广义特征值问题的道路跟踪算法

A Path-Following Algorithm to a Kind of Problem of Generatized Eigenvalues

针对古典广义特征值问题：（λＢ—Ａ）Ｘ＝０．（１）其中，Ａ为对称方阵，Ｂ为对称正定方阵，提出了一种保稀疏性、保序性（特征值），不需化为标准特征问题的道路跟踪算法．其思想是从一平凡问题的解出发，沿着光滑道路跟踪到所论问题（１）的解．此算法尤其适合于大型稀疏问题和当Ｂ求送病态的问题．最后，通过例子验证了算法的有效性．

This paper focuses on the problem of a kind of classical generatized eigenvalnes:(λB- A)x= 0, these A and B are symmetric matrices and B is positive definite.We advance a path-following algorithm,this algorithm has the advantages of spare-preserving(the-preserving(the order of eigenvalue) and not needing to be turned into the problem of standard eigenvalue. The main points are to begin from an ordinary problem and follow a smooth path to pursue the solution which has been mentioning. This mathod can especially be suitable for large-scale spare matrix and the problem in the situation of finding inverse of B being ill conditioned. Finally,we have carried on effective test and verification to them.