juliaapproximation / algebraiccurveorthogonalpolynomials.jl Goto Github PK

Could we bump the dependencies for AlgebraicPolys from

ClassicalOPs v0.8.1 -> v0.9.0
SemiclassicalOPs v0.3.4 -> 0.3.5

so that we can start using Cholesky for ZernikeAnnulus? Can we also make a release for AlgebraicCurveOrthogonalPolynomials.jl? @dlfivefifty

I am trying to get for instance

ZernikeAnnulus(0.5,1,1) \ Weighted(ZernikeAnnulus(0.5,1,1))

working. If I uncomment lines 126-132 in src/annulus.jl I get the error (just to get the \ command working, I know the matrix output is incorrect) then I get the error:

ERROR: ArgumentError: Override == to compare bases of type Weighted{Float64, ZernikeAnnulus{Float64}} and ZernikeAnnulus{Float64}
Stacktrace:
 [1] _equals(#unused#::ContinuumArrays.WeightedBasisLayout{ContinuumArrays.BasisLayout}, #unused#::ContinuumArrays.BasisLayout, A::Weighted{Float64, ZernikeAnnulus{Float64}}, B::ZernikeAnnulus{Float64})
   @ ContinuumArrays C:\Users\john.papad\.julia\packages\ContinuumArrays\qOVxC\src\bases\bases.jl:47
 [2] ==
   @ C:\Users\john.papad\.julia\packages\QuasiArrays\hPWyG\src\abstractquasiarray.jl:521 [inlined]
 [3] copy
   @ C:\Users\john.papad\.julia\packages\ContinuumArrays\qOVxC\src\bases\bases.jl:73 [inlined]
 [4] materialize
   @ C:\Users\john.papad\.julia\packages\ArrayLayouts\DX7ic\src\ldiv.jl:22 [inlined]
 [5] ldiv
   @ C:\Users\john.papad\.julia\packages\ArrayLayouts\DX7ic\src\ldiv.jl:86 [inlined]
 [6] \(A::Weighted{Float64, ZernikeAnnulus{Float64}}, B::ZernikeAnnulus{Float64})
   @ QuasiArrays C:\Users\john.papad\.julia\packages\QuasiArrays\hPWyG\src\matmul.jl:34
 [7] top-level scope
   @ REPL[96]:1

An idea occurred to me: Let P(x,y) be a family of OPs on the circle w.r.t. w(x) and Q(x,y) w.r.t to sqrt(1-x^2)*w. Let y = (1-x^2)^(1/4). Then can't we write quartic OPs in terms of P(x,y^2) and y*Q(x,y^2)?

@MikaelSlevinsky is there a way to do a Q&D transform for OPs in the annulus?

I've started thinking about curvature flow in 2D. We will need a procedure to expand the components of κ*n into HermLaurent as they are typically not identically polynomial.

The best way may be to first support general matrix-spaces. This will likely need a new type called HvcatInterlace or similar...

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@MikaelSlevinsky's transform work by first expanding in a tensor product basis, which contains non-polynomial terms, before transforming to a polynomial basis. For example, to compute OPs on the triangle we expand in the basis

P_k(x) P_j(y/(1-x))

(I'm using OPs on 0..1 here) which includes non-polynomial terms y/(1-x)^j, before converting to the polynomial basis

P_{n-k}^(2k+1,0)(x)(1-x)^kP_k(y/(1-x))

The point is polynomials in x and y are a subspace of the non-polynomial basis.

Does this technique translate to OPs on the disk (a la Dunkl & Xu, not Zernike)

C_{n-k}^(k)(x)*ρ(x)^k*T_k(y/ρ(x))

where ρ(x) = sqrt(1-x^2)? A tensor basis like

T_k(x) T_j(y/ρ(x))

does not seem to work here... seems like we'd need a sum-space frame that also includes ρ(x)*T_k(x)*T_j(y/ρ(x))....

What about ρ(x) = (1-x^4)^(1/4)? Now we wouldn't know the OP basis but perhaps we can represent it by the conversion to a bigger basis a la LanczosPolynomial.

I want to make this a package, to help prevent making breaking changes. First step is to rename it

A question of performance (for code that I wrote). I have noticed that creating finite subsections of matrices can sometimes be slow:

julia> Z = ZernikeAnnulus(0.5,1,1); wZ  = Weighted(Z);
julia> @time Δ = Z \ (Laplacian(axes(Z,1)) * wZ);
  0.002635 seconds (2.40 k allocations: 240.312 KiB)
julia> @time Δ[Block.(1:20),Block.(1:20)];
  0.185276 seconds (498.23 k allocations: 26.108 MiB, 8.23% gc time)

versus

julia> @time L = Z \ wZ;
  0.000005 seconds (2 allocations: 256 bytes)
julia> @time L[Block.(1:20), Block.(1:20)];
  5.623841 seconds (7.67 M allocations: 605.328 MiB, 1.50% gc time)

Even if I rewrite Z \ wZ using BroadcastVector, I get similar timings.. Any ideas @dlfivefifty?

I think we can do any star like geometry as long as we can do the boundary: consider it as

p(x/z,y/z) = 1

for z = 0..1 (disk being classic example). Then we can construct OPs in 3-variables x,y,z from boundary OPs in 2 variables Y_{m,k}(x,y) as

Q_{m,k,i}(x,y,z) = P_k^(0,2m+1)(2z-1) * z^m * Y_{m,i}(x/z,y/z)

Claims to be double checked:

1. This spans all polynomials in x,y,z mod the constraint p(x/z,y/z) =1, which is in fact a polynomial constraint: if p is degree d just multiply through by z^d.
2. This basis is orthogonal w.r.t. \int_0^1 \int_{z B} f(x,y,z) g(x,y,z) dx dy dz

To get back to OPs in 2-variables we would then construct the connection matrix. Since Q_{m,k,i} spans all polynomials, it contains 2-variable polynomials as a sub space. We can compute this connection matrix by Lanczos (that is, multiply by x and y).