Technical Report CS0821

TR#:CS0821
Class:CS
Title: FAST DIAGONALIZATION OF LARGE AND DENSE COMPLEX SYMMETRIC MATRICES, WITH APPLICATIONS TO QUANTUM REACTION DYNAMICS.
Authors: I. Bar-On and V. Ryaboy
PDFCS0821.pdf
Abstract:

We present a new fast and efficient algorithm for computing the eigenvalues and eigenvectors of large size complex symmetric dense matrices. The principal new idea is to reduce the matrix to a tridiagonal complex symmetric form. We can then compute the eigenvalues, very fast, using a complex version of the QL algorithm for tridiagonal symmetric matrices. The corresponding eigenvectors are similarly computed using a complex version of the inverse iteration algorithm. We show that the new algorithm is faster by an order of magnitude than the corresponding EISPACK routines that are currently used for such problems. We present also similar methods for skew symmetric matrices, and for complex Hermitian matrices, the last being twice as fast as the corresponding EISPACK routine. The new algorithm is also highly suitable for modern parallel and vector supercomputers. The complex scaled discrete variable representation (DVR) method for quantum chemical reactive problems relies heavily on the efficient diagonalization of complex symmetric matrices. We present an application to this problem which has motivated our work. We finally present new perturbation bounds for the Householder transformation which improve those results obtained before by Wilkinson.

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