不同类型的“追赶”法及其稳定性判别

“追赶”法是求解差分方程两点边值问题或条形矩阵(即只有几条对角线上的元素不为零的矩阵)线代数问题的有效解法。“追赶”法的主要问题是稳定性问题。在早期的工作1,2]中,利用主对角线占优的性质证明了“追赶”法的稳定性。后来“追赶”法利用到求解比较一般的差分方程两点边值问题,并利用差分方程中系数矩阵的特征值性质证明了稳定性。在3]中证明了:当差分方程两点边值问题是C-良态的,则正交“追赶”法是稳定的。直接利用问题的性态证明“追赶”法的稳定性是有意义的,因为有些差分方程

VARIOUS TYPES OF DOUBLE SWEEP METHODS AND THEIR STABILITY

Three types of double sweep methods for solving the two-point boundary value pro-blems of difference equations are discussed. The forward sweep procedures in these me-thods are the same, but the backward sweep procedures are different. In the first typemethods, some relations for solutions are set up in the backward sweep by using only theboundary conditions at one point as in the forward sweep. In the second type methodsthese relat ons are set up by using also the relations obtained in the forward sweep. In thethird type methods, after the forward sweep some relations for solutions are set up byusing difference equations and the relations obtained in the forward sweep. Stsbility theorems for methods of these three types are proved by using directly thel~2-well conditionedness of the two-point boundary value problems, without requiring thediagonal dominance property.