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Granat, Robert
Publications (10 of 16) Show all publications
Granat, R., Kagstrom, B., Kressner, D. & Shao, M. (2015). Algorithm 953: Parallel Library Software for the Multishift QR Algorithm with Aggressive Early Deflation. ACM Transactions on Mathematical Software, 41(4), Article ID 29.
Open this publication in new window or tab >>Algorithm 953: Parallel Library Software for the Multishift QR Algorithm with Aggressive Early Deflation
2015 (English)In: ACM Transactions on Mathematical Software, ISSN 0098-3500, E-ISSN 1557-7295, Vol. 41, no 4, article id 29Article in journal (Refereed) Published
Abstract [en]

Library software implementing a parallel small-bulge multishift QR algorithm with Aggressive Early Deflation (AED) targeting distributed memory high-performance computing systems is presented. Starting from recent developments of the parallel multishift QR algorithm [Granat et al., SIAM J. Sci. Comput. 32(4), 2010], we describe a number of algorithmic and implementation improvements. These include communication avoiding algorithms via data redistribution and a refined strategy for balancing between multishift QR sweeps and AED. Guidelines concerning several important tunable algorithmic parameters are provided. As a result of these improvements, a computational bottleneck within AED has been removed in the parallel multishift QR algorithm. A performance model is established to explain the scalability behavior of the new parallel multishift QR algorithm. Numerous computational experiments confirm that our new implementation significantly outperforms previous parallel implementations of the QR algorithm.

Keywords
Algorithms, Performance, Multishift QR algorithm, aggressive early deflation, parallel algorithms, stributed memory architectures
National Category
Computer Sciences
Identifiers
urn:nbn:se:umu:diva-111765 (URN)10.1145/2699471 (DOI)000363733000007 ()2-s2.0-84943313977 (Scopus ID)
Available from: 2015-11-25 Created: 2015-11-23 Last updated: 2018-06-07Bibliographically approved
Granat, R. & Kågström, B. (2010). Algorithm 904: the SCASY library - parallel solvers for Sylvester-type matrix equations with applications in condition estimation, part II. ACM Transactions on Mathematical Software, 37(3), Article ID 33.
Open this publication in new window or tab >>Algorithm 904: the SCASY library - parallel solvers for Sylvester-type matrix equations with applications in condition estimation, part II
2010 (English)In: ACM Transactions on Mathematical Software, ISSN 0098-3500, E-ISSN 1557-7295, Vol. 37, no 3, article id 33Article in journal (Refereed) Published
Abstract [en]

We continue our presentation of parallel ScaLAPACK-style algorithms for solving Sylvester-type matrix equations. In Part II, we present SCASY (SCAlable SYlvester solvers), a state-of-the-art HPC software library for solving 44 sign and transpose variants of eight common standard and generalized Sylvester-type matrix equations.

Place, publisher, year, edition, pages
New York: Association for Computing Machinery (ACM), 2010
Keywords
Algorithms, Performance, Reliability, Parallel computing, parallel algorithms, Eigenvalue problems, condition estimation, Sylvester matrix equations
National Category
Computer and Information Sciences
Identifiers
urn:nbn:se:umu:diva-51035 (URN)10.1145/1824801.1824811 (DOI)000282761200010 ()
External cooperation:
Funder
Swedish Research Council, 70625701
Available from: 2012-01-09 Created: 2012-01-09 Last updated: 2018-06-08Bibliographically approved
Granat, R. & Kågström, B. (2010). Parallel Solvers for Sylvester-type Matrix Equations with Applications in Condition Estimation, Part I: Theory and Algorithms. ACM Transactions on Mathematical Software, 37(3), 32:1-32:32
Open this publication in new window or tab >>Parallel Solvers for Sylvester-type Matrix Equations with Applications in Condition Estimation, Part I: Theory and Algorithms
2010 (English)In: ACM Transactions on Mathematical Software, ISSN 0098-3500, E-ISSN 1557-7295, Vol. 37, no 3, p. 32:1-32:32Article in journal (Refereed) Published
Abstract [en]

Parallel ScaLAPACK-style algorithms for solving eight common standard and generalized Sylvester-type 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 wavefront-like 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 − 1-estimators are developed. Experimental results from three parallel platforms, including results from a mixed OpenMP/MPI platform, 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.

