非线性方程求解的若干研究
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摘要
本文主要研究了非线性方程求解中的一些问题,研究主要针对以下几个方面,这几个方面恰恰是研究非线性方程求解问题中非常重要的领域.一是在给定的一定量信息的情况下构造具有最大收敛阶迭代法;二是在一定理论框架下研究迭代法的半局部收敛性;三是研究迭代法对多项式的整体行为.
在研究迭代法的构造时,每一迭代步所用到的信息是一个很重要的关键,信息通常由前面一些迭代近似点上的函数值和导数值给定.我们先对信息加以筛选,给出了标准信息的定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}.考虑到信息的获取是要付出一定的代价的,因而我们这里用到的信息是N(x_n~s,…,x_(n-l+1)~s,x_(n-l)~(s');f).qd商差表法是一种基于Pade有理逼近的求根方法,对于给定的标准信息,我们将Pade有理逼近进行了推广,得到了基于标准信息的有理逼近,这些有理逼近对我们构造迭代法很有帮助.
最高效的方法就是拥有最大收敛阶的迭代法,我们给出了两族具最大收敛阶的迭代法H_(p,s,f)和M_(p,s,f),它们分别是Halley迭代族以及M(?)ller法的自然推广,并且具有非常好的收敛性质.其中g(·)=1/f(·)且0＜s'≤s,l≥0.则若x_i(i=-l,…,-1,0)与函数f的根z~*足够接近时,当n→∞时,两族迭代法产生的x_n收敛到与x_0最接近的函数f的根z~*,且它的收敛阶是多项式t~(l+1)-(?)-s'的唯一正实根.
上述两族迭代法是基于标准信息N(x_n~s,,…,x_(n-l=1)~s,x_(n-l)~(s');f)的具最高收敛阶的迭代法,我们利用代数组合学的基本工具Fa(?)di公式,部分Bell多项式以及对称群循环指标给出了H_(p,s,f)和M_(p,s,f)的显示表达式.
对迭代法性质的研究最主要的是研究它的收敛性行为,在本文中我们给出了迭代族H_(p,s,f)的半局部收敛性定理.我们是在区域B(x_0,r)上的算子类S_γ~((k))(Ω)中研究半局部收敛性的,函数f属于区域B(x_0,r)上的算子类S_γ~((k))(Ω)是指对于k是正整数,设0＜r＜(?),且f满足其中Ω:[0,(?)]→[0,(?)]在区间[0,r]上存在k阶单调递增的连续导数.
区域B(x_0,r)上的算子类S_γ~((k))(Ω)首先是由王兴华提出的,王兴华首先将Smale的解析条件弱化提出了弱条件,然后再进一步地,提出了更加一般的区域B(x_0,r)上的算子类S_γ~((k))(Ω),在这种算子类下,半局部收敛性仍存在普适常数b,而r_0满足只要函数f属于区域B(x_0,r)上的算子类S_γ~((k))(Ω),初始条件β=‖f'(x_0)~(-1)f(x_0)‖≤b,x_i∈B(x_0,r),i=-1,…,-l,迭代法就收敛并且有相应的误差估计.这里得到的常数b同王兴华已经证明的对Halley族以及Euller族普遍适用的常数是相同的,从而推广了普适常数的适用范围.
对大多数迭代法,例如Euler族和Halley族的迭代,当f是多项式时都是有理迭代,从离散动力系统的观点看,关于迭代法整体行为的首要问题是:是否存在一般收敛的单点定常迭代算法?McMullen在1987年否定地回答了这个问题,证明了当多项式的次数大于3时,一般收敛的定常迭代法不存在.同时他也给出了一个对三次多项式一般收敛的有理迭代,设p(z)=z~3+az+b,则对三次多项式一般收敛.
自然地,是否存在对次数小于等于3次的多项式一般收敛且能用来作为一般函数求根算子的有理算子呢?我们给出了满足这样条件的一个有理算子.为方便起见,设p(z)是一个第d-1次项系数为零的d次多项式,则我们证明了迭代算子T_p对二次、三次多项式一般收敛,而且对次数d≥4的多项式是二阶收敛的,对于非多项式函数d可以任意选取.进一步地,我们可以固定d=3,这时不管p是多项式还是非多项式,T_p是三阶收敛的.同时,迭代法T_p具有如下性质:
1.T_p(z)在复平面C上的不动点是超吸引的或排斥的,也就是说,p的所有单根是超吸引的,复平面C上的所有额外不动点是排斥的.
