Theoretical insights on the residual transformation from bi-conjugate gradient into bi-conjugate residual via a smoothing scheme
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| author | Kawase, Arisa Aihara, Kensuke |
| author_facet | Kawase, Arisa Aihara, Kensuke |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_18159 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Theoretical insights on the residual transformation from bi-conjugate gradient into bi-conjugate residual via a smoothing scheme Kawase, Arisa Aihara, Kensuke Numerical Analysis 65F10 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. |
| title | Theoretical insights on the residual transformation from bi-conjugate gradient into bi-conjugate residual via a smoothing scheme |
| topic | Numerical Analysis 65F10 |
| url | https://arxiv.org/abs/2602.18159 |