大型矩阵特征值问题并行计算研究概况

摘要

The matrix eigenvalue problem can be used to solve directly a lot of mathematical problems such as nonlinear programming， optimization， ordinary differential equations, and computational methods. Moreover, it plays an important role in structure mechanics， engineering designment， computational physics, and quantum mechanics. Now, applications of the matrix eigenvalue problem mostly originate from equations of mathematical physics， difference equations， Markov process,and so on. Because of its great importance and wide applications, the eigenvalue problem becomes one of the main computational tasks for high performance computers all over the world.%矩阵特征值问题不仅可直接解决数学中诸如非线性规划、优化、常微分方程，以及各类数学计算问题，而且在结构力学、工程设计、计算物理和量子力学中具有重要作用，目前矩阵特征值问题的应用大多来自于解数学物理方程、差分方程、Markov过程等。正因为它具有重要意义和广泛的应用，所以矩阵特征值问题是当前国内外高性能计算机的主要计算任务之一。本文概括介绍了当前并行求解大型矩阵特征值问题的计算方法，特别介绍了分治算法、同伦连续算法、并行块消去迭代法和谱分解算法的基本思想及其实现方法。