Analysis of a novel preconditioner for a class of p-level lower rank extracted systems

This paper proposes and studies the performance of a preconditioner suitable for solving a class of symmetric positive definite systems, Âx = b, which we call p-level lower rank extracted systems (p-level LRES), by the preconditioned conjugate gradient method. The study of these systems is motivated by the numerical approximation of integral equations with convolution kernels defined on arbitrary p-dimensional domains. This is in contrast to p-level Toeplitz systems which only apply to rectangular domains. The coefficient matrix, Â, is a principal submatrix of a p-level Toeplitz matrix, A, and the preconditioner for the preconditioned conjugate gradient algorithm is provided in terms of the inverse of a p-level circulant matrix constructed from the elements of A. The preconditioner is shown to yield clustering in the spectrum of the preconditioned matrix which leads to a substantial reduction in the computational cost of solving LRE systems.