latexify docs

Author: MikaelSlevinsky

docs/src/index.md

@@ -51,6 +51,30 @@ paduapoints
 
 ## Internal Methods
 
+```@docs
+FastTransforms.half
+```
+
+```@docs
+FastTransforms.two
+```
+
+```@docs
+FastTransforms.δ
+```
+
+```@docs
+FastTransforms.Λ
+```
+
+```@docs
+FastTransforms.pochhammer
+```
+
+```@docs
+FastTransforms.stirlingseries
+```
+
 ```@docs
 FastTransforms.clenshawcurtis
 ```
@@ -76,25 +100,33 @@ FastTransforms.fejerweights2
 ```
 
 ```@docs
-FastTransforms.half
+FastTransforms.chebyshevjacobimoments1
 ```
 
 ```@docs
-FastTransforms.two
+FastTransforms.chebyshevjacobimoments2
 ```
 
 ```@docs
-FastTransforms.δ
+FastTransforms.incrementα!
 ```
 
 ```@docs
-FastTransforms.Λ
+FastTransforms.incrementβ!
 ```
 
 ```@docs
-FastTransforms.pochhammer
+FastTransforms.incrementαβ!
 ```
 
 ```@docs
-FastTransforms.stirlingseries
+FastTransforms.decrementα!
+```
+
+```@docs
+FastTransforms.decrementβ!
+```
+
+```@docs
+FastTransforms.decrementαβ!
 ```

src/PaduaTransform.jl

@@ -41,7 +41,7 @@ Inverse Padua Transform maps the 2D Chebyshev coefficients to the values of the
 ipaduatransform(v::AbstractVector,lex...) = plan_ipaduatransform!(v,lex...)*copy(v)
 
 doc"""
-Creates (n+2)x(n+1) Chebyshev coefficient matrix from triangle coefficients.
+Creates ``(n+2)x(n+1)`` Chebyshev coefficient matrix from triangle coefficients.
 """
 function trianglecfsmat(P::IPaduaTransformPlan{true},cfs::AbstractVector)
     N=length(cfs)
@@ -146,7 +146,7 @@ Padua Transform maps from interpolant values at the Padua points to the 2D Cheby
 paduatransform(v::AbstractVector,lex...) = plan_paduatransform!(v,lex...)*copy(v)
 
 doc"""
-Creates (n+2)x(n+1) matrix of interpolant values on the tensor grid at the (n+1)(n+2)/2 Padua points.
+Creates ``(n+2)x(n+1)`` matrix of interpolant values on the tensor grid at the ``(n+1)(n+2)/2`` Padua points.
 """
 function paduavalsmat(P::PaduaTransformPlan,v::AbstractVector)
     N=length(v)
@@ -166,7 +166,7 @@ function paduavalsmat(P::PaduaTransformPlan,v::AbstractVector)
 end
 
 doc"""
-Creates length (n+1)(n+2)/2 vector from matrix of triangle Chebyshev coefficients.
+Creates length ``(n+1)(n+2)/2`` vector from matrix of triangle Chebyshev coefficients.
 """
 function trianglecfsvec!(v,P::PaduaTransformPlan{true},cfs::Matrix)
     m=size(cfs,2)
@@ -195,7 +195,7 @@ function trianglecfsvec!(v,P::PaduaTransformPlan{false},cfs::Matrix)
 end
 
 doc"""
-Returns coordinates of the (n+1)(n+2)/2 Padua points.
+Returns coordinates of the ``(n+1)(n+2)/2`` Padua points.
 """
 function paduapoints{T}(::Type{T},n::Integer)
     N=div((n+1)*(n+2),2)

src/cjt.jl

@@ -114,25 +114,31 @@ end
 
 doc"""
 Computes the Chebyshev expansion coefficients
-given the Jacobi expansion coefficients ``c`` with parameters ``α`` and ``β``.
