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  • 1.
    Andersson, Per
    et al.
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Institutionen för datavetenskap.
    Granat, Robert
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Institutionen för datavetenskap. Umeå universitet, Teknisk-naturvetenskaplig fakultet, HPC2N (Högpresterande beräkningscentrum norr).
    Jonsson, Isak
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Institutionen för datavetenskap. Umeå universitet, Teknisk-naturvetenskaplig fakultet, HPC2N (Högpresterande beräkningscentrum norr).
    Kågström, Bo
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Institutionen för datavetenskap. Umeå universitet, Teknisk-naturvetenskaplig fakultet, HPC2N (Högpresterande beräkningscentrum norr).
    Parallel Algorithms for Triangular Periodic Sylvester-Type Matrix Equations2008Ingår i: Euro-Par 2008 - Parallel Processing: 14th International Euro-Par Conference Conference Location: Las Palmas de Gran Canaria, SPAIN, Springer , 2008, s. 780-789Konferensbidrag (Refereegranskat)
    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.

  • 2.
    Granat, Robert
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för datavetenskap.
    Algorithms and Library Software for Periodic and Parallel Eigenvalue Reordering and Sylvester-Type Matrix Equations with Condition Estimation2007Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
    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.

  • 3.
    Granat, Robert
    et al.
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Institutionen för datavetenskap. Umeå universitet, Teknisk-naturvetenskaplig fakultet, HPC2N (Högpresterande beräkningscentrum norr).
    Jonsson, Isak
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Institutionen för datavetenskap.
    Kågström, Bo
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Institutionen för datavetenskap. Umeå universitet, Teknisk-naturvetenskaplig fakultet, HPC2N (Högpresterande beräkningscentrum norr).
    Combining Explicit and Recursive Blocking for Solving Triangular Sylvester-Type Matrix Equations on Distributed Memory Platforms2004Ingår i: Euro-Par 2004 Parallel Processing: 10th International Euro-Par Conference, Springer , 2004, s. 742-750Konferensbidrag (Refereegranskat)
  • 4.
    Granat, Robert
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för datavetenskap. Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Högpresterande beräkningscentrum norr (HPC2N).
    Jonsson, Isak
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för datavetenskap. Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Högpresterande beräkningscentrum norr (HPC2N).
    Kågström, Bo
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för datavetenskap. Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Högpresterande beräkningscentrum norr (HPC2N).
    RECSY and SCASY library software: recursive blocked and parallel algorithms for Sylvester-type matrix equations with some applications2009Ingår i: Parallel scientific computing and optimization: advances and applications / [ed] Raimondas Čiegis, David Henty, Bo Kågström, Julius Žilinskas, Springer-Verlag New York, 2009, s. 3-24Kapitel i bok, del av antologi (Övrigt vetenskapligt)
  • 5.
    Granat, Robert
    et al.
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Institutionen för datavetenskap. Umeå universitet, Teknisk-naturvetenskaplig fakultet, HPC2N (Högpresterande beräkningscentrum norr).
    Jonsson, Isak
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Institutionen för datavetenskap. Umeå universitet, Teknisk-naturvetenskaplig fakultet, HPC2N (Högpresterande beräkningscentrum norr).
    Kågström, Bo
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Institutionen för datavetenskap. Umeå universitet, Teknisk-naturvetenskaplig fakultet, HPC2N (Högpresterande beräkningscentrum norr).
    Recursive Blocked Algorithms for Solving Periodic Triangular Sylvester-Type Matrix Equations2007Ingår i: Applied Parallel Computing - State of the Art in Scientific Computing: 8th International Workshop, PARA 2006, Springer , 2007, s. 531-539Konferensbidrag (Refereegranskat)
    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.

  • 6.
    Granat, Robert
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för datavetenskap. Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Högpresterande beräkningscentrum norr (HPC2N).
    Kagstrom, Bo
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Högpresterande beräkningscentrum norr (HPC2N). Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för datavetenskap.
    Kressner, Daniel
    Shao, Meiyue
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för datavetenskap. Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Högpresterande beräkningscentrum norr (HPC2N).
    Algorithm 953: Parallel Library Software for the Multishift QR Algorithm with Aggressive Early Deflation2015Ingår i: ACM Transactions on Mathematical Software, ISSN 0098-3500, E-ISSN 1557-7295, Vol. 41, nr 4, artikel-id 29Artikel i tidskrift (Refereegranskat)
    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.

