# ClassicalOrthogonalPolynomials.jl

A Julia package for classical orthogonal polynomials and expansions

This package implements classical orthogonal polynomials.

```
julia> using ClassicalOrthogonalPolynomials, ContinuumArrays
julia> chebyshevt.(0:5,0.1)
6-element Array{Float64,1}:
1.0
0.1
-0.98
-0.296
0.9208
0.48016
```

Other examples include `chebyshevu`

, `legendrep`

, `jacobip`

, `ultrasphericalc`

, `hermiteh`

and `laguerrel`

.

For expansion, it supports usafe as quasi-arrays where one one axes is continuous and the other axis is discrete (countably infinite), as implemented in QuasiArrays.jl and ContinuumArrays.jl.

```
julia> P = Legendre(); # Legendre polynomials
julia> size(P) # uncountable ∞ x countable ∞
(ℵ₁, ∞)
julia> axes(P) # essentially (-1..1, 1:∞), Inclusion plays the same role as Slice
(Inclusion(-1.0..1.0 (Chebyshev)), OneToInf())
julia> P[0.1,1:10] # [P_0(0.1), …, P_9(0.1)]
10-element Array{Float64,1}:
1.0
0.1
-0.485
-0.14750000000000002
0.3379375
0.17882875
-0.2488293125
-0.19949294375000004
0.180320721484375
0.21138764183593753
julia> @time P[range(-1,1; length=10_000), 1:10_000]; # construct 10_000^2 Vandermonde matrix
1.624796 seconds (10.02 k allocations: 1.491 GiB, 6.81% gc time)
```

This also works for associated Legendre polynomials as weighted Ultraspherical polynomials:

```
julia> associatedlegendre(m) = ((-1)^m*prod(1:2:(2m-1)))*(UltrasphericalWeight((m+1)/2).*Ultraspherical(m+1/2))
associatedlegendre (generic function with 1 method)
julia> associatedlegendre(2)[0.1,1:10]
10-element Array{Float64,1}:
2.9699999999999998
1.4849999999999999
-6.9052500000000006
-5.041575
10.697754375
10.8479361375
-13.334647528125
-18.735466024687497
13.885467170308594
28.220563705988674
```

## p-Finite Element Method

The language of quasi-arrays gives a natural framework for constructing p-finite element methods. The convention is that adjoint-products are understood as inner products over the axes with uniform weight. To solve Poisson's equation using its weak formulation with Dirichlet conditions we can thus expand in a weighted Jacobi basis:

```
julia> P¹¹ = Jacobi(1.0,1.0); # Quasi-matrix of Jacobi polynomials
julia> w = JacobiWeight(1.0,1.0); # quasi-vector correspoinding to (1-x^2)
julia> w[0.1] ≈ (1-0.1^2)
true
julia> S = w .* P¹¹; # Quasi-matrix of weighted Jacobi polynomials
julia> D = Derivative(axes(S,1)); # quasi-matrix corresponding to derivative
julia> Δ = (D*S)'*(D*S) # weak laplacian corresponding to inner products of weighted Jacobi polynomials
∞×∞ LazyArrays.ApplyArray{Float64,2,typeof(*),Tuple{Adjoint{Int64,BandedMatrices.BandedMatrix{Int64,Adjoint{Int64,InfiniteArrays.InfStepRange{Int64,Int64}},InfiniteArrays.OneToInf{Int64}}},LazyArrays.BroadcastArray{Float64,2,typeof(*),Tuple{LazyArrays.BroadcastArray{Float64,1,typeof(/),Tuple{Int64,InfiniteArrays.InfStepRange{Int64,Int64}}},BandedMatrices.BandedMatrix{Int64,Adjoint{Int64,InfiniteArrays.InfStepRange{Int64,Int64}},InfiniteArrays.OneToInf{Int64}}}}}} with indices OneToInf()×OneToInf():
2.66667 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ …
⋅ 6.4 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ 10.2857 ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ 14.2222 ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ 18.1818 ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ 22.1538 ⋅ ⋅ …
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 26.1333 ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 30.1176
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ …
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋮ ⋮ ⋱
```

## Related Packages

**HarmonicOrthogonalPolynomials.jl**: adds support for spherical harmonic expansions and harmonic polynomials in balls.

**MultivariateOrthogonalPolynomials.jl**: adds support for multivariate orthogonal polynomials on disks, spheres, triangles, and other simple geometries

**SemiclassicalOrthogonalPolynomials.jl**: adds support for semiclassical orthogonal polynomials such as Semiclassical Jacobi polynomials with weights `x^a * (1-x)^b * (t-x)^c`

.

**PiecewiseOrthogonalPolynomials.jl**: adds support for piecewise orthogonal polynomials which can be used in p-FEM.