Flexible fixed-point iteration and its applications for nonsymmetric algebraic Riccati equations
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | |
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| _version_ | 1866914067590938624 |
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| author | Guo, Zhen-Chen Liang, Xin |
| author_facet | Guo, Zhen-Chen Liang, Xin |
| contents | In this paper, we reveal the intrinsic Toeplitz structure in the unique stabilizing solution for nonsymmetric algebraic Riccati equations by employing a shift-involved fixed-point iteration, and propose an RADI-type method for computing this solution for large-scale equations of this type with sparse and low-rank structure by incorporating flexible shifts into the fixed-point iteration. We present a shift-selection strategy, termed Leja shifts, based on rational approximation theory, which is incorporated into the RADI-type method. We further discuss important implementation aspects for the method, such as low-rank factorization of residuals, implicit update of large-scale sparse matrices, real arithmetics with complex shifts, and related equations of other type.
Numerical experiments demonstrate the efficiency of both the proposed method and the introduced shift-selection strategy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_25942 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Flexible fixed-point iteration and its applications for nonsymmetric algebraic Riccati equations Guo, Zhen-Chen Liang, Xin Numerical Analysis 65F45, 15A24, 49N10, 93B52 In this paper, we reveal the intrinsic Toeplitz structure in the unique stabilizing solution for nonsymmetric algebraic Riccati equations by employing a shift-involved fixed-point iteration, and propose an RADI-type method for computing this solution for large-scale equations of this type with sparse and low-rank structure by incorporating flexible shifts into the fixed-point iteration. We present a shift-selection strategy, termed Leja shifts, based on rational approximation theory, which is incorporated into the RADI-type method. We further discuss important implementation aspects for the method, such as low-rank factorization of residuals, implicit update of large-scale sparse matrices, real arithmetics with complex shifts, and related equations of other type. Numerical experiments demonstrate the efficiency of both the proposed method and the introduced shift-selection strategy. |
| title | Flexible fixed-point iteration and its applications for nonsymmetric algebraic Riccati equations |
| topic | Numerical Analysis 65F45, 15A24, 49N10, 93B52 |
| url | https://arxiv.org/abs/2509.25942 |