Yongzhong Song

Constructing higher-order methods for obtaining the multiple roots of nonlinear equations

This paper concentrates on iterative methods for obtaining the multiple roots of nonlinear equations. Using the computer algebra system Mathematica, we construct an iterative scheme and discuss the conditions to obtain fourth-order methods from it. All the presented fourth-order methods require one-function and two-derivative evaluation per iteration, and are optimal higher-order iterative methods for obtaining multiple roots. We present some special methods from the iterative scheme, including some known already. Numerical examples are also given to show their performance.

Families of third and fourth order methods for multiple roots of nonlinear equations

This paper presents two families of higher-order iterative methods for solving multiple roots of nonlinear equations. One is of order three and the other is of order four. The presented iterative families all require two evaluations of the function and one evaluation of its first derivative, thus the latter is of optimal order. The third-order family contains several iterative methods known already. And, different from the optimal fourth-order methods for multiple roots known already, the presented fourth-order family use the modified Newtons method as its first step. Local convergence analyses and some special cases of the presented families are given. We also carry out some numerical examples to show their performance.

Trigonometrically fitted explicit Numerov-type method for periodic IVPs with two frequencies

Abstract In this paper, new trigonometrically fitted Numerov type methods for the periodic initial problems are proposed. These methods are based on the original Numerov-type sixth order method with fifth internal stages motivated by Tsitouras (see [Ch. Tsitouras, Explicit Numerov type methods with reduced number of stages, Comput. Math. Appl. 45 (2003) 37–42]). Some parameters are added to these methods so that they can integrate exactly the combination of trigonometrically functions with two frequencies. Numerical stability and phase properties of the new methods are analyzed. Numerical experiments are carried out to show the efficiency and robustness of our new methods in comparison with the well known codes proposed in the scientific literature.

Semiconvergence of extrapolated iterative methods for singular linear systems

Abstract In this paper, we discuss convergence of the extrapolated iterative methods for solving singular linear systems. A general principle of extrapolation is presented. The semiconvergence of an extrapolated method induced by a regular splitting and a nonnegative splitting is proved whenever the coefficient matrix A is a singular M-matrix with ‘property c’ and an irreducible singular M-matrix, respectively. Since the (generalized, block) JOR and AOR methods are respectively the extrapolated methods of the (generalized, block) Jacobi and SOR methods, so the semiconvergence of the (generalized, block) JOR and AOR methods for solving general singular systems are proved. Furthermore, the semiconvergence of the extrapolated power method, the (block) JOR, AOR and SOR methods for solving Markov chains are discussed.

Two stage waveform relaxation method for the initial value problems of differential-algebraic equations

In this paper, we consider a two stage strategy for waveform relaxation (WR) iterations, applied to initial value problems for differential-algebraic equations (DAEs) in the form A y ? ( t ) + B y ( t ) = f ( t ) . Outer iterations of TSWR are defined by M A y ? ( k + 1 ) ( t ) + M 1 y ( k + 1 ) ( t ) = N 1 y ( k ) ( t ) + N A y ? ( k ) ( t ) + f ( t ) , where A = M A - N A , B = M 1 - N 1 , and each iteration y ( k + 1 ) ( t ) is computed using an inner iterative process, based on another splitting M 1 = M 2 - N 2 . Meanwhile, by the means of the Theta method, the discretized TSWR of DAEs is realized. Furthermore, when M A is an Hermitian positive semi-definite matrix with P -regular splittings, the convergence and the comparison theorems of TSWR are analyzed. Finally, the numerical experiments are presented.

Semiconvergence of block SOR method for singular linear systems with p -cyclic matrices

In this paper, we discuss semiconvergence of the block SOR method for solving singular linear systems with p-cyclic matrices. Some sufficient conditions for the semiconvergence of the block SOR method for solving a general p-cyclic singular system are proved.

A successive quadratic approximations method for nonlinear eigenvalue problems

Numerical methods for matrix eigenvalue problems that are nonlinear in the eigenvalue parameter are discussed. We propose a successive quadratic approximations method, which reduces the nonlinear eigenvalue problem into a sequence of quadratic problems. The convergence for the new method is investigated. Numerical experiments illustrate the effectiveness of the method.

On the convergence radius of the modified Newton method for multiple roots under the center---Hölder condition

It is very important to enlarge the convergence ball of an iterative method. Recently, the convergence radius of the modified Newton method for finding multiple roots of nonlinear equations has been presented by Ren and Argyros when the involved function is Hölder and center–Hölder continuous. Different from the technique and the hypothesis used by them, in this paper, we also investigate the convergence radius of the modified Newton method under the condition that the derivative

Convergence radius of Osada's method under center-Hölder continuous condition

f^{(m)}

A new local energy-preserving algorithm for the BBM equation

of function f satisfies the center–Hölder continuous condition. The radius given here is larger than that given by Ren and Argyros. The uniqueness ball of solution is also discussed. Some examples are given to show applications of our theorem.