scholarly journals Two new effective iteration methods for nonlinear systems with complex symmetric Jacobian matrices

2021 ◽  
Vol 40 (3) ◽  
Author(s):  
Lv Zhang ◽  
Qing-Biao Wu ◽  
Min-Hong Chen ◽  
Rong-Fei Lin
Keyword(s):  
Nonlinear Systems ◽  
Iterative Methods ◽  
Newton Method ◽  
Convergence Properties ◽  
Jacobian Matrices ◽  
New Methods ◽  
Modified Newton Method ◽  
Iteration Methods ◽  
Complex Symmetric ◽  
Inner Iteration

AbstractIn this paper, we mainly discuss the iterative methods for solving nonlinear systems with complex symmetric Jacobian matrices. By applying an FPAE iteration (a fixed-point iteration adding asymptotical error) as the inner iteration of the Newton method and modified Newton method, we get the so–called Newton-FPAE method and modified Newton-FPAE method. The local and semi-local convergence properties under Lipschitz condition are analyzed. Finally, some numerical examples are given to expound the feasibility and validity of the two new methods by comparing them with some other iterative methods.

scholarly journals MN-PGSOR Method for Solving Nonlinear Systems with Block Two-by-Two Complex Symmetric Jacobian Matrices

2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Yu-Ye Feng ◽  
Qing-Biao Wu
Keyword(s):  
Nonlinear Systems ◽  
Iteration Method ◽  
Computing Time ◽  
Coefficient Matrix ◽  
New Method ◽  
Jacobian Matrices ◽  
Modified Newton Method ◽  
Preconditioned Matrix ◽  
Outer Iteration ◽  
Inner Iteration

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.

scholarly journals New Families of Third-Order Iterative Methods for Finding Multiple Roots

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
R. F. Lin ◽  
H. M. Ren ◽  
Z. Šmarda ◽  
Q. B. Wu ◽  
Y. Khan ◽  
...  
Keyword(s):  
Iterative Methods ◽  
Newton Method ◽  
Multiple Roots ◽  
Third Order ◽  
Cubic Convergence ◽  
Numerical Examples ◽  
Mild Conditions ◽  
First Derivative ◽  
Modified Newton Method ◽  
Iteration Methods

Two families of third-order iterative methods for finding multiple roots of nonlinear equations are developed in this paper. Mild conditions are given to assure the cubic convergence of two iteration schemes (I) and (II). The presented families include many third-order methods for finding multiple roots, such as the known Dong's methods and Neta's method. Some new concrete iterative methods are provided. Each member of the two families requires two evaluations of the function and one of its first derivative per iteration. All these methods require the knowledge of the multiplicity. The obtained methods are also compared in their performance with various other iteration methods via numerical examples, and it is observed that these have better performance than the modified Newton method, and demonstrate at least equal performance to iterative methods of the same order.

Accelerating Vector Iteration Methods

1986 ◽  
Vol 53 (2) ◽  
pp. 291-297 ◽  
Author(s):  
M. Papadrakakis
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Iterative Methods ◽  
Conjugate Gradient ◽  
Test Problems ◽  
Convergence Properties ◽  
Dynamic Relaxation ◽  
Element Level ◽  
Iteration Methods ◽  
Partial Elimination

This paper describes a technique for accelerating the convergence properties of iterative methods for the solution of large sparse symmetric linear systems that arise from the application of finite element method. The technique is called partial preconditioning process (PPR) and can be combined with pure vector iteration methods, such as the conjugate gradient, the dynamic relaxation, and the Chebyshev semi-iterative methods. The proposed triangular splitting preconditioner combines Evans’ SSOR preconditioner with a drop-off tolerance criterion. The (PPR) is attractive in a FE framework because it is simple and can be implemented at the element level as opposed to incomplete Cholesky preconditioners, which require a sparse assembly. The method, despite its simplicity, is shown to be more efficient on a set of test problems for certain values of the drop-off tolerance parameter than the partial elimination method.

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

2016 ◽  
Vol 6 (4) ◽  
pp. 367-383 ◽  
Author(s):  
Min-Li Zeng ◽  
Guo-Feng Zhang
Keyword(s):  
Nonlinear Systems ◽  
Iteration Method ◽  
Large Scale ◽  
Positive Definite ◽  
Weakly Nonlinear ◽  
Large Scale Systems ◽  
New Methods ◽  
Iteration Methods ◽  
Optimal Iterative Parameters ◽  
Two Parameter

AbstractIn this paper, we first construct a preconditioned two-parameter generalized Hermitian and skew-Hermitian splitting (PTGHSS) iteration method based on the two-parameter generalized Hermitian and skew-Hermitian splitting (TGHSS) iteration method for non-Hermitian positive definite linear systems. Then a class of PTGHSS-based iteration methods are proposed for solving weakly nonlinear systems based on separable property of the linear and nonlinear terms. The conditions for guaranteeing the local convergence are studied and the quasi-optimal iterative parameters are derived. Numerical experiments are implemented to show that the new methods are feasible and effective for large scale systems of weakly nonlinear systems.

scholarly journals Modified Newton method to determine multiple zeros of nonlinear equations

2016 ◽  
Vol 11 (10) ◽  
pp. 5774-5780
Author(s):  
Rajinder Thukral
Keyword(s):  
Iterative Method ◽  
Nonlinear Equations ◽  
Newton Method ◽  
New Method ◽  
Convergence Order ◽  
Multiple Zeros ◽  
Numerical Comparisons ◽  
Modified Newton Method ◽  
Iteration Methods ◽  
Better Than

New one-point iterative method for solving nonlinear equations is constructed.  It is proved that the new method has the convergence order of three. Per iteration the new method requires two evaluations of the function.  Kung and Traub conjectured that the multipoint iteration methods, without memory based on n evaluations, could achieve maximum convergence order2n-1  but, the new method produces convergence order of three, which is better than expected maximum convergence order of two.  Hence, we demonstrate that the conjecture fails for a particular set of nonlinear equations. Numerical comparisons are included to demonstrate exceptional convergence speed of the proposed method using only a few function evaluations.

scholarly journals Derivative-Free Iterative Methods with Some Kurchatov-Type Accelerating Parameters for Solving Nonlinear Systems

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 943
Author(s):  
Xiaofeng Wang ◽  
Yingfanghua Jin ◽  
Yali Zhao
Keyword(s):  
Nonlinear Systems ◽  
Iterative Methods ◽  
Numerical Experiments ◽  
Order Of Convergence ◽  
Nonlinear Ordinary Differential Equations ◽  
New Methods ◽  
Derivative Free ◽  
Initial Scheme ◽  
With Memory ◽  
Theoretical Results

Some Kurchatov-type accelerating parameters are used to construct some derivative-free iterative methods with memory for solving nonlinear systems. New iterative methods are developed from an initial scheme without memory with order of convergence three. New methods have the convergence order 2+5≈4.236 and 5, respectively. The application of new methods can solve standard nonlinear systems and nonlinear ordinary differential equations (ODEs) in numerical experiments. Numerical results support the theoretical results.

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