A note on the O(n)-storage implementation of the GKO algorithm

TL;DR

This paper introduces an O(n)-space implementation of the GKO-Cauchy algorithm that improves cache efficiency and outperforms existing methods for large matrices, with applications to special Cauchy-like matrices.

Contribution

It presents a novel O(n)-space algorithm for the GKO-Cauchy method, enhancing computational efficiency and cache utilization for large matrices.

Findings

Outperforms existing algorithms for matrices larger than 500-1000 in size.

Efficiently handles Cauchy-like matrices with non-reconstructible diagonals.

Adapts to basic linear algebra operations using low displacement-rank generators.

Abstract

We propose a new O(n)-space implementation of the GKO-Cauchy algorithm for the solution of linear systems with Cauchy-like matrix. Despite its slightly higher computational cost, this new algorithm makes a more efficient use of the processor cache memory. Thus, for matrices of size larger than about 500-1000, it outperforms the existing algorithms. We present an applicative case of Cauchy-like matrices with non-reconstructible main diagonal. In this special instance, the O(n) space algorithms can be adapted nicely to provide an efficient implementation of basic linear algebra operations in terms of the low displacement-rank generators.