Gradient-descent iterative algorithm for solving a class of linear matrix equations with applications to heat and Poisson equations

Abstract

Abstract In this paper, we introduce a new iterative algorithm for solving a generalized Sylvester matrix equation of the form $\sum_{t=1}^{p}A_{t}XB_{t}=C$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mo>∑</mml:mo> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi>p</mml:mi> </mml:msubsup> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mi>X</mml:mi> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>C</mml:mi> </mml:math> 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.