Abstract
For solving the large sparse linear systems with 2 × 2 block structure, the generalized successive overrelaxation (GSOR) iteration method is an efficient iteration method. Based on the GSOR method, the PGSOR method introduces a preconditioned matrix with a new parameter for the coefficient matrix which can enhance the efficiency. To solve the nonlinear systems in which the Jacobian matrices are complex and symmetric with the block two-by-two form, we try to use the PGSOR method as an inner iteration, with the help of the modified Newton method as an efficient outer iteration method. This new method is called the modified Newton-PGSOR (MN-PGSOR) method. Local convergence properties of the MN-PGSOR are analyzed under the Hölder condition. Finally, we give the comparison of our new method with some previous methods in the numerical results. The MN-PGSOR method is superior in both iteration steps and computing time.
Highlights
IntroductionWhere F: D ⊂ C2n ⟶ C2n is a nonlinear and continuously differentiable function. Here, the Jacobian matrix of F(x) is large and sparse, which has the block form
We consider solving the nonlinear system F(x) 0, (1)where F: D ⊂ C2n ⟶ C2n is a nonlinear and continuously differentiable function
We apply the modified Newton method as the outer iteration method. en, we obtain a new method to solve the nonlinear systems with the block two-by-two complex symmetric Jacobian matrices, and we denote this new approach as MN-preconditioned GSOR (PGSOR) method
Summary
Where F: D ⊂ C2n ⟶ C2n is a nonlinear and continuously differentiable function. Here, the Jacobian matrix of F(x) is large and sparse, which has the block form. To solve the nonlinear systems (1) faster, of which the Jacobian matrix is complex symmetric with block two-by-two form, we focus on finding an efficient inner iteration method to solve the block complex linear system:. In 2020, Qi et al [25] proposed the modified Newton-AGSOR method, which can solve nonlinear systems with complex and symmetric Jacobian matrices of block form efficiently. We focus on improving the inner iteration method of the modified Newton method to solve the nonlinear systems with complex symmetric Jacobian matrices of the block form faster. En, we obtain a new method to solve the nonlinear systems with the block two-by-two complex symmetric Jacobian matrices, and we denote this new approach as MN-PGSOR method. We give some summaries of the results we have obtained
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