Best polynomial approximation for non-autonomous linear ODEs in the *-product framework
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Sep 21, 2024
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Abstract
We present the first formulation of the optimal polynomial approximation of the solution of linear non-autonomous systems of ODEs in the framework of the so-called *-product. This product is the basis of new approaches for the solution of such ODEs, both in the analytical and the numerical sense. The paper shows how to formally state the problem and derives upper bounds for its error.
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Pozza, S.
(2024).
Best polynomial approximation for non-autonomous linear ODEs in the *-product framework.
Proceedings Of The Conference Algoritmy, , 36 - 45.
Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/2142/1025
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References
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[13] S. Pozza, A new closed-form expression for the solution of ODEs in a ring of distributions and its connection with the matrix algebra, Linear Multilinear Algebra, to appear. arXiv:2302.11375 [math.NA]
[14] S. Pozza and V. Simoncini, Inexact Arnoldi residual estimates and decay properties for functions of non-Hermitian matrices, Bit Numer. Math., 59 (2019), pp. 969–986.
[15] S. Pozza and N. Van Buggenhout, A new Legendre polynomial-based approach for non-autonomous linear ODEs, Electron. Trans. Numer. Anal., to appear. arXiv:2303.11284 [math.NA]
[16] S. Pozza and N. Van Buggenhout, A new matrix equation expression for the solution of non-autonomous linear systems of ODEs, Proc. Appl. Math. Mech., 22 (2023), p. e202200117.
[17] S. Pozza and N. Van Buggenhout, A ⋆-product solver with spectral accuracy for non-autonomous ordinary differential equations, Proc. Appl. Math. Mech., 23 (2023), p. e202200050.
[18] M. Ryckebusch, A Fréchet-Lie group on distributions, arXiv preprint arXiv:2307.09037, (2023).
[2] M. Benzi, V. Simoncini, editors, Exploiting hidden structure in matrix computations: Algorithms and applications, Springer; 2016.
[3] C. Bonhomme, S. Pozza, and N. V. Buggenhout, A new fast numerical method for the generalized Rosen-Zener model, arXiv preprint arXiv:2311.04144, (2023).
[4] L. Dieci and T. Eirola, On smooth decompositions of matrices, SIAM J. Matrix Anal. Appl., 20 (1999), pp. 800–19.
[5] P.-L. Giscard, On the solutions of linear Volterra equations of the second kind with sum kernels, J. Integral Equations Applications, 32 (2020), p. 429–445.
[6] P.-L. Giscard and C. Bonhomme, Dynamics of quantum systems driven by time-varying Hamiltonians: Solution for the Bloch-Siegert Hamiltonian and applications to NMR, Phys. Rev. Research, 2 (2020), p. 023081.
[7] P.-L. Giscard, K. Lui, S. J. Thwaite, and D. Jaksch, An exact formulation of the time-ordered exponential using path-sums, J. Math. Phys., 56 (2015), p. 053503.
[8] P.-L. Giscard and S. Pozza, Lanczos-like algorithm for the time-ordered exponential: The ∗-inverse problem, Appl. Math., 65 (2020), pp. 807–827.
[9] P.-L. Giscard and S. Pozza, A Lanczos-like method for non-autonomous linear ordinary differential equations, Boll. Unione Mat. Ital., 16 (2023), pp. 81–102.
[10] T. Kato, Perturbation Theory for Linear Operators, Second ed., Springer-Verlag, Berlin, 1976.
[11] J. Liesen and Z. Strakoš, Krylov subspace methods: principles and analysis, Oxford University Press, 2012.
[12] G. Meinardus, Approximation of Functions: Theory and Numerical Methods, Springer Tracts Natur. Philos., vol. 13, Springer, New York, 1967.
[13] S. Pozza, A new closed-form expression for the solution of ODEs in a ring of distributions and its connection with the matrix algebra, Linear Multilinear Algebra, to appear. arXiv:2302.11375 [math.NA]
[14] S. Pozza and V. Simoncini, Inexact Arnoldi residual estimates and decay properties for functions of non-Hermitian matrices, Bit Numer. Math., 59 (2019), pp. 969–986.
[15] S. Pozza and N. Van Buggenhout, A new Legendre polynomial-based approach for non-autonomous linear ODEs, Electron. Trans. Numer. Anal., to appear. arXiv:2303.11284 [math.NA]
[16] S. Pozza and N. Van Buggenhout, A new matrix equation expression for the solution of non-autonomous linear systems of ODEs, Proc. Appl. Math. Mech., 22 (2023), p. e202200117.
[17] S. Pozza and N. Van Buggenhout, A ⋆-product solver with spectral accuracy for non-autonomous ordinary differential equations, Proc. Appl. Math. Mech., 23 (2023), p. e202200050.
[18] M. Ryckebusch, A Fréchet-Lie group on distributions, arXiv preprint arXiv:2307.09037, (2023).