求解非线性方程的某些高阶迭代方法的收敛性分析
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摘要
我们在利用数学手段研究自然现象和社会现象,或解决工程技术问题时,常常会把不少实际问题归结为Banach空间中形如F(x)=0的非线性方程的求解。而求解非线性方程的一个最有效方法则是迭代法。因此迭代法的研究有着十分重要的科学价值和实际意义。
这篇论文共分为五个部分。本文在第一章中给出了全文要用到的一些基本概念和记号,总结了证明各种迭代法收敛性的技巧以及几个著名迭代法的收敛条件。
在第二章中,针对Sharma提出的一个三阶收敛复合型Newton-Steffensen迭代方法,在假设算子的一阶导数满足(K,p)-Holder连续条件时进行了讨论,得到了其半局部收敛定理,同时在相应条件下给出了其局部收敛定理。
在第三章中,研究了复合型Newton-Steffensen迭代方法在算子的二阶导数满足Lipschitz连续条件时的半局部收敛性,并利用优函数技巧得到了其在该条件下的误差估计,最后给出了这个定理的一个应用。
在第四章中,对于近年来出现的一类特殊高阶牛顿型迭代方法,我们对其由来给出了一个统一构造技巧,并用这一技巧在著名Halley迭代基础上构造了一个高阶Halley型迭代方法,并证得其局部收敛阶为五。最后通过几个数值例子将该方法与经典迭代法进行比较,结果表明新Halley型迭代法具有很好的可行性。
在第五章中,通过优序列技巧,对第四章得到的新Halley型迭代方法在算子二阶导数满足广义Lipschitz条件时,给出了其半局部收敛定理。
When we use mathematical methods to study natural and social phenomena or to solve engineering technique problems, we often regard the solutions of most practical problems as the ones of nonlinear equations in form ofF(x) = 0in Banach space. While iterative methods are the most efficient algorithms for solving nonlinear equations. So it is very important and meaningful to do the research of iterative methods.
This thesis consists of five chapters. In Chapter 1, we give some basic definitions and notations which will be used throughout the whole thesis; summarize the techniques in proving iterative methods' convergence theorems and the convergence conditions of several famous iterative methods.
In Chapter 2, under the (K,p)-Ho|¨lder continuous condition of the first Fréchet derivative, we discuss the convergence property of a third order composite Newton-Steffensen method, which was introduced by Sharma , and obtain a semilocal convergence theorem for it. Meanwhile, a local convergence theorem is also given under the similar conditions.
In Chapter 3, we study the semilocal convergence for the method appeared in Chapter 2 under Lipschitz continuous condition of the second Frechet derivative, and get the corresponding error estimation by majoriz-ing function technique. And also, an application of the theorem to a non- linear equation is given.
In Chapter 4, we present a(?)united structural approach for a special type of high-order iterative methods which were proposed in recent years. By this means, we deduce a new high-order Halley type method based on the famous Halley's method, and proves that its local convergence order is five. Finally, we give some numerical results with a number of function tests to show that the new method performs better in efficiency comparing to some classical methods.
In Chapter 5, by using the majorizing function technique, we establish a semilocal convergence theorem for the method obtained in Chapter 4 under generalized Lipschitz continuous condition of the second Fréchet derivative.
