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An inverse free parallel spectral divide and conquer algorithm for nonsymmetric eigenproblems
, 1997
"... We discuss an inversefree, highly parallel, spectral divide and conquer algorithm. It can compute either an invariant subspace of a nonsymmetric matrix A, or a pair of left and right deflating subspaces of a regular matrix pencil A − λB. This algorithm is based on earlier ones of Bulgakov, Godunov ..."
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Cited by 69 (10 self)
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We discuss an inversefree, highly parallel, spectral divide and conquer algorithm. It can compute either an invariant subspace of a nonsymmetric matrix A, or a pair of left and right deflating subspaces of a regular matrix pencil A − λB. This algorithm is based on earlier ones of Bulgakov, Godunov and Malyshev, but improves on them in several ways. This algorithm only uses easily parallelizable linear algebra building blocks: matrix multiplication and QR decomposition, but not matrix inversion. Similar parallel algorithms for the nonsymmetric eigenproblem use the matrix sign function, which requires matrix inversion and is faster but can be less stable than the new algorithm.
The spectral decomposition of nonsymmetric matrices on distributed memory parallel computers
 SIAM J. Sci. Comput
, 1997
"... Abstract. The implementation and performance of a class of divideandconquer algorithms for computing the spectral decomposition of nonsymmetric matrices on distributed memory parallel computers are studied in this paper. After presenting a general framework, we focus on a spectral divideandconqu ..."
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Cited by 36 (10 self)
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Abstract. The implementation and performance of a class of divideandconquer algorithms for computing the spectral decomposition of nonsymmetric matrices on distributed memory parallel computers are studied in this paper. After presenting a general framework, we focus on a spectral divideandconquer (SDC) algorithm with Newton iteration. Although the algorithm requires several times as many floating point operations as the best serial QR algorithm, it can be simply constructed from a small set of highly parallelizable matrix building blocks within Level 3 basic linear algebra subroutines (BLAS). Efficient implementations of these building blocks are available on a wide range of machines. In some illconditioned cases, the algorithm may lose numerical stability, but this can easily be detected and compensated for. The algorithm reached 31 % efficiency with respect to the underlying PUMMA matrix multiplication and 82 % efficiency with respect to the underlying ScaLAPACK matrix inversion on a 256 processor Intel Touchstone Delta system, and 41 % efficiency with respect to the matrix multiplication in CMSSL on a 32 node Thinking Machines CM5 with vector units. Our performance model predicts the performance reasonably accurately. To take advantage of the geometric nature of SDC algorithms, we have designed a graphical user interface to let the user choose the spectral decomposition according to specified regions in the complex plane.
Computation of error bounds in linear least squares problems with equality constraints and generalized linear model problems
, 1997
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Inverse Free Parallel Spectral Divide and Conquer Algorithms for Nonsymmetric Eigenproblems
, 1994
"... We discuss two inverse free, highly parallel, spectral divide and conquer algorithms: one for computing an invariant subspace of a nonsymmetric matrix and another one for computing left and right deflating subspaces of a regular matrix pencil A 0 B. These two closely related algorithms are based on ..."
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We discuss two inverse free, highly parallel, spectral divide and conquer algorithms: one for computing an invariant subspace of a nonsymmetric matrix and another one for computing left and right deflating subspaces of a regular matrix pencil A 0 B. These two closely related algorithms are based on earlier ones of Bulgakov, Godunov and Malyshev, but improve on them in several ways. These algorithms only use easily parallelizable linear algebra building blocks: matrix multiplication and QR decomposition. The existing parallel algorithms for the nonsymmetric eigenproblem use the matrix sign function, which is faster but can be less stable than the new algorithm. Appears also as ffl Research Report 9401, Department of Mathematics, University of Kentucky. ffl University of California at Berkeley, Computer Science Division Report UCB//CSD94793. It is also available by anonymous ftp on trftp.cs.berkeley.edu, in directory pub/techreports/cs/csd94 793. ffl Lawrence Berkeley Laboratory...
HPCx: TOWARDS CAPABILITY COMPUTING
 SUBMITTED TO CONCURRENCY AND COMPUTATION: PRACTICE AND EXPERIENCE
"... We introduce HPCx the UK's new National HPC Service which aims to deliver a worldclass service for capability computing to the UK scientific community. HPCx is targeting an environment that will both result in worldleading science and address the challenges involved in scaling existing code ..."
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We introduce HPCx the UK's new National HPC Service which aims to deliver a worldclass service for capability computing to the UK scientific community. HPCx is targeting an environment that will both result in worldleading science and address the challenges involved in scaling existing codes to the capability levels required. Close working relationships with scientific consortia and user groups throughout the research process will be a central feature of the service. A significant number of key user applications have already been ported to the system. We present initial benchmark results from this process and discuss the optimisation of the codes and the performance levels achieved on HPCx in comparison with other systems. We find a range of performance with some algorithms scaling far better than others.
Improved Conditions for the Existence and Uniqueness of Solutions to the General Equality Constrained Quadratic Programming Problem
, 2012
"... This paper presents an approach that directly utilizes the Hessian matrix to investigate the existence and uniqueness of global solutions for the ECQP problem. The novel features of this proposed algorithm are its uniqueness and faster rate of convergence to the solution. The merit of this algorithm ..."
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This paper presents an approach that directly utilizes the Hessian matrix to investigate the existence and uniqueness of global solutions for the ECQP problem. The novel features of this proposed algorithm are its uniqueness and faster rate of convergence to the solution. The merit of this algorithm is base on cost, accuracy and number of operations.