An Improvement on a Class of Fixed Point Iterative Methods for Solving Absolute Value Equations

Abstract

Abstract We consider a class of iterative methods based on block splitting (BBS) to solve absolute value equations A ⁢ x - | x | = b Ax-\lvert x\rvert=b . Recently, several works were devoted to deriving sufficient conditions for the convergence of iterative methods of this type under certain assumptions including ν := ∥ A - 1 ∥ < 1 \nu:=\lVert A^{-1}\rVert<1 . However, the BBS-type iterative methods tend to converge slowly when 𝜈 is very close to one (i.e., ν ≈ 1 \nu\approx 1 ). In this paper, using an auxiliary matrix, we develop a new approach by first rewriting the main problem into a new equivalent block system having shifted ( 1 , 1 ) (1,1) -block and then constructing a fixed point iteration. The exploited strategy can significantly improve the convergence speed of the BBS-type iterative methods when ν ≈ 1 \nu\approx 1 . Numerical experiments are reported to demonstrate the superiority of the new modified iterative scheme over the existing original form of BBS-type methods in the literature.