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add docstrings for Jacobi and Legendre (#215)
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‎docs/make.jl

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using Documenter, ClassicalOrthogonalPolynomials
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DocMeta.setdocmeta!(ClassicalOrthogonalPolynomials,
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:DocTestSetup,
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:(using ClassicalOrthogonalPolynomials))
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makedocs(
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modules = [ClassicalOrthogonalPolynomials],
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sitename="ClassicalOrthogonalPolynomials.jl",

‎docs/src/index.md

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```@docs
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ClassicalOrthogonalPolynomials.Chebyshev
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```
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```@docs
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ClassicalOrthogonalPolynomials.chebyshevt
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```
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```@docs
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ClassicalOrthogonalPolynomials.chebyshevu
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```
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```@docs
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ClassicalOrthogonalPolynomials.Legendre
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ClassicalOrthogonalPolynomials.legendrep
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```
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```@docs
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ClassicalOrthogonalPolynomials.Jacobi
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ClassicalOrthogonalPolynomials.jacobip
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```
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```@docs
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```
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### Weights
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```@docs
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ClassicalOrthogonalPolynomials.Weighted
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```
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```@docs
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ClassicalOrthogonalPolynomials.LegendreWeight
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ClassicalOrthogonalPolynomials.ChebyshevWeight
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ClassicalOrthogonalPolynomials.JacobiWeight
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```
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```@docs
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ClassicalOrthogonalPolynomials.LaguerreWeight
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ClassicalOrthogonalPolynomials.HalfWeighted
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```
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### Affine-mapped
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```@docs
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ClassicalOrthogonalPolynomials.legendre
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ClassicalOrthogonalPolynomials.jacobi
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ClassicalOrthogonalPolynomials.legendreweight
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ClassicalOrthogonalPolynomials.jacobiweight
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```
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### Recurrences
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```
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### Internal
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```@docs
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ClassicalOrthogonalPolynomials.ShuffledIR2HC
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```
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```@docs
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ClassicalOrthogonalPolynomials.ShuffledR2HC
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```
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```@docs
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ClassicalOrthogonalPolynomials.ShuffledFFT
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ClassicalOrthogonalPolynomials.ShuffledIFFT
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ClassicalOrthogonalPolynomials.ShuffledR2HC
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ClassicalOrthogonalPolynomials.ShuffledIR2HC
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```
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```@docs
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ClassicalOrthogonalPolynomials.qr_jacobimatrix
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```
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```@docs
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ClassicalOrthogonalPolynomials.MappedOPLayout
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```
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```@docs
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ClassicalOrthogonalPolynomials.cholesky_jacobimatrix
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```
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```@docs
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ClassicalOrthogonalPolynomials.AbstractNormalizedOPLayout
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```
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```@docs
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ClassicalOrthogonalPolynomials.ShuffledFFT
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ClassicalOrthogonalPolynomials.MappedOPLayout
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ClassicalOrthogonalPolynomials.WeightedOPLayout
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```
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```@docs
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ClassicalOrthogonalPolynomials.legendre_grammatrix
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ClassicalOrthogonalPolynomials.weightedgrammatrix
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```
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```@docs
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ClassicalOrthogonalPolynomials.WeightedOPLayout
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```
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```@docs
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ClassicalOrthogonalPolynomials.interlace!
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```
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```@docs

