resolvant-approx

Approximating for several complex poles and right hand sides b.

This project provides a matlab function resolvantApprox(K, M, b, xis, tols, cheb_max_order) and its dependencies that allows approximating the solutions to complex-shifted linear systems with Chebyshev approximation. Each system solution is approximated up to a specified tolerance error.

Installing

Simply launch the compile.m matlab file, which should compile the ./src/chebyProj.cpp file into a ./chebyProj.* matlab routine.

How to use it

Here is an example for 3 complex poles and a uniform tolerance of 10^-9, a discretized Laplacian operator, and one vector right hand side b.

J = 15; h = 2/J;
K = 0.02*gallery('poisson',J-1)/h^2;            % *negative* 2D Laplacian (positive spectrum)
n = length(K);
M = 3*speye(n, n);                              % symmetric definite positive matrix

b = rand(n, 1); b = b/norm(b);                  % normalized right hand side
xis = [ -9.5-2.0i, -11.7+0.5i, -8.7+0.1i];      % 3 poles
tols = ones(length(xis), 1) * 1e-9;             % uniform 1e-9 tolerance
cheb_max_order = 25;                            % maximum chebyshev order

out = resolvantApprox(K, M, b, xis, tols, cheb_max_order);

How it works

The method used for constructing the approximation along with an API reference will be detailed in approximantConstruction.pdf.