juliaapproximation / algebraiccurveorthogonalpolynomials.jl Goto Github PK
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License: MIT License
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:
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
.\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
).
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