Modified Newton-DSS method for solving a class of systems of nonlinear equations with complex symmetric Jacobian matrice
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Modified Newton-DSS method for solving a class of systems of nonlinear equations with complex symmetric Jacobian matrices Fang Xie1 · Rong-Fei Lin2 · Qing-Biao Wu1 Received: 26 December 2018 / Accepted: 6 November 2019 / © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract Double-step scale splitting (DSS) iteration method is proved to be an unconditionally convergent iteration method, which is also efficient and robust for solving a class of large sparse complex symmetric systems of linear equations. In this paper, by making use of the DSS iteration technique as the inner solver to approximately solve the Newton equations, we establish a new modified Newton-DSS method for solving systems of nonlinear equations whose Jacobian matrices are large, sparse, and complex symmetric. Subsequently, we investigate the local and semilocal convergence properties of our method under some proper assumptions. Finally, numerical results on some problems illustrate the superiority of our method over some previous methods. Keywords Complex nonlinear system · Double-step scale splitting · Modified Newton method · Convergence analysis
Fang Xie
[email protected] Rong-Fei Lin
[email protected] Qing-Biao Wu
[email protected] 1
School of Mathematical Sciences, Zhejiang University, Hangzhou, 310027 Zhejiang, People’s Republic of China
2
Department of Mathematics, Taizhou University, Linhai, 317000 Zhejiang, People’s Republic of China
Numerical Algorithms
1 Introduction We assume that F : D ⊂ Cn → Cn is a continuously differentiable mapping defined on an open convex subset of n-dimensional complex linear space Cn and consider the iteration solution of the large sparse system of nonlinear equations: F (x) = 0.
(1.1)
The Jacobian matrix of F (x) is large, sparse, and complex symmetric, i.e., F (x) = W (x) + iT (x)
(1.2)
satisfies that matrices W (x) and T (x) are both symmetric and real√positive definite, which implies that the complex matrix F (x) is nonsingular. i = −1 is the imaginary unit. Actually, such nonlinear equations can be derived in many practical cases, such as nonlinear waves, quantum mechanics, chemical oscillations, and turbulence (see [1–4]). To our knowledge, inexact Newton method [5] is the most classic and popular iteration method for solving the system of nonlinear equations, which can be formulated as: F (xk )sk = −F (xk ) + rk , with xk+1 := xk + sk , where x0 ∈ D is a given initial vector and rk is a residual yielded by the inner iteration. Obviously, it is the variant of Newton’s method where the so-called Newton equation F (xk )sk = −F (xk ) is solved approximately at each iteration. In particular, when the scale of problems is large, linear iterative methods are commonly applied to compute the approximation solution. For example, the Newton-Krylov subspace methods [6], which make use of Krylov subspace methods as inner iterations to solve the Newton equations, have been widely studied and successfully used. Recently, based on the Hermitian and skew-Hermitian spli
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