On tridiagonalizing and diagonalizing symmetric matrices with repeated eigenvalues
We describe a divide-and-conquer tridiagonalization approach for matrices with repeated eigenvalues. Our algorithm hinges on the fact that, under easily constructively verifiable conditions, a symmetric matrix with band width b and k distinct eigenvalues must be block diagonal with diagonal blocks of size at most bk. A slight modification of the usual orthogonal band-reduction algorithm allows us to reveal this structure, which then leads to potential parallelism in the form of independent diagonal blocks. Compared to the usual Householder reduction algorithm, the new approach exhibits improved data locality, significantly more scope for parallelism, and the potential to reduce arithmetic complexity by close to 50% for matrices that have only two numerically distinct eigenvalues. The actual improvement depends to a large extent on the number of distinct eigenvalues and a good estimate thereof. However, at worst the algorithms behave like a successive band-reduction approach to tridiagonalization. Moreover, we provide a numerically reliable and effective algorithm for computing the eigenvalue decomposition of a symmetric matrix with two numerically distinct eigenvalues. Such matrices arise, for example, in invariant subspace decomposition approaches to the symmetric eigenvalue problem.
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