求解矩阵极大特征值问题的保守BFGS算法

Cautious BFGS Method for Matrix Largest Eigenvalue Problem

基于求解无约束优化问题,本文提出求解大型对称正定矩阵极大特征值问题的保守BFGS算法.所提算法有效地避免了求解大型Hessian矩阵逆的问题.同时,在一些合理的条件下,建立了所提算法的全局收敛性.最后,将所提算法和EIGS(Matlab内部计算矩阵极大特征值的命令)进行了对比测试.数据结果表明,本文所提算法快速、高效、稳定.

Based on solving the unconstrained optimization problems, we propose a cautious BFGS method for solving the extreme eigenvalue problems of large scale symmetric and positive definite matrices. The method effectively avoids the problem of solving the inverse problem of the large scale Hession matrix. Then, we prove the global convergence of the algorithm under some reasonable conditions. Finally, we compare our method with EIGS （a matlab implementation for computig the extreme eigenvalue of matrix）. The numerical experiments show that the proposed method is fast, efficient and stable.