On the Forsythe conjecture

TL;DR

This paper revisits Forsythe's 1968 conjecture on the asymptotic behavior of the restarted conjugate gradient method, extending it from symmetric positive definite matrices to symmetric and nonsymmetric matrices using modern iterative techniques.

Contribution

It translates Forsythe's original results into modern terminology and generalizes the conjecture to broader classes of matrices, introducing a two-sided iteration approach.

Findings

Proves new results on the limiting behavior of the generalized iteration.

Extends the conjecture to nonsymmetric matrices.

The conjecture remains largely open despite these advances.

Abstract

Forsythe formulated a conjecture about the asymptotic behavior of the restarted conjugate gradient method in 1968. We translate several of his results into modern terms, and generalize the conjecture (originally formulated only for symmetric positive definite matrices) to symmetric and nonsymmetric matrices. Our generalization is based on a two-sided or cross iteration with the given matrix and its transpose, which is based on the projection process used in the Arnoldi (or for symmetric matrices the Lanczos) algorithm. We prove several new results about the limiting behavior of this iteration, but the conjecture still remains largely open.