这篇论文共分为五个部分。本文在第一章中给出了全文要用到的一些基本概念和记号,总结了证明各种迭代法收敛性的技巧以及几个著名迭代法的收敛条件。
在第二章中,针对Sharma提出的一个三阶收敛复合型Newton-Steffensen迭代方法,在假设算子的一阶导数满足(K,p)-Holder连续条件时进行了讨论,得到了其半局部收敛定理,同时在相应条件下给出了其局部收敛定理。
在第三章中,研究了复合型Newton-Steffensen迭代方法在算子的二阶导数满足Lipschitz连续条件时的半局部收敛性,并利用优函数技巧得到了其在该条件下的误差估计,最后给出了这个定理的一个应用。
在第四章中,对于近年来出现的一类特殊高阶牛顿型迭代方法,我们对其由来给出了一个统一构造技巧,并用这一技巧在著名Halley迭代基础上构造了一个高阶Halley型迭代方法,并证得其局部收敛阶为五。最后通过几个数值例子将该方法与经典迭代法进行比较,结果表明新Halley型迭代法具有很好的可行性。
在第五章中,通过优序列技巧,对第四章得到的新Halley型迭代方法在算子二阶导数满足广义Lipschitz条件时,给出了其半局部收敛定理。
When we use mathematical methods to study natural and social phenomena or to solve engineering technique problems, we often regard the solutions of most practical problems as the ones of nonlinear equations in form ofF(x) = 0in Banach space. While iterative methods are the most efficient algorithms for solving nonlinear equations. So it is very important and meaningful to do the research of iterative methods.
This thesis consists of five chapters. In Chapter 1, we give some basic definitions and notations which will be used throughout the whole thesis; summarize the techniques in proving iterative methods' convergence theorems and the convergence conditions of several famous iterative methods.
In Chapter 2, under the (K,p)-Ho|¨lder continuous condition of the first Fréchet derivative, we discuss the convergence property of a third order composite Newton-Steffensen method, which was introduced by Sharma , and obtain a semilocal convergence theorem for it. Meanwhile, a local convergence theorem is also given under the similar conditions.
In Chapter 3, we study the semilocal convergence for the method appeared in Chapter 2 under Lipschitz continuous condition of the second Frechet derivative, and get the corresponding error estimation by majoriz-ing function technique. And also, an application of the theorem to a non- linear equation is given.
In Chapter 4, we present a(?)united structural approach for a special type of high-order iterative methods which were proposed in recent years. By this means, we deduce a new high-order Halley type method based on the famous Halley's method, and proves that its local convergence order is five. Finally, we give some numerical results with a number of function tests to show that the new method performs better in efficiency comparing to some classical methods.
In Chapter 5, by using the majorizing function technique, we establish a semilocal convergence theorem for the method obtained in Chapter 4 under generalized Lipschitz continuous condition of the second Fréchet derivative.
引文
[1]J.R.Sharma.A composite third order Newton-Steffensen method for solving nonlinear equations[J].Appl.Math.Comput.,2005,169:242-246.
[2]李庆扬,莫孜中,祁力群.非线性方程组的数值解法[M].北京:科学出版社,1999.
[3]J.M.Ortega,W.C.Rheinbolbt.多元非线性方程组迭代解法[M].朱季纳.北京:科学出版社,1983.
[4]蒋尔雄,赵风光.数值逼近[M].上海:复旦大学出版社,1996.
[5]王兴华,韩丹夫.Newton迭代的区域估计和点估计[J].计算数学,1990,1:47-53.
[6]王兴华,韩丹夫.弱条件下的α判据和Newton法[J].计算数学,1997,2:103-111.
[7]T.Yamamoto.A method for finding sharp error bounds for Newton's method under the Kantorovich assumptions[J].Numer.Math.,1988,49:203-220.
[8]J.M.Gutiérrez,M.A.Hernández.A family of Chebyshev-Halley type methods in Banach spaces[J].Bull.Austral.Math.Soc.,1997,55:113-130.
[9]I.K.Argyros,D.Chen.Results on the Chebyshev method in Banach spaces[J].Proyecciones,1993,12:119-128.
[10]J.M.Gutiérrez,M.A.Hernández.New recurrence relations for Chebyshev method[J].Appl.Math.Lett.,1997,10:63-65.
[11]W.Gander.On Halley's iteration method[J].Amer.Math.Monthly,1985,92:131-134.
[12]M.A.Hernández,M.A.Salanova.Indices of convexity and concanity:application to Halley method[J].Appl.Math.Comput.,1999,103:27-49.
