Algebraic and FFT-Based Methods for Discrete-Time Matrix Convolutions with Applications to Semi-Markov Models
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918534959857664 |
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| author | Kordalis, L. Trevezas, S. |
| author_facet | Kordalis, L. Trevezas, S. |
| contents | We study the convolution product of matrix-valued sequences and its role in the computation of Markov renewal equations. Explicit representations and recursive formulae for the convolutional inverse are derived and used to construct FFT-accelerated convolution and Newton-type inversion schemes, together with a Gauss--Jordan alternative in truncated power-series rings. The proposed framework is also applied to discrete approximations of matrix Stieltjes convolutions, which arise in continuous-time semi-Markov models. These tools are then used for the numerical evaluation of semi-Markov reliability and availability functions. The numerical results show substantial reductions in runtime, while preserving close agreement with exact benchmark solutions, direct computations, and Monte Carlo simulations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_30379 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Algebraic and FFT-Based Methods for Discrete-Time Matrix Convolutions with Applications to Semi-Markov Models Kordalis, L. Trevezas, S. Numerical Analysis Probability We study the convolution product of matrix-valued sequences and its role in the computation of Markov renewal equations. Explicit representations and recursive formulae for the convolutional inverse are derived and used to construct FFT-accelerated convolution and Newton-type inversion schemes, together with a Gauss--Jordan alternative in truncated power-series rings. The proposed framework is also applied to discrete approximations of matrix Stieltjes convolutions, which arise in continuous-time semi-Markov models. These tools are then used for the numerical evaluation of semi-Markov reliability and availability functions. The numerical results show substantial reductions in runtime, while preserving close agreement with exact benchmark solutions, direct computations, and Monte Carlo simulations. |
| title | Algebraic and FFT-Based Methods for Discrete-Time Matrix Convolutions with Applications to Semi-Markov Models |
| topic | Numerical Analysis Probability |
| url | https://arxiv.org/abs/2605.30379 |