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17
Block algorithms for reordering standard and generalized Schur forms
 ACM Transactions on Mathematical Software
, 2006
"... Abstract. Block algorithms for reordering a selected set of eigenvalues in a standard or generalized Schur form are proposed. Efficiency is achieved by delaying orthogonal transformations and (optionally) making use of level 3 BLAS operations. Numerical experiments demonstrate that existing algorith ..."
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Cited by 10 (6 self)
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Abstract. Block algorithms for reordering a selected set of eigenvalues in a standard or generalized Schur form are proposed. Efficiency is achieved by delaying orthogonal transformations and (optionally) making use of level 3 BLAS operations. Numerical experiments demonstrate that existing algorithms, as currently implemented in LAPACK, are outperformed by up to a factor of four. Key words. Schur form, reordering, invariant subspace, deflating subspace. AMS subject classifications. 65F15, 65Y20. 1. Introduction. Applying
Parallel Solvers for Sylvestertype Matrix Equations with Applications in Condition Estimation, Part I: Theory and Algorithms
, 2007
"... Parallel ScaLAPACKstyle algorithms for solving eight common standard and generalized Sylvestertype matrix equations and various sign and transposed variants are presented. All algorithms are blocked variants based on the Bartels–Stewart method and involve four major steps: reduction to triangular ..."
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Cited by 10 (5 self)
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Parallel ScaLAPACKstyle algorithms for solving eight common standard and generalized Sylvestertype matrix equations and various sign and transposed variants are presented. All algorithms are blocked variants based on the Bartels–Stewart method and involve four major steps: reduction to triangular form, updating the right hand side with respect to the reduction, computing the solution to the reduced triangular problem and transforming the solution back to the original coordinate system. Novel parallel algorithms for solving reduced triangular matrix equations based on wavefrontlike traversal of the right hand side matrices are presented together with a generic scalability analysis. These algorithms are used in condition estimation and new robust parallel sep−1estimators are developed. Experimental results from three parallel platforms are presented and analyzed using several performance and accuracy metrics. The analysis includes results regarding general and triangular parallel solvers as well as parallel condition estimators.
Implicit QR Algorithms for Palindromic and Even Eigenvalue Problems
, 2008
"... In the spirit of the Hamiltonian QR algorithm and other bidirectional chasing algorithms, a structurepreserving variant of the implicit QR algorithm for palindromic eigenvalue problems is proposed. This new palindromic QR algorithm is strongly backward stable and requires less operations than the s ..."
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Cited by 9 (1 self)
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In the spirit of the Hamiltonian QR algorithm and other bidirectional chasing algorithms, a structurepreserving variant of the implicit QR algorithm for palindromic eigenvalue problems is proposed. This new palindromic QR algorithm is strongly backward stable and requires less operations than the standard QZ algorithm, but is restricted to matrix classes where a preliminary reduction to structured Hessenberg form can be performed. By an extension of the implicit Q theorem, the palindromic QR algorithm is shown to be equivalent to a previously developed explicit version. Also, the classical convergence theory for the QR algorithm can be extended to prove local quadratic convergence. We briefly demonstrate how even eigenvalue problems can be addressed by similar techniques.
A novel parallel QR algorithm for hybrid distributed memory HPC systems, Technical report 200915, Seminar for applied mathematics
, 2009
"... Abstract. A novel variant of the parallel QR algorithm for solving dense nonsymmetric eigenvalue problems on hybrid distributed high performance computing (HPC) systems is presented. For this purpose, we introduce the concept of multiwindow bulge chain chasing and parallelize aggressive early defla ..."
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Cited by 8 (3 self)
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Abstract. A novel variant of the parallel QR algorithm for solving dense nonsymmetric eigenvalue problems on hybrid distributed high performance computing (HPC) systems is presented. For this purpose, we introduce the concept of multiwindow bulge chain chasing and parallelize aggressive early deflation. The multiwindow approach ensures that most computations when chasing chains of bulges are performed in level 3 BLAS operations, while the aim of aggressive early deflation is to speed up the convergence of the QR algorithm. Mixed MPIOpenMP coding techniques are utilized for porting the codes to distributed memory platforms with multithreaded nodes, such as multicore processors. Numerous numerical experiments confirm the superior performance of our parallel QR algorithm in comparison with the existing ScaLAPACK code, leading to an implementation that is one to two orders of magnitude faster for sufficiently large problems, including a number of examples from applications.
Parallel Variants of the Multishift QZ Algorithm with Advanced Deflation Techniques
 In PARA’06  State of the Art in Scientific and Parallel Computing. Vol. 4699. Lecture Notes in Computer Science
, 2007
"... Abstract. The QZ algorithm reduces a regular matrix pair to generalized Schur form, which can be used to address the generalized eigenvalue problem. This paper summarizes recent work on improving the performance of the QZ algorithm on serial machines and work in progress on a novel parallel implemen ..."
