非线性方程求解的理论和方法及其在边界层问题中的应用
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摘要
近几十年来，随着数学研究本身的发展和大型计算机的出现及完善，各种非线性问题日益引起科学家和工程技术人员的兴趣和重视。特别是在近代物理和科学工程计算中的一些关键问题，归根结底都依赖于某些特定的非线性方程的求解。所以无论在理论研究方面，还是在实际应用中，非线性方程的求解都占有非常重要的地位。
本文由两大部分组成，分为六章。
从第一章到第四章为第一部分。
第一章 在各种收敛性判据下讨论Hansen和Patrick迭代族的半局部收敛性，建立了相应的收敛性定理；
第二章 讨论Hansen和Patrick迭代法的种种变形，以适应实际应用的需要；
第三章 深入地讨论了弦截法的局部收敛性，比较了在各种收敛性定理中弦截法的收敛球半径，建立了具有最大收敛球半径的弦截法收敛性定理；
第四章 应用离散动力系统的观点讨论了一族带参数迭代法的整体行为，并且给出了定常弦截法的某些动力行为和性质。
从第五章到第六章为第二部分。
第五章 从计算效能的角度来讨论奇异摄动问题的求解，构造了简洁和方便的直接求解法；
第六章 讨论奇异摄动问题的有限差分数值方法。分别在Bakhvalov-Shishkin网格和修正Bakhvalov-Shishkin网格两种特殊的加密网格上构造相应的差分格式，得到关于摄动参数ε一致收敛的离散数值解。
This dissertation mainly consists of two parts.
There are four chapters in first part.
·Analhsis of the convergence for Hansen and patrick iterative methods
·Deformed Hansen and Patrick iterative methods
·The local convergence of Secant method
·Global behavior for algorithms with one parameter
There are two chapters in second part.
·The direct solving of singularly perturbed problems
·The finite difference method of singularly perturbed problems
本文由两大部分组成，分为六章。
从第一章到第四章为第一部分。
第一章 在各种收敛性判据下讨论Hansen和Patrick迭代族的半局部收敛性，建立了相应的收敛性定理；
第二章 讨论Hansen和Patrick迭代法的种种变形，以适应实际应用的需要；
第三章 深入地讨论了弦截法的局部收敛性，比较了在各种收敛性定理中弦截法的收敛球半径，建立了具有最大收敛球半径的弦截法收敛性定理；
第四章 应用离散动力系统的观点讨论了一族带参数迭代法的整体行为，并且给出了定常弦截法的某些动力行为和性质。
从第五章到第六章为第二部分。
第五章 从计算效能的角度来讨论奇异摄动问题的求解，构造了简洁和方便的直接求解法；
第六章 讨论奇异摄动问题的有限差分数值方法。分别在Bakhvalov-Shishkin网格和修正Bakhvalov-Shishkin网格两种特殊的加密网格上构造相应的差分格式，得到关于摄动参数ε一致收敛的离散数值解。
This dissertation mainly consists of two parts.
There are four chapters in first part.
·Analhsis of the convergence for Hansen and patrick iterative methods
·Deformed Hansen and Patrick iterative methods
·The local convergence of Secant method
·Global behavior for algorithms with one parameter
There are two chapters in second part.
·The direct solving of singularly perturbed problems
·The finite difference method of singularly perturbed problems
引文
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[10] Shiming Zheng & Desmond Robbie 1995 A Note on the Convergence of Halley's Method for Solving Operator Equations. The Journal of the Australian Mathematical Society 37(B), 16-25．
[11] Huang Zhengda 1993 A Note on the Kantorovich Theorem for Newton Iteration. J Comp and Appl Math 47, 211-217．
[12] Han Danfu & Wang Xinghua 1997 The Error Estimate of Halley's Method. Numerical Mathematic (In Chinese) 6, 231-240．
[13] 王兴华1979几个数值求根方法的误差估计，数学学报 22, 638-642．
[14] 王兴华1975一个迭代过程的收敛，科学通报 20, 558-559; 1977杭州大学学报(2) ，16-42．
[15] 梁克维1999 Hansen和Patrick方法的收敛性．杭州大学学报(自然科学版)26(1)，25-35．
[16] Smale S. 1986 Newton's method estimates from data at one point. The Merging of Disci plines: New Directions in Pure, Applied and Computational Mathematics(R. Ewing, K. Gross and C. Martin eds). New York: Spring, pp 185-196.
[17] Smale S. 1987 Algorithms for Solving Equations. Proceeding of the International Congress of Mathematics(Berkedey 1986), Amer Math Soc. 1,172-195.
[18] 王兴华&韩丹夫1989点估计中的优序列方法以及Smale定理的条件和结论的最优化．中国科学(A辑)9，905-913．
[19] Wang Xinghua 1993 Some result relevant to Smale's reports. From Topology to Computation: Proceedings of the Smalefest (M. Hirsch, J. Marsden and M. Shub eds). New York: Springer, pp 456-465.
[20] 王兴华，韩丹夫&孙方裕1990若干变形Newton迭代的点估计．计算数学2，145-156．
[21] Wang Xinghua, Zheng Shiming & Han Danfu 1990 On convergence of Euler series, Euler's iterative family and Halley's iterative family from data at one point. Acta Math. Sinica (in Chinese) 33(6), 721-73S.
[22] 梁克维1999非线性方程的迭代解法——Hansen和Patrick迭代方法．杭州大学大学生科技活动中心课题．
[23] 傅晓阳1990 Hansen和Patrick方法的点估计．计算数学4，376-382．
[24] Wang Xinghua & Han Danfu 1997 Criterion a and Newton method in weak conditions. Math. Numerica Sinica 19(1), 103-114; Chinese J. Numer. and Appl. Math. 19(2), 96-105.
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