‎src/classical/jacobi.jl

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@@ -21,13 +21,48 @@ broadcasted(::LazyQuasiArrayStyle{1}, ::typeof(sqrt), w::AbstractJacobiWeight) =
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broadcasted(::LazyQuasiArrayStyle{1}, ::typeof(Base.literal_pow), ::Base.RefValue{typeof(^)}, w::AbstractJacobiWeight, ::Base.RefValue{Val{k}}) where k =
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JacobiWeight(k * w.a, k * w.b)
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"""
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JacobiWeight{T}(a,b)
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JacobiWeight(a,b)
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The quasi-vector representing the Jacobi weight function ``(1-x)^a (1+x)^b`` on ``[-1,1]``. See also [`jacobiweight`](@ref) and [`Jacobi`](@ref).
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# Examples
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```jldoctest
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julia> J=JacobiWeight(1.0,1.0)
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(1-x)^1.0 * (1+x)^1.0 on -1..1
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julia> J[0.5]
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0.75
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julia> axes(J)
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(Inclusion(-1.0 .. 1.0 (Chebyshev)),)
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```
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"""
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struct JacobiWeight{T} <: AbstractJacobiWeight{T}
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a::T
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b::T
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JacobiWeight{T}(a, b) where T = new{T}(convert(T,a), convert(T,b))
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end
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JacobiWeight(a::V, b::T) where {T,V} = JacobiWeight{promote_type(T,V)}(a,b)
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"""
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jacobiweight(a,b, d::AbstractInterval)
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The [`JacobiWeight`](@ref) affine-mapped to interval `d`.
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# Examples
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```jldoctest
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julia> J = jacobiweight(1, 1, 0..1)
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(1-x)^1 * (1+x)^1 on -1..1 affine mapped to 0 .. 1
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julia> axes(J)
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(Inclusion(0 .. 1),)
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julia> J[0.5]
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1.0
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```
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"""
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jacobiweight(a,b, d::AbstractInterval{T}) where T = JacobiWeight(a,b)[affine(d,ChebyshevInterval{T}())]
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AbstractQuasiArray{T}(w::JacobiWeight) where T = JacobiWeight{T}(w.a, w.b)
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singularitiesbroadcast(::typeof(*), ::LegendreWeight, b::AbstractJacobiWeight) = b
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singularitiesbroadcast(::typeof(*), a::AbstractJacobiWeight, ::LegendreWeight) = a
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"""
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Jacobi{T}(a,b)
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Jacobi(a,b)
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The quasi-matrix representing Jacobi polynomials, where the first axes represents the interval and the second axes represents the polynomial index (starting from 1). See also [`jacobi`](@ref), [`jacobip`](@ref) and [`JacobiWeight`](@ref).
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The eltype, when not specified, will be converted to a floating point data type.
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# Examples
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```jldoctest
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julia> J=Jacobi(0,0) # The eltype will be converted to float
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Jacobi(0.0, 0.0)
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julia> axes(J)
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(Inclusion(-1.0 .. 1.0 (Chebyshev)), OneToInf())
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julia> J[0,:] # Values of polynomials at x=0
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ℵ₀-element view(::Jacobi{Float64}, 0.0, :) with eltype Float64 with indices OneToInf():
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1.0
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0.0
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-0.5
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-0.0
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0.375
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0.0
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-0.3125
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-0.0
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0.2734375
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0.0
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⋮
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julia> J0=J[:,1]; # J0 is the first Jacobi polynomial which is constant.
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julia> J0[0],J0[0.5]
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(1.0, 1.0)
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```
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"""
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struct Jacobi{T} <: AbstractJacobi{T}
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a::T
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b::T
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AbstractQuasiArray{T}(w::Jacobi) where T = Jacobi{T}(w.a, w.b)
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AbstractQuasiMatrix{T}(w::Jacobi) where T = Jacobi{T}(w.a, w.b)
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"""
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jacobi(a,b, d::AbstractInterval)
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The [`Jacobi`](@ref) polynomials affine-mapped to interval `d`.
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# Examples
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```jldoctest
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julia> J = jacobi(1, 1, 0..1)
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Jacobi(1.0, 1.0) affine mapped to 0 .. 1
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julia> axes(J)
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(Inclusion(0 .. 1), OneToInf())
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julia> J[0,:]
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ℵ₀-element view(::Jacobi{Float64}, -1.0, :) with eltype Float64 with indices OneToInf():
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1.0
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-2.0
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3.0
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-4.0
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5.0
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-6.0
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7.0
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-8.0
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9.0
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-10.0
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⋮
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```
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"""
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jacobi(a,b) = Jacobi(a,b)
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jacobi(a,b, d::AbstractInterval{T}) where T = Jacobi{float(promote_type(eltype(a),eltype(b),T))}(a,b)[affine(d,ChebyshevInterval{T}()), :]
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‎src/classical/legendre.jl