[13]V.Candela,A.Marquina.Recurrence relations for.rational cubic mehtods I:the Halley method[J].Computing,1990,44,169-184.
[14]J.M.Gutiérrez,M.A.Hernández.Convex acceleraion of Newton's method[C].In:Pub.Sem.Galdeano Serie Ⅱ Section I No.8.University of Zaragoza,1995.
[15]J.M.Gutiérrez,M.A.Hernández.An acceleration of Newton's method:Supper-Halley method[J].Appl.Math.Comput.,2001,117:223-239.
[16]L.V.Kantorovich,G.P.Akilov.Functional Analysis[M].New York:Pergamon,1982.
[17]R.A.Tapia.The Kantorovich theorem for Newton's method[J].Amer.Math.Monthly,1971,78:389-392.
[18]J.Rokne.Newton's method under mild differenfiability condition with error analysis[J].Numer.Math.,1972,18:401-412.
[19]J.Appell.New results on Newton-Kantorovich approximations with applications to non-linear integral equations[J].Monatschefte Math.,1990,101:175-193.
[20]M.A.Hernádez.Relaxing convergence conditions for Newton's method[J].J.Math.Anal.Appl.,2000,249(2):463-475.
[21]Z.D.Huang.A note on the Kantorovich theorem for Newton's method[J].J.Comput.Appl.Math.,1993,47:211-217.
[22]J.M.Gutiérrez.A new semilocal convergence theorem for Newton's method[J].J.Comput.Appl.Math.,1997,79:131-145.
[23]S.Smale.The function theory for solving equations[C].Pro.International Congress of Mathematicians.Providence:AMS,1986.185-196.
[24]X.H.Wang.Convergence of Newton's method and inverse function in Banach spaces[J].Math.Comput.,1999,68:169-186.
[25]I.K.Argyros.A newton-Kantorovich theorem for equation involving m-Fréchet differentiable operators and applications in radiative transfer[J],J.Comput.Appl.Math.,2001,131:149-159.
[26]I.K.Argyros.Caoncerning the "terra incogrito" between convergence regions of two Newton methods[J].Nonlinear Anal.,2005,62:179-194.
[27]J.Traub,H.Wozrtiakowski.Convergence and complexity of Newton iteration for operator equation[J].J.Assoc.Comput.Mech.,1979,26:250-258,
[28]X.H.Wang.关于牛顿法的收敛性[J].科学通报,1980,25:36-37.
[29]S.Smale.Newton's method estimates from data at one point[C].In:R.Ewing,K.Gross,C.Martin.The Merging of Disciplines:New Directions in Pure,Applied and Computational Mathematicas.New York:Springer,1986.185-196.
[30]X.H.Wang.Convergence of Newton's method and uniqueness of the solution of equations in Banach space[J].IMA Journal of Numerical Analysis,2000,19(2):123-134.
[31]X.H.Wang,C.Li.Convergence of Newton's method and uniqueness of the solution of equations in Banach spaces Ⅱ[J].Acta Mathematica Sinica,English Series,2003,19(2):405-412.
[32]Z.D.Huang.The convergence ball of Newton's method and the uniqueness ball of equations under Holder-type continuous derivatives[J].Computo Math.Appl.,2004,47:247-251.
[33]D.Han,X.H.Wang.The error estimates of Halley's method[J].Numer.Math.,1995,6(2):231-240.
[34]D.Hart.The convergence on a family of iterations with cubic order[J].J.Comput.Math.,2001,19(5):467-474.
[35]X.H.Wang,C.Li.On the united theory of the family of Euler-Halley type methods with cubical convergence in Banach spaces[J].J.Comput.Math.,2003,21(2):195-200.
[36]J.A.Ezquerro,M.A.Hernández.On the R-order of the Halley method[J].J.Comput.Appl.Math.,2005,303:591-601.
[37]J.A.Ezquerro,M.A.Hernández.Halley's method for operator with unbounded second derivative[J].Appl.Numer.Math.,2007,57:354-360.