2.p的重数为n_i≥3的重根均为T_p(z)的排斥不动点,且乘子为1+(?).p的两重根不是T_p(z)的不动点.
3.当d≥4并且z~(d-2)项的系数不为0时,则∞是T_p(z)的一个吸引不动点且乘子为1-(?).因此,T_p(z)的Julia集是有界的.这一工作是对McMullen工作的重要补充
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 x_0 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,f)in operator class S_γ~((k))(Ω) of domain B(x_0, r). A function f is in the operator class S_γ~((k))(Ω)of domain B(x_0, 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(x_0, 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(x_0,r). The universal constant also existsunder this kind of condition, that iswhere r_0 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(x_0,r) and initial condi-tionsβ=‖f'(x_0)~(-1)f(x_0)‖≤b,x_i∈B(x_0,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) = z~3+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.
在研究迭代法的构造时,每一迭代步所用到的信息是一个很重要的关键,信息通常由前面一些迭代近似点上的函数值和导数值给定.我们先对信息加以筛选,给出了标准信息的定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}.考虑到信息的获取是要付出一定的代价的,因而我们这里用到的信息是N(x_n~s,…,x_(n-l+1)~s,x_(n-l)~(s');f).qd商差表法是一种基于Pade有理逼近的求根方法,对于给定的标准信息,我们将Pade有理逼近进行了推广,得到了基于标准信息的有理逼近,这些有理逼近对我们构造迭代法很有帮助.
最高效的方法就是拥有最大收敛阶的迭代法,我们给出了两族具最大收敛阶的迭代法H_(p,s,f)和M_(p,s,f),它们分别是Halley迭代族以及M(?)ller法的自然推广,并且具有非常好的收敛性质.其中g(·)=1/f(·)且0＜s'≤s,l≥0.则若x_i(i=-l,…,-1,0)与函数f的根z~*足够接近时,当n→∞时,两族迭代法产生的x_n收敛到与x_0最接近的函数f的根z~*,且它的收敛阶是多项式t~(l+1)-(?)-s'的唯一正实根.
上述两族迭代法是基于标准信息N(x_n~s,,…,x_(n-l=1)~s,x_(n-l)~(s');f)的具最高收敛阶的迭代法,我们利用代数组合学的基本工具Fa(?)di公式,部分Bell多项式以及对称群循环指标给出了H_(p,s,f)和M_(p,s,f)的显示表达式.
对迭代法性质的研究最主要的是研究它的收敛性行为,在本文中我们给出了迭代族H_(p,s,f)的半局部收敛性定理.我们是在区域B(x_0,r)上的算子类S_γ~((k))(Ω)中研究半局部收敛性的,函数f属于区域B(x_0,r)上的算子类S_γ~((k))(Ω)是指对于k是正整数,设0＜r＜(?),且f满足其中Ω:[0,(?)]→[0,(?)]在区间[0,r]上存在k阶单调递增的连续导数.
区域B(x_0,r)上的算子类S_γ~((k))(Ω)首先是由王兴华提出的,王兴华首先将Smale的解析条件弱化提出了弱条件,然后再进一步地,提出了更加一般的区域B(x_0,r)上的算子类S_γ~((k))(Ω),在这种算子类下,半局部收敛性仍存在普适常数b,而r_0满足只要函数f属于区域B(x_0,r)上的算子类S_γ~((k))(Ω),初始条件β=‖f'(x_0)~(-1)f(x_0)‖≤b,x_i∈B(x_0,r),i=-1,…,-l,迭代法就收敛并且有相应的误差估计.这里得到的常数b同王兴华已经证明的对Halley族以及Euller族普遍适用的常数是相同的,从而推广了普适常数的适用范围.
对大多数迭代法,例如Euler族和Halley族的迭代,当f是多项式时都是有理迭代,从离散动力系统的观点看,关于迭代法整体行为的首要问题是:是否存在一般收敛的单点定常迭代算法?McMullen在1987年否定地回答了这个问题,证明了当多项式的次数大于3时,一般收敛的定常迭代法不存在.同时他也给出了一个对三次多项式一般收敛的有理迭代,设p(z)=z~3+az+b,则对三次多项式一般收敛.
