Non-recursive equivalent of the conjugate gradient method without the need to restart

TL;DR

This paper introduces a stable, non-recursive alternative to the conjugate gradient method that does not require restarts, simplifies implementation, and maintains efficiency with only one matrix-vector multiplication per iteration.

Contribution

The paper presents a novel iterative algorithm derived from the Ritz method that avoids conjugacy and restart procedures, enhancing stability and simplicity over traditional CG methods.

Findings

More stable than conjugate gradient method.

Eliminates the need for restart techniques.

Maintains efficiency with a single matrix-vector multiplication per step.

Abstract

A simple alternative to the conjugate gradient(CG) method is presented; this method is developed as a special case of the more general iterated Ritz method (IRM) for solving a system of linear equations. This novel algorithm is not based on conjugacy, i.e. it is not necessary to maintain overall orthogonalities between various vectors from distant steps. This method is more stable than CG, and restarting techniques are not required. As in CG, only one matrix-vector multiplication is required per step with appropriate transformations. The algorithm is easily explained by energy considerations without appealing to the A-orthogonality in n-dimensional space. Finally, relaxation factor and preconditioning-like techniques can be adopted easily.