+given the Jacobi expansion coefficients ``c`` with parameters ``\alpha`` and ``\beta``:
 
-See also [`icjt`](#method__icjt.1) and [`jjt`](#method__jjt.1).
+```math
+{\rm CJT} : \sum_{n=0}^N c_n^{\rm jac}P_n^{(\alpha,\beta)}(x) \to \sum_{n=0}^N c_n^{\rm cheb}T_n(x).
+```
 """
 cjt(c,α,β) = plan_cjt(c,α,β)*c
 
 doc"""
-Computes the Jacobi expansion coefficients with parameters ``α`` and ``β``
-given the Chebyshev expansion coefficients ``c``.
+Computes the Jacobi expansion coefficients with parameters ``\alpha`` and ``\beta``
+given the Chebyshev expansion coefficients ``c``:
 
-See also [`cjt`](#method__cjt.1) and [`jjt`](#method__jjt.1).
+```math
+{\rm iCJT} : \sum_{n=0}^N c_n^{\rm cheb}T_n(x) \to \sum_{n=0}^N c_n^{\rm jac}P_n^{(\alpha,\beta)}(x).
+```
 """
 icjt(c,α,β) = plan_icjt(c,α,β)*c
 
 doc"""
-Computes the Jacobi expansion coefficients with parameters ``γ`` and ``δ``
-given the Jacobi expansion coefficients ``c`` with parameters ``α`` and ``β``.
+Computes the Jacobi expansion coefficients with parameters ``\gamma`` and ``\delta``
+given the Jacobi expansion coefficients ``c`` with parameters ``\alpha`` and ``\beta``:
 
-See also [`cjt`](#method__cjt.1) and [`icjt`](#method__icjt.1).
+```math
+{\rm JJT} : \sum_{n=0}^N c_n^{\rm jac}P_n^{(\alpha,\beta)}(x) \to \sum_{n=0}^N c_n^{\rm jac}P_n^{(\gamma,\delta)}(x).
+```
 """
 function jjt(c,α,β,γ,δ)
     if isapprox(α,γ) && isapprox(β,δ)
@@ -149,7 +155,7 @@ arrays, normalization constants, and recurrence coefficients for a forward Cheby
 
 ``c`` is the vector of coefficients; and,
 
-``α`` and ``β`` are the Jacobi parameters.
+``\alpha`` and ``\beta`` are the Jacobi parameters.
 
 Optionally:
 
@@ -163,7 +169,7 @@ arrays, normalization constants, and recurrence coefficients for an inverse Cheb
 
 ``c`` is the vector of coefficients; and,
 
-``α`` and ``β`` are the Jacobi parameters.
+``\alpha`` and ``\beta`` are the Jacobi parameters.
 
 Optionally:
 

src/gaunt.jl

@@ -1,15 +1,18 @@
 doc"""
 Calculates the Gaunt coefficients, defined by:
 
-    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.
-
+```math
+a(m,n,\mu,\nu,q) = \frac{2(n+\nu-2q)+1}{2} \frac{(n+\nu-2q-m-\mu)!}{(n+\nu-2q+m+\mu)!} \int_{-1}^{+1} P_m^n(x) P_\nu^\mu(x) P_{n+\nu-2q}^{m+\mu}(x) {\rm\,d}x.
+```
 or defined by:
 
-    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)
+```math
+P_n^m(x) P_\nu^\mu(x) = \sum_{q=0}^{q_{\rm max}} a(m,n,\mu,\nu,q) P_{n+\nu-2q}^{m+\mu}(x)
+```
 
 This is a Julia implementation of the stable recurrence described in:
 
-    Y.-l. Xu, "Fast evaluation of Gaunt coefficients: recursive approach", J. Comp. Appl. Math., 85:53–65, 1997.
+Y.-l. Xu, Fast evaluation of Gaunt coefficients: recursive approach, *J. Comp. Appl. Math.*, **85**:53–65, 1997.
 """
