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An Extended Set of Fortran Basic Linear Algebra Subprograms
 ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE
, 1986
"... This paper describes an extension to the set of Basic Linear Algebra Subprograms. The extensions are targeted at matrixvector operations which should provide for efficient and portable implementations of algorithms for high performance computers. ..."
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Cited by 450 (71 self)
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This paper describes an extension to the set of Basic Linear Algebra Subprograms. The extensions are targeted at matrixvector operations which should provide for efficient and portable implementations of algorithms for high performance computers.
ALGORITHM 656  An Extended Set of Basic Linear Algebra . . .
, 1988
"... ... Subprograms (Level 2 BLAS). Level 2 BLAS are targeted at matrixvector operations with the aim of providing more efficient, but portable, implementations of algorithms on highperformance computers. The model implementation provides a portable set of FORTRAN 77 Level 2 BLAS for machines where sp ..."
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Cited by 46 (9 self)
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... Subprograms (Level 2 BLAS). Level 2 BLAS are targeted at matrixvector operations with the aim of providing more efficient, but portable, implementations of algorithms on highperformance computers. The model implementation provides a portable set of FORTRAN 77 Level 2 BLAS for machines where specialized implementations do not exist or are not required. The test software aims to verify that specialized implementations meet the specification of Level 2 BLAS and that implementations are correctly installed.
Stability of Block Algorithms with Fast Level 3 BLAS
 ACM Trans. Math. Soft
, 1992
"... . Block algorithms are becoming increasingly popular in matrix computations. Since their basic unit of data is a submatrix rather than a scalar they have a higher level of granularity than point algorithms, and this makes them wellsuited to highperformance computers. The numerical stability of the ..."
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Cited by 37 (15 self)
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. Block algorithms are becoming increasingly popular in matrix computations. Since their basic unit of data is a submatrix rather than a scalar they have a higher level of granularity than point algorithms, and this makes them wellsuited to highperformance computers. The numerical stability of the block algorithms in the new linear algebra program library LAPACK is investigated here. It is shown that these algorithms have backward error analyses in which the backward error bounds are commensurate with the error bounds for the underlying level 3 BLAS (BLAS3). One implication is that the block algorithms are as stable as the corresponding point algorithms when conventional BLAS3 are used. A second implication is that the use of BLAS3 based on fast matrix multiplication techniques affects the stability only insofar as it increases the constant terms in the normwise backward error bounds. For linear equation solvers employing LU factorization it is shown that fixed precision iterative re...
A Parallel Implementation of the Nonsymmetric QR Algorithm for Distributed Memory Architectures
 SIAM J. SCI. COMPUT
, 2002
"... One approach to solving the nonsymmetric eigenvalue problem in parallel is to parallelize the QR algorithm. Not long ago, this was widely considered to be a hopeless task. Recent efforts have led to significant advances, although the methods proposed up to now have suffered from scalability problems ..."
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Cited by 36 (3 self)
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One approach to solving the nonsymmetric eigenvalue problem in parallel is to parallelize the QR algorithm. Not long ago, this was widely considered to be a hopeless task. Recent efforts have led to significant advances, although the methods proposed up to now have suffered from scalability problems. This paper discusses an approach to parallelizingthe QR algorithm that greatly improves scalability. A theoretical analysis indicates that the algorithm is ultimately not scalable, but the nonscalability does not become evident until the matrix dimension is enormous. Experiments on the Intel Paragon system, the IBM SP2 supercomputer, the SGI Origin 2000, and the Intel ASCI Option Red supercomputer are reported.
New Serial and Parallel Recursive QR Factorization Algorithms for SMP Systems
, 1998
"... . We present a new recursive algorithm for the QR factorization of an m by n matrix A. The recursion leads to an automatic variable blocking that allow us to replace a level 2 part in a standard block algorithm by level 3 operations. However, there are some additional costs for performing the update ..."
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Cited by 31 (6 self)
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. We present a new recursive algorithm for the QR factorization of an m by n matrix A. The recursion leads to an automatic variable blocking that allow us to replace a level 2 part in a standard block algorithm by level 3 operations. However, there are some additional costs for performing the updates which prohibits the efficient use of the recursion for large n. This obstacle is overcome by using a hybrid recursive algorithm that outperforms the LAPACK algorithm DGEQRF by 78% to 21% as m = n increases from 100 to 1000. A successful parallel implementation on a PowerPC 604 based IBM SMP node based on dynamic load balancing is presented. For 2, 3, 4 processors and m = n = 2000 it shows speedups of 1.96, 2.99, and 3.92 compared to our uniprocessor algorithm. 1 Introduction LAPACK algorithm DGEQRF requires more floating point operations than LAPACK algorithm DGEQR2, see [1]. Yet, DGEQRF outperforms DGEQR2 on a RS/6000 workstation by nearly a factor of 3 on large matrices. Dongarra, Kaufm...
QR factorization for the Cell Broadband Engine
, 2009
"... The QR factorization is one of the most important operations in dense linear algebra, offering a numerically stable method for solving linear systems of equations including overdetermined and underdetermined systems. Modern implementations of the QR factorization, such as the one in the LAPACK libra ..."
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Cited by 5 (5 self)
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The QR factorization is one of the most important operations in dense linear algebra, offering a numerically stable method for solving linear systems of equations including overdetermined and underdetermined systems. Modern implementations of the QR factorization, such as the one in the LAPACK library, suffer from performance limitations due to the use of matrixâ€“vector type operations in the phase of panel factorization. These limitations can be remedied by using the idea of updating of QR factorization, rendering an algorithm, which is much more scalable and much more suitable for implementation on a multicore processor. It is demonstrated how the potential of the cell broadband engine can be utilized to the fullest by employing the new algorithmic approach and successfully exploiting the capabilities of the chip in terms of single instruction multiple data parallelism, instruction level parallelism and threadlevel parallelism.
Numerical Algorithms Group, Ltd. and
"... Subprograms (Level 2 BLAS). Level 2 BLAS are targeted at matrixvector operations with the aim of providing more efficient, but portable, implementations of algorithms on highperformance computers. The model implementation provides a portable set of FORTRAN 77 Level 2 BLAS for machines where speci ..."
Abstract
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Subprograms (Level 2 BLAS). Level 2 BLAS are targeted at matrixvector operations with the aim of providing more efficient, but portable, implementations of algorithms on highperformance computers. The model implementation provides a portable set of FORTRAN 77 Level 2 BLAS for machines where specialized implementations do not exist or are not required. The test software aims to verify that specialized implementations meet the specification of Level 2 BLAS and that implementations are correctly installed. Categories and Subject Descriptors: F.2.1 [Analysis of Algorithms and Problem Complexity]: Numerical Algorithms and Problemscompututiotts on matrices; G.l.O [Numerical Analysis]: