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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.

We consider the two-dimensional space-time fractional differential equation with the Caputo's time derivative and the Riemann-Liouville space derivatives on bounded domains. The equation is subjected to the zero Dirichlet boundary condition and the zero initial condition. We discretize the equation by finite difference schemes based on Grünwald-Letnikov approximation. Then we linearize the discretized equations into a sparse linear system. To solve such linear system, we propose a gradient-descent iterative algorithm with a sequence of optimal convergence factor aiming to minimize the error occurring at each iteration. The convergence analysis guarantees the capability of the algorithm as long as the coefficient matrix is invertible. In addition, the convergence rate and error estimates are provided. Numerical experiments demonstrate the efficiency, the accuracy and the performance of the proposed algorithm.

Abstract This paper proposes an effective gradient-descent iterative algorithm for solving a generalized Sylvester-transpose equation with rectangular matrix coefficients. The algorithm is applicable for the equation and its interesting special cases when the associated matrix has full column-rank. The main idea of the algorithm is to have a minimum error at each iteration. The algorithm produces a sequence of approximated solutions converging to either the unique solution, or the unique least-squares solution when the problem has no solution. The convergence analysis points out that the algorithm converges fast for a small condition number of the associated matrix. Numerical examples demonstrate the efficiency and effectiveness of the algorithm compared to renowned and recent iterative methods.

Abstract In this paper, we introduce a new iterative algorithm for solving a generalized Sylvester matrix equation of the form ∑ t = 1 p A t X B t = C $\sum_{t=1}^{p}A_{t}XB_{t}=C$ 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.

Abstract We introduce an effective iterative method for solving rectangular linear systems, based on gradients along with the steepest descent optimization. We show that the proposed method is applicable with any initial vectors as long as the coefficient matrix is of full column rank. Convergence analysis produces error estimates and the asymptotic convergence rate of the algorithm, which is governed by the term 1 − κ − 2 $\sqrt {1-\kappa^{-2}}$ , where κ is the condition number of the coefficient matrix. Moreover, we apply the proposed method to a sparse linear system arising from a discretization of the one-dimensional Poisson equation. Numerical simulations illustrate the capability and effectiveness of the proposed method in comparison to the well-known and recent methods.

We propose a new iterative method for solving a generalized Sylvester matrix equation A1XA2+A3XA4=E with given square matrices A1,A2,A3,A4 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.