 function gaunt{T}(::Type{T},m::Int,n::Int,μ::Int,ν::Int;normalized::Bool=false)
     if normalized

src/specialfunctions.jl

@@ -21,13 +21,17 @@ two(x::Number) = oftype(x,2)
 two{T<:Number}(::Type{T}) = convert(T,2)
 
 doc"""
-The Kronecker δ function.
+The Kronecker ``\delta`` function:
+
+```math
+\delta_{k,j} = \left\{\begin{array}{ccc} 1 & {\rm for} & k = j,\\ 0 & {\rm for} & k \ne j.\end{array}\right.
+```
 """
 δ(k::Integer,j::Integer) = k == j ? 1 : 0
 
 
 doc"""
-Pochhammer symbol $(x)_n = \frac{\Gamma(x+n)}{\Gamma(x)}$ for the rising factorial.
+Pochhammer symbol ``(x)_n = \frac{\Gamma(x+n)}{\Gamma(x)}`` for the rising factorial.
 """
 function pochhammer(x::Number,n::Integer)
     ret = one(x)
@@ -58,7 +62,7 @@ function pochhammer{T<:Real}(x::Number,n::UnitRange{T})
 end
 
 doc"""
-Stirling series for Γ(z).
+Stirling's asymptotic series for ``\Gamma(z)``.
 """
 stirlingseries(z) = gamma(z)*sqrt((z/π)/2)*exp(z)/z^z
 
@@ -136,13 +140,13 @@ Anαβ{T<:Integer}(n::AbstractMatrix{T},α::Number,β::Number) = [ Anαβ(n[i,j]
 
 
 doc"""
-The Lambda function Λ(z) = Γ(z+½)/Γ(z+1) for the ratio of gamma functions.
+The Lambda function ``\Lambda(z) = \frac{\Gamma(z+\frac{1}{2})}{\Gamma(z+1)}`` for the ratio of gamma functions.
 """
 Λ(z::Number) = exp(lgamma(z+half(z))-lgamma(z+one(z)))
 doc"""
-For 64-bit floating-point arithmetic, the Lambda function uses the asymptotic series for τ in Appendix B of
+For 64-bit floating-point arithmetic, the Lambda function uses the asymptotic series for ``\tau`` in Appendix B of
 
-    I. Bogaert and B. Michiels and J. Fostier, 𝒪(1) computation of Legendre polynomials and Gauss–Legendre nodes and weights for parallel computing, SIAM J. Sci. Comput., 34:C83–C101, 2012.
+I. Bogaert and B. Michiels and J. Fostier, 𝒪(1) computation of Legendre polynomials and Gauss–Legendre nodes and weights for parallel computing, *SIAM J. Sci. Comput.*, **34**:C83–C101, 2012.
 """
 function Λ(x::Float64)
     if x > 9.84475
@@ -153,9 +157,8 @@ function Λ(x::Float64)
     end
 end
 
-
 doc"""
-The Lambda function Λ(z,λ₁,λ₂) = Γ(z+λ₁)/Γ(z+λ₂) for the ratio of gamma functions.
+The Lambda function ``\Lambda(z,λ₁,λ₂) = \frac{\Gamma(z+\lambda_1)}{Γ(z+\lambda_2)}`` for the ratio of gamma functions.
 """
 Λ(z::Number,λ₁::Number,λ₂::Number) = exp(lgamma(z+λ₁)-lgamma(z+λ₂))
 function Λ(x::Float64,λ₁::Float64,λ₂::Float64)
@@ -393,8 +396,9 @@ end
 doc"""
 Modified Chebyshev moments of the first kind with respect to the Jacobi weight:
 
-    ∫₋₁⁺¹ T_n(x) (1-x)^α(1+x)^β dx.
-
+```math
+    \int_{-1}^{+1} T_n(x) (1-x)^\alpha(1+x)^\beta{\rm\,d}x.
+```
 """
 function chebyshevjacobimoments1{T<:AbstractFloat}(N::Int,α::T,β::T)
     μ = zeros(T,N)
@@ -411,8 +415,9 @@ end
 doc"""
 Modified Chebyshev moments of the second kind with respect to the Jacobi weight:
 
