Chansangiam, Pattrawut

Preferred name

Alternative Name

Chansangiam, P.

Main Affiliation

School of Science

pattrawut.ch@kmitl.ac.th

Now showing 1 - 10 of 14

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Least-squares solutions of generalized linear systems and the matrix equation AXB = C under the general semi-tensor products
(2026-01-01)
Jaiprasert, Janthip
;
Phoonphiphat, Thanaphon
;
Chansangiam, Pattrawut
;
Zhang, Yang
We investigate least-squares (LS) solutions of Sylvester-type matrix equations formulated via the general semi-tensor product (GSTP) of matrices. In particular, we consider generalized linear systems of the form A (Formula presented) x = B, where A and B are given rectangular matrices and x is an unknown column vector, with k denoting the GSTP that extends both the conventional matrix product and the semi-tensor product. By analyzing the derivative of the LS error associated with the equation, we show that LS solutions can be obtained by solving an equivalent linear system under the usual matrix product. Using matrix partitioning techniques, these results are further extended to several Sylvester-type equations, including A (Formula presented) X = B, X (Formula presented) A = B. and A (Formula presented) X x B = C, where X is an unknown matrix of compatible size. This framework unifies the classical and semi-tensor product cases under a generalized algebraic setting. Furthermore, we develop a gradient-descent iterative algorithm to compute approximate LS solutions efficiently. Numerical experiments confirm the convergence, capability, and effectiveness of the proposed method.
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Riccati equation and metric geometric means of positive semidefinite matrices involving semi-tensor products
(2023-01-01)
Chansangiam, Pattrawut
;
Ploymukda, Arnon
We investigate the Riccati matrix equation XA<sup>−1</sup>X = B in which the conventional matrix products are generalized to the semi-tensor products ⋉. When A and B are positive definite matrices satisfying the factor-dimension condition, this equation has a unique positive definite solution, which is defined to be the metric geometric mean of A and B. We show that this geometric mean is the maximum solution of the Riccati inequality. We then extend the notion of the metric geometric mean to positive semidefinite matrices by a continuity argument and investigate its algebraic properties, order properties and analytic properties. Moreover, we establish some equations and inequalities of metric geometric means for matrices involving cancellability, positive linear map and concavity. Our results generalize the conventional metric geometric means of matrices.
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Solving the Sylvester-Transpose Matrix Equation under the Semi-Tensor Product
(2022-06-01)
Jaiprasert, Janthip
;
Chansangiam, Pattrawut
This paper investigates the Sylvester-transpose matrix equation A ⋉ X + X<sup>T</sup> ⋉ B = C, where all mentioned matrices are over an arbitrary field. Here, ⋉ is the semi-tensor product, which is a generalization of the usual matrix product defined for matrices of arbitrary dimensions. For matrices of compatible dimensions, we investigate criteria for the equation to have a solution, a unique solution, or infinitely many solutions. These conditions rely on ranks and linear dependence. Moreover, we find suitable matrix partitions so that the matrix equation can be transformed into a linear system involving the usual matrix product. Our work includes the studies of the equation A ⋉ X = C, the equation X ⋉ B = C, and the classical Sylvester-transpose matrix equation.
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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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Sylvester matrix equation under the semi-tensor product of matrices
(2022-01-01)
Chansangiam, Pattrawut
;
Sabau, Sorin V.
We investigate the Sylvester matrix equation in which the product is given by the semi-tensor product, and all involved matrices are matrices over an arbitrary field. We discuss necessary/sufficient condition(s) for the matrix equation to have a solution or a unique solution, or infinitely many solutions. These conditions concern ranks and linear independence. Moreover, we apply a certain kind of vectorization and matrix partitioning to transform the Sylvester equation into an equivalent linear system with respect to the conventional matrix product. Our study includes the Lyapunov equation and the equation A ⋉ X = C as special cases.
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The Nature and Significance of Mathematics from Contemporary Viewpoints
(2025-01-01)
Boonnam, Nathaphon
;
Hama, Rattanasak
;
Chansangiam, Pattrawut
;
Sabau, Sorin V.
This paper explores the nature and significance of mathematics, presenting it as a systematic and logical study of patterns in nature. Mathematics can be conceptualized as a pyramid consisting of three layers. The first layer is pure mathematics, which focuses on the study of abstract objects and concepts. The second layer is applied mathematics, dedicated to the development and application of mathematical methods to address specific problems. The final layer involves the applications of mathematics, where established results from pure or applied mathematics are utilized to solve concrete, real-world problems. A deep appreciation of the importance and beauty of abstract patterns requires engaging in research within pure or applied mathematics. To undertake such research, the fundamentals of pure mathematics are essential. Advances in mathematical research often lead to the development of new theories or innovative techniques for problem-solving.
