{"id":20500585,"url":"https://github.com/juliaapproximation/algebraiccurveorthogonalpolynomials.jl","last_synced_at":"2025-03-05T19:33:55.307Z","repository":{"id":40370221,"uuid":"244042734","full_name":"JuliaApproximation/AlgebraicCurveOrthogonalPolynomials.jl","owner":"JuliaApproximation","description":null,"archived":false,"fork":false,"pushed_at":"2024-07-19T21:10:21.000Z","size":24945,"stargazers_count":1,"open_issues_count":11,"forks_count":3,"subscribers_count":4,"default_branch":"master","last_synced_at":"2025-02-21T01:45:45.798Z","etag":null,"topics":[],"latest_commit_sha":null,"homepage":null,"language":"Mathematica","has_issues":true,"has_wiki":null,"has_pages":null,"mirror_url":null,"source_name":null,"license":"mit","status":null,"scm":"git","pull_requests_enabled":true,"icon_url":"https://github.com/JuliaApproximation.png","metadata":{"files":{"readme":"README.md","changelog":null,"contributing":null,"funding":null,"license":"LICENSE","code_of_conduct":null,"threat_model":null,"audit":null,"citation":null,"codeowners":null,"security":null,"support":null,"governance":null,"roadmap":null,"authors":null,"dei":null,"publiccode":null,"codemeta":null}},"created_at":"2020-02-29T21:16:44.000Z","updated_at":"2024-01-12T18:12:10.000Z","dependencies_parsed_at":"2023-09-30T15:16:03.654Z","dependency_job_id":"ad98128e-fc8e-4422-9386-23f350d9cfa2","html_url":"https://github.com/JuliaApproximation/AlgebraicCurveOrthogonalPolynomials.jl","commit_stats":null,"previous_names":[],"tags_count":2,"template":false,"template_full_name":null,"repository_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/JuliaApproximation%2FAlgebraicCurveOrthogonalPolynomials.jl","tags_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/JuliaApproximation%2FAlgebraicCurveOrthogonalPolynomials.jl/tags","releases_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/JuliaApproximation%2FAlgebraicCurveOrthogonalPolynomials.jl/releases","manifests_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/JuliaApproximation%2FAlgebraicCurveOrthogonalPolynomials.jl/manifests","owner_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/owners/JuliaApproximation","download_url":"https://codeload.github.com/JuliaApproximation/AlgebraicCurveOrthogonalPolynomials.jl/tar.gz/refs/heads/master","host":{"name":"GitHub","url":"https://github.com","kind":"github","repositories_count":242092290,"owners_count":20070495,"icon_url":"https://github.com/github.png","version":null,"created_at":"2022-05-30T11:31:42.601Z","updated_at":"2022-07-04T15:15:14.044Z","host_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub","repositories_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories","repository_names_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repository_names","owners_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/owners"}},"keywords":[],"created_at":"2024-11-15T18:21:45.861Z","updated_at":"2025-03-05T19:33:55.274Z","avatar_url":"https://github.com/JuliaApproximation.png","language":"Mathematica","funding_links":[],"categories":[],"sub_categories":[],"readme":"## AlgebraicCurveOrthogonalPolynomials.jl\nA Julia package for orthogonal polynomials on algebraic curves\n\n\nThis package contains ongoing research on orthogonal polynomials on\nalgebraic curves. That is, given an algebraic curve in 2D defined by\n`S = {(x,y) : p(x,y) = 0}` for some bivariate polynomial `p` we wish\nto construct polynomials orthogonal with respect to an inner product\nsupported on a  subset `Ω ⊆ S`. Note to avoid degenerecies  these polynomials need to\n be thought of as polynomials modulo the vanishing ideal `I(S)` associated to `S`.\nThe general theory and construction is not at yet possible, so we\noutline some specific cases that we have implemented below. \n\nThis is funded by a Leverhulme Trust Research Project Grant on\n\"Constructive approximation on algebraic curves and surfaces\".\n\n# Arc\n\nWe can construct orthogonal polynomials on an arc, that is, \n`Ω = {(cos(θ), sin(θ)) : a ≤ θ ≤ b}`, which is a subset of \nthe circle `{(x,y) : x^2 + y^2 = 1}`. \nWe parameterise points on the circle by angle, using a special\ntype `CircleCoordinate(θ)`:\n```julia\njulia\u003e CircleCoordinate(0.1)\n2-element CircleCoordinate{Float64} with indices SOneTo(2):\n 0.9950041652780258\n 0.09983341664682815\n```\nFor now we only support the half circle `y ≥ 0` with the \nweight `y^a`, which we construct via `UltrasphericalArc(a)`,\nwhich is implemented in the framework of ContinuumArrays.jl:\n```julia\njulia\u003e P = UltrasphericalArc() # uniform weight on the arc\nUltrasphericalArc(0.0)\n\njulia\u003e P[CircleCoordinate(0.1),1:5] # first 5 polynomials\n5-element Array{Float64,1}:\n 1.0\n 0.9950041652780258\n 0.7024490016371341\n 2.030105652576658\n 0.06160390817639964\n```\nNote there are two (and only two) degree-`d` polynomials\napart from `d = 1`. This is accessible as the columns of `P`\nare blocked a la BlockArrays.jl:\n```julia\njulia\u003e P[CircleCoordinate(0.1), Block.(1:3)]\n3-blocked 5-element PseudoBlockArray{Float64,1,Array{Float64,1},Tuple{BlockedUnitRange{StepRange{Int64,Int64}}}}:\n 1.0                \n ───────────────────\n 0.9950041652780258 \n 0.7024490016371341 \n ───────────────────\n 2.030105652576658  \n 0.06160390817639964\n```","project_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fjuliaapproximation%2Falgebraiccurveorthogonalpolynomials.jl","html_url":"https://awesome.ecosyste.ms/projects/github.com%2Fjuliaapproximation%2Falgebraiccurveorthogonalpolynomials.jl","lists_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fjuliaapproximation%2Falgebraiccurveorthogonalpolynomials.jl/lists"}