-    ∫₋₁⁺¹ U_n(x) (1-x)^α(1+x)^β dx.
-
+```math
+    \int_{-1}^{+1} U_n(x) (1-x)^\alpha(1+x)^\beta{\rm\,d}x.
+```
 """
 function chebyshevjacobimoments2{T<:AbstractFloat}(N::Int,α::T,β::T)
     μ = zeros(T,N)
@@ -427,7 +432,7 @@ function chebyshevjacobimoments2{T<:AbstractFloat}(N::Int,α::T,β::T)
 end
 
 doc"""
-Compute Jacobi expansion coefficients in Pₙ^(α+1,β) given Jacobi expansion coefficients in Pₙ^(α,β) in-place.
+Compute Jacobi expansion coefficients in ``P_n^{(\alpha+1,\beta)}(x)`` given Jacobi expansion coefficients in ``P_n^{(\alpha,\beta)}(x)`` in-place.
 """
 function incrementα!(c::AbstractVector,α,β)
     αβ,N = α+β,length(c)
@@ -438,7 +443,7 @@ function incrementα!(c::AbstractVector,α,β)
 end
 
 doc"""
-Compute Jacobi expansion coefficients in Pₙ^(α,β+1) given Jacobi expansion coefficients in Pₙ^(α,β) in-place.
+Compute Jacobi expansion coefficients in ``P_n^{(\alpha,\beta+1)}(x)`` given Jacobi expansion coefficients in ``P_n^{(\alpha,\beta)}(x)`` in-place.
 """
 function incrementβ!(c::AbstractVector,α,β)
     αβ,N = α+β,length(c)
@@ -449,7 +454,7 @@ function incrementβ!(c::AbstractVector,α,β)
 end
 
 doc"""
-Compute Jacobi expansion coefficients in Pₙ^(α+1,α+1) given Jacobi expansion coefficients in Pₙ^(α,α) in-place.
+Compute Jacobi expansion coefficients in ``P_n^{(\alpha+1,\alpha+1)}(x)`` given Jacobi expansion coefficients in ``P_n^{(\alpha,\alpha)}(x)`` in-place.
 """
 function incrementαβ!(c::AbstractVector,α,β)
     @assert α == β
@@ -462,7 +467,7 @@ function incrementαβ!(c::AbstractVector,α,β)
 end
 
 doc"""
-Compute Jacobi expansion coefficients in Pₙ^(α-1,β) given Jacobi expansion coefficients in Pₙ^(α,β) in-place.
+Compute Jacobi expansion coefficients in ``P_n^{(\alpha-1,\beta)}(x)`` given Jacobi expansion coefficients in ``P_n^{(\alpha,\beta)}(x)`` in-place.
 """
 function decrementα!(c::AbstractVector,α,β)
     αβ,N = α+β,length(c)
@@ -473,7 +478,7 @@ function decrementα!(c::AbstractVector,α,β)
 end
 
 doc"""
-Compute Jacobi expansion coefficients in Pₙ^(α,β-1) given Jacobi expansion coefficients in Pₙ^(α,β) in-place.
+Compute Jacobi expansion coefficients in ``P_n^{(\alpha,\beta-1)}(x)`` given Jacobi expansion coefficients in ``P_n^{(\alpha,\beta)}(x)`` in-place.
 """
 function decrementβ!(c::AbstractVector,α,β)
     αβ,N = α+β,length(c)
@@ -484,7 +489,7 @@ function decrementβ!(c::AbstractVector,α,β)
 end
 
 doc"""
-Compute Jacobi expansion coefficients in Pₙ^(α-1,α-1) given Jacobi expansion coefficients in Pₙ^(α,α) in-place.
+Compute Jacobi expansion coefficients in ``P_n^{(\alpha-1,\alpha-1)}(x)`` given Jacobi expansion coefficients in ``P_n^{(\alpha,\alpha)}(x)`` in-place.
 """
 function decrementαβ!(c::AbstractVector,α,β)
     @assert α == β