[38]Z.D.Huang.On a family of Chebyshev-Halley type methods in Banach Space under weaker Smale condition[J].Numer.Math.,2000,9:37-44.
[39]王兴华,李冲.解方程算法的局部行为和整体行为[J].科学通报,2001,6(26):444-450.
[40]I.K.Argyros.On the approximate solutions of non-linear functional equations under mild differentiability conditions[J].Acta Mathematica Sinica,1991,58:3-7.
[41]F.A.Potra,V.Pták.Nondiscrete induction and iterative processes[C].In:Research Notes in Mathematics.Boston:Pitman,1984,vol.103.
[42]M.Grau,M.Noguera.A variant of Cauchy's method with accelerated fifth-order convergence[J].Appl.Math.Lett.,2004,17:509-517.
[43]M.Grau,José Luis,Díaz-Barrero.An improvement of the Euler-Chebyshev iterative method[J].J.Math.Anal.Appl.,2006,315:1-7.
[44]M.Frontini,E.Sormani.Some variant of Newton's method with cubic convergence[J].Appl.Math.Comput.,2003,140:419-426.
[45]H.H.H.Homeier.On Newton-type methods with cubic convergence[J].J.Comput.Appl.Math.,2005,176:425-432.
[46]L.B.Rail.Quadratic equations in Banach spaces[J].Rend.Circ.Mat.Palermo.,1961,2(10):314-332.
[47]S.Weerakoon,T.G.I.Femando.A variant of Newton's method with accelerated third-order convergence[J].Appl.Math.Lett.,2000,13:87-93.
[48]J.F.Traub.Iterative methods for the solution of equations[M].Prentice-Hall,1964.
[49]S.Busquier,S.Amat,J.M.Gutiérrez.Geometric constructions of iterative functions to solve nonlinear equations[J].Appl.Math.Lett.,2000,13:87-93.
[50]J.A.Ezquerro,M.A.Hernández.Avoiding the computation of the second Fréchet-derivative inthe convex acceleration of Newton's method[J].J.Comput.Appl.Math.,1998,96:1-12.
[51]M,A.Hernández.Second-derivative-free variant of the Chebyshev method for nonlinear equations[J].J.Optim.Theory Appl.,2000,104(3):501-515.
[52]M.A.Hernández.Chebyshev's approximation algorithms and applications[J].Comput.Math.Appl.,2001,41:433-445.
[53]Z.Zhang.Deformation of Halley method under hypotheses on the first Fréchetderivative[J].J.Zhejiang Univ.,2003,30(3):260-262.
[54]X.T.Ye,C.Li.Convergence of the family of the deformed Euler-Halley iterations under the Holder condition of the second derivative[J].J.Comput.Appl.Math.,2006,194:294-308.
[55]C.Li,X.T.Tao,W.P.Shen.Convergence of the variants of the Chebyshev-Halley iteration family under the Holder condition of the first derivative[J].J.Comput.Appl.Math.,2007,203:279-288.
[56]J.M.Gutierrez,M.A.Hernández.A family of Chebyshev-Halley type methods in Banach spaces[J].Bull.Aust.Math.Soc.,1997,55:113-130.
[2]李庆扬,莫孜中,祁力群.非线性方程组的数值解法[M].北京:科学出版社,1999.
[3]J.M.Ortega,W.C.Rheinbolbt.多元非线性方程组迭代解法[M].朱季纳.北京:科学出版社,1983.
[4]蒋尔雄,赵风光.数值逼近[M].上海:复旦大学出版社,1996.
[5]王兴华,韩丹夫.Newton迭代的区域估计和点估计[J].计算数学,1990,1:47-53.
[6]王兴华,韩丹夫.弱条件下的α判据和Newton法[J].计算数学,1997,2:103-111.
[7]T.Yamamoto.A method for finding sharp error bounds for Newton's method under the Kantorovich assumptions[J].Numer.Math.,1988,49:203-220.
