The steepest descent of gradient-based iterative method for solving rectangular linear systems with an application to Poisson�s equation

Abstract

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 $\sqrt {1-\kappa^{-2}}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msqrt><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:msup><mml:mi>κ</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msqrt></mml:math> , 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.