Weihong Bi

A new family of eighth-order iterative methods for solving nonlinear equations

A family of eighth-order iterative methods with four evaluations for the solution of nonlinear equations is presented. Kung and Traub conjectured that an iteration method without memory based on n evaluations could achieve optimal convergence order 2^n^-^1. The new family of eighth-order methods agrees with the conjecture of Kung-Traub for the case n=4. Therefore this family of methods has efficiency index equal to 1.682. Numerical comparisons are made with several other existing methods to show the performance of the presented methods.

A CLASS OF TWO-STEP STEFFENSEN TYPE METHODS WITH FOURTH-ORDER CONVERGENCE

Abstract Based on Steffensen’s method, we derive a one-parameter class of fourth-order methods for solving nonlinear equations. In the proposed methods, an interpolating polynomial is used to get a better approximation to the derivative of the given function. Each member of the class requires three evaluations of the given function per iteration. Therefore, this class of methods has efficiency index which equals 1.587. Kung and Traub conjectured an iteration using n evaluations of f or its derivatives without memory is of convergence order at most 2 n - 1 . The new class of fourth-order methods agrees with the conjecture of Kung–Traub for the case n = 3 . Numerical comparisons are made to show the performance of the presented methods.

New variants of Jarratt’s method with sixth-order convergence

In this paper, by using the two-variable Taylor expansion formula, we introduce some new variants of Jarratt’s method with sixth-order convergence for solving univariate nonlinear equations. The proposed methods contain some recent improvements of Jarratt’s method. Furthermore, a new variant of Jarratt’s method with sixth-order convergence for solving systems of nonlinear equations is proposed only with an additional evaluation for the involved function, and not requiring the computation of new inverse. Numerical comparisons are made to show the performance of the presented methods.

New family of seventh-order methods for nonlinear equations

Abstract A family of seventh-order iterative methods for the solution of nonlinear equations is presented. The new methods are based on King’s fourth-order methods and without using the second derivatives. Per iteration the new methods require three evaluations of the function and one evaluation of its first derivative. Therefore, this family of methods has efficiency index equal to 1.627. Numerical comparisons are made with several other existing methods to show the performance of the presented methods.

On convergence of a new secant-like method for solving nonlinear equations

In this paper, we prove that the order of a new secant-like method presented recently with the so-called order of 2.618 is only 2.414. Some mistakes in the derivation leading to such a conclusion are pointed out. Meanwhile, under the assumption that the second derivative of the involved function is bounded, the convergence radius of the secant-like method is given, and error estimates matching its convergence order are also provided by using a generalized Fibonacci sequence.

Convergence of the modified Halley's method for multiple zeros under Hölder continuous derivative

In this paper, the estimate of the radius of the convergence ball of the modified Halley’s method for finding multiple zeros of nonlinear equations is provided under the hypotheses that the derivative f(m + 1) of function f is Hölder continuous, and f(m + 1) is bounded. The uniqueness ball of solution is also established. Finally, some examples are provided to show applications of our theorem.

Convergence ball and error analysis of Müller’s method

Abstract Under the hypotheses that the second-order and third-order derivative of function f are bounded, an estimate of the radius of the convergence ball of Muller’s method is obtained, an error analysis is given which matches the convergence order of the method.

Solving nonlinear equations system via an efficient genetic algorithm with symmetric and harmonious individuals

We present an efficient genetic algorithm as a general tool for solving optimum problems. As a specialized application this algorithm can be used to approximate a solution of a system of nonlinear equations on n-dimensional Euclidean space setting. The new idea involves the introduction of some pairs of symmetric and harmonious individuals for the generation of a genetic algorithm. The population diversity is maintained better this way. The elitist model is used to ensure the convergence. Simulation results show that the new method is indeed very effective.

A new semilocal convergence theorem of Müller's method

Abstract A semilocal convergence theorem of Muller’s method is established under the γ -condition. An error estimation is given which matches the convergence order of the method. Numerical examples are provided to show that our results apply, where earlier ones fail.

The convergence ball of Wang's method for finding a zero of a derivative

Under the hypotheses that the third-order and fourth-order derivatives of function f are bounded, an estimate of the radius of the convergence ball of Wangs method is obtained. The error analysis is also given. Finally, two numerical examples are provided to show applications of our theorem.