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Cited by 6 (4 self)
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Abstract. The QZ algorithm reduces a regular matrix pair to generalized Schur form, which can be used to address the generalized eigenvalue problem. This paper summarizes recent work on improving the performance of the QZ algorithm on serial machines and work in progress on a novel parallel implementation. In both cases, the QZ iterations are based on chasing chains of tiny bulges. This allows to formulate the majority of the computation in terms of matrixmatrix multiplications, resulting in natural parallelism and better performance on modern computing systems with memory hierarchies. In addition, advanced deflation strategies are used, specifically the so called aggressive early deflation, leading to a considerable convergence acceleration and consequently to a reduction of floating point operations and computing time. 1
Parallel eigenvalue reordering in real Schur forms
"... A parallel variant of the standard eigenvalue reordering method for the real Schur form is presented and discussed. The novel parallel algorithm adopts computational windows and delays multiple outsidewindow updates until each window has been completely reordered locally. By using multiple concurr ..."
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Cited by 6 (3 self)
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A parallel variant of the standard eigenvalue reordering method for the real Schur form is presented and discussed. The novel parallel algorithm adopts computational windows and delays multiple outsidewindow updates until each window has been completely reordered locally. By using multiple concurrent windows the parallel algorithm has a high level of concurrency, and most work is level 3 BLAS operations. The presented algorithm is also extended to the generalized real Schur form. Experimental results for ScaLAPACKstyle Fortran 77 implementations on a Linux cluster confirm the efficiency and scalability of our algorithms in terms of more than 16 times of parallel speedup using 64 processor for large scale problems. Even on a single processor our implementation is demonstrated to perform significantly better compared to the stateoftheart serial implementation.
Perturbation, Computation and Refinement of Invariant Pairs for Matrix Polynomials
, 2009
"... Generalizing the notion of an eigenvector, invariant subspaces are frequently used in the context of linear eigenvalue problems, leading to conceptually elegant and numerically stable formulations in applications that require the computation of several eigenvalues and/or eigenvectors. Similar benefi ..."
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Cited by 3 (2 self)
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Generalizing the notion of an eigenvector, invariant subspaces are frequently used in the context of linear eigenvalue problems, leading to conceptually elegant and numerically stable formulations in applications that require the computation of several eigenvalues and/or eigenvectors. Similar benefits can be expected for polynomial eigenvalue problems, for which the concept of invariant subspaces needs to be replaced by the concept of invariant pair. Little is known so far about numerical aspects of such invariant pairs. The aim of this paper is to fill this gap. The behavior of invariant pairs under perturbations of the matrix polynomial is studied and a firstorder perturbation expansion is given. From a computational point of view, we investigate how to best extract invariant pairs from a linearization of the matrix polynomial. Moreover, we describe efficient refinement procedures directly based on the polynomial formulation. Numerical experiments with matrix polynomials from a number of applications demonstrate the effectiveness of our extraction and refinement procedures.
Perturbation, Extraction and Refinement of Invariant Pairs for Matrix Polynomials
, 2010
"... Generalizing the notion of an eigenvector, invariant subspaces are frequently used in the context of linear eigenvalue problems, leading to conceptually elegant and numerically stable formulations in applications that require the computation of several eigenvalues and/or eigenvectors. Similar benefi ..."
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Cited by 3 (2 self)
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Generalizing the notion of an eigenvector, invariant subspaces are frequently used in the context of linear eigenvalue problems, leading to conceptually elegant and numerically stable formulations in applications that require the computation of several eigenvalues and/or eigenvectors. Similar benefits can be expected for polynomial eigenvalue problems, for which the concept of an invariant subspace needs to be replaced by the concept of an invariant pair. Little has been known so far about numerical aspects of such invariant pairs. The aim of this paper is to fill this gap. The behavior of invariant pairs under perturbations of the matrix polynomial is studied and a firstorder perturbation expansion is given. From a computational point of view, we investigate how to best extract invariant pairs from a linearization of the matrix polynomial. Moreover, we describe efficient refinement procedures directly based on the polynomial formulation. Numerical experiments with matrix polynomials from a number of applications demonstrate the effectiveness of our extraction and refinement procedures. 1
Prospectus for the Next LAPACK and ScaLAPACK Libraries
"... Dense linear algebra (DLA) forms the core of many scientific computing applications. Consequently, there is continuous interest and demand for the development of increasingly better algorithms in the field. Here ’better ’ has a broad meaning, and includes improved reliability, accuracy, robustness, ..."
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Cited by 2 (0 self)
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Dense linear algebra (DLA) forms the core of many scientific computing applications. Consequently, there is continuous interest and demand for the development of increasingly better algorithms in the field. Here ’better ’ has a broad meaning, and includes improved reliability, accuracy, robustness, ease of use, and
The Multishift QZ Algorithm with Aggressive
"... Abstract. Recent improvements to the QZ algorithm for solving generalized eigenvalue problems are summarized. Among the major modifications are novel multishift QZ iterations based on chasing chains of tiny bulges and an extension of the so called aggressive early deflation strategy. The former modi ..."
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Cited by 1 (1 self)
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Abstract. Recent improvements to the QZ algorithm for solving generalized eigenvalue problems are summarized. Among the major modifications are novel multishift QZ iterations based on chasing chains of tiny bulges and an extension of the so called aggressive early deflation strategy. The former modification aims to improve the execution time of the QZ algorithm on modern computing systems without changing the number of floating point operations (flops) significantly. In contrast, the new deflation strategy results in a considerable convergence acceleration and consequently in a reduction of both, flops and computing time. 1