Banded Eigenvalue Solvers on Vector Machines

Abstract

Most algorithms associated with finding eigenvalues of banded matrices do not have many, if any, obvious vector operations. On vector machmes it is not immediately obvious, if sufficient memory is available, whether one should treat banded matrices as dense matrices and use algorithms that have vector operations, or take advantage of the band structure of the problem. Most of the algorithms for banded matrices are chasing schemes in which unwanted elements are flushed out the bottom of the matrix using planar similarity transformations. We show that it is possible to chase several elements simultaneously down the matrix and to use vectorhke operations to compute the planar transformations and to apply the transformations. The length of the vectors depends on the ratio of the dimension of the problem and the bandwidth. For large problems with narrow bandwidths, it is definitely advantageous to use these new algorithms. © 1984, ACM. All rights reserved.