特征值问题的预变换方法(I):杨辉三角阵变换与二阶PDE特征多项式

On pre-transformed methods for eigen-problems, I: Yanghui-triangle trans-form and 2nd order PDE eigen-problems

本文提出一类求解特征值问题的下三角预变换方法,目标是通过相似变换后矩阵下三角元素平方和明显减少、且变换后的特征值及其特征向量较易求解,使变换后的对角线可作为全体特征值很好的一组初值,其作用如同对于解方程组找到好的预条件子,加速迭代收敛.以二阶PDE数值计算为例,对于以Laplace方程为代表的特征波向量组及正交多项式组有广泛的应用前景.杨辉三角是我国古代数学家的一项重要成就.本文引入杨辉三角矩阵作为预变换子,给出一般矩阵用杨辉三角矩阵作为左、右预变换子时变为上三角矩阵的充要条件,给出了元素为行指标二次多项式的两个矩阵类(三对角线阵与五对角线阵)中特征值何时保持二次多项式的充要条件,并应用于构造新的二元PDE正交多项式.

A so-called pre-transformed method for solving eigen-problems is proposed in this paper. The aim is to reduce the total sum of off-diagonal entries in lower triangular of T-1AT much smaller than the original one. Finding a good pre-transformer, just like a good pre-conditioner in solving linear system, may accelerate the eigen-solver iteration. In this paper, we take the pre-transformer T as a special elementary unit triangular, which is called Yanghui matrix.Yanghui triangle was found in China much earlier than Pascal triangle in abroad. Some suffcient and necessary conditions, with which a matrix can be reduced to an upper triangular form through similar transforming with Yanghui matrix, are given. As an application, the existence of a class of 2-D second order PDE eigen-polynomial problems is proved.