【作者】 高峰；

【导师】 王仁宏；

【作者基本信息】 大连理工大学 ， 计算数学， 2004， 博士

【摘要】 代数函数论是一古老的数学分枝。在18世纪后半期，曾是许多最卓越的数学家的研究重点。在沉寂了一段时间之后，又以现代的形式复兴起来。并且，牵连着一些新的重要问题。古典代数函数论的研究对象，是以代数关系式 f(x，y)=0 (1)联系着的两个变量(x，y)的有理函数φ(x，y)。其中f(x，y)=0是这两个变量(x，y)的多项式。代数函数论在历史上是由企图把形如 ∫φ(x，y)dx (2)的积分(阿贝尔积分)用有限的形式积出而产生的。古典的代数函数论可以看作是在克莱茵(F．Klein)意义下的一种几何学系统。 本文着重研究代数函数理论在计算数学领域中的问题和应用。文中在某种意义上推广了Padé逼近的定义，给出了任意一个解析函数在一点处的[n，m]级代数函数逼近的定义，并且研究了这种逼近式存在的充分必要条件，以及它与Padé逼近式的关系。本文研究了满足某些特定条件的exp(z)的[n，m]级代数函数逼近式应具有的形式，并给出了exp(z)的多种代数函数逼近式。并且估计其可以达到的逼近阶。 本文利用exp(z)的多个代数函数逼近式来构造常微分方程初值问题及某些偏微分方程定解问题的若干线性及非线性差分格式。并分析其收敛性及稳定性。 对常微分方程初值问题(3)的另外一种要求是考察当t→∞时，解的状态如何。因此，(3)可以看成一个动力系统。本文指出利用exp(z)的代数函数逼近式得到的许多差分格式，在一定程度上可以避免出现伪周期轨道。即其中的若干算法为2-正则算法(R-算法)。 本文利用exp(z)的代数函数逼近式来构造线性Hamilton系统的的辛(Symlectic)数值算法并且利用对角Padé逼近和代数函数逼近，给出了刚性常微分方程组的几个A-稳定的显式算法，并给出了相应的数值例子。 在数值逼近中，有时需考虑函数的最佳一致逼近。任意给定一个闭区间[a，b]上的连续函数，其在Pn(n次多项式空间)中的最佳一致逼近是存在且唯一的。在侧n，m)(分子为n次分母为m次的有理多项式空间)中的最佳一致逼近也是存在且唯一的。本文考虑闭区间[a，司上的任一个连续函数的[n，2!级代数函数最佳一致逼近，并证明了其存在性.另外研究了卜，2!级代数曲线插值间题.证明了平面上满足一定条件的任意5个结点，可用唯一一条代数曲线插值.并给出了数值例子。更多还原

【Abstract】 The theory of algebaric functions is an old branch of mathemathics. The therory attracted many excellent mathematicians in the late 18th century. But from that time it seemed to have been forgotten for a long time untill it arised again and taking a new form , and being connected with many other important mathematical problems. The classical theory of algebaric function studies the rational function ψ(x, y) of x and y which are connected by the equationf(x,y)=0, (1)where f(x, y) is a polynomial of x and y . In the history, mathematicians were motivated to develop the theory by their tring to give the result ofψ(x,y) (2)in finite terms. The key to the problem is to choose proper transforms likex=ψ1(u,v),y=ψ2(u,v) (3)where ψ1,ψ2 are both rational functions, so that the calculus (2) becomes a calculus of a single variable. The property of calculus (3) that it can be calculated in finite terms is indepedent of the tansforms (3). From this problem, an idea initiated, that is to sdudy the invariant properties of tansforms like (3). An important case is when tansforms (3) and their inverse tansforms are both rational functions. This kind of transforms are called birational transforms. So the classical theory of algebaic functions can be seen as an geometric system in Klein sense. In numerical mathematics, it is interesting and of importance to study the theory of algebraic functions and its applications. But so far this kind of study seems to be neglected. It is known that polynomials or spline functions are often used as approximation functions. Some times, rational functions and rational splines are also used as appoximation tools. However algebaric functions is a kind of generalization of rational functions because in some cases, algebaric functions reduce to rational functions. In the thesis, we study the applications of algebaric approximation in numerical mathematics, especially in the numerical solution of differential equations. the main results are as follows:1.The definition of algebraic approximant of order [n, m] to a given analyticfunction is given. The conditions for its existence as well as its connection with Pade approximant are also studied.2.The exponential function exp(z) satisfies the conditionexp(a+b) = exp(a) exp(b) (4)so it has good properties in practical application. In the thesis, we give the algebraic approximants to exp(z) of order [1,2] , [2,2] , [1,3] that satisfies (4), and give the error estimates of these approximants.3.The approximation to exp(z) is of great importance in numerical solution of differential equations. A approximation to exp(z) often leads to a difference scheme for differential equations. In the thesis, we give many difference schemes for the initial problemby the use of algebraic approximants to exp(z). Some of these schems are nonlinear multistep schemes. The covergence and stability of these schems are studied. we also give many difference schemes for some partial differential equations.4.Another study for problem (5) is to observe what its solution would be like when t So problem (5) can be seen as a dynamic system. In the thesis, we give some numerical methods for these dynamic systems by using the results of algebraic approximations. and we prove that these methods do not give rise to spurious orbits. Some numerical experiments are also given.5.In mathematical physics, the Hamiltonian system plays a important role.In the thesis, we put forward some symplectic difference schemes for numerically solving linear Hamiltonian systems by using the algebraic approximants to exp(z).7. The Numerical approximation to the solution of stiff ODE equations is an very important problem in numerical mathematics that far from being solved. In the thesis, we develop some explicit and A-stable methods for the problem. Numerical experiments are also given.8.We prove the existence of the b更多还原