A preconditioner for systems with symmetric Toeplitz blocks

This paper proposes and studies the performance of a preconditioner used in the preconditioned conjugate gradient method for solving a class of symmetric positive definite systems, A_px = b, which we call Lower Rank Extracted Systems (LRES). These systems correspond to integral equations with convolution kernels defined on a union of many line segments in contrast to only one line segment in the case of Toeplitz systems. The p × p matrix, A_p, is shown to be a principal submatrix of a larger N × N Toeplitz matrix, A_N. The preconditioner is provided in terms of the inverse of a 2N × 2N circular matrix constructed from the elements of A_N. The preconditioner is shown to yield clustering in the spectrum of preconditioned matrix similar to the clustering results in iterative algorithms used to solve Toeplitz systems. The analysis further demonstrates that the computational expense to solve LRE systems is reduced to O(N log N).