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  • 1.
    Bujanović, Zvonimir
    et al.
    Department of Mathematics, Faculty of Science, University of Zagreb, Zagreb, Croatia.
    Karlsson, Lars
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Kressner, Daniel
    Institute of Mathematics, EPFL, Lausanne, Switzerland.
    A Householder-Based Algorithm for Hessenberg-Triangular Reduction2018In: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, E-ISSN 1095-7162, Vol. 39, no 3, p. 1270-1294Article in journal (Refereed)
    Abstract [en]

    The QZ algorithm for computing eigenvalues and eigenvectors of a matrix pencil $A - \lambda B$ requires that the matrices first be reduced to Hessenberg-triangular (HT) form. The current method of choice for HT reduction relies entirely on Givens rotations regrouped and accumulated into small dense matrices which are subsequently applied using matrix multiplication routines. A nonvanishing fraction of the total flop-count must nevertheless still be performed as sequences of overlapping Givens rotations alternately applied from the left and from the right. The many data dependencies associated with this computational pattern leads to inefficient use of the processor and poor scalability. In this paper, we therefore introduce a fundamentally different approach that relies entirely on (large) Householder reflectors partially accumulated into block reflectors, by using (compact) WY representations. Even though the new algorithm requires more floating point operations than the state-of-the-art algorithm, extensive experiments on both real and synthetic data indicate that it is still competitive, even in a sequential setting. The new algorithm is conjectured to have better parallel scalability, an idea which is partially supported by early small-scale experiments using multithreaded BLAS. The design and evaluation of a parallel formulation is future work.

  • 2.
    Dmytryshyn, Andrii
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).
    Johansson, Stefan
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).
    Kågström, Bo
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).
    Canonical structure transitions of system pencils2017In: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, E-ISSN 1095-7162, Vol. 38, no 4, p. 1249-1267Article in journal (Refereed)
    Abstract [en]

    We investigate the changes of the canonical structure information under small perturbations for a system pencil associated with a (generalized) linear time-invariant state-space system. The equivalence class of the pencil is taken with respect to feedback-injection equivalence transformations. The results allow us to track possible changes of important linear system characteristics under small perturbations.

  • 3.
    Dmytryshyn, Andrii
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå Univ, HPC2N, SE-90187 Umeå, Sweden.
    Kågstrom, Bo
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå Univ, HPC2N, SE-90187 Umeå, Sweden.
    Coupled Sylvester-type Matrix Equations and Block Diagonalization2015In: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, E-ISSN 1095-7162, Vol. 36, no 2, p. 580-593Article in journal (Refereed)
    Abstract [en]

    We prove Roth-type theorems for systems of matrix equations including an arbitrary mix of Sylvester and $\star$-Sylvester equations, in which the transpose or conjugate transpose of the unknown matrices also appear. In full generality, we derive consistency conditions by proving that such a system has a solution if and only if the associated set of $2 \times 2$ block matrix representations of the equations are block diagonalizable by (linked) equivalence transformations. Various applications leading to several particular cases have already been investigated in the literature, some recently and some long ago. Solvability of these cases follow immediately from our general consistency theory. We also show how to apply our main result to systems of Stein-type matrix equations.

  • 4.
    Dmytryshyn, Andrii
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Kågstrom, Bo
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).
    Orbit closure hierarchies of skew-symmetric matrix pencils2014In: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, E-ISSN 1095-7162, Vol. 35, no 4, p. 1429-1443Article in journal (Refereed)
    Abstract [en]

    We study how small perturbations of a skew-symmetric matrix pencil may change its canonical form under congruence. This problem is also known as the stratification problem of skew-symmetric matrix pencil orbits and bundles. In other words, we investigate when the closure of the congruence orbit (or bundle) of a skew-symmetric matrix pencil contains the congruence orbit (or bundle) of another skew-symmetric matrix pencil. The developed theory relies on our main theorem stating that a skew-symmetric matrix pencil A - lambda B can be approximated by pencils strictly equivalent to a skew-symmetric matrix pencil C - lambda D if and only if A - lambda B can be approximated by pencils congruent to C - lambda D.

  • 5.
    Elmroth, Erik
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Johansson, Stefan
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Kågström, Bo
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Compting Center North (HPC2N).
    Stratification of controllability and observability pairs: theory and use in applications2009In: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, E-ISSN 1095-7162, Vol. 31, no 2, p. 203-226Article in journal (Refereed)
    Abstract [en]

    Cover relations for orbits and bundles of controllability and observability pairs associated with linear time-invariant systems are derived. The cover relations are combinatorial rules acting on integer sequences, each representing a subset of the Jordan and singular Kronecker structures of the corresponding system pencil. By representing these integer sequences as coin piles, the derived stratification rules are expressed as minimal coin moves between and within these piles, which satisfy and preserve certain monotonicity properties. The stratification theory is illustrated with two examples from systems and control applications, a mechanical system consisting of a thin uniform platform supported at both ends by springs, and a linearized Boeing 747 model. For both examples, nearby uncontrollable systems are identified as subsets of the complete closure hierarchy for the associated system pencils.

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  • 6.
    Granat, Robert
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, HPC2N (High Performance Computing Centre North).
    Kågström, Bo
    Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, HPC2N (High Performance Computing Centre North).
    Direct Eigenvalue Reordering in a Product of Matrices in Periodic Schur Form2006In: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, E-ISSN 1095-7162, Vol. 28, no 1, p. 285-300Article in journal (Refereed)
    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.

  • 7.
    Shao, Meiyue
    et al.
    Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N). Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Gao, Weiguo
    Xue, Jungong
    Aggressively truncated Taylor series method for accurate computation of exponentials of essentially nonnegative matrices2014In: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, E-ISSN 1095-7162, Vol. 35, no 2, p. 317-338Article in journal (Refereed)
    Abstract [en]

    Small relative perturbations to the entries of an essentially nonnegative matrix introduce small relative errors to entries of its exponential. It is thus desirable to compute the exponential with high componentwise relative accuracy. Taylor series approximation coupled with scaling and squaring is used to compute the exponential of an essentially nonnegative matrix. An a priori componentwise relative error bound of truncation is established, from which one can choose the degree of Taylor series expansion and the scale factor so that the exponential is computed with desired componentwise relative accuracy. To reduce the computational cost, the degree of the Taylor series expansion is chosen small, while the scale factor is chosen sufficiently large to achieve the desired accuracy. The rounding errors in the squaring stage are not serious as squaring is forward stable for nonnegative matrices. We also establish a posteriori componentwise error bounds and derive a novel interval algorithm for the matrix exponential of an essentially nonnegative matrix. Rounding error analysis and numerical experiments demonstrate the efficiency and accuracy of the proposed methods.

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