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Cobra: Parallel path following for computing the matrix pseudospectrum
 Parallel Computing
"... The construction of an accurate approximation of the fflpseudospectrum of a matrix by means of the standard grid method is a demanding computational task, even for matrices of medium size. In this paper we describe Cobra, a new domainbased method for the computation of pseudospectra that is a hybr ..."
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Cited by 10 (4 self)
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The construction of an accurate approximation of the fflpseudospectrum of a matrix by means of the standard grid method is a demanding computational task, even for matrices of medium size. In this paper we describe Cobra, a new domainbased method for the computation of pseudospectra that is a hybrid of the path following based algorithm of Bruhl and the standard grid approach. We implement Cobra using standard LAPACK components and show that it offers large and medium grain parallelism and that it is faster and more robust than the original path following algorithm. The algorithm is also combined with a partial SVD algorithm to produce an effective parallel method for computing the matrix pseudospectrum.
Towards the Effective Parallel Computation of Matrix Pseudospectra
 In Proceedings of the International Conference on Supercomputing (ICS
, 2001
"... Given a matrix A, the computation of its pseudospectrum (A) is a far more expensive task than the computation of characteristics such as the condition number and the matrix spectrum. As research of the last 15 years has shown, however, the matrix pseudospectrum provides valuable information that i ..."
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Cited by 7 (2 self)
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Given a matrix A, the computation of its pseudospectrum (A) is a far more expensive task than the computation of characteristics such as the condition number and the matrix spectrum. As research of the last 15 years has shown, however, the matrix pseudospectrum provides valuable information that is not included in other indicators. So, ask how to compute it eciently and build a tool that would facilitate engineers and scientists to make such analyses? In this paper we focus on parallel algorithms for computing pseudospectra. The most widely used algorithm for computing pseudospectra is embarassingly parallel; nevertheless, it is extremely costly and one cannot hope to achieve absolute high performance with it. We describe algorithms that have drastically improved performance while maintaining a high degree of large grain parallelism. We evaluate the eectiveness of these methods in the context of a MATLABbased environment for parallel programming using MPI on small, otheshelf parallel systems. Keywords Pseudospectra, MPI, MATLAB, NOWs 1.
Spectral Portraits for Matrix Pencils
, 1996
"... Spectral portraits and pseudospectra have recently attracted the attention of many scientists as a tool of choice for investigating spectral properties of nonnormal matrices. Until now, most of the papers devoted to this topic were generally dealing with the standard eigenproblem Ax = x. However in ..."
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Cited by 7 (0 self)
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Spectral portraits and pseudospectra have recently attracted the attention of many scientists as a tool of choice for investigating spectral properties of nonnormal matrices. Until now, most of the papers devoted to this topic were generally dealing with the standard eigenproblem Ax = x. However in practice, physical applications often give rise to generalized eigenproblems Ax = Bx. It is therefore natural to extend the notion of "pseudospectrum and spectral portrait to matrix pencils (A; B). In this paper, we explain how to define and compute spectral portraits for matrix pencils and we illustrate their importance on and example in Computational Fluid Dynamics. Keywords : generalized eigenproblem, pseudospectrum, spectral portrait, nonnormality. 1 Introduction Eigenvalues of nonsymmetric matrices play an essential role in physical problems. These eigenvalues may be used either for their physical meaning (vibration modes, stability region,: : : ) or for their numerical meaning (conv...
The Power of Backward Error Analysis
"... par Val'erie FRAYSS 'E CERFACS devant le Jury compos'e de: ..."
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par Val'erie FRAYSS 'E CERFACS devant le Jury compos'e de:
Polynomial Acceleration For Large Nonsymmetric Eigenproblems
, 1997
"... In this thesis, we propose a highly efficient accelerating method for the restarted Arnoldi iteration to compute the eigenvalues of a large nonsymmetric matrix. Its effectiveness is proved by various numerical experiments and comparisons with other approaches. Several new results on the characterist ..."
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In this thesis, we propose a highly efficient accelerating method for the restarted Arnoldi iteration to compute the eigenvalues of a large nonsymmetric matrix. Its effectiveness is proved by various numerical experiments and comparisons with other approaches. Several new results on the characteristics of the polynomial acceleration are also reported. The Arnoldi iteration has been the most popular method for nonsymmetric large eigenproblems. Its defect of increasing computational complexity per iteration step was improved with the explicitly restarting technique, by which the dimensions of the Krylov subspaces are kept modest. Although the restarted Arnoldi iteration is a quite effective approach, the dimension of the subspace becomes inevitably large, especially when the required eigenvalues are clustered. Furthermore, it favors the convergence on the envelope of the spectrum. In this paper, we seek a polynomial such that the components in the direction of unwanted eigenvectors are d...
Parallel computation of pseudospectra of large sparse matrices
, 2000
"... The parallel computation of the pseudospectrum is presented. The Parallel Path following Algorithm using Triangles (PPAT) is based on the Path following Algorithm using Triangles (PAT). This algorithm offers total reliability and can handle singular points along the level curve without difficulty. F ..."
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The parallel computation of the pseudospectrum is presented. The Parallel Path following Algorithm using Triangles (PPAT) is based on the Path following Algorithm using Triangles (PAT). This algorithm offers total reliability and can handle singular points along the level curve without difficulty. Furthermore, PPAT offers a guarantee of termination even in the presence of roundoff errors and makes use of the large granularity for parallelism in PAT. This results in large speedups and high efficiency. The PPAT is able to trace multiple level curves simultaneously and takes into account the symmetry of the pseudospectrum in the case
EVALUATION OF ACCELERATION TECHNIQUES FOR THE RESTARTED ARNOLDI METHOD
"... Abstract. We present an approach for the acceleration of the restarted Arnoldi iteration for the computation of a number of eigenvalues of the standardeigenproblem $Ax=\lambda x $. This study applies the Chebyshev polynomial to the restarted Arnoldi iteration and proves that it computes necessary e ..."
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Abstract. We present an approach for the acceleration of the restarted Arnoldi iteration for the computation of a number of eigenvalues of the standardeigenproblem $Ax=\lambda x $. This study applies the Chebyshev polynomial to the restarted Arnoldi iteration and proves that it computes necessary eigenvalues with far less complexity than the QR method. We also discuss the dependence of the convergence rate of the restarted Arnoldi iteration on the distribution of spectrum. This research aims to compare this algorithm with other stateoftheart approaches.