SingularIntegralEquations.jl
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JuliaApproximation
Julia package for solving singular integral equations
This suggests about a 2x speed increase: julia> r=rand(10000000);@time sqrt(r); 0.049254 seconds (6 allocations: 76.294 MB, 11.57% gc time) julia> r=rand(10000000);@time AppleAccelerate.sqrt(r); 0.026610 seconds (8 allocations: 76.294 MB, 21.70% gc...
One approach is to embed the domain between an inner and outer ellipse (or in the simplest case, an annulus). Then (mapped) Laurent series can be used instead of the...
If the Hilbert transform of the second partial derivative is diagonal in some basis, then time-evolution could be taken care of using a fourth-order time-stepper for stiff PDEs, like [this](https://people.maths.ox.ac.uk/trefethen/publication/PDF/2005_111.pdf).
Should higher order singular integral operators be implemented as (normal) derivatives of log|y-x|?
One could then resolve the fundamental solution as A*log|y-x| + B, take the normal derivative wrt y, and this would be ∂A \* log|y-x| + A \* ∂(log|y-x|) + ∂B....
Should OffHilbert be multiplied by -π and renamed Stieltjes? This matches the function call stieltjes which is now overwritten.
This would be useful for fractal screens of constant coefficient self-adjoint PDOs. (Only half the inner products are needed and a simpler factorization can occur as Schur complements are symmetric.)
and have its own variant of linsolve. 'nuff sed.
Banded linear algebra is arguably inapplicable to these operators, setting them on the path to being amenable to hierarchical solvers. Should we also combine them with Hilbert/SingularIntegral and have them...
It's currently covered by `logkernel(Fun(u,Fourier),z)`, but it would be better to have both formulae.
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Julia package for solving singular integral equations