A general approach for improving the Padé iterations for the matrix sign function
- Authors
- Jung, D.[Jung, Dohee]; Chun, C.[Chun, Changbum]
- Issue Date
- 15-Jan-2024
- Publisher
- Elsevier B.V.
- Keywords
- Iterative methods; Matrix sign function; Padé iterations; Stability
- Citation
- Journal of Computational and Applied Mathematics, v.436
- Indexed
- SCIE
SCOPUS
- Journal Title
- Journal of Computational and Applied Mathematics
- Volume
- 436
- URI
- https://scholarworks.bwise.kr/skku/handle/2021.sw.skku/107018
- DOI
- 10.1016/j.cam.2023.115348
- ISSN
- 0377-0427
- Abstract
- The Padé family of iterations is a well-known set of methods used to compute the matrix sign function, which includes classical methods such as Newton's method, the Newton–Schultz iteration, and Halley's method. In this paper, we present a general approach to enhance the Padé iterations by choosing an arbitrary three-point family of methods based on weight functions. We determine the weight functions in a way that, for a complex quadratic with distinct roots, the three-point methods are conformally conjugate to a complex polynomial with as many parameters as desired. This approach leads to a multi-parameter family of iterations for the matrix sign function, which allows us to discover many new methods, including the Padé family of iterations as a special case. We provide a convergence and stability analysis of the multi-parameter family and conduct numerical experiments to confirm the improved performance of the new methods. Although the three-point family of methods is arbitrarily chosen, our approach can be easily extended to any other multipoint methods. © 2023 Elsevier B.V.
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