自然地,是否存在对次数小于等于3次的多项式一般收敛且能用来作为一般函数求根算子的有理算子呢?我们给出了满足这样条件的一个有理算子.为方便起见,设p(z)是一个第d-1次项系数为零的d次多项式,则我们证明了迭代算子T_p对二次、三次多项式一般收敛,而且对次数d≥4的多项式是二阶收敛的,对于非多项式函数d可以任意选取.进一步地,我们可以固定d=3,这时不管p是多项式还是非多项式,T_p是三阶收敛的.同时,迭代法T_p具有如下性质:
1.T_p(z)在复平面C上的不动点是超吸引的或排斥的,也就是说,p的所有单根是超吸引的,复平面C上的所有额外不动点是排斥的.
2.p的重数为n_i≥3的重根均为T_p(z)的排斥不动点,且乘子为1+(?).p的两重根不是T_p(z)的不动点.
3.当d≥4并且z~(d-2)项的系数不为0时,则∞是T_p(z)的一个吸引不动点且乘子为1-(?).因此,T_p(z)的Julia集是有界的.这一工作是对McMullen工作的重要补充
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 x_0 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,f)in operator class S_γ~((k))(Ω) of domain B(x_0, r). A function f is in the operator class S_γ~((k))(Ω)of domain B(x_0, 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(x_0, 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(x_0,r). The universal constant also existsunder this kind of condition, that iswhere r_0 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(x_0,r) and initial condi-tionsβ=‖f'(x_0)~(-1)f(x_0)‖≤b,x_i∈B(x_0,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) = z~3+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.
引文
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[43] Petkovic M S. Iterative methods for simultaneous inclusion of polynomial zeros. Lecture Notes in Math, Vol 1387, Berlin, Springer, 1989.
[44] Wang X H, Zheng S M. A family of parallel and interval iterations for finding all roots of polynomail simultaneously with rapid convergence. JCM,1984, 2(1):70-76.
[45] Wang X H, Zheng S M, Shen G X. Bell's disk polynomials and parallel disk iteration. Numer. Math. J. Chinese Univ., 1987, 9(4):328-346.
[46] Wang X H, Han D F. A general parallel iterative method for finding all the roots of a polynomial simultaneously. (Chinese)Kexue Tongbao(Chinese),1996, 41(12):1057-1060.
[47] Farmer M R, Loizou G. A class of iteration functions for improving, simultaneously, approximations to the zeros of a polynomial. BIT, 1975,15:250-258.
[48] Gerlach J. Accelerated convergence in Newton's method. SIAM Rev., 1996,38:658-659.
[49] Petkovic M S, Herceg D. On rediscovered iteration methods for solving equations. J. Comput. Appl. Math., 1999, 107:275-284.
[50] Kalantari B, Gerlach J. Newton's method and generation of a determinantal family of iteration functions. J. Comput. Appl. Math., 2000, 116:195-200.
[51] Argyros I K. The super-Halley method using divided differences. Appl.Math. Lett., 1997, 10:91-95.
[52] Argyros I K. On a two-point Newton-like method of convergent order two.Int. J. Comput. Math., 2005, 88:219-234.
[53] Rutishauser. Der Quotienten-Differenzen-Algorithmus. Birkhauser Verlag,Basel, 1957.
[54] Henrici, Peter. The Quotient-Difference Algorithm. Nat. Bur. Standards Appl. Math. Ser., 1958, 49:23-46.
[55] 徐献瑜,李家楷,徐国良.Pade逼近概论.上海科学技术出版社,上海,1990.
[56] Baker G A, Jr, Gammel J L(eds.) . The Fade approximation in theoretical physics. Academie press, New York, 1970.
[57] Baker G A, Jr. Essentiables of Fade approximants. Academic press, London, 1975.
[58] Bernoulli D. Comm. Acad. Sci. Imper. Petropolitanae. 1732, 3:62-69,Petropolis.
[59] Schr(?)der E. (?)ber unendlich viele Algorithmen zur Aufl(?)sung der Gleichungen. Math. Ann., 1870, 2:317-365.
[60] Wozniakowski H. Generalized information and maximal order of iteration for operator equations. SIAM J. Numer. Anal., 1975, 12:121-135.