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Numerical solutions of the space-time fractional diffusion equation via a gradient-descent iterative procedure
(2023-01-01)
Tansri, Kanjanaporn
;
Kittisopaporn, Adisorn
;
Chansangiam, Pattrawut
A one-dimensional space-time fractional diffusion equation describes anomalous diffusion on fractals in one dimension. In this paper, this equation is discretized by finite difference schemes based on the Grünwald-Letnikov approximation for Riemann-Liouville and Caputo’s fractional derivatives. It turns out that the discretized equations can be put into a compact form, i.e., a linear system with a block lower-triangular coefficient matrix. To solve the linear system, we formulate a matrix iterative algorithm based on gradient-descent technique. In particular, we work out for the space fractional diffusion equation. Theoretically, the proposed solver is always applicable with satisfactory convergence rate and error estimates. Simulations are presented numerically and graphically to illustrate the accuracy, the efficiency, and the performance of the algorithm, compared to other iterative procedures for linear systems.
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Gradient-descent iterative algorithm for solving exact and weighted least-squares solutions of rectangular linear systems
(2023-01-01)
Tansri, Kanjanaporn
;
Chansangiam, Pattrawut
Consider a linear system Ax = b where the coefficient matrix A is rectangular and of full-column rank. We propose an iterative algorithm for solving this linear system, based on gradient-descent optimization technique, aiming to produce a sequence of well-approximate least-squares solutions. Here, we consider least-squares solutions in a full generality, that is, we measure any related error through an arbitrary vector norm induced from weighted positive definite matrices W. It turns out that when the system has a unique solution, the proposed algorithm produces approximated solutions converging to the unique solution. When the system is inconsistent, the sequence of residual norms converges to the weighted least-squares error. Our work includes the usual least-squares solution when W = I. Numerical experiments are performed to validate the capability of the algorithm. Moreover, the performance of this algorithm is better than that of recent gradient-based iterative algorithms in both iteration numbers and computational time.
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Metric geometric means with arbitrary weights of positive definite matrices involving semi-tensor products
(2023-01-01)
Ploymukda, Arnon
;
Chansangiam, Pattrawut
We extend the notion of classical metric geometric mean (MGM) for positive definite matrices of the same dimension to those of arbitrary dimensions, so that usual matrix products are replaced by semi-tensor products. When the weights are arbitrary real numbers, the weighted MGMs possess not only nice properties as in the classical case, but also affine change of parameters, exponential law, and cancellability. Moreover, when the weights belong to the unit interval, the weighted MGM has remarkable properties, namely, monotonicity and continuity from above. Then we apply a continuity argument to extend the weighted MGM to positive semidefinite matrices, here the weights belong to the unit interval. It turns out that this matrix mean posses rich algebraic, order, and analytic properties, such as, monotonicity, continuity from above, congruent invariance, permutation invariance, affine change of parameters, and exponential law. Furthermore, we investigate certain equations concerning weighted MGMs of positive definite matrices. It turns out that such equations are always uniquely solvable with explicit solutions. The notion of MGMs can be applied to solve certain symmetric word equations in two letters.
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Conjugate Gradient Algorithm for Least-Squares Solutions of a Generalized Sylvester-Transpose Matrix Equation
(2022-09-01)
Tansri, Kanjanaporn
;
Chansangiam, Pattrawut
We derive a conjugate-gradient type algorithm to produce approximate least-squares (LS) solutions for an inconsistent generalized Sylvester-transpose matrix equation. The algorithm is always applicable for any given initial matrix and will arrive at an LS solution within finite steps. When the matrix equation has many LS solutions, the algorithm can search for the one with minimal Frobenius-norm. Moreover, given a matrix Y, the algorithm can find a unique LS solution closest to Y. Numerical experiments show the relevance of the algorithm for square/non-square dense/sparse matrices of medium/large sizes. The algorithm works well in both the number of iterations and the computation time, compared to the direct Kronecker linearization and well-known iterative methods.

Chansangiam, Pattrawut

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