Place, publisher, year, edition, pages
New York: ACM Press, 2010
National Category
Mathematics
Identifiers
urn:nbn:se:umu:diva-2708 (URN)10.1145/1824801.1824810 (DOI)000282761200009 ()
Note

Artikelnummer/article number: 32

Available from: 2007-11-01 Created: 2007-11-01 Last updated: 2018-06-09Bibliographically approved
Granat, R., Kågström, B. & Kressner, D. (2009). Parallel Eigenvalue Reordering in Real Schur Forms. Concurrency and Computation, 21(9), 1225-1250
Open this publication in new window or tab >>Parallel Eigenvalue Reordering in Real Schur Forms
2009 (English)In: Concurrency and Computation, ISSN 1532-0626, E-ISSN 1532-0634, Vol. 21, no 9, p. 1225-1250Article in journal (Refereed) Published
Abstract [en]

A parallel algorithm for reordering the eigenvalues in the real Schur form of a matrix is presented and discussed. Our novel approach adopts computational windows and delays multiple outside-window 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 ScaLAPACK-style 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 processors for large-scale problems. Even on a single processor our implementation is demonstrated to perform significantly better compared with the state-of-the-art serial implementation.

Keywords
parallel algorithms;eigenvalue problems;invariant subspaces, direct reordering, Sylvester matrix equations, condition number estimates
National Category
Computer Sciences Software Engineering
Identifiers
urn:nbn:se:umu:diva-24699 (URN)10.1002/cpe.1386 (DOI)
Available from: 2009-07-10 Created: 2009-07-10 Last updated: 2018-06-08Bibliographically approved
Granat, R., Jonsson, I. & Kågström, B. (2009). RECSY and SCASY library software: recursive blocked and parallel algorithms for Sylvester-type matrix equations with some applications. In: Raimondas Čiegis, David Henty, Bo Kågström, Julius Žilinskas (Ed.), Parallel scientific computing and optimization: advances and applications (pp. 3-24). Springer-Verlag New York
Open this publication in new window or tab >>RECSY and SCASY library software: recursive blocked and parallel algorithms for Sylvester-type matrix equations with some applications
2009 (English)In: Parallel scientific computing and optimization: advances and applications / [ed] Raimondas Čiegis, David Henty, Bo Kågström, Julius Žilinskas, Springer-Verlag New York, 2009, p. 3-24Chapter in book (Other academic)
Place, publisher, year, edition, pages
Springer-Verlag New York, 2009
Series
Springer optimization and its applications, ISSN 1931-6828 ; 27
National Category
Computer Sciences
Identifiers
urn:nbn:se:umu:diva-24697 (URN)978-0-387-09707-7 (ISBN)
Available from: 2009-07-09 Created: 2009-07-09 Last updated: 2018-06-08Bibliographically approved
Granat, R., Kågström, B. & Kressner, D. (2008). A Parallel Schur Method for Solving Continuous-Time Algebraic Riccati Equations. In: Conference Name: 2008 IEEE Conference on Computer-Aided Control System Design (CACSD) Conference Location: San Antonio, TX (pp. 51-56).
Open this publication in new window or tab >>A Parallel Schur Method for Solving Continuous-Time Algebraic Riccati Equations
2008 (English)In: Conference Name: 2008 IEEE Conference on Computer-Aided Control System Design (CACSD) Conference Location: San Antonio, TX, 2008, p. 51-56Conference paper, Published paper (Refereed)
Abstract [en]

Numerical algorithms for solving the continuous-time algebraic Riccati matrix equation on a distributed memory parallel computer are considered. In particular, it is shown that the Schur method, based on computing the stable invariant subspace of a Hamiltonian matrix, can be parallelized in an efficient and scalable way. Our implementation employs the state-of-the-art library ScaLAPACK as well as recently developed parallel methods for reordering the eigenvalues in a real Schur form. Some experimental results are presented, confirming the scalability of our implementation and comparing it with an existing implementation of the matrix sign iteration from the PLiCOC library.