[8]J.M.Gutiérrez,M.A.Hernández.A family of Chebyshev-Halley type methods in Banach spaces[J].Bull.Austral.Math.Soc.,1997,55:113-130.
[9]I.K.Argyros,D.Chen.Results on the Chebyshev method in Banach spaces[J].Proyecciones,1993,12:119-128.
[10]J.M.Gutiérrez,M.A.Hernández.New recurrence relations for Chebyshev method[J].Appl.Math.Lett.,1997,10:63-65.
[11]W.Gander.On Halley's iteration method[J].Amer.Math.Monthly,1985,92:131-134.
[12]M.A.Hernández,M.A.Salanova.Indices of convexity and concanity:application to Halley method[J].Appl.Math.Comput.,1999,103:27-49.
[13]V.Candela,A.Marquina.Recurrence relations for.rational cubic mehtods I:the Halley method[J].Computing,1990,44,169-184.
[14]J.M.Gutiérrez,M.A.Hernández.Convex acceleraion of Newton's method[C].In:Pub.Sem.Galdeano Serie Ⅱ Section I No.8.University of Zaragoza,1995.
[15]J.M.Gutiérrez,M.A.Hernández.An acceleration of Newton's method:Supper-Halley method[J].Appl.Math.Comput.,2001,117:223-239.
[16]L.V.Kantorovich,G.P.Akilov.Functional Analysis[M].New York:Pergamon,1982.
[17]R.A.Tapia.The Kantorovich theorem for Newton's method[J].Amer.Math.Monthly,1971,78:389-392.
[18]J.Rokne.Newton's method under mild differenfiability condition with error analysis[J].Numer.Math.,1972,18:401-412.
[19]J.Appell.New results on Newton-Kantorovich approximations with applications to non-linear integral equations[J].Monatschefte Math.,1990,101:175-193.
[20]M.A.Hernádez.Relaxing convergence conditions for Newton's method[J].J.Math.Anal.Appl.,2000,249(2):463-475.
[21]Z.D.Huang.A note on the Kantorovich theorem for Newton's method[J].J.Comput.Appl.Math.,1993,47:211-217.
[22]J.M.Gutiérrez.A new semilocal convergence theorem for Newton's method[J].J.Comput.Appl.Math.,1997,79:131-145.
[23]S.Smale.The function theory for solving equations[C].Pro.International Congress of Mathematicians.Providence:AMS,1986.185-196.
[24]X.H.Wang.Convergence of Newton's method and inverse function in Banach spaces[J].Math.Comput.,1999,68:169-186.
[25]I.K.Argyros.A newton-Kantorovich theorem for equation involving m-Fréchet differentiable operators and applications in radiative transfer[J],J.Comput.Appl.Math.,2001,131:149-159.
[26]I.K.Argyros.Caoncerning the "terra incogrito" between convergence regions of two Newton methods[J].Nonlinear Anal.,2005,62:179-194.
[27]J.Traub,H.Wozrtiakowski.Convergence and complexity of Newton iteration for operator equation[J].J.Assoc.Comput.Mech.,1979,26:250-258,
[28]X.H.Wang.关于牛顿法的收敛性[J].科学通报,1980,25:36-37.
[29]S.Smale.Newton's method estimates from data at one point[C].In:R.Ewing,K.Gross,C.Martin.The Merging of Disciplines:New Directions in Pure,Applied and Computational Mathematicas.New York:Springer,1986.185-196.
[30]X.H.Wang.Convergence of Newton's method and uniqueness of the solution of equations in Banach space[J].IMA Journal of Numerical Analysis,2000,19(2):123-134.
[31]X.H.Wang,C.Li.Convergence of Newton's method and uniqueness of the solution of equations in Banach spaces Ⅱ[J].Acta Mathematica Sinica,English Series,2003,19(2):405-412.
[32]Z.D.Huang.The convergence ball of Newton's method and the uniqueness ball of equations under Holder-type continuous derivatives[J].Computo Math.Appl.,2004,47:247-251.
