• Metadata Files:
  • Readme: README.md
  • License: LICENSE

Awesome Lists containing this project

README

          
# HarmonicOrthogonalPolynomials.jl

A Julia package for working with spherical harmonic expansions and

harmonic polynomials in  balls.

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

[![codecov](https://codecov.io/gh/JuliaApproximation/HarmonicOrthogonalPolynomials.jl/branch/master/graph/badge.svg)](https://codecov.io/gh/JuliaApproximation/HarmonicOrthogonalPolynomials.jl)

A [harmonic polynomial](https://en.wikipedia.org/wiki/Harmonic_polynomial) is a multivariate polynomial that solves Laplace's equation. 

[Spherical harmonics](https://en.wikipedia.org/wiki/Spherical_harmonics) are restrictions of harmonic polynomials to the sphere. Importantly they

are orthogonal. This package is primarily an implementation of spherical harmonics (in 2D and 3D) but exploiting their 

polynomial features.

Currently this package focusses on support for

3D spherical harmonics. We use the convention of [FastTransforms](https://mikaelslevinsky.github.io/FastTransforms/transforms.html) for real spherical harmonics:

```julia

julia> θ,φ = 0.1,0.2 # θ is polar, φ is azimuthal (physics convention)

julia> sphericalharmonicy(ℓ, m, θ, φ)

0.07521112971423363 + 0.015246050775019674im

```

But we also allow function approximation, building on top of  [ContinuumArrays.jl](https://github.com/JuliaApproximation/ContinuumArrays.jl) and [ClassicalOrthogonalPolynomials.jl](https://github.com/JuliaApproximation/ClassicalOrthogonalPolynomials.jl):

```julia

julia> S = SphericalHarmonic() # A quasi-matrix representation of spherical harmonics

SphericalHarmonic{Complex{Float64}}

julia> S[SphericalCoordinate(θ,φ),Block(ℓ+1)] # evaluate all spherical harmonics with specified ℓ

5-element Array{Complex{Float64},1}:

 0.003545977402630546 - 0.0014992151996309556im

  0.07521112971423363 - 0.015246050775019674im

    0.621352880681805 + 0.0im

  0.07521112971423363 + 0.015246050775019674im

 0.003545977402630546 + 0.0014992151996309556im

julia> 𝐱 = axes(S,1) # represent the unit sphere as a quasi-vector

Inclusion(the 3-dimensional unit sphere)

julia> f = 𝐱 -> ((x,y,z) = 𝐱; exp(x)*cos(y*sin(z))); # function to be approximation

julia> S \ f.(𝐱) # expansion coefficients, adaptively computed

∞-blocked ∞-element BlockedArray{Complex{Float64},1,LazyArrays.CachedArray{Complex{Float64},1,Array{Complex{Float64},1},Zeros{Complex{Float64},1,Tuple{InfiniteArrays.OneToInf{Int64}}}},Tuple{BlockedOneTo{Int,ArrayLayouts.RangeCumsum{Int64,InfiniteArrays.InfStepRange{Int64,Int64}}}}}:

        4.05681442931116 + 0.0im                   

 ──────────────────────────────────────────────────

      1.5777291816142751 + 3.19754060061646e-16im  

  -8.006900295635809e-17 + 0.0im                   

      1.5777291816142751 - 3.539535261006306e-16im 

 ──────────────────────────────────────────────────

      0.3881560551355611 + 5.196884701505137e-17im 

  -7.035627371746071e-17 + 2.5784941810054987e-18im

    -0.30926350498081934 + 0.0im                   

   -6.82462130695514e-17 - 3.515332651034677e-18im 

      0.3881560551355611 - 6.271963079558218e-17im 

 ──────────────────────────────────────────────────

     0.06830566496722756 - 8.852861226980248e-17im 

 -2.3672451919730833e-17 + 2.642173739237023e-18im 

     -0.0514592471634392 - 1.5572791163000952e-17im

  1.1972144648274198e-16 + 0.0im                   

    -0.05145924716343915 + 1.5264133695821818e-17im

                         ⋮

julia> f̃ = S * (S \ f.(𝐱)); # expansion of f in spherical harmonics

julia> f̃[SphericalCoordinate(θ,φ)] # approximates f

1.1026374731849062 + 4.004893695029451e-16im

```