Identifiers
urn:nbn:se:umu:diva-23268 (URN)
Available from: 2009-06-09 Created: 2009-06-09 Last updated: 2018-06-08
Andersson, P., Granat, R., Jonsson, I. & Kågström, B. (2008). Parallel Algorithms for Triangular Periodic Sylvester-Type Matrix Equations. In: Euro-Par 2008 - Parallel Processing: 14th International Euro-Par Conference Conference Location: Las Palmas de Gran Canaria, SPAIN (pp. 780-789). Springer
Open this publication in new window or tab >>Parallel Algorithms for Triangular Periodic Sylvester-Type Matrix Equations
2008 (English)In: Euro-Par 2008 - Parallel Processing: 14th International Euro-Par Conference Conference Location: Las Palmas de Gran Canaria, SPAIN, Springer , 2008, p. 780-789Conference paper, Published paper (Refereed)
Abstract [en]

We present parallel algorithms for triangular periodic Sylvester-type matrix equations, conceptually being the third step of a periodic Bartels-Stewart-like solution method for general periodic Sylvester-type matrix equations based on variants of the periodic Schur decomposition. The presented algorithms are designed and implemented in the framework of the recently developed HPG library SCASY and are based on explicit blocking, 2-dimensional block cyclic data distribution and a wavefront-like traversal of the right hand side matrices. High performance is obtained by rich usage of level 3 BLAS operations. It is also demonstrated how several important key concepts of SCASY regarding communications and the treatment of quasi-triangular coefficient matrices are generalized to the periodic case. Some experimental results from a distributed memory Linux cluster demonstrate are also presented.

Place, publisher, year, edition, pages
Springer, 2008
Series
Lecture Notes in Computer Science ; LNCS 5168
Identifiers
urn:nbn:se:umu:diva-23235 (URN)
Available from: 2009-06-08 Created: 2009-06-08 Last updated: 2018-06-08
Granat, R. (2007). Algorithms and Library Software for Periodic and Parallel Eigenvalue Reordering and Sylvester-Type Matrix Equations with Condition Estimation. (Doctoral dissertation). Umeå: Datavetenskap
Open this publication in new window or tab >>Algorithms and Library Software for Periodic and Parallel Eigenvalue Reordering and Sylvester-Type Matrix Equations with Condition Estimation
2007 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This Thesis contains contributions in two different but closely related subfields of Scientific and Parallel Computing which arise in the context of various eigenvalue problems: periodic and parallel eigenvalue reordering and parallel algorithms for Sylvestertype matrix equations with applications in condition estimation.

Many real world phenomena behave periodically, e.g., helicopter rotors, revolving satellites and dynamic systems corresponding to natural processes, like the water flow in a system of connected lakes, and can be described in terms of periodic eigenvalue problems. Typically, eigenvalues and invariant subspaces (or, specifically, eigenvectors) to certain periodic matrix products are of interest and have direct physical interpretations. The eigenvalues of a matrix product can be computed without forming the product explicitly via variants of the periodic Schur decomposition. In the first part of the Thesis, we propose direct methods for eigenvalue reordering in the periodic standard and generalized real Schur forms which extend earlier work on the standard and generalized eigenvalue problems. The core step of the methods consists of solving periodic Sylvester-type equations to high accuracy. Periodic eigenvalue reordering is vital in the computation of periodic eigenspaces corresponding to specified spectra. The proposed direct reordering methods rely on orthogonal transformations and can be generalized to more general periodic matrix products where the factors have varying dimensions and ±1 exponents of arbitrary order.

In the second part, we consider Sylvester-type matrix equations, like the continuoustime Sylvester equation AX −XB =C, where A of size m×m, B of size n×n, and C of size m×n are general matrices with real entries, which have applications in many areas. Examples include eigenvalue problems and condition estimation, and several problems in control system design and analysis. The parallel algorithms presented are based on the well-known Bartels–Stewart’s method and extend earlier work on triangular Sylvester-type matrix equations resulting in a novel software library SCASY. The parallel library provides robust and scalable software for solving 44 sign and transpose variants of eight common Sylvester-type matrix equations. SCASY also includes a parallel condition estimator associated with each matrix equation.

In the last part of the Thesis, we propose parallel variants of the direct eigenvalue reordering method for the standard and generalized real Schur forms. Together with the existing and future parallel implementations of the non-symmetric QR/QZ algorithms and the parallel Sylvester solvers presented in the Thesis, the developed software can be used for parallel computation of invariant and deflating subspaces corresponding to specified spectra and associated reciprocal condition number estimates.

