求解广义特征值问题的并行保域多分法

近年来，随着并行机的发展，提出了代数特征值问题的并行多分法，但国内外的研究工作迄今仅限于对称三对角矩阵的标准特征值问题。在科学与工程众多领域内有着重要应用的广义特征值问题的多分法，因难度大等方面原因尚无人研究。本文提出广义特征值问题的并行保域多分法，该算法适用于大型稀疏实对称矩阵广义特征值问题的求解，它克服了传统的广义特征值问题的对分法（行列式查找法）出现的漏根或迭代不收敛等缺点，并保持其优点。作者在ＹＨ－１向量机上对这一算法进行了数值实验，并与并行保域行列式查找法作了比较。数值结果表明，该算法具有较高的加速比，当系统自由度为２１１４、求解特征对个数为３时，加速比可达７．７；且当问题规模较大时，并行保域多分法优于并行保域行列式查找法。

PARALLEL REGION PRESERVING MULTISECTION METHOD FOR SOLVING GENERALIZED EIGENPROBLEM

The parallel multisection method for solving algebraic eigenproblem has been presented in recent years with the development of the parallel computers, but all the research work is limited in standard eigenproblems of symmetric tridiagonal matrix. The multisection method for solving the generalized eigenproblem applied significantly in many science and engineering domains has not been studied. The parallel region preserving multisection method (PRM for short) for solving generalized eigenproblems of large sparse and real symmetric matrix is presented in this paper. This method not only retains the advantages of the conventional determinant search method (DS for short), but also overcomes its disadvantages such as leaking roots and disconvergence. We have tested the method on the YH 1 vector computer, and compared it with the parallel region preserving determinant search method the parallel region preserving bisection method (PRB for short). The numerical results show that PRM has a higher speed up, for instance, it attains the speed up of 7.7 when the scale of the problem is 2 114 and the eigenpair found is 3, and PRM is superior to PRB when the scale of the problem is large.