Abstract
In this paper, we suggest and analyze an improved generalized Newton method for solving the NP-hard absolute value equations Ax-|x|=b when the singular values of A exceed 1. We show that the global and local quadratic convergence of the proposed method. Numerical experiments show the efficiency of the method and the high accuracy of calculation.
Highlights
BackgroundWhere A ∈ Rn×n, b ∈ Rn , and |x| denotes a vector in Rn, whose i-th component is |xi|
We consider the absolute value equations (AVEs): Ax − |x| = b, (1)where A ∈ Rn×n, b ∈ Rn, and |x| denotes a vector in Rn, whose i-th component is |xi|
Based on the linear complementarity problems (LCPs) reformulation, sufficient conditions for the existence and nonexistence of solutions are given in this paper
Summary
Where A ∈ Rn×n, b ∈ Rn , and |x| denotes a vector in Rn, whose i-th component is |xi|. Lemma 4 Let the singular values of A exceed 1, the sequence {xk } generated by the improved generalized Newton method (6) is bounded, and there exists an accumulation point xsuch that (A − D )x = b + af (x) for some diagonal matrixes Dwith diagonal elements of ±1 or 0. D with diagonal elements of ±1 or 0, the improved generalized Newton method (6) converges linearly from any starting point x0 to a solution xfor any solvable AVEs (1). Theorem 3 (Locally quadratic convergence) If A − D is nonsingular for any diagonal matrix D with diagonal elements of ±1 or 0, the sequence {xk } from improved generalized Newton’s method (7) converges to xand xk+1 − x = O( xk − x 2) Proof This theorem can be proved in a similar way as Theorem 2 by Qi and Sun (1993).
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