Chansangiam, Pattrawut

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Chansangiam, Pattrawut
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Chansangiam, P.
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pattrawut.ch@kmitl.ac.th

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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.
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    Gradient iterative method with optimal convergent factor for solving a generalized sylvester matrix equation with applications to diffusion equations
    (2020-10-01)
    Boonruangkan, Nunthakarn
    ;
    Chansangiam, Pattrawut
    We introduce a gradient iterative scheme with an optimal convergent factor for solving a generalized Sylvester matrix equation ∑<sup>p</sup>i=1<sup>A</sup> i XB<inf>i</inf> = F, where A<inf>i</inf>, B<inf>i</inf> and F are conformable rectangular matrices. The iterative scheme is derived from the gradients of the squared norm-errors of the associated subsystems for the equation. The convergence analysis reveals that the sequence of approximated solutions converge to the exact solution for any initial value if and only if the convergent factor is chosen properly in terms of the spectral radius of the associated iteration matrix. We also discuss the convergent rate and error estimations. Moreover, we determine the fastest convergent factor so that the associated iteration matrix has the smallest spectral radius. Furthermore, we provide numerical examples to illustrate the capability and efficiency of this method. Finally, we apply the proposed scheme to discretized equations for boundary value problems involving convection and diffusion.
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    Modified jacobi-gradient iterative method for generalized sylvester matrix equation
    (2020-11-01)
    Sasaki, Nopparut
    ;
    Chansangiam, Pattrawut
    We propose a new iterative method for solving a generalized Sylvester matrix equation A<inf>1</inf> XA<inf>2</inf> + A<inf>3</inf> XA<inf>4</inf> = E with given square matrices A<inf>1</inf>, A<inf>2</inf>, A<inf>3</inf>, A<inf>4</inf> and an unknown rectangular matrix X. The method aims to construct a sequence of approximated solutions converging to the exact solution, no matter the initial value is. We decompose the coefficient matrices to be the sum of its diagonal part and others. The recursive formula for the iteration is derived from the gradients of quadratic norm-error functions, together with the hierarchical identification principle. We find equivalent conditions on a convergent factor, relied on eigenvalues of the associated iteration matrix, so that the method is applicable as desired. The convergence rate and error estimation of the method are governed by the spectral norm of the related iteration matrix. Furthermore, we illustrate numerical examples of the proposed method to show its capability and efficacy, compared to recent gradient-based iterative methods.
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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.