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‎.gitignore

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*.jl.cov
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*.jl.*.cov
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*.jl.mem
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site

‎builddocs

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cd docs
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julia build.jl
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cd ..
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mkdocs build --clean
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(sleep 1 && open -a Safari http://127.0.0.1:8000)& # waits for mkdoc server to start up
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mkdocs serve

‎docs/api/FastTransforms.md

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# FastTransforms
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## Exported
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---
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<a id="method__cjt.1" class="lexicon_definition"></a>
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#### cjt(c, α, β) [](#method__cjt.1)
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Computes the Chebyshev expansion coefficients
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given the Jacobi expansion coefficients ``c`` with parameters ``α`` and ``β``.
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See also [`icjt`](#method__icjt.1) and [`jjt`](#method__jjt.1).
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*source:*
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[FastTransforms/src/cjt.jl:127](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/b1dce339ed19119766d44b5ec44d954932da58d3/src/cjt.jl#L127)
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---
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<a id="method__gaunt.1" class="lexicon_definition"></a>
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#### gaunt(m::Int64, n::Int64, μ::Int64, ν::Int64) [](#method__gaunt.1)
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Calculates the Gaunt coefficients in 64-bit floating-point arithmetic.
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*source:*
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[FastTransforms/src/gaunt.jl:24](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/b1dce339ed19119766d44b5ec44d954932da58d3/src/gaunt.jl#L24)
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---
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<a id="method__gaunt.2" class="lexicon_definition"></a>
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#### gaunt{T}(::Type{T}, m::Int64, n::Int64, μ::Int64, ν::Int64) [](#method__gaunt.2)
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Calculates the Gaunt coefficients, defined by:
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a(m,n,μ,ν,q) = (2(n+ν-2q)+1)/2 (n+ν-2q-m-μ)!/(n+ν-2q+m+μ)! ∫₋₁⁺¹ P_m^n(x) P_ν^μ(x) P_{n+ν-2q}^{m+μ}(x) dx.
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or defined by:
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P_n^m(x) P_ν^μ(x) = ∑_{q=0}^{q_{max}} a(m,n,μ,ν,q) P_{n+ν-2q}^{m+μ}(x)
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This is a Julia implementation of the stable recurrence described in:
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Y.-l. Xu, "Fast evaluation of Gaunt coefficients: recursive approach", J. Comp. Appl. Math., 85:53–65, 1997.
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*source:*
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[FastTransforms/src/gaunt.jl:14](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/b1dce339ed19119766d44b5ec44d954932da58d3/src/gaunt.jl#L14)
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---
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<a id="method__icjt.1" class="lexicon_definition"></a>
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#### icjt(c, α, β) [](#method__icjt.1)
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Computes the Jacobi expansion coefficients with parameters ``α`` and ``β``
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given the Chebyshev expansion coefficients ``c``.
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See also [`cjt`](#method__cjt.1) and [`jjt`](#method__jjt.1).
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*source:*
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[FastTransforms/src/cjt.jl:135](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/b1dce339ed19119766d44b5ec44d954932da58d3/src/cjt.jl#L135)
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---
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<a id="method__jjt.1" class="lexicon_definition"></a>
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#### jjt(c, α, β, γ, δ) [](#method__jjt.1)
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Computes the Jacobi expansion coefficients with parameters ``γ`` and ``δ``
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given the Jacobi expansion coefficients ``c`` with parameters ``α`` and ``β``.
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See also [`cjt`](#method__cjt.1) and [`icjt`](#method__icjt.1).
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*source:*
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[FastTransforms/src/cjt.jl:143](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/b1dce339ed19119766d44b5ec44d954932da58d3/src/cjt.jl#L143)
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---
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<a id="method__plan_cjt.1" class="lexicon_definition"></a>
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#### plan_cjt(c::AbstractArray{T, 1}, α, β) [](#method__plan_cjt.1)
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Pre-plan optimized DCT-I and DST-I plans and pre-allocate the necessary
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arrays, normalization constants, and recurrence coefficients for a forward Chebyshev—Jacobi transform.
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``c`` is the vector of coefficients; and,
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``α`` and ``β`` are the Jacobi parameters.
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Optionally:
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``M`` determines the number of terms in Hahn's asymptotic expansion.
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*source:*
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[FastTransforms/src/cjt.jl:157](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/b1dce339ed19119766d44b5ec44d954932da58d3/src/cjt.jl#L157)
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---
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<a id="method__plan_icjt.1" class="lexicon_definition"></a>
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#### plan_icjt(c::AbstractArray{T, 1}, α, β) [](#method__plan_icjt.1)
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Pre-plan optimized DCT-I and DST-I plans and pre-allocate the necessary
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arrays, normalization constants, and recurrence coefficients for an inverse Chebyshev—Jacobi transform.
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``c`` is the vector of coefficients; and,
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``α`` and ``β`` are the Jacobi parameters.
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Optionally:
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``M`` determines the number of terms in Hahn's asymptotic expansion.
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*source:*
