A class of iteration methods based on the HSS for Toeplitz systems of weakly nonlinear equations

Abstract

For Toeplitz systems of weakly nonlinear equations, combining the separability and strong dominance between the linear and the nonlinear terms with the Hermitian and skew-Hermitian splitting (HSS) iteration technique, we establish two nonlinear composite iteration schemes, called Picard-cSSS and nonlinear cSSS-like iteration methods, which are based on a special case of the HSS, where the symmetric part H = 1 2 (A + A T ) is a centrosymmetric matrix and the skew-symmetric part H = 1 2 (A − A T ) is a skew-centrosymmetric matrix. The advantages of these methods are that they can transfer the linear sub-systems involved in inner iteration to two linear systems of half an order, besides, fast methods are available for computing the two half-steps involved in the inner iteration. Numerical results are provided, to further show that both Picard-cSSS and nonlinear cSSS-like iteration methods are feasible and effective.

Gradient Iterative Method with Optimal Convergent Factor for Solving a Generalized Sylvester Matrix Equation with Applications to Diffusion Equations

“…The convergence of Method 3 implies that (10) holds. Then, Method 3 has the convergence rate as the same to the linear iteration (12), and thus, it is governed by the spectral radius ρ [T] in (11). The fastest convergence rate is equivalent to the smallest of ρ [T].…”

“…Many researchers have developed such iterative methods for solving a class of matrix Equations (1)- (4); see e.g., [4][5][6][7][8][9][10]. One of an interesting iterative method, called the Hermitian and skew Hermitian splitting iterative method (HSS), was investigated by many authors, e.g., [11][12][13][14]. Gradient-based iterative methods were firstly introduced by Ding and Chen for solving (1), (2) and (4).…”

A Class of Preconditioned TGHSS-Based Iteration Methods for Weakly Nonlinear Systems

“…It's known that the Newton iteration method has quadratic convergence speed if a good initial guess x (0) for the equation F(x) = 0 is obtained. To simplify or avoid computation of the Jacobian matrix and reduce the cost of the function evaluation, one can consider many variants in terms of approximate, quasiupdate, inner/outer or inexact Newton methods, see [3,5,7,10,26].…”

“…Obviously, the disadvantages of the Newton-HSS iteration methods are often practically prohibitive due to limited computer memory and the admissible computing time. Therefore, authors of [1,17,21,[24][25][26] made further generalizations.…”