  • 7.
    Granat, Robert
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för datavetenskap.
    Kågström, Bo
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för datavetenskap.
    Algorithm 904: the SCASY library - parallel solvers for Sylvester-type matrix equations with applications in condition estimation, part II2010Ingår i: ACM Transactions on Mathematical Software, ISSN 0098-3500, E-ISSN 1557-7295, Vol. 37, nr 3, artikel-id 33Artikel i tidskrift (Refereegranskat)
    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.

  • 8.
    Granat, Robert
    et al.
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Institutionen för datavetenskap. Umeå universitet, Teknisk-naturvetenskaplig fakultet, HPC2N (Högpresterande beräkningscentrum norr).
    Kågström, Bo
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Institutionen för datavetenskap. Umeå universitet, Teknisk-naturvetenskaplig fakultet, HPC2N (Högpresterande beräkningscentrum norr).
    Direct Eigenvalue Reordering in a Product of Matrices in Periodic Schur Form2006Ingår i: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, E-ISSN 1095-7162, Vol. 28, nr 1, s. 285-300Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    A direct method for eigenvalue reordering in a product of a K-periodic matrix sequence in periodic or extended periodic real Schur form is presented and analyzed. Each reordering of two adjacent sequences of diagonal blocks is performed tentatively to guarantee backward stability and involves solving a K-periodic Sylvester equation (PSE) and constructing a K-periodic sequence of orthogonal transformation matrices. An error analysis of the direct reordering method is presented, and results from computational experiments confirm the stability and accuracy of the method for well-conditioned as well as ill-conditioned problems. These include matrix sequences with fixed and time-varying dimensions, and sequences of small and large periodicity.

  • 9.
    Granat, Robert
    et al.
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Institutionen för datavetenskap. Umeå universitet, Teknisk-naturvetenskaplig fakultet, HPC2N (Högpresterande beräkningscentrum norr).
    Kågström, Bo
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Institutionen för datavetenskap. Umeå universitet, Teknisk-naturvetenskaplig fakultet, HPC2N (Högpresterande beräkningscentrum norr).
    Evaluating Parallel Algorithms for Solving Sylvester-Type Matrix Equations: Direct Transformation-Based versus Iterative Matrix-Sign-Function-Based Methods.2006Ingår i: Applied Parallel Computing - State of the Art in Scientific Computing: 7th International Workshop, PARA 2004, Springer , 2006, s. 717-729Konferensbidrag (Refereegranskat)
  • 10.
    Granat, Robert
    et al.
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Institutionen för datavetenskap. Umeå universitet, Teknisk-naturvetenskaplig fakultet, HPC2N (Högpresterande beräkningscentrum norr).
    Kågström, Bo
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Institutionen för datavetenskap. Umeå universitet, Teknisk-naturvetenskaplig fakultet, HPC2N (Högpresterande beräkningscentrum norr).
    Parallel Algorithms and Condition Estimators for Standard and Generalized Triangular Sylvester-Type Matrix Equations2006Ingår i: Applied Parallel Computing - State of the Art in Scientific Computing: 7th International Workshop, PARA 2004, Springer , 2006, s. 127-136Konferensbidrag (Refereegranskat)
    Abstract [en]

    We discuss parallel algorithms for solving eight common standard and generalized triangular Sylvester-type matrix equation. Our parallel algorithms are based on explicit blocking, 2D block-cyclic data distribution of the matrices and wavefront-like traversal of the right hand side matrices while solving small-sized matrix equations at different nodes and updating the rest of the right hand side using level 3 operations. We apply the triangular solvers in condition estimation, developing parallel sep(-1)-estimators. Some experimental results are presented.

  • 11.
    Granat, Robert
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för datavetenskap.
    Kågström, Bo
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för datavetenskap.
    Parallel Solvers for Sylvester-type Matrix Equations with Applications in Condition Estimation, Part I: Theory and Algorithms2010Ingår i: ACM Transactions on Mathematical Software, ISSN 0098-3500, E-ISSN 1557-7295, Vol. 37, nr 3, s. 32:1-32:32Artikel i tidskrift (Refereegranskat)
    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.

