Jisheng Kou

An improvement of the Jarratt method

In this paper, we present a variant of Jarratt method for solving non-linear equations. Per iteration the new method adds the evaluation of the function at another point in the procedure iterated by Jarratt method. As a consequence, the local order of convergence is improved from four for Jarratt method to six for the new method. The new multistep iteration scheme, based on the new method, is developed and numerical tests verifying the theory are also given.

The improvements of modified Newton’s method

Abstract In this paper, we improve some third-order modifications of Newton’s method and obtain many new methods for solving non-linear equations. The new methods have the order of convergence five or six. Per iteration these methods require two evaluations of the function and two evaluations of its first derivative and therefore the efficiency of the new methods may also be improved. These methods can compete with Newton’s method, as we show in some examples.

A variant of Jarratt method with sixth-order convergence

In this paper, we present a variant of Jarratt method for solving non-linear equations. Per iteration the new method adds the evaluation of the function at another point in the procedure iterated by Jarratt method. As a consequence, the local order of convergence is improved from four for Jarratt method to six for the new method. A new multistep iteration scheme, based on the new method, is developed and numerical tests verifying the theory are also given.

Modified Jarratt method with sixth-order convergence

In this paper, we present a variant of Jarratt method with order of convergence six for solving non-linear equations. Per iteration the method requires two evaluations of the function and two of its first derivatives. The new multistep iteration scheme, based on the new method, is developed and numerical tests verifying the theory are also given.

A family of modified super-Halley methods with fourth-order convergence

In this paper, we present a family of modified super-Halley methods for solving non-linear equations. Analysis of convergence shows that the methods have fourth-order convergence. The superiority of the new methods is that they require no additional evaluations of the function, the first derivative or second derivative as compared with the classical third-order methods although their order is improved. Numerical results show that the new methods can be efficient.

On Chebyshev–Halley methods with sixth-order convergence for solving non-linear equations

Abstract In this paper, we present a family of new variants of Chebyshev–Halley methods. The new methods have sixth-order convergence although they only add one evaluation of the function at the point iterated by Chebyshev–Halley methods. The numerical results presented show that the new methods work better not only in order but in efficiency.

Some improvements of Ostrowski's method

Abstract In this paper, we present some new variants of Ostrowski’s method with order of convergence eight. For each iteration the new methods require three evaluations of the function and one evaluation of its first derivative and therefore they have the efficiency index equal to 1.682. Numerical tests verifying the theory are also given.

Some new sixth-order methods for solving non-linear equations

Abstract In this paper, we present some new sixth-order methods based on the Chebyshev–Halley methods. Per iteration the new methods require an additional evaluation of the function compared with Chebyshev–Halley methods. Analysis of efficiency shows that the new methods can compete with Newton’s method and the classical third-order methods, as we show in numerical results.

Fourth-order variants of Cauchy’s method for solving non-linear equations

Abstract In this paper, we present some new variants of Cauchy’s method, in which the second derivative is replaced by a finite difference between the first derivatives. Analysis of convergence shows that the new methods have fourth-order convergence. Per iteration the new methods require one evaluations of the function and two of its first derivative. Numerical results show that the new methods are efficient.

Second-derivative-free variants of Cauchy’s method

Abstract In this paper, we present some new variants of Cauchy’s method free from the second derivative for solving non-linear equations. Per iteration the new methods require two evaluations of the function and one of its first derivative. The new methods have fourth-order convergence. Numerical tests verifying the theory are also given.