On the Behavior of the Residual in Conjugate Gradient Method

TL;DR

This paper investigates the behavior of residuals in the conjugate gradient method, focusing on how finite arithmetic affects the accuracy of the residuals and establishing bounds related to digit loss.

Contribution

It provides a detailed analysis of the lower bounds of residual differences caused by digit loss in finite arithmetic within the conjugate gradient method.

Findings

Residuals diverge from true values due to digit loss.

Lower bounds of residual differences are characterized.

Analysis enhances understanding of numerical stability in CG.

Abstract

In conjugate gradient method, it is well known that the recursively computed residual differs from true one as the iteration proceeds in finite arithmetic. Some work have been devoted to analyze this be-havior and to evaluate the lower and the upper bounds of the difference. This paper focuses on the behavior of these two kinds of residuals, especially their lower bounds caused by the loss of trailing digit, respectively.