Scaling-and-squaring method for computing the inverses of matrix $φ$-functions
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866913654668001280 |
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| author | Aceto, Lidia Gemignani, Luca |
| author_facet | Aceto, Lidia Gemignani, Luca |
| contents | This paper aims to develop efficient numerical methods for computing the inverse of matrix $φ$-functions, $ψ_\ell(A) := (φ_\ell(A))^{-1}$, for $\ell =1,2,\ldots,$ when $A$ is a large and sparse matrix with eigenvalues in the open left half-plane. While $φ$-functions play a crucial role in the analysis and implementation of exponential integrators, their inverses arise in solving certain direct and inverse differential problems with non-local boundary conditions. We propose an adaptation of the standard scaling-and-squaring technique for computing $ψ_\ell(A)$, based on the Newton-Schulz iteration for matrix inversion. The convergence of this method is analyzed both theoretically and numerically. In addition, we derive and analyze Padé approximants for approximating $ψ_1(A/2^s)$, where $s$ is a suitably chosen integer, necessary at the root of the squaring process. Numerical experiments demonstrate the effectiveness of the proposed approach. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_10028 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Scaling-and-squaring method for computing the inverses of matrix $φ$-functions Aceto, Lidia Gemignani, Luca Numerical Analysis This paper aims to develop efficient numerical methods for computing the inverse of matrix $φ$-functions, $ψ_\ell(A) := (φ_\ell(A))^{-1}$, for $\ell =1,2,\ldots,$ when $A$ is a large and sparse matrix with eigenvalues in the open left half-plane. While $φ$-functions play a crucial role in the analysis and implementation of exponential integrators, their inverses arise in solving certain direct and inverse differential problems with non-local boundary conditions. We propose an adaptation of the standard scaling-and-squaring technique for computing $ψ_\ell(A)$, based on the Newton-Schulz iteration for matrix inversion. The convergence of this method is analyzed both theoretically and numerically. In addition, we derive and analyze Padé approximants for approximating $ψ_1(A/2^s)$, where $s$ is a suitably chosen integer, necessary at the root of the squaring process. Numerical experiments demonstrate the effectiveness of the proposed approach. |
| title | Scaling-and-squaring method for computing the inverses of matrix $φ$-functions |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2501.10028 |