On Tridiagonalizing and Diagonalizing Symmetric Matrices with Repeated Eigenvalues

We describe a divide-and-conquer tridiagonalizationapproach for matrices with repeated eigenvalues. Our algorithmhinges on the fact that, under easily constructivelyveriiable conditions,a symmetricmatrix with bandwidth b and k distinct eigenvalues must be block diagonal with diagonal blocks of size at most bk. A slight modiication 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 with the usual Householder reduction algorithm, the new approach exhibits improved data locality, signiicantly 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 algorithm behaves like a successive bandreduction approach to tridiagonalization. Moreover, we provide a numerically reliable and eeective 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.