A fast solver for linear systems with displacement structure

Abstract

We describe a fast solver for linear systems with reconstructible Cauchy-like structure, which requires O(rn2) floating point operations and O(rn) memory locations, where n is the size of the matrix and r its displacement rank. The solver is based on the application of the generalized Schur algorithm to a suitable augmented matrix, under some assumptions on the knots of the Cauchy-like matrix. It includes various pivoting strategies, already discussed in the literature, and a new algorithm, which only requires reconstructibility. We have developed a software package, written inMatlab and C-MEX, which provides a robust implementation of the above method. Our package also includes solvers for Toeplitz(+Hankel)-like and Vandermonde-like linear systems, as these structures can be reduced to Cauchy-like by fast and stable transforms. Numerical experiments demonstrate the effectiveness of the software

A note on the O(n)-storage implementation of the GKO algorithm and its adaptation to Trummer-like matrices

“…More recently, a deeper analysis and a ready-to-use Matlab implementation were provided by Aricò and Rodriguez [1].…”

“…k = 1 being the step that zeroes out all the elements of the f irst column but the f irst. During the algorithm, 1. The (i, j) entry of the (1,1) block is updated at all steps k with k < min(i, j + 1).…”

“…During the algorithm, 1. The (i, j) entry of the (1,1) block is updated at all steps k with k < min(i, j + 1). After its last update, it contains U i, j .…”

“…. , B (1) , which were computed (and then discarded) in the first phase of the algorithm. I.e., at each step we "downdate" B to its previous value, reversing the GKO step.…”

“…In Sections 3 and 4 we will study respectively the original GKO algorithm and its first O(n)-space variant due to Rodriguez [1,18]. In Section 5 we will introduce and analyze our new O(n)-space variant.…”