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David S Watkins - One of the best experts on this subject based on the ideXlab platform.

  • the Matrix Eigenvalue Problem gr and krylov subspace methods
    2007
    Co-Authors: David S Watkins
    Abstract:

    This book presents the first in-depth, complete, and unified theoretical discussion of the two most important classes of algorithms for solving Matrix Eigenvalue Problems: QR-like algorithms for dense Problems and Krylov subspace methods for sparse Problems. The author discusses the theory of the generic GR algorithm, including special cases (for example, QR, SR, HR), and the development of Krylov subspace methods. Also addressed are a generic Krylov process and the Arnoldi and various Lanczos algorithms, which are obtained as special cases. The chapter on product Eigenvalue Problems provides further unification, showing that the generalized Eigenvalue Problem, the singular value decomposition Problem, and other product Eigenvalue Problems can all be viewed as standard Eigenvalue Problems. The author provides theoretical and computational exercises in which the student is guided, step by step, to the results. Some of the exercises refer to a collection of MATLAB programs compiled by the author that are available on a Web site that supplements the book. Audience: Readers of this book are expected to be familiar with the basic ideas of linear algebra and to have had some experience with Matrix computations. This book is intended for graduate students in numerical linear algebra. It will also be useful as a reference for researchers in the area and for users of Eigenvalue codes who seek a better understanding of the methods they are using. Contents: Preface; Chapter 1: Preliminary Material; Chapter 2: Basic Theory of Eigensystems; Chapter 3: Elimination; Chapter 4: Iteration; Chapter 5: Convergence; Chapter 6: The Generalized Eigenvalue Problem; Chapter 7: Inside the Bulge; Chapter 8: Product Eigenvalue Problems; Chapter 9: Krylov Subspace Methods; Bibliography; Index.

  • structured Eigenvalue methods for the computation of corner singularities in 3d anisotropic elastic structures
    Computer Methods in Applied Mechanics and Engineering, 2002
    Co-Authors: Thomas Apel, Volker Mehrmann, David S Watkins
    Abstract:

    This paper is concerned with the computation of three-dimensional vertex singularities of anisotropic elastic fields. The singularities are described by eigenpairs of a corresponding operator pencil on a subdomain of the sphere. The solution approach is to introduce a modified quadratic variational boundary Eigenvalue Problem which consists of two self-adjoint, positive definite sesquilinear forms and a skew-Hermitian form. This Eigenvalue Problem is discretized by the finite element method. The resulting quadratic Matrix Eigenvalue Problem is then solved with the skew Hamiltonian implicitly restarted Arnoldi method which is specifically adapted to the structure of this Problem. Some numerical examples are given that show the performance of this approach.

  • convergence of algorithms of decomposition type for the Eigenvalue Problem
    Linear Algebra and its Applications, 1991
    Co-Authors: David S Watkins, Ludwig Elsner
    Abstract:

    We develop the theory of convergence of a generic GR algorithm for the Matrix Eigenvalue Problem that includes the QR,LR,SR, and other algorithms as special cases. Our formulation allows for shifts of origin and multiple GR steps. The convergence theory is based on the idea that the GR algorithm performs nested subspace iteration with a change of coordinate system at each step. Thus the convergence of the GR algorithm depends on the convergence of certain sequences of subspaces. It also depends on the quality of the coordinate transformation matrices, as measured by their condition numbers. We show that with a certain obvious shifting strategy the GR algorithm typically has a quadratic asymptotic convergence rate. For matrices possessing certain special types of structure, cubic convergence can be achieved.

Franz F. Schöberl - One of the best experts on this subject based on the ideXlab platform.

  • Spinless Salpeter equation as a simple Matrix Eigenvalue Problem.
    Physical Review D, 1992
    Co-Authors: Wolfgang Lucha, Heinz Rupprecht, Franz F. Schöberl
    Abstract:

    We propose a new method for solving the spinless Salpeter equation. Choosing a special set of orthonormal basis functions we are able to calculate analytically the integral over the corresponding kernel as well as the Matrix elements of a class of potentials. In this way the Problem can be reduced to the solution of a simple Matrix Eigenvalue Problem with explicitly known matrices, which can be solved very fast numerically. The method is demonstrated in detail for {ital S} waves using a typical interquark potential as an example.

Wolfgang Lucha - One of the best experts on this subject based on the ideXlab platform.

  • Spinless Salpeter equation as a simple Matrix Eigenvalue Problem.
    Physical Review D, 1992
    Co-Authors: Wolfgang Lucha, Heinz Rupprecht, Franz F. Schöberl
    Abstract:

    We propose a new method for solving the spinless Salpeter equation. Choosing a special set of orthonormal basis functions we are able to calculate analytically the integral over the corresponding kernel as well as the Matrix elements of a class of potentials. In this way the Problem can be reduced to the solution of a simple Matrix Eigenvalue Problem with explicitly known matrices, which can be solved very fast numerically. The method is demonstrated in detail for {ital S} waves using a typical interquark potential as an example.

Jnis Priede - One of the best experts on this subject based on the ideXlab platform.

  • capacitance Matrix technique for avoiding spurious eigenmodes in the solution of hydrodynamic stability Problems by chebyshev collocation method
    Journal of Computational Physics, 2013
    Co-Authors: Jonathan Hagan, Jnis Priede
    Abstract:

    We present a simple technique for avoiding physically spurious eigenmodes that often occur in the solution of hydrodynamic stability Problems by the Chebyshev collocation method. The method is demonstrated on the solution of the Orr-Sommerfeld equation for plane Poiseuille flow. Following the standard approach, the original fourth-order differential equation is factorised into two second-order equations using a vorticity-type auxiliary variable with unknown boundary values which are then eliminated by a capacitance Matrix approach. However the elimination is constrained by the conservation of the structure of Matrix Eigenvalue Problem, it can be done in two basically different ways. A straightforward application of the method results in a couple of physically spurious Eigenvalues which are either huge or close to zero depending on the way the vorticity boundary conditions are eliminated. The zero Eigenvalues can be shifted to any prescribed value and thus removed by a slight modification of the second approach.

Ke Wu - One of the best experts on this subject based on the ideXlab platform.

  • finite difference frequency domain algorithm for modeling guided wave properties of substrate integrated waveguide
    IEEE Transactions on Microwave Theory and Techniques, 2003
    Co-Authors: Feng Xu, Yulin Zhang, Wei Hong, Ke Wu
    Abstract:

    In multilayer microwave integrated circuits such as low-temperature co-fired ceramics or multilayered printed circuit boards, waveguide-like structures can be fabricated by using periodic metallic via-holes referred to as substrate integrated waveguide (SIW). Such SIW structures can largely preserve the advantages of conventional rectangular waveguides such as high-Q factor and high power capacity. However, they are subject to leakage due to periodic gaps, which potentially results in wave attenuation. Therefore, such a guided-wave modeling Problem becomes a very complicated complex Eigenvalue Problem. Since the SIW are bilaterally unbounded, absorbing boundary conditions should be deployed in numerical algorithms. This often leads to a difficult complex root-extracting Problem of a transcend equation. In this paper, we present a novel finite-difference frequency-domain algorithm with a perfectly matched layer and Floquet's theorem for the analysis of SIW guided-wave Problems. In this scheme, the Problem is converted into a generalized Matrix Eigenvalue Problem and finally transformed to a standard Matrix Eigenvalue Problem that can be solved with efficient subroutines available. This approach has been validated by experiment.