10 research outputs found
A fast and well-conditioned spectral method for singular integral equations
We develop a spectral method for solving univariate singular integral
equations over unions of intervals by utilizing Chebyshev and ultraspherical
polynomials to reformulate the equations as almost-banded infinite-dimensional
systems. This is accomplished by utilizing low rank approximations for sparse
representations of the bivariate kernels. The resulting system can be solved in
operations using an adaptive QR factorization, where is
the bandwidth and is the optimal number of unknowns needed to resolve the
true solution. The complexity is reduced to operations by
pre-caching the QR factorization when the same operator is used for multiple
right-hand sides. Stability is proved by showing that the resulting linear
operator can be diagonally preconditioned to be a compact perturbation of the
identity. Applications considered include the Faraday cage, and acoustic
scattering for the Helmholtz and gravity Helmholtz equations, including
spectrally accurate numerical evaluation of the far- and near-field solution.
The Julia software package SingularIntegralEquations.jl implements our method
with a convenient, user-friendly interface
Polynomial and rational measure modifications of orthogonal polynomials via infinite-dimensional banded matrix factorizations
We describe fast algorithms for approximating the connection coefficients
between a family of orthogonal polynomials and another family with a
polynomially or rationally modified measure. The connection coefficients are
computed via infinite-dimensional banded matrix factorizations and may be used
to compute the modified Jacobi matrices all in linear complexity with respect
to the truncation degree. A family of orthogonal polynomials with modified
classical weights is constructed that support banded differentiation matrices,
enabling sparse spectral methods with modified classical orthogonal
polynomials