Approximated least-squares solutions of a generalized Sylvester-transpose matrix equation via gradient-descent iterative algorithm

Abstract

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.

Keywords

• Generalized Sylvester-transpose matrix equation
• Gradient descent
• Iterative method
• Least-squares solution