[33]D.Han,X.H.Wang.The error estimates of Halley's method[J].Numer.Math.,1995,6(2):231-240.
[34]D.Hart.The convergence on a family of iterations with cubic order[J].J.Comput.Math.,2001,19(5):467-474.
[35]X.H.Wang,C.Li.On the united theory of the family of Euler-Halley type methods with cubical convergence in Banach spaces[J].J.Comput.Math.,2003,21(2):195-200.
[36]J.A.Ezquerro,M.A.Hernández.On the R-order of the Halley method[J].J.Comput.Appl.Math.,2005,303:591-601.
[37]J.A.Ezquerro,M.A.Hernández.Halley's method for operator with unbounded second derivative[J].Appl.Numer.Math.,2007,57:354-360.
[38]Z.D.Huang.On a family of Chebyshev-Halley type methods in Banach Space under weaker Smale condition[J].Numer.Math.,2000,9:37-44.
[39]王兴华,李冲.解方程算法的局部行为和整体行为[J].科学通报,2001,6(26):444-450.
[40]I.K.Argyros.On the approximate solutions of non-linear functional equations under mild differentiability conditions[J].Acta Mathematica Sinica,1991,58:3-7.
[41]F.A.Potra,V.Pták.Nondiscrete induction and iterative processes[C].In:Research Notes in Mathematics.Boston:Pitman,1984,vol.103.
[42]M.Grau,M.Noguera.A variant of Cauchy's method with accelerated fifth-order convergence[J].Appl.Math.Lett.,2004,17:509-517.
[43]M.Grau,José Luis,Díaz-Barrero.An improvement of the Euler-Chebyshev iterative method[J].J.Math.Anal.Appl.,2006,315:1-7.
[44]M.Frontini,E.Sormani.Some variant of Newton's method with cubic convergence[J].Appl.Math.Comput.,2003,140:419-426.
[45]H.H.H.Homeier.On Newton-type methods with cubic convergence[J].J.Comput.Appl.Math.,2005,176:425-432.
[46]L.B.Rail.Quadratic equations in Banach spaces[J].Rend.Circ.Mat.Palermo.,1961,2(10):314-332.
[47]S.Weerakoon,T.G.I.Femando.A variant of Newton's method with accelerated third-order convergence[J].Appl.Math.Lett.,2000,13:87-93.
[48]J.F.Traub.Iterative methods for the solution of equations[M].Prentice-Hall,1964.
[49]S.Busquier,S.Amat,J.M.Gutiérrez.Geometric constructions of iterative functions to solve nonlinear equations[J].Appl.Math.Lett.,2000,13:87-93.
[50]J.A.Ezquerro,M.A.Hernández.Avoiding the computation of the second Fréchet-derivative inthe convex acceleration of Newton's method[J].J.Comput.Appl.Math.,1998,96:1-12.
[51]M,A.Hernández.Second-derivative-free variant of the Chebyshev method for nonlinear equations[J].J.Optim.Theory Appl.,2000,104(3):501-515.
[52]M.A.Hernández.Chebyshev's approximation algorithms and applications[J].Comput.Math.Appl.,2001,41:433-445.
[53]Z.Zhang.Deformation of Halley method under hypotheses on the first Fréchetderivative[J].J.Zhejiang Univ.,2003,30(3):260-262.
[54]X.T.Ye,C.Li.Convergence of the family of the deformed Euler-Halley iterations under the Holder condition of the second derivative[J].J.Comput.Appl.Math.,2006,194:294-308.
[55]C.Li,X.T.Tao,W.P.Shen.Convergence of the variants of the Chebyshev-Halley iteration family under the Holder condition of the first derivative[J].J.Comput.Appl.Math.,2007,203:279-288.
[56]J.M.Gutierrez,M.A.Hernández.A family of Chebyshev-Halley type methods in Banach spaces[J].Bull.Aust.Math.Soc.,1997,55:113-130.