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A block diagonal generalised eigenvalue solver for parallel systems

A generalised eigenvalue problem Az = lambda Bz arising from an ideal linear 3-dimensional magnetohydrodynamics (MHD) stability problem was 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, for L less than, or equal to, 0.5 M. For L = 0.5 M, the matrices degenerate to tridiagonal block structures. For L less than 0.5 M, there are M-2L nonoverlapping 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 B and can best be realised on a parallel system by performing the following steps:
perform the eigenvalue shift A - lambda-0;
eliminate y in the decomposition A = LDL(T);
eliminate x in the decomposition A = LDL(T);
backsubstitute for x;
backsubstitute for y.

The first two steps and the last step can be performed entirely in parallel for all N blocks. The elimination of x and its backsubstitution, are more difficult to parallelise.

Reported by

Ecole Polytechnique Federale de Lausanne
1015 Lausanne
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