1 Backward

Abstract

We present a multigrid algorithm to solve linear sistems whose coefficient metrices belongs to circulant, Hartley or τ multilevel algebras and are generated by a nonnegative multivariate polinomial f. It is known that these matrices are banded (with respect to their multilevel structure) and their eigenvalues are obtained by sampling f on uniform meshes, so they are ill-conditioned (or singular, and need some corrections) whenever f takes the zero value. We prove the proposed metod to be optimal even in presence of ill-conditioning: if the multilevel coefficient matrix has dimension ni at level i, i = 1,..., d, then only N(n) = �d i=1 ni operations are required on each iteration, but the convergence rate keeps constant with respect to N(n) as it depends only on f. Some numerical experiments on PDEs are considered and discussed