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Author: search.filters.author.Kittisopaporn, AdisornSubject: search.filters.subject.Generalized Sylvester matrix equation

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    Convergence analysis of gradient-based iterative algorithms for a class of rectangular Sylvester matrix equations based on Banach contraction principle
    (2021-12-01)
    Kittisopaporn, Adisorn
    ;
    Chansangiam, Pattrawut
    ;
    Lewkeeratiyutkul, Wicharn
    We derive an iterative procedure for solving a generalized Sylvester matrix equation AXB+ CXD= E, where A, B, C, D, E are conforming rectangular matrices. Our algorithm is based on gradients and hierarchical identification principle. We convert the matrix iteration process to a first-order linear difference vector equation with matrix coefficient. The Banach contraction principle reveals that the sequence of approximated solutions converges to the exact solution for any initial matrix if and only if the convergence factor belongs to an open interval. The contraction principle also gives the convergence rate and the error analysis, governed by the spectral radius of the associated iteration matrix. We obtain the fastest convergence factor so that the spectral radius of the iteration matrix is minimized. In particular, we obtain iterative algorithms for the matrix equation AXB= C, the Sylvester equation, and the Kalman–Yakubovich equation. We give numerical experiments of the proposed algorithm to illustrate its applicability, effectiveness, and efficiency.
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    Gradient-descent iterative algorithm for solving a class of linear matrix equations with applications to heat and Poisson equations
    (2020-12-01)
    Kittisopaporn, Adisorn
    ;
    Chansangiam, Pattrawut
    In this paper, we introduce a new iterative algorithm for solving a generalized Sylvester matrix equation of the form ∑t=1pAtXBt=C which includes a class of linear matrix equations. The objective of the algorithm is to minimize an error at each iteration by the idea of gradient-descent. We show that the proposed algorithm is widely applied to any problems with any initial matrices as long as such problem has a unique solution. The convergence rate and error estimates are given in terms of the condition number of the associated iteration matrix. Furthermore, we apply the proposed algorithm to sparse systems arising from discretizations of the one-dimensional heat equation and the two-dimensional Poisson’s equation. Numerical simulations illustrate the capability and effectiveness of the proposed algorithm comparing to well-known methods and recent methods.