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[FastTransforms/src/cjt.jl:176](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/b1dce339ed19119766d44b5ec44d954932da58d3/src/cjt.jl#L176)
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## Internal
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---
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<a id="method__chebyshevjacobimoments1.1" class="lexicon_definition"></a>
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#### chebyshevjacobimoments1{T<:AbstractFloat}(N::Int64, α::T<:AbstractFloat, β::T<:AbstractFloat) [](#method__chebyshevjacobimoments1.1)
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Modified Chebyshev moments of the first kind with respect to the Jacobi weight:
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∫₋₁⁺¹ T_n(x) (1-x)^α(1+x)^β dx.
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*source:*
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[FastTransforms/src/specialfunctions.jl:362](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/b1dce339ed19119766d44b5ec44d954932da58d3/src/specialfunctions.jl#L362)
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---
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<a id="method__chebyshevjacobimoments2.1" class="lexicon_definition"></a>
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#### chebyshevjacobimoments2{T<:AbstractFloat}(N::Int64, α::T<:AbstractFloat, β::T<:AbstractFloat) [](#method__chebyshevjacobimoments2.1)
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Modified Chebyshev moments of the second kind with respect to the Jacobi weight:
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∫₋₁⁺¹ U_n(x) (1-x)^α(1+x)^β dx.
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*source:*
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[FastTransforms/src/specialfunctions.jl:380](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/b1dce339ed19119766d44b5ec44d954932da58d3/src/specialfunctions.jl#L380)
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---
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<a id="method__clenshawcurtisweights.1" class="lexicon_definition"></a>
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#### clenshawcurtisweights{T<:AbstractFloat}(N::Int64, α::T<:AbstractFloat, β::T<:AbstractFloat) [](#method__clenshawcurtisweights.1)
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Compute weights of the Clenshaw—Curtis quadrature rule with a Jacobi weight.
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*source:*
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[FastTransforms/src/clenshawcurtis.jl:12](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/b1dce339ed19119766d44b5ec44d954932da58d3/src/clenshawcurtis.jl#L12)
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---
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<a id="method__clenshawcurtis.1" class="lexicon_definition"></a>
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#### clenshawcurtis{T<:AbstractFloat}(N::Int64, α::T<:AbstractFloat, β::T<:AbstractFloat) [](#method__clenshawcurtis.1)
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Compute nodes and weights of the Clenshaw—Curtis quadrature rule with a Jacobi weight.
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*source:*
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[FastTransforms/src/clenshawcurtis.jl:6](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/b1dce339ed19119766d44b5ec44d954932da58d3/src/clenshawcurtis.jl#L6)
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---
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<a id="method__fejer1.1" class="lexicon_definition"></a>
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#### fejer1{T<:AbstractFloat}(N::Int64, α::T<:AbstractFloat, β::T<:AbstractFloat) [](#method__fejer1.1)
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Compute nodes and weights of Fejer's first quadrature rule with a Jacobi weight.
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*source:*
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[FastTransforms/src/fejer.jl:7](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/b1dce339ed19119766d44b5ec44d954932da58d3/src/fejer.jl#L7)
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---
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<a id="method__fejer2.1" class="lexicon_definition"></a>
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#### fejer2{T<:AbstractFloat}(N::Int64, α::T<:AbstractFloat, β::T<:AbstractFloat) [](#method__fejer2.1)
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Compute nodes and weights of Fejer's second quadrature rule with a Jacobi weight.
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*source:*
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[FastTransforms/src/fejer.jl:12](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/b1dce339ed19119766d44b5ec44d954932da58d3/src/fejer.jl#L12)
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---
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<a id="method__fejerweights1.1" class="lexicon_definition"></a>
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#### fejerweights1{T<:AbstractFloat}(N::Int64, α::T<:AbstractFloat, β::T<:AbstractFloat) [](#method__fejerweights1.1)
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Compute weights of Fejer's first quadrature rule with a Jacobi weight.
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*source:*
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[FastTransforms/src/fejer.jl:21](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/b1dce339ed19119766d44b5ec44d954932da58d3/src/fejer.jl#L21)
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---
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<a id="method__fejerweights2.1" class="lexicon_definition"></a>
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#### fejerweights2{T<:AbstractFloat}(N::Int64, α::T<:AbstractFloat, β::T<:AbstractFloat) [](#method__fejerweights2.1)
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Compute weights of Fejer's second quadrature rule with a Jacobi weight.
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*source:*
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[FastTransforms/src/fejer.jl:26](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/b1dce339ed19119766d44b5ec44d954932da58d3/src/fejer.jl#L26)
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---
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<a id="method__pochhammer.1" class="lexicon_definition"></a>
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#### pochhammer(x::Number, n::Integer) [](#method__pochhammer.1)
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Pochhammer symbol (x)_n = Γ(x+n)/Γ(x) for the rising factorial.
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*source:*
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[FastTransforms/src/specialfunctions.jl:13](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/b1dce339ed19119766d44b5ec44d954932da58d3/src/specialfunctions.jl#L13)
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‎docs/api/README.md

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Files in this directory are generated using the `build.jl` script. Make
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all changes to the originating docstrings/files rather than these ones.
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Documentation should *only* be built directly on the `master` branch.
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Source links would otherwise become unavailable should a branch be
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deleted from the `origin`. This means potential pull request authors
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*should not* run the build script when filing a PR.

