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

Arisa Kawase, Kensuke Aihara

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.

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

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.
Paper Structure (13 sections, 4 theorems, 37 equations, 1 figure, 4 algorithms)

This paper contains 13 sections, 4 theorems, 37 equations, 1 figure, 4 algorithms.

Table of Contents

  1. Introduction
  2. Motivation and objective
  3. Notation
  4. Organization
  5. Residual smoothing techniques
  6. Generation of the Bi-CR residuals
  7. Bi-CG and Bi-CR algorithms
  8. Transformation from Bi-CG residuals into Bi-CR residuals
  9. Prove for bi-orthogonal properties
  10. Preparations and observations
  11. Main theorem
  12. Concise algorithm and numerical example
  13. Concluding remarks

Key Result

Proposition 1

In Algorithm Bi-CR_2, the $k$th residual $\bm{r}_k$ and search direction $\bm{p}_k$ satisfy the following bi-orthogonal properties:

Figures (1)

  • Figure 1: Convergence histories of the relative residual norms of Algorithms \ref{['Bi-CR']}--\ref{['Bi-CR_3']} for the Toeplitz matrix.

Theorems & Definitions (9)

  • Proposition 1
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Remark 1