一阶和二阶数值微分的Lanczos方法  

LANCZOS' METHOD FOR NUMERICAL DIFFERENTIATION OF THE FIRST AND SECOND ORDER

作　　者：王泽文[1] 温荣生[1] 

机构地区：[1]东华理工大学理学院,南昌330013

出　　处：《高等学校计算数学学报》2012年第2期160-178,共19页Numerical Mathematics A Journal of Chinese Universities

基　　金：国家自然科学基金(11161002);江西省自然科学基金(2009GZS0001);东华理工大学校长基金.

摘　　要：数值微分问题是由未知函数的已知分布或未知函数的近似来确定其导数的问题，是一类典型的不适定问题．数值微分在图像处理、偏微分方程反问题、医学成像以及金融工程等领域都有着广泛的应用，因此对该类问题进行深入的研究是非常有意义的．众所周知，数值微分的不适定性主要体现在计算的不稳定性上，即输入数据的微小扰动将造成数值微分的急剧变化，从而使得数值解毫无意义．那么，从未知函数的已知分布（即离散点上的近似值）或近似函数重建其导数时，得到的近似导数将与真正的导数相差甚远，导致不可用；对于高阶导数更是如此．因此，在数值微分中往往需要引入必要的正则化方法，从而获得稳定的正则化解．This paper mainly studies a numerical differential problem, which aims to compute the first and second order derivatives of an unknown function from its approximate data. New schemes approached by Lanczos＇ method are proposed to compute the first and the second order derivatives numerically. And, we prove that the high-precision Lanczos＇ derivative proposed in this paper is indeed a generalized derivative. When the unknown function is 5 times continuously differentiable and its fifth order derivative is bound, the convergence rate of the numerical deriva- tive of first order is O （δ4/5）. While the unknown function is 6 times continuously differentiable and its sixth order derivative is bound, the convergence rate of the numerical derivative of second order is O （δ2/3）. Moreover, we discuss the numerical differentiation problems for a function of multiple variables, and propose a Lanc-zos＇ generalized Laplace operator to approximate △u（x）=n∑i=1δ^2u/δx^2 Numericalexamples show that the presented methods are stable and efficient for computing the approximate derivatives at internal points of an interval. Our work is a gener- alization of Lanczos＇ derivatives.

关 键 词：数值微分 LANCZOS方法 偏微分方程反问题 高阶导数 二阶 一阶 未知函数 不稳定性 

分 类 号：O241.2[理学—计算数学] O241.4[理学—数学]