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A Framework for Symmetric Band Reduction
, 1999
"... this paper, we generalize the ideas behind the RSalgorithms 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 alg ..."
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Cited by 29 (6 self)
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this paper, we generalize the ideas behind the RSalgorithms 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 MHLalgorithm; 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
Parallel Tridiagonalization through TwoStep Band Reduction
 In Proceedings of the Scalable HighPerformance Computing Conference
, 1994
"... We present a twostep variant of the "successive band reduction" paradigm for the tridiagonalization of symmetric matrices. Here we reduce a full matrix first to narrowbanded form and then to tridiagonal form. The first step allows easy exploitation of block orthogonal transformations. In the secon ..."
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Cited by 23 (12 self)
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We present a twostep variant of the "successive band reduction" paradigm for the tridiagonalization of symmetric matrices. Here we reduce a full matrix first to narrowbanded 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 BLAS3 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...
On Tridiagonalizing and Diagonalizing Symmetric Matrices with Repeated Eigenvalues
 PREPRINT ANL/MCSP54541095, MATHEMATICS AND COMPUTER SCIENCE DIVISION, ARGONNE NATIONAL LABORATORY
, 1995
"... We describe a divideandconquer 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 ..."
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Cited by 2 (2 self)
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We describe a divideandconquer 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 bandreduction 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...
A Framework for Symmetric Band Reduction
"... this paper, we generalize the ideas behind the RSalgorithms 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 alg ..."
Abstract
 Add to MetaCart
this paper, we generalize the ideas behind the RSalgorithms 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 MHLalgorithm; 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