Place, publisher, year, edition, pages
Umeå: Datavetenskap, 2007. p. 44
Series
Report / UMINF, ISSN 0348-0542 ; 07.21
Keywords
periodic eigenvalue problems, product eigenvalue problems, periodic Schur form, periodic eigenvalue reordering, periodic eigenspaces, parallel algorithms, Sylvester-type matrix equations, parallel eigenvalue reordering, condition estimation
National Category
Computer Sciences
Identifiers
urn:nbn:se:umu:diva-1415 (URN)978-91-7264-410-6 (ISBN)
External cooperation:
Public defence
2007-11-23, MA121, MIT-huset, Umeå Universitet, UMEÅ, 10:00 (English)
Opponent
Supervisors
Available from: 2007-11-01 Created: 2007-11-01 Last updated: 2018-06-09Bibliographically approved
Granat, R., Kågström, B. & Kressner, D. (2007). Computing Periodic Deflating Subspaces Associated with a Specified Set of Eigenvalues. BIT Numerical Mathematics, 47(4), 763-791
Open this publication in new window or tab >>Computing Periodic Deflating Subspaces Associated with a Specified Set of Eigenvalues
2007 (English)In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 47, no 4, p. 763-791Article in journal (Refereed) Published
Abstract [en]

We present a direct method for reordering eigenvalues in the generalized periodic real Schur form of a regular K-cyclic matrix pair sequence (A (k) ,E (k) ). Following and generalizing existing approaches, reordering consists of consecutively computing the solution to an associated Sylvester-like equation and constructing K pairs of orthogonal matrices. These pairs define an orthogonal K-cyclic equivalence transformation that swaps adjacent diagonal blocks in the Schur form. An error analysis of this swapping procedure is presented, which extends existing results for reordering eigenvalues in the generalized real Schur form of a regular pair (A,E). Our direct reordering method is used to compute periodic deflating subspace pairs corresponding to a specified set of eigenvalues. This computational task arises in various applications related to discrete-time periodic descriptor systems. Computational experiments confirm the stability and reliability of the presented eigenvalue reordering method.

Keywords
generalized product of a K-cyclic matrix pair sequence, generalized periodic real Schur form, eigenvalue reordering, periodic generalized coupled Sylvester equation, K-cyclic equivalence transformation, generalized periodic eigenvalue problem
Identifiers
urn:nbn:se:umu:diva-21939 (URN)10.1007/s10543-007-0143-y (DOI)0006-3835 (ISBN)
Note
Granat, R. Kagstroem, B. Kressner, D.Available from: 2009-04-21 Created: 2009-04-21 Last updated: 2018-06-08
Granat, R., Jonsson, I. & Kågström, B. (2007). Recursive Blocked Algorithms for Solving Periodic Triangular Sylvester-Type Matrix Equations. In: Applied Parallel Computing - State of the Art in Scientific Computing: 8th International Workshop, PARA 2006 (pp. 531-539). Springer
Open this publication in new window or tab >>Recursive Blocked Algorithms for Solving Periodic Triangular Sylvester-Type Matrix Equations
2007 (English)In: Applied Parallel Computing - State of the Art in Scientific Computing: 8th International Workshop, PARA 2006, Springer , 2007, p. 531-539Conference paper, Published paper (Refereed)
Abstract [en]

Recently, recursive blocked algorithms for solving triangular one-sided and two-sided Sylvester-type equations were introduced by Jonsson and Kagstrom. This elegant yet simple technique enables an automatic variable blocking that has the potential of matching the memory hierarchies of today's HPC systems. The main parts of the computations are performed as level 3 general matrix multiply and add (GEMM) operations. We extend and apply the recursive blocking technique to solving periodic Sylvester-type matrix equations. Successive recursive splittings are performed on 3-dimensional arrays, where the third dimension represents the periodicity of a matrix equation.

Place, publisher, year, edition, pages
Springer, 2007
Series
Lecture Notes in Computer Science ; LNCS 4699
Identifiers
urn:nbn:se:umu:diva-23266 (URN)
Available from: 2009-06-09 Created: 2009-06-09 Last updated: 2018-06-08
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