Structure of the polynomials in preconditioned BiCG algorithms and the switching direction of preconditioned systems

TL;DR

This paper analyzes the polynomial structures and direction switching in preconditioned BiCG algorithms, providing theoretical insights and extending the understanding to bi-Lanczos-type methods.

Contribution

It introduces a theorem characterizing the direction of preconditioned systems and extends it to bi-Lanczos algorithms, highlighting how direction switching occurs.

Findings

Direction of preconditioned systems can be switched by construction.

Polynomial structures of four preconditioned BiCG algorithms are compared.

Initial shadow residual vector settings influence the system direction.

Abstract

We present a theorem that defines the direction of a preconditioned system for the bi-conjugate gradient (BiCG) method, and we extend it to preconditioned bi-Lanczos-type algorithms. We show that the direction of a preconditioned system is switched by construction and by the settings of the initial shadow residual vector. We analyze and compare the polynomial structures of four preconditioned BiCG algorithms.