Tridiagonal matrix algorithm

Abstract: In this paper we present a new divide-and-conquer parallel algorithm to compute the eigenvalues of symmetric tridiagonal matrices. This algorithm combines the use of rank-one modifications in the division phase and the application of the Laguerre iteration in the updating phase. Our method is compared with one based on the same scheme but using rank-two modifications. A thorough experimental analysis in the Cray T3D parallel computer has been carried out. Special emphasis has been put on analysing the influence of the deflation phenomena on the computational cost of this kind of algorithm. Experimental results show that an adequate exploitation of the inherent parallelism in the divide-and-conquer scheme produces very efficient parallel algorithms. The obtained speedups clearly improve the best sequential algorithm, including the standard implementation of QR iteration in LAPACK.

Abstract: A scalable algorithm for the reduction to tridiagonal form of symmetric matrices is developed. It uses one-sided rotations instead of similarity transforms. This allows a data distribution independent implementation with low communication volume. Timings on the Fujitsu AP 1000 and VPP 500 show good performance. >

Abstract: QL(QR) method is an efficient method to find eigenvalues of a matrix. Especially we use QL(QR) method to find eigenvalues of a symmetric tridiagonal matrix. In this case it only costs O(n 2) flops, to find all eigenvalues. So it is one of the most efficient method for symmetric tridiagonal matrices. Many experts have researched it. Even the method is mature, it still has many problems need to be researched. We put forward five problems here. They are: (1) Convergence and convergence rate; (2) The convergence of diagonal elements; (3) Shift designed to produce the eigenvalues in monotone order; (4) QL algorithm with multi-shift; (5) Error bound. We intoduce our works on these problems, some of them were published and some are new.