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

In this paper, we discuss work in progress on a complete eigensolver based on the Invariant Subspace Decomposition Algorithm for dense symmetric matrices (SYISDA). We describe a recently developed acceleration technique that substantially reduces the overall work required by this algorithm and review the algorithmic highlights of a distributed-memory implementation of this approach. These include a fast matrix-matrix multiplication algorithm, a new approach to parallel band reduction and tridiagonalization, and a harness for coordinating the divide-and-conquer parallelism in the problem. We present performance results for the dominant kernel, dense matrix multiplication, as well as for the overall SYISDA implementation on the Intel Touchstone Delta and the Intel Paragon. 1. Introduction Computation of eigenvalues and eigenvectors is an essential kernel in many applications, and several promising parallel algorithms have been investigated [26, 3, 28, 22, 25, 6]. The work presented in t...

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this paper. Their singleprecision twins are identical except for a leading "S" instead of "D" in the routine's name and REAL instead of DOUBLE PRECISION scalars and arrays in the parameter list.

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In this paper, we give an overview of the Invariant Subspace Decomposition Algorithm for banded symmetric matrices and describe a sequential implementation of this algorithm. Our implementation uses a specialized routine for performing banded matrix multiplication together with successive band reduction, yielding a sequential algorithm that is competitive for large problems with the LAPACK QR code in computing all of the eigenvalues and eigenvectors of a dense symmetric matrix. Performance results are given on a variety of machines. 1 Introduction Computation of eigenvalues and eigenvectors is an essential kernel in many applications, and several promising parallel algorithms have been investigated [8, 11, 7]. The work presented in this paper is part of the PRISM (Parallel Research on Invariant Subspace Methods) Project, which involves researchers from Argonne National Laboratory, the Supercomputing Research Center, the University of California at Berkeley, and the University of Kent...

We describe two techniques for speeding up eigenvalue and singular value computations on shared memory parallel computers. Depending on the information that is required, different steps in the overall process can be made more efficient. If only the eigenvalues or singluar values are sought then the reduction to condensed form may be done in two or more steps to make best use of optimized level-3 BLAS. If eigenvectors and/or singular vectors are required, too, then their accumulation can be sped up by another blocking technique. The efficiency of the blocked algorithms depends heavily on the values of certain control parameters. We also present a very simple performance model that allows selecting these parameters automatically. Keywords: Linear algebra; Eigenvalues and singular values; Reduction to condensed form; Hessenberg QR iteration; Blocked algorithms. 1 Introduction The problem of determining eigenvalues and associated eigenvectors (or singular values and vectors) of a matrix ...

This paper introduces a novel implementation in reducing a symmetric dense matrix to tridiagonal form, which is the preprocessing step toward solving symmetric eigenvalue problems. Based on tile algorithms, the reduction follows a two-stage approach, where the tile matrix is first reduced to symmetric band form prior to the final condensed structure. The challenging trade-off between algorithmic performance and task granularity has been tackled through a grouping technique, which consists of aggregating fine-grained and memory-aware computational tasks during both stages, while sustaining the applications overall high performance. A dynamic runtime environment system then schedules the different tasks in an out-of-order fashion. The performance for the tridiagonal reduction reported in this paper is unprecedented. Our implementation results in up to 50-fold and 12-fold improvement (130 Gflop/s) compared to the equivalent routines from LAPACK V3.2 and Intel MKL V10.3, respectively, on an eight socket hexa-core AMD Opteron multicore shared-memory system with a matrix size of 24000 × 24000. 1.

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...

We present an overview of the banded Invariant Subspace Decomposition Algorithm for symmetric matrices and describe a parallel implementation of this algorithm. The algorithm described here is a promising variant of the Invariant Subspace Decomposition Algorithm for dense symmetric matrices (SYISDA) that retains the property of using scalable primitives, while requiring significantly less overall computation than SYISDA. 1 Introduction Computation of eigenvalues and eigenvectors is an essential kernel in many applications, and several promising parallel algorithms have been investigated. The work presented in this paper is part of the PRISM (Parallel Research on Invariant Subspace Methods) Project, which involves researchers from Argonne National Laboratory, the Supercomputing Research Center, the University of California at Berkeley, and the University of Kentucky. The goal of the PRISM project is the development of algorithms and software for solving large-scale eigenvalue problems ...