‎docs/build.jl

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# Adapted from https://github.com/MichaelHatherly/Docile.jl
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# (C) Michael Hatherly, under MIT-license
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using Lexicon, FastTransforms
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const api_directory = "api"
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const modules = [FastTransforms]
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cd(dirname(@__FILE__)) do
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# Run the doctests *before* we start to generate *any* documentation.
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for m in modules
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failures = failed(doctest(m))
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if !isempty(failures.results)
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println("\nDoctests failed, aborting commit.\n")
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display(failures)
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exit(1) # Bail when doctests fail.
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end
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end
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# also execute examples
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#include("../examples/ex1.jl")
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# Generate and save the contents of docstrings as markdown files.
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index = Index()
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for mod in modules
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update!(index, save(joinpath(api_directory, "$(mod).md"), mod))
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end
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#save(joinpath(api_directory, "index.md"), index; md_subheader = :category)
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# Add a reminder not to edit the generated files.
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open(joinpath(api_directory, "README.md"), "w") do f
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print(f, """
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Files in this directory are generated using the `build.jl` script. Make
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all changes to the originating docstrings/files rather than these ones.
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Documentation should *only* be built directly on the `master` branch.
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Source links would otherwise become unavailable should a branch be
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deleted from the `origin`. This means potential pull request authors
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*should not* run the build script when filing a PR.
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""")
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end
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#info("Adding all documentation changes in $(api_directory) to this commit.")
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#success(`git add $(api_directory)`) || exit(1)
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end

‎docs/index.md

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# FastTransforms.jl
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[![Build Status](https://travis-ci.org/MikaelSlevinsky/FastTransforms.jl.svg?branch=master)](https://travis-ci.org/MikaelSlevinsky/FastTransforms.jl)
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The aim of this package is to provide a new class of fast transforms
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based on the use of asymptotic formulae to relate the transforms to a small
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number of fast Fourier transforms. This new class of fast transforms does not
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require large pre-computation for fast execution, and they are designed
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to work on expansions of functions with any degree of regularity.
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The Chebyshev—Jacobi transform and its inverse are implemented. This
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allows the fast conversion of Chebyshev expansion coefficients to Jacobi expansion
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coefficients and back.
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```julia
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julia> Pkg.add("FastTransforms")
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julia> using FastTransforms
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julia> c = rand(10001);
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julia> @time norm(icjt(cjt(c,0.1,-0.2),0.1,-0.2)-c,Inf)
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0.435853 seconds (507 allocations: 5.366 MB)
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1.4830359162942841e-12
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julia> p1 = plan_cjt(c,0.1,-0.2);
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julia> p2 = plan_icjt(c,0.1,-0.2);
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julia> @time norm(p2*(p1*c)-c,Inf)
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0.396803 seconds (101 allocations: 473.281 KB)
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1.4830359162942841e-12
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```
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The design and implementation is analogous to FFTW: there is a type `ChebyshevJacobiPlan`
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that stores pre-planned optimized DCT-I and DST-I plans, recurrence coefficients,
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and temporary arrays to allow the execution of either the `cjt` or the `icjt` allocation-free.
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This type is constructed with either `plan_cjt` or `plan_icjt`. Composition of transforms
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allows the Jacobi—Jacobi transform, computed via `jjt`. The remainder in Hahn's asymptotic expansion
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is valid for the half-open square `(α,β) ∈ (-1/2,1/2]^2`. Therefore, the fast transform works best
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when the parameters are inside. If the parameters `(α,β)` are not exceptionally beyond the square,
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then increment/decrement operators are used with linear complexity (and linear conditioning) in the degree.
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# References:
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1. N. Hale and A. Townsend. <a href="http://dx.doi.org/10.1137/130932223">A fast, simple, and stable Chebyshev—Legendre transform using and asymptotic formula</a>, SIAM J. Sci. Comput., 36:A148—A167, 2014.
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2. R. M. Slevinsky. <a href="http://arxiv.org/abs/1602.02618">On the use of Hahn's asymptotic formula and stabilized recurrence for a fast, simple, and stable Chebyshev—Jacobi transform</a>, arXiv:1602.02618, 2016.

‎mkdocs.yml

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site_name: FastTransforms.jl
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repo_url: https://github.com/MikaelSlevinsky/FastTransforms.jl
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site_description: Julia package for a new class of fast transforms based on asymptotic formulae
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site_author: Richard Mikael Slevinsky
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theme: readthedocs
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markdown_extensions: [tables, fenced_code]
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pages:
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- Introduction: index.md
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- API: api/FastTransforms.md

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