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On the orthogonality of eigenvectors computed by divide-and-conquer techniques. (English) Zbl 0743.65039

For the symmetric eigenvalue problem a new algorithm featuring a divide- and-conquer strategy has emerged. It is especially significant on parallel machines, but is also in serial mode faster than previous methods. It has to splice the divided problem together again by computing the eigenvalues and -vectors of a rank 1 perturbation of a diagonal matrix. Here a nonlinear equation (the secular equation) has to be solved to high relative accuracy in order to keep the updated eigenvectors orthogonal.
Two methods are proposed to simulate doubled precision in normal working precision for the critical part of the algorithm. Unfortunately these approaches are not portable to all hardware environments. Numerical examples are provided to illustrate the discussion.

MSC:

65F15 Numerical computation of eigenvalues and eigenvectors of matrices
65Y05 Parallel numerical computation
65G50 Roundoff error

Software:

EISPACK
PDFBibTeX XMLCite
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