Some Investigations On Solving Nonlinear Equation

This dissertation mainly studies some problems in solving nonlinear equations. The study aims at following aspects which are very important in studyingof nonlinear equation solving. One is construction of maximal order iterationmethod based on some mount of information, one is semi-local behavior of iteration method with some kind of theory frame, the other is global behavior forpolynomials.Information that used in every step is very important when we constructan iteration method. Generally, information is determined by values of function and derivations of function at previous approximate points. For all kinds ofinformation, we choose the most natural one and give a definition of standardinformation N(x_n^sn,x_n-1^sn-1,…,x_n-l^sn-l;f)={f^(k)(x_j):k=0,…,s_j-1,j=n-l,…,n}. Since it is veryhard to achieve information, the standard information used in this paper isN(x_n^s,…,x_n-l+1^s,x_n-l^s';f).We know that qd method is a kind of root finding method basedof Pade approximation. For some mount of standard information, we obtain ageneralization of Pade approximation. It is a rational approximation which isvery important in construction of iteration method based on standard information.Given standard information, the most efficient iteration method is maximalorder method. We construct two kinds of iteration methods H_p,s,f and M_p,s,f.They are maximal order iteration methods based on standard information. Further more, they are a natural generalization of Halley's family and M(?)ller method,respectively, with great convergence property. where g(·)=1/f(·) and 0＜s'≤s,l≥0.whereThen if x_i(i=-l,…,-1,0) are close enough to the root z^* of f, x_n generated bythe two iteration methods will converge to the root nearest to x₀ of f as n→∞.Their order of convergence is the unique real positive root of the polynomial(?).The two iteration methods are both iteration methods with maximal orderbased on standard information N(x_n^s,,…,x_n-l+1^s,x_n-l^s';f). By using partial Faadi formula, Bell polynomial and cycle index of symmetric group in combinatoricswe obtain an explicit expression of both iteration method H_p,s,f and M_p,s,f.In this dissertation we also have studied semi-local behavior, which is veryimportant in the research of convergence property of iteration method, of iteration method H_p,s,f. We study the semi-local behavior of iteration method H_p,s,fin operator class S_γ^(k)(Ω) of domain B(x₀, r). A function f is in the operator class S_γ^(k)(Ω)of domain B(x₀, r) if k is a positive integer, 0＜r＜(?) and f satisfieswhereΩ:[0,(?)]→[0,(?)] has continuous monotone increasing derivative of order kin [0, r]The operator class S_γ^(k)(Ω) of domain B(x₀, r) was first obtained by XinghuaWang. Wang first gave weak condition which is a much weaker condition thanSmale's analytic condition. Furthermore, he constructed a very general conditionoperator class S_γ^(k)(Ω) of domain B(x₀,r). The universal constant also existsunder this kind of condition, that iswhere r₀ satisfiesThe iteration method H_p,s,f will convergent and has an error estimate when thefunction f is in the operator class S_γ^(k)(Ω) of domain B(x₀,r) and initial condi-tionsβ=‖f'(x₀)^-1f(x₀)‖≤b,x_i∈B(x₀,r),i=-1,…,-l are satisfied. Theconstant b obtained in our dissertation is the same as that of Halley's family andEuler's family, whose universal constant is obtained by Xinghua Wang. Therefore, the universal constant can be applied to much wider iteration methods.Many iterations, such as Halley's family and Euler's family, are rationalwhen f is a polynomial. In the viewpoint of discrete dynamical system, the mostimportant problem is that whether there exists any generally convergent onepoint stationary algorithm. McMullen given a negative answer to this questionin 1987. He proved that there does not exist any generally convergent onepoint stationary algorithm when the degree of polynomial is greater than 3. Furthermore, he also gave a generally convergent rational iteration method forpolynomials of degree 3. Let p(z) = z³+az + b,is generally convergent for polynomials of degree 3.Therefore, we derive a question that does there exist any rational operatorwhich is generally convergent for polynomials of degree less than or equal to 3and can also be used to as root finding algorithm for general functions. We givea positive answer and construct an algorithm. Let p(z) be a polynomial of degreed with the coefficient of z^d-1 vanishes for simplicity. LetWe have proved that T_p is generally convergent for quadratic and cubic polynomials and it is an order 2 algorithm for polynomials of degree d≥4. Forany non-polynomials, d can be chosen arbitrarily. Furthermore, T_p is an order 3algorithm for every function no matter it is a polynomial or not, when we fixedd by 3.The method T_p has the following properties:1. The fixed points of T_p(z) in C are either superattracting or repelling. Thatis, the simple roots of p are superattracting and the extraneous fixed pointsin C are all repelling.2. The roots of p with multiplicities n_i≥3 are all repelling fixed points ofT_p(z) with multipliers 1+(?). The double roots are not fixed points ofT_p(z).3. For d≥4 and the coefficient of z^d-2 does not vanish, then∞is an attractingfixed point of T_p(z) with multiplier 1 -(?). Hence the Julia set of T_p(z) isbounded.It is an important implement of McMullen's result and meaningful in dynamics.