The WY representation for products of householder matrices
SIAM Journal on Scientific and Statistical Computing - Papers from the Second Conference on Parallel Processing for Scientific Computin
A fully parallel algorithm for the symmetric eigenvalue problem
SIAM Journal on Scientific and Statistical Computing
The algebraic eigenvalue problem
The algebraic eigenvalue problem
A Stable and Efficient Algorithm for the Rank-One Modification of the Symmetric Eigenproblem
SIAM Journal on Matrix Analysis and Applications
A Divide-and-Conquer Algorithm for the Symmetric TridiagonalEigenproblem
SIAM Journal on Matrix Analysis and Applications
Applied numerical linear algebra
Applied numerical linear algebra
ScaLAPACK user's guide
Algorithm 582: The Gibbs-Poole-Stockmeyer and Gibbs-King Algorithms for Reordering Sparse Matrices
ACM Transactions on Mathematical Software (TOMS)
A framework for symmetric band reduction
ACM Transactions on Mathematical Software (TOMS)
Multi-sweep Algorithms for the Symmetric Eigenproblem
VECPAR '98 Selected Papers and Invited Talks from the Third International Conference on Vector and Parallel Processing
Computing Approximate Eigenpairs of Symmetric Block Tridiagonal Matrices
SIAM Journal on Scientific Computing
Block tridiagonalization of "effectively" sparse symmetric matrices
ACM Transactions on Mathematical Software (TOMS)
High performance parallel approximate eigensolver for real symmetric matrices
High performance parallel approximate eigensolver for real symmetric matrices
A parallel symmetric block-tridiagonal divide-and-conquer algorithm
ACM Transactions on Mathematical Software (TOMS)
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Two parallel block tridiagonalization algorithms and implementations for dense real symmetric matrices are presented. Block tridiagonalization is a critical pre-processing step for the block tridiagonal divide-and-conquer algorithm for computing eigensystems and is useful for many algorithms desiring the efficiencies of block structure in matrices. For an ''effectively'' sparse matrix, which frequently results from applications with strong locality properties, a heuristic parallel algorithm is used to transform it into a block tridiagonal matrix such that the eigenvalue errors remain bounded by some prescribed accuracy tolerance. For a dense matrix without any usable structure, orthogonal transformations are used to reduce it to block tridiagonal form using mostly level 3 BLAS operations. Numerical experiments show that block tridiagonal structure obtained from this algorithm directly affects the computational complexity of the parallel block tridiagonal divide-and-conquer eigensolver. Reduction to block tridiagonal form provides significantly lower execution times, as well as memory traffic and communication cost, over the traditional reduction to tridiagonal form for eigensystem computations.