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    Conjugate gradient algorithm for consistent generalized Sylvester-transpose matrix equations
    (2022-01-01)
    Tansri, Kanjanaporn
    ;
    Choomklang, Sarawanee
    ;
    Chansangiam, Pattrawut
    We develop an effective algorithm to find a well-approximate solution of a generalized Sylvester-transpose matrix equation where all coefficient matrices and an unknown matrix are rectangular. The algorithm aims to construct a finite sequence of approximated solutions from any given initial matrix. It turns out that the associated residual matrices are orthogonal, and thus, the desire solution comes out in the final step with a satisfactory error. We provide numerical experiments to show the capability and performance of the algorithm.
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    Convergence analysis of a gradient iterative algorithm with optimal convergence factor for a generalized sylvester-transpose matrix equation
    (2021-01-01)
    Boonruangkan, Nunthakarn
    ;
    Chansangiam, Pattrawut
    Consider a generalized Sylvester-transpose matrix equation with rectangular coefficient matrices. Based on gradients and hierarchical identification principle, we derive an iterative algorithm to produce a sequence of approximated solutions with a reasonable stopping rule concerning a relative norm-error. A convergence analysis via Banach fixed-point theorem reveals the sequence converges to a unique solution of the matrix equation for any given initial matrix if and only if the convergence factor is chosen appropriately in a certain range. The performance of algorithm is theoretically analysed through the convergence rate and error estimations. The optimal convergence factor is chosen to attain the fastest asymptotic behaviour. Finally, numerical experiments are provided to illustrate the capability and efficiency of the proposed algorithm, compared to recent gradient-based iterative algorithms.
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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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    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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