Some Numerical Methods For Solving Nonlinear Equations And Their Theoretical Analysis

The study of nonlinear problems is an active field in modern mathematical research. Dealing with a large amount of nonlinear problems, such as nonlinear finite element prob-lems,economic and nonlinear programming problems, along with many pivotal problems in physics,chemistry and hydrodynamics,it is often needed to solve nonlinear equations. Moreover,solving nonlinear equations is also a common and important problem in scien-tific and engineering computation. But it is difficult to solve nonlinear equations by direct method except a few special cases. In many cases of the practice, it is not necessory to find the accurate solution of nonlinear equation. where an approximate solution should be suited. Of course, the error between approximate solutions and accurate solutions should be tolerated by the problem in practice. Since the approximate solution can be obtained by numerical methods, it is of important theoretical significance and practical utility to study numerical methods for solving non-linear equations.This paper consists of seven chapters.In the first chapter, we summarize the research actuality and our main work.In the second chapter, a family of third-order method with parameters is studied. This family includes the classical Cauchy's method, Chebyshev's method and Halley's method. So we call it Cauchy-Chebyshev-Halley methods. Based on this family, we further present a class of modifed Cauchy-Chebyshev-Halley methods. The total number of function and derivative evaluations required by each iteration of the modified methods is the same as Cauchy's, Chebyshev's, Halley's and super-Halley methods, but the order of convergence is increased from three to four. Therefore, the effciency index of the modified methods is largely improved. Some numerical results show the better performance of the modified methods.In the third chapter, based on Newton's irrational method, we present a family of multi-step methods. Per iteration the proposed methods add one evaluation of the function at another point obtained in the procedure iterated by Newton's irrational method. So each iteration of the proposed methods require two function evaluations, one first derivative evaluation and one second derivative evaluation, but the order of convergence is increased from three for the old method to five for the proposed method. Compared to Newton's irrational method, the effciency index of the modified method is higher. Numerical results show that the modified methods are efficient. In the fourth chapter, an acceleration technique is developed to modify Newton's method, and as a result, a class of methods with the asymptotic convergent order 1+(?) is obtained. Per iteration the new methods require the same evaluations of the function and its first derivative as Newton's method. Therefore the effciency index of these methods is larger than the one of Newton's method. Furthermore, we present a class of iterative method free from the derivative for solving nonlinear equations. Theoretical analysis shows that the asymptotic order of convergence of this family is 1+(?)-It is worthy to note that only two evaluations of the function are required by per iteration of this family, but it doesnot require any derivatives. This family of method can save the computational cost when the computational cost of the derivative is expensive, and thus the computational efficiency is enhanced. Numerical results show that the two classes of methods are efficient.In the fifth chapter, we study semilocal convergence of a class of multipoint fourth-order super-Halley method in banach spaces. Using recurrence relations, we make an attempt to prove that the sequence generated by the methods converge to a solution of the equation under certain conditions. An existence-uniqueness theorem is given to establish the R-order of the methods to be four and a priori error bounds is also obtained. Finally, we apply a special one of this family to solve the integral equation, and numerical results show this method is efficient.In the sixth chapter, based on the Jarratt method, we present a modified Jarratt method.Per iteration the new method adds an evaluation of the function at another point in the procedure iterated by Jarratt method.We also extend the modified method to Banach spaces and make an attempt to establish the semilocal convergence of this method by using recurrence relations. We also prove that the R-order of the method is six.Numerical results show the rationality of our recurrence relations.In the seventh chapter, we consider the application of some nonlinear iterative solvers to flash calculation, which is a basic and interesting ingredient of chemical engeering. It is a key part of flash calculation to solve the Rachford-Rice equation, which is a nonlinear scalar equation. Newton's method is often used to solve this equation. Note that the calculation of deratives is more expensive than that of functions. In this chapter, we employ some proposed methods to solve this equation and compare them with many classical methods including Newton's method. Numerical results demonstrate that for most tested cases, the sixth-order variant of Jarratt's method and accerated Newton's method with the order of 1+(?) perfom better than Jarratt's method and Newton's method, and the modified secant-like method with the order of 1+(?) is the best choice among the tested methods, which does not require any derivative.