Improved Seed Methods for Symmetric Positive Definite Linear Equations with Multiple Right-hand Sides

TL;DR

This paper enhances the seed conjugate gradient method for symmetric positive definite systems with multiple right-hand sides by proposing modifications that improve convergence and reduce rounding error effects.

Contribution

It introduces three key modifications to the seed conjugate gradient method, including using only the first right-hand side for seeding, solving it past convergence, and controlling roundoff error with reorthogonalization.

Findings

Using only the first right-hand side for seeding yields better results.

Controlling rounding errors enables rapid convergence for subsequent systems.

The proposed method improves efficiency in solving multiple right-hand side systems.

Abstract

We consider symmetric positive definite systems of linear equations with multiple right-hand sides. The seed conjugate gradient method solves one right-hand side with the conjugate gradient method and simultaneously projects over the Krylov subspace thus developed for the other right-hand sides. Then the next system is solved and used to seed the remaining ones. Rounding error in the conjugate gradient method limits how much the seeding can improve convergence. We propose three changes to the seed conjugate gradient method: only the first right-hand side is used for seeding, this system is solved past convergence, and the roundoff error is controlled with some reorthogonalization. We will show that results are actually better with only one seeding, even in the case of related right-hand sides. Controlling rounding error gives the potential for rapid convergence for the second and subsequent right-hand sides.