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"""
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LegendreWeight{T}()
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LegendreWeight()
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The quasi-vector representing the Legendre weight function (const 1) on [-1,1]. See also [`legendreweight`](@ref) and [`Legendre`](@ref).
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# Examples
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```jldoctest
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julia> w = LegendreWeight()
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LegendreWeight{Float64}()
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julia> w[0.5]
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1.0
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julia> axes(w)
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(Inclusion(-1.0 .. 1.0 (Chebyshev)),)
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```
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"""
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struct LegendreWeight{T} <: AbstractJacobiWeight{T} end
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LegendreWeight() = LegendreWeight{Float64}()
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"""
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legendreweight(d::AbstractInterval)
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The [`LegendreWeight`](@ref) affine-mapped to interval `d`.
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# Examples
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```jldoctest
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julia> w = legendreweight(0..1)
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LegendreWeight{Float64}() affine mapped to 0 .. 1
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julia> axes(w)
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(Inclusion(0 .. 1),)
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julia> w[0]
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1.0
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```
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"""
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legendreweight(d::AbstractInterval{T}) where T = LegendreWeight{float(T)}()[affine(d,ChebyshevInterval{T}())]
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AbstractQuasiArray{T}(::LegendreWeight) where T = LegendreWeight{T}()
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basis_singularities(::LegendreWeight{T}) where T = Legendre{T}()
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basis_singularities(v::SubQuasiArray) = view(basis_singularities(parent(v)), parentindices(v)[1], :)
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"""
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Legendre{T=Float64}(a,b)
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The quasi-matrix representing Legendre polynomials, where the first axes represents the interval and the second axes represents the polynomial index (starting from 1). See also [`legendre`](@ref), [`legendrep`](@ref), [`LegendreWeight`](@ref) and [`Jacobi`](@ref).
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# Examples
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```jldoctest
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julia> P = Legendre()
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Legendre()
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julia> typeof(P) # default eltype
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Legendre{Float64}
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julia> axes(P)
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(Inclusion(-1.0 .. 1.0 (Chebyshev)), OneToInf())
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julia> P[0,:] # Values of polynomials at x=0
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ℵ₀-element view(::Legendre{Float64}, 0.0, :) with eltype Float64 with indices OneToInf():
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1.0
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0.0
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-0.5
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-0.0
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0.375
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0.0
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-0.3125
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-0.0
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0.2734375
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0.0
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⋮
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julia> P₀=P[:,1]; # P₀ is the first Legendre polynomial which is constant.
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julia> P₀[0],P₀[0.5]
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(1.0, 1.0)
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```
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"""
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struct Legendre{T} <: AbstractJacobi{T} end
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Legendre() = Legendre{Float64}()
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weighted(P::SubQuasiArray{<:Any,2,<:Legendre}) = P
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weighted(P::SubQuasiArray{<:Any,2,<:Normalized{<:Any,<:Legendre}}) = P
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"""
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legendre(d::AbstractInterval)
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The [`Legendre`](@ref) polynomials affine-mapped to interval `d`.
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# Examples
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```jldoctest
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julia> P = legendre(0..1)
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Legendre() affine mapped to 0 .. 1
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julia> axes(P)
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(Inclusion(0 .. 1), OneToInf())
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julia> P[0.5,:]
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ℵ₀-element view(::Legendre{Float64}, 0.0, :) with eltype Float64 with indices OneToInf():
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1.0
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0.0
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-0.5
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-0.0
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0.375
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0.0
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-0.3125
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-0.0
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0.2734375
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0.0
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⋮
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```
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"""
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legendre() = Legendre()
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legendre(d::AbstractInterval{T}) where T = Legendre{float(T)}()[affine(d,ChebyshevInterval{T}()), :]
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legendre(d::ChebyshevInterval{T}) where T = Legendre{float(T)}()

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