[61] Kacewicz B. An integral-interpolation iterative method for the solution of scalar equations. Numer. Math., 1976, 26:355-365.
[62] Wang X H, Wang C J. Maximal stationary iterative methods for the solution of f~((k))(x) = 0. submitted to Sci. China Ser. A.
[63] Wozniakowski H. Maximal stationary iterative methods for the solution of operator equations. SIAM J. Numer. Anal, 1974, 11:934-949.
[64] Kung H T, Traub J F. Optimal-order of one point and multipoint iteration.J. Assoc. Comput. Mach., 1974, 21:643-651.
[65] Rissanen J. On optimum root-finding algorithms. J. Math. Anal. Appl.,1971, 36:220-225.
[66] Wang X H. On the Hermite interpolation. Sci. China Ser. A, 2007, 50:1651-1660.
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[68] Abramowitz M, Stegun I A. Handbook of Mathematical Functions with Formulas. Graphs, and Mathematical Tables, Dover Publications, Inc.,New York, 1972.
[69] Comtet L. Analyse combinatoire. Tomes Ⅰ et Ⅱ (French). Presses Universitaires de France, Paris, 1970.
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[71] Ortega J M, Rheinboldt W C. Iterative solution of nonlinear equations in several variables. Academic Press, New York, 1970.
[72] Ostrowski A. Solution of systems of equations. Academic Press, New York,1966.
[73] Rafiq A, Hussain S, Ahmad F, Awais M, Zafar F. An efficient three-step iterative method with sixthorder convergence for solving nonlinear equations.Int. J. Comput. Math., 2007, 84:369-375.
[74] Salkuyeh D K. A family of Newton-type methods for solving nonlinear equations. Int. J. Comput. Math., 2007, 84:411-419.
[75] Grau M. An improvement to the computing of nonlinear equation solutions.Numer. Algorithms, 2003, 34:1-12,.
[76] 王兴华,李冲.解方程算法的局部行为和整体行为.科学通报,46(6),(2001),444-451.
[77] Fatou P. Sur les solutions uniforms de certains e'quations fonctionnelle.C. R. Acad. Sci. Paris, 1906, 143:546-548.
[78] Fatou P. Sur les e'quations fonctionnelles. Bull. Soc. Math. France, 1919,47:161-271; 1920, 48:33-94, 208-314.
[79] Julia G. Memoire sur l'iteration de fonctions rationelles. J. Math. Pure ppl, 1918, 8:47-245.
[80] 任福尧.复解析动力系统.复旦大学出版社,上海,1996.
[81] Beardon A F. Iteration of Rational Functions. New York, Berlin: Springer-Verlag, 1991.
[82] Blanchard P. Complex analytic dynamics on the Riemann sphere. Bull.(New Ser.) Amer. Math. Soc, 1984,11(1):85 - 141.
[83] Montel P. Legons sur les families normales de fonctions analytiques et leurs applications. Gauthier-Villars, Pairs, 1927.
[84] 石钟慈,袁亚湘.奇效的计算湖南科学技术出版社,长沙,1998.
[85] Sullivan D. Quasiconformal homeomorphisms and dynamics I: Solution of the Fatou-Julia problem on wandering domains. Ann. Math., 1985,122:401-418.
[86] Buff X, Henriksen C. On K(?)nig' s rootfinding algorithms. Nonlinearlity,2003, 16(3):989 -1015.
[87] Wang X H. Definite version on precise point estimate. Progr. in Natur.Sci. (English Ed.), 1998, 8(2):152-158.
[88] Drakopoulos V. Generalized computation of Schroder iteration functions to motivate families of Julia and Mandelbrotlike sets. SIAM J. Numer. Anal.,1999, 36(2):417-435.
[89] Vrscay E R. Julia sets and mandelbrot-like sets associated with higher order Schr(?)der rational iteration functions: A computer assisted study. Math.Comp., 1986, 46:151-169.
[90] Vrscay E R, Gilbert W. Extraneous fixed points, basin boundaries and chaotic dynamics for Sch(?)der and K(?)nig rational itertion functions. Numer. Math., 1988, 52(1):1-16.
[91] Drakopoulos V. Are there any Julia sets for the Laguerre iteration function?. Comput. Math. Appl, 2003, 46(8-9):1201-1210.