Orthogonal projection method for eigenpair derivatives of large symmetric matrices

Abstract

In this paper, orthogonal projection method (OPM) is proposed to calculate a few eigenpair derivatives of large real symmetric matrices, if the eigenvalues are simple. By projecting large matrix-valued functions to small subspaces dependent on parameter, linear systems of equations for eigenpair derivatives are greatly reduced from the original matrix size. Error bounds on the eigenpairs and their derivatives computed by OPM are established with trivial conditions, which is one of the main contributions of this paper. It is shown that if the deviations from small subspaces to invariant subspace corresponding to the desired eigenvalues and the derivatives of residuals of the computed eigenpairs tend to zero, then the computed eigenpairs and their derivatives converge to the desired eigenpairs and their derivatives. Next, a strategy to generate small subspaces in OPM is given based on the implicitly restarted Lanczos process (IRLP) for symmetric matrix-valued function. Convergence of the IRLP-based OPM is analysed in detail, which is the main emphasis of this method. Finally some numerical experiments are reported to show the efficiency of our method.

Keywords:

• eigenvector derivative
• eigenvalue derivative
• large matrix
• orthogonal projection method

2000 AMS Subject Classifications:

• 65F15
• 15A18

Acknowledgements

The authors would like to thank referees and editor for their valuable suggestions, which resulted in a number of improvements to the paper. This research was supported by the National Natural Science Foundation of China (No. 11001079) and the Fundamental Research Funds for the Central Universities.