On C-To-R-Based Iteration Methods for a Class of Complex Symmetric Weakly Nonlinear Equations

Min-Li Zeng,Guo-Feng Zhang

doi:10.3390/math8020208

Abstract

To avoid solving the complex systems, we first rewrite the complex-valued nonlinear system to real-valued form (C-to-R) equivalently. Then, based on separable property of the linear and the nonlinear terms, we present a C-to-R-based Picard iteration method and a nonlinear C-to-R-based splitting (NC-to-R) iteration method for solving a class of large sparse and complex symmetric weakly nonlinear equations. At each inner process iterative step of the new methods, one only needs to solve the real subsystems with the same symmetric positive and definite coefficient matrix. Therefore, the computational workloads and computational storage will be saved in actual implements. The conditions for guaranteeing the local convergence are studied in detail. The quasi-optimal parameters are also proposed for both the C-to-R-based Picard iteration method and the NC-to-R iteration method. Numerical experiments are performed to show the efficiency of the new methods.

Highlights

We consider the iterative solutions of nonlinear system of equations in the following form, Au = φ(u), orF (u) = Au − φ(u) = 0, (1)where A = W + iT ∈ Cn×n is a large, sparse, complex symmetric matrix, with W ∈ Rn×n andT ∈ Rn×n being the real parts and the imaginary parts of the coefficient matrix A, respectively.Here, we assume that W and T are both symmetric positive and semidefinite (SPSD) and at least one of them being symmetric positive and definite (SPD)
We assume that W and T are both symmetric positive and semidefinite (SPSD) and at least one of them being symmetric positive and definite (SPD)
When the linear term Au is strongly dominant over the nonlinear term φ(u) in certain norm [1], we say that the system of nonlinear Equation (1) is weakly nonlinear

Summary

Introduction

To improve the efficiency of the Newton iteration method, Bai and Guo [4] use the Hermitian and skew-Hermitian splitting (HSS) method to solve approximately the Newton Equations (2), called the Newton-HSS method and Guo and Duff [22] analyze the Kantorovich-type semilocal convergence. Bai and Yang [1] present the nonlinear HSS-like iteration method based on the Hermitian and skew-Hermitian (HS) splitting of the non-Hermitian coefficient matrix of the linear term Au. Some variants of the HSS-based methods for nonlinear equations can be found in references, e.g., the lopsided preconditioned modified HSS (LPMHSS) iteration method [23], the Newton-MHSS method [24], the accelerated Newton-GPSS iteration method [25], the preconditioned modified.

The C-To-R-Based Picard Iteration Method

The Nonlinear C-to-R-Based Splitting Iteration Method

Numerical Experiments

Concluding Remarks