  • 12.
    Granat, Robert
    et al.
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Institutionen för datavetenskap. Umeå universitet, Teknisk-naturvetenskaplig fakultet, HPC2N (Högpresterande beräkningscentrum norr).
    Kågström, Bo
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Institutionen för datavetenskap. Umeå universitet, Teknisk-naturvetenskaplig fakultet, HPC2N (Högpresterande beräkningscentrum norr).
    Kressner, Daniel
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Institutionen för datavetenskap. Umeå universitet, Teknisk-naturvetenskaplig fakultet, HPC2N (Högpresterande beräkningscentrum norr).
    A Parallel Schur Method for Solving Continuous-Time Algebraic Riccati Equations2008Ingår i: Conference Name: 2008 IEEE Conference on Computer-Aided Control System Design (CACSD) Conference Location: San Antonio, TX, 2008, s. 51-56Konferensbidrag (Refereegranskat)
    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.

  • 13.
    Granat, Robert
    et al.
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Institutionen för datavetenskap. Umeå universitet, Teknisk-naturvetenskaplig fakultet, HPC2N (Högpresterande beräkningscentrum norr).
    Kågström, Bo
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Institutionen för datavetenskap. Umeå universitet, Teknisk-naturvetenskaplig fakultet, HPC2N (Högpresterande beräkningscentrum norr).
    Kressner, Daniel
    ETH Zürich.
    Computing Periodic Deflating Subspaces Associated with a Specified Set of Eigenvalues2007Ingår i: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 47, nr 4, s. 763-791Artikel i tidskrift (Refereegranskat)
    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.

  • 14.
    Granat, Robert
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för datavetenskap. Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Högpresterande beräkningscentrum norr (HPC2N).
    Kågström, Bo
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för datavetenskap. Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Högpresterande beräkningscentrum norr (HPC2N).
    Kressner, Daniel
    ETH, Zürich.
    Parallel Eigenvalue Reordering in Real Schur Forms2009Ingår i: Concurrency and Computation, ISSN 1532-0626, E-ISSN 1532-0634, Vol. 21, nr 9, s. 1225-1250Artikel i tidskrift (Refereegranskat)
    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.

  • 15.
    Granat, Robert
    et al.
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Institutionen för datavetenskap. Umeå universitet, Teknisk-naturvetenskaplig fakultet, HPC2N (Högpresterande beräkningscentrum norr).
    Kågström, Bo
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Institutionen för datavetenskap. Umeå universitet, Teknisk-naturvetenskaplig fakultet, HPC2N (Högpresterande beräkningscentrum norr).
    Kressner, Daniel
    ETH, Zürich.
    Reordering the Eigenvalues of a Periodic Matrix Pair with Applications in Control2006Ingår i: 2006 IEEE Conferences on Computer Aided Control System Design (CACSD), 2006, s. 25-30Konferensbidrag (Refereegranskat)
  • 16.
    Granat, Robert
    et al.
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Institutionen för datavetenskap. Umeå universitet, Teknisk-naturvetenskaplig fakultet, HPC2N (Högpresterande beräkningscentrum norr).
    Kågström, Bo
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Institutionen för datavetenskap. Umeå universitet, Teknisk-naturvetenskaplig fakultet, HPC2N (Högpresterande beräkningscentrum norr).
    Poromaa, Peter
    Umeå universitet, Teknisk-naturvetenskaplig fakultet, Institutionen för datavetenskap. Umeå universitet, Teknisk-naturvetenskaplig fakultet, HPC2N (Högpresterande beräkningscentrum norr).
    Parallel ScaLAPACK-Style Algorithms for Solving Continuous-Time Sylvester Matrix Equations2003Ingår i: Euro-Par 2003 Parallel Processing: Conference Name: 9th International Euro-Par Conference Conference Location: Klagenfurt, Austria, Springer , 2003, s. 800-809Konferensbidrag (Refereegranskat)
    Abstract [en]

    An implementation of a parallel ScaLAPACK-style solver for the general Sylvester equation, op(A)X - Xop(B) = C, where op(A) denotes A or its transpose AT, is presented. The parallel algorithm is based on explicit blocking of the Bartels-Stewart method. An initial transformation of the coefficient matrices A and B to Schur form leads to a reduced triangular matrix equation. We use different matrix traversing strategies to handle the transposes in the problem to solve, leading to different new parallel wave-front algorithms. We also present a strategy to handle the problem when 2 x 2 diagonal blocks of the matrices in Schur form, corresponding to complex conjugate pairs of eigenvalues, are split between several blocks in the block partitioned matrices. Finally, the solution of the reduced matrix equation is transformed back to the originally coordinate system. The implementation acts in a ScaLAPACK environment using 2-dimensional block cyclic mapping of the matrices onto a rectangular grid of processes. Real performance results are presented which verify that our parallel algorithms are reliable and scalable.

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