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Axel Ruhe - One of the best experts on this subject based on the ideXlab platform.

  • RATIONAL KRYLOV FOR REAL PENCILS WITH COMPLEX EigenvalueS
    Taiwanese Journal of Mathematics, 2010
    Co-Authors: Axel Ruhe
    Abstract:

    A rational Krylov algorithm for Eigenvalue Computation is described. It is usable on a real matrix pencil with complex Eigenvalues and builds up a real basis. The main purpose is to get real reduced models of a real linear dynamic system. Two variants are described, one where two real vectors are added to the Krylov space in each step and another where just one real vector is added in each step. Results are reported from one small example that has been used earlier and where the solution is known, and one more realistic example, a linear descriptor system from a Computational fluid dynamics application.

  • Rational Krylov for Eigenvalue Computation and model order reduction
    Bit Numerical Mathematics, 2006
    Co-Authors: K. Henrik A. Olsson, Axel Ruhe
    Abstract:

    A rational Krylov algorithm for Eigenvalue Computation and model order reduction is described. It is shown how to implement it as a modified shift-and-invert spectral transformation Arnoldi decomposition. It is shown how to do deflation, locking converged Eigenvalues and purging irrelevant approximations. Computing reduced order models of linear dynamical systems by moment matching of the transfer function is considered. Results are reported from one illustrative toy example and one practical example, a linear descriptor system from a Computational fluid dynamics application.

  • RATIONAL KRYLOV FOR Eigenvalue Computation AND MODEL ORDER
    2006
    Co-Authors: Axel Ruhe
    Abstract:

    A rational Krylov algorithm for Eigenvalue Computation and model order reduction is described. It is shown how to implement it as a modified shift-and-invert spectral transformation Arnoldi decomposition. It is shown how to do deflation, locking converged Eigenvalues and purging irrelevant approximations. Computing reduced order models of linear dynamical systems by moment matching of the transfer function is considered. Results are reported from one illustrative toy example and one practical example, a linear descriptor system from a Computational fluid dynamics application.

  • Rational Krylov for Model Order Reduction and Eigenvalue Computation
    2005
    Co-Authors: K. Henrik A. Olsson, Axel Ruhe
    Abstract:

    Rational Krylov methods for model order reduction are studied. A dual rational Arnoldi method for model order reduction and a rational Krylov method for model order reduction and Eigenvalue Computation have been implemented. It is shown how to deflate redundant or unwanted vectors and how to obtain moment matching. Both methods are designed for generalised state space systems---the former for multiple-input-multiple-output (MIMO) systems from finite element discretisations and the latter for single-input-single-output (SISO) systems---and applied to relevant test problems. The dual rational Arnoldi method is designed for generating real reduced order systems using complex shift points and stabilising a system that happens to be unstable. For the rational Krylov method, a forward error in the recursion and an estimate of the error in the approximation of the transfer function are studie. A stability analysis of a heat exchanger model is made. The model is a nonlinear partial differential-algebraic equation (PDAE). Its well-posedness and how to prescribe boundary data is investigated through analysis of a linearised PDAE and numerical experiments on a nonlinear DAE. Four methods for generating reduced order models are applied to the nonlinear DAE and compared: a Krylov based moment matching method, balanced truncation, Galerkin projection onto a proper orthogonal decomposition (POD) basis, and a lumping method.

  • rational krylov algorithms for Eigenvalue Computation and model reduction
    Parallel Computing, 1998
    Co-Authors: Axel Ruhe, Daniel Skoogh
    Abstract:

    Rational Krylov is an extension of the Lanczos or Arnoldi Eigenvalue algorithm where several shifts (matrix factorizations) are performed in one run. A variant has been developed, where these factorizations are performed in parallel.

François Morain - One of the best experts on this subject based on the ideXlab platform.

Daniel Skoogh - One of the best experts on this subject based on the ideXlab platform.

Luca Gemignani - One of the best experts on this subject based on the ideXlab platform.

  • Implicit QR for companion-like pencils
    Mathematics of Computation, 2016
    Co-Authors: Paola Boito, Yuli Eidelman, Luca Gemignani
    Abstract:

    A fast implicit QR algorithm for Eigenvalue Computation of low rank corrections of unitary matrices is adjusted to work with matrix pencils arising from polynomial zero-finding problems. The modified QZ algorithm computes the generalized Eigenvalues of certain $ N\times N$ rank structured matrix pencils using $ O(N^2)$ flops and $ O(N)$ memory storage. Numerical experiments and comparisons confirm the effectiveness and the stability of the proposed method. - See more at: http://www.ams.org/journals/mcom/2016-85-300/S0025-5718-2015-03020-8/#sthash.JxbTFYBc.dpuf

  • Implicit QR for rank-structured matrix pencils
    BIT Numerical Mathematics, 2014
    Co-Authors: Paola Boito, Yuli Eidelman, Luca Gemignani
    Abstract:

    A fast implicit QR algorithm for Eigenvalue Computation of low rank corrections of Hermitian matrices is adjusted to work with matrix pencils arising from zerofinding problems for polynomials expressed in Chebyshev-like bases. The modified QZ algorithm computes the generalized Eigenvalues of certain $$N\times N$$ N × N rank structured matrix pencils using $$O(N^2)$$ O ( N 2 ) flops and $$O(N)$$ O ( N ) memory storage. Numerical experiments and comparisons confirm the effectiveness and the stability of the proposed method.

  • Implicit QR for rank-structured matrix pencils
    BIT Numerical Mathematics, 2014
    Co-Authors: Paola Boito, Yuli Eidelman, Luca Gemignani
    Abstract:

    A fast implicit QR algorithm for Eigenvalue Computation of low rank corrections of Hermitian matrices is adjusted to work with matrix pencils arising from zerofinding problems for polynomials expressed in Chebyshev-like bases. The modified QZ algorithm computes the generalized Eigenvalues of certain N×N rank structured matrix pencils using O(N2) flops and O(N) memory storage. Numerical experiments and comparisons confirm the effectiveness and the stability of the proposed method.

  • Efficient Eigenvalue Computation for quasiseparable Hermitian matrices under low rank perturbations
    Numerical Algorithms, 2008
    Co-Authors: Yuli Eidelman, Luca Gemignani, Israel Gohberg
    Abstract:

    In this paper we address the problem of efficiently computing all the Eigenvalues of a large N × N Hermitian matrix modified by a possibly non Hermitian perturbation of low rank. Previously proposed fast adaptations of the QR algorithm are considerably simplified by performing a preliminary transformation of the matrix by similarity into an upper Hessenberg form. The transformed matrix can be specified by a small set of parameters which are easily updated during the QR process. The resulting structured QR iteration can be carried out in linear time using linear memory storage. Moreover, it is proved to be backward stable. Numerical experiments show that the novel algorithm outperforms available implementations of the Hessenberg QR algorithm already for small values of N .

  • ISSAC - Structured matrix methods for polynomial root-finding
    Proceedings of the 2007 international symposium on Symbolic and algebraic computation - ISSAC '07, 2007
    Co-Authors: Luca Gemignani
    Abstract:

    In this paper we discuss the use of structured matrix methods for the numerical approximation of the zeros of a univariate polynomial. In particular, it is shown that root-finding algorithms based on floating-point Eigenvalue Computation can benefit from the structure of the matrix problem to reduce their complexity and memory requirements by an order of magnitude.

Pierrick Gaudry - One of the best experts on this subject based on the ideXlab platform.