Theoretical insights on the residual transformation from bi-conjugate gradient into bi-conjugate residual via a smoothing scheme

TL;DR

This paper provides a theoretical analysis of transforming residuals from the bi-conjugate gradient method to the bi-conjugate residual method using a smoothing scheme, establishing their bi-orthogonal properties.

Contribution

It offers a rigorous proof of the residual transformation's theoretical validity and introduces a more concise transformation algorithm.

Findings

Proved the transformed algorithm retains bi-orthogonal properties.

Established the theoretical validity of residual transformation.

Presented a numerical example demonstrating the transformation.

Abstract

Bi-conjugate gradient (Bi-CG) and bi-conjugate residual (Bi-CR) methods are underlying iterative solvers for linear systems with nonsymmetric matrices. Residual smoothing is a standard technique for obtaining smooth convergence behavior of residual norms; additionally, it represents the transformation between iterative methods. For example, the residuals of the CR method can be obtained by applying a smoothing scheme to those of the CG method for symmetric linear systems. Based on this relationship, the transformation from Bi-CG residuals to Bi-CR residuals using a smoothing scheme was examined in our previous study [Kawase, A., Aihara, K.: Transformation from Bi-CG into Bi-CR Using a Residual Smoothing-like Scheme. AIP Conference Proceedings (2026)]; however, we only provided heuristic and experimental observations. In the present study, we provide a detailed discussion on the theoretical aspects of these transformations. Specifically, we prove that the resulting algorithm transformed from the Bi-CG method using the residual smoothing technique has the same bi-orthogonal properties as those of the original Bi-CR method. We also present a more concise transformation algorithm and its numerical example. These analyses complement our previous study and provide theoretical validity of the residual transformation between the Bi-CG and Bi-CR methods.