A block-diagonal generalized eigenvalue solver for parallel systems
A generalised eigenvalue problem Az = lambda Bz arising from an ideal linear 3D MHD stability problem is considered. The matrices A and B are symmetric and B positive definite. They have a pentadiagonal full block structure. There are N of such blocks of size M. The blocks overlap within L rows, where L is less than or equal to M/2. For L equal to M/2, the matrices degenerate to tridiagonal block structures. For L less than M/2, there are M-2L non-overlapping rows. To solve this eigenvalue problem, an inverse vector iteration is performed, preceded by a spectral shift of lambda(0). The iteration procedure converges towards the eigenvalue lying closest to lambda(0) and can best be realised on a parallel system by performing the following steps: (1) perform the eigenvalue shift A - lambda(0)B; (2) eliminate y in the decomposition A = LDL(T); (3) eliminate x in the decomposition A = LDL(T); (4) backsubstitute for x, and (5) backsubstitute for y. The steps (1), (2) and (5) can be performed entirely in parallel for all N blocks. The elimination of x and its backsubstitution, steps (3) and (4) are more difficult to parallelise.
Bibliographic Reference: Article: Proceedings of Physics Computing '91 Conference, San Jose (US), June 10-14, 1991
Record Number: 199111172 / Last updated on: 1994-12-02
Original language: en
Available languages: en