Matrix multisplitting Picard-iterative method for solving generalized absolute value matrix equation

Abstract

The absolute value equation appears in various fields of applied mathematics such as operational research. Here we consider its generalized version AX+B|X|=C, where A,B,C∈Cn×n are given, |X|=(|xi,j|) and X∈Cn×n is an unknown matrix that must be determined. In this investigation, based on the Picard matrix splitting iteration method, we applied a matrix splitting method for solving it. We will see that under the condition σmin(A)>nσmax(|B|), this method is convergent, where σmax(|B|) denotes the largest singular value of matrix |B| and σmin(A) denotes the smallest singular value of matrix A. Then we give some convergence theorems for our new method and analyze this procedure in detail. Then we consider a p-step iteration method for solving this equation and analyze this procedure. Numerical experiment results show the efficiency of the method.