this paper, we generalize the ideas behind the RS-algorithms and the MHLalgorithm. We develop a band reduction algorithm that eliminates d subdiagonals of a symmetric banded matrix with semibandwidth b (d < b), in a fashion akin to the MHL tridiagonalization algorithm. Then, like the Rutishauser algorithm, the band reduction algorithm is repeatedly used until the reduced matrix is tridiagonal. If d = b 1, it is the MHL-algorithm; and if d = 1 is used for each reduction step, it results in the Rutishauser algorithm. However, d need not be chosen this way; indeed, exploiting the freedom we have in choosing d leads to a class of algorithms for banded reduction and tridiagonalization with favorable computational properties. In particular, we can derive algorithms with

We present a two-step variant of the "successive band reduction" paradigm for the tridiagonalization of symmetric matrices. Here we reduce a full matrix first to narrow-banded form and then to tridiagonal form. The first step allows easy exploitation of block orthogonal transformations. In the second step, we employ a new blocked version of a banded matrix tridiagonalization algorithm by Lang. In particular, we are able to express the update of the orthogonal transformation matrix in terms of block transformations. This expression leads to an algorithm that is almost entirely based on BLAS-3 kernels and has greatly improved data movement and communication characteristics. We also present some performance results on the Intel Touchstone DELTA and the IBM SP1. 1 Introduction Reduction to tridiagonal form is a major step in eigenvalue computations for symmetric matrices. If the matrix is full, the conventional Householder tridiagonalization approachthereof [8] is the method of This work...

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 bandwidth 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 with 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 algorithm behaves like a successive bandreduction approach to tridia...

this paper, we generalize the ideas behind the RS-algorithms and the MHLalgorithm. We develop a band reduction algorithm that eliminates d subdiagonals of a symmetric banded matrix with semibandwidth b (d ! b), in a fashion akin to the MHL tridiagonalization algorithm. Then, like the Rutishauser algorithm, the band reduction algorithm is repeatedly used until the reduced matrix is tridiagonal. If d = b \Gamma 1, it is the MHL-algorithm; and if d = 1 is used for each reduction step, it results in the Rutishauser algorithm. However, d need not be chosen this way; indeed, exploiting the freedom we have in choosing d leads to a class of algorithms for banded reduction and tridiagonalization with favorable computational properties. In particular, we can derive algorithms with