Rayleigh quotient algorithms for nonsymmetric matrix pencils

A classical Rayleigh-quotient iterative algorithm (known as “broken iteration”) for finding eigenvalues and eigenvectors is applied to semisimple regular matrix pencils A − λB. It is proved that cubic convergence is attained for eigenvalues and superlinear convergence of order three for eigenvectors. Also, each eigenvalue has a local basin of attraction. A closely related Newton algorithm is examined. Numerical examples are included. Dedicated to the memory of Gene H. Golub.     

On perturbations of matrix pencils with real spectra. II

A well-known result on spectral variation of a Hermitian matrix due to Mirsky is the following: Let and be two Hermitian matrices, and let and be their eigenvalues arranged in ascending order. Then for any unitarily invariant norm . In this paper, we generalize this to the perturbation theory for diagonalizable matrix pencils with real spectra. The much studied case of definite pencils is included in this.

     

On perturbations of matrix pencils with real spectra, a revisit

This paper continues earlier studies by Bhatia and Li on eigenvalue perturbation theory for diagonalizable matrix pencils having real spectra. A unifying framework for creating crucial perturbation equations is developed. With the help of a recent result on generalized commutators involving unitary matrices, new and much sharper bounds are obtained.

     

Circular trichotomy of the spectrum of regular matrix pencils

We introduce a method for approximating the right and left deflating subspaces of a regular matrix pencil corresponding to the eigenvalues inside, on and outside the unit circle. The method extends the iteration used in the context of spectral dichotomy, where the assumption on the absence of eigenvalues on the unit circle is removed. It constructs two matrix sequences whose null spaces and the null space of their sum lead to approximations of the deflating subspaces corresponding to the eigenvalues of modulus less than or equal to 1, equal to 1 and larger than or equal to 1. An orthogonalization process is then used to extract the desired delating subspaces. The resulting algorithm is an inverse free, easy to implement, and sufficiently fast. The derived convergence estimates reveal the key parameters, which determine the rate of convergence. The method is tested on several numerical examples.     

Generalized principal lattices and cubic pencils

Given a cubic pencil, an addition of lines can be defined in order to construct generalized principal lattices. In this paper we show the converse: the lines defining a generalized principal lattice belong to the same cubic pencil, which is unique for degrees ≥ 4. Partially supported by the Spanish Research Grant MTM2006-03388, by Gobierno de Aragón and Fondo Social Europeo.     

On computing eigenvalues of second-order linear pencils

^** Email: mhannaby{at}yahoo.com^*** Email: zahraa26{at}yahoo.com In this paper, we use sinc techniques to compute the eigenvaluesof a second-order operator pencil of the form Q – P approximately.Here Q and P are self-adjoint differential operators of thesecond and first order, respectively. Also the eigenparameterappears in the boundary conditions linearly.     

A non-Euclidean gradient descent method with sketching for unconstrained matrix minimization

We propose a method that incorporates a non-Euclidean gradient descent step with a generic matrix sketching procedure, for solving unconstrained, nonconvex, matrix optimization problems, in which the decision variable cannot be saved in memory due to its size, and the objective function is the composition of a vector function on a linear operator. The method updates the sketch directly without updating or storing the decision variable. Subsequence convergence, global convergence under the Kurdyka–Lojasiewicz property, and rate of convergence, are established.     

Implicit QR for rank-structured matrix pencils

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 $N\times N$ 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.     

Definitizable hermitian matrix pencils

Summary An hermitian matrix pencilA – B withA nonsingular is called strongly definitizable ifAp(A ^–1 B) is positive definite for some polynomialp. We present three characterizations of strongly definitizable pencils, which generalize the classical results for definite pencils. They are, in particular, stably simultaneously diagonable. We also discuss this form of stability with respect to an open subset of the real line. Implications for some quadratic eigenvalue problems are included.Research supported in part by the National Sciences and Engineering Research Council of Canada.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth.     

A generalized global Arnoldi method for ill-posed matrix equations

This paper discusses the solution of large-scale linear discrete ill-posed problems with a noise-contaminated right-hand side. Tikhonov regularization is used to reduce the influence of the noise on the computed approximate solution. We consider problems in which the coefficient matrix is the sum of Kronecker products of matrices and present a generalized global Arnoldi method, that respects the structure of the equation, for the solution of the regularized problem. Theoretical properties of the method are shown and applications to image deblurring are described.     

Notes on linear factor polynomial deflation in polynomial bases

Scalar polynomials as approximations to more general scalar functions lead to the study of scalar polynomials represented in a variety of classical systems of polynomials, including orthogonal systems and Lagrange polynomials, for example. This article, motivated in part by analogy with the existing methods for linear factor polynomial deflation in the monomial basis, finds forward and backward deflation formulae for several such representations. It also finds the sensitivity factor of the deflation process for each representation.     

Efficient orthogonal matrix polynomial based method for computing matrix exponential

The matrix exponential plays a fundamental role in the solution of differential systems which appear in different science fields. This paper presents an efficient method for computing matrix exponentials based on Hermite matrix polynomial expansions. Hermite series truncation together with scaling and squaring and the application of floating point arithmetic bounds to the intermediate results provide excellent accuracy results compared with the best acknowledged computational methods. A backward-error analysis of the approximation in exact arithmetic is given. This analysis is used to provide a theoretical estimate for the optimal scaling of matrices. Two algorithms based on this method have been implemented as MATLAB functions. They have been compared with MATLAB functions funm and expm obtaining greater accuracy in the majority of tests. A careful cost comparison analysis with expm is provided showing that the proposed algorithms have lower maximum cost for some matrix norm intervals. Numerical tests show that the application of floating point arithmetic bounds to the intermediate results may reduce considerably computational costs, reaching in numerical tests relative higher average costs than expm of only 4.43% for the final Hermite selected order, and obtaining better accuracy results in the 77.36% of the test matrices. The MATLAB implementation of the best Hermite matrix polynomial based algorithm has been made available online.     

Partial inverse nodal problems for differential pencils on a star-shaped graph

In this paper, the authors study partial inverse nodal problems for differential pencils on a star-shaped graph. We firstly show that the potential on each edge can be uniquely determined by twin-dense nodal subsets on some interior intervals under certain conditions. Without any nodal information on some potential on the fixed edge, we may add some spectral information to guarantee these uniqueness theorems. We still consider the case of arbitrary intervals having the internal vertex. In particular, we pose and solve a new partial inverse nodal problem for differential pencils on the star-shaped graph from the Weyl m-function and the theory concerning densities of zeros of entire functions.     

The clique numbers of regular graphs of matrix algebras are finite

Let F be a field, char(F)≠2, and SGL_n(F), where n is a positive integer. In this paper we show that if for every distinct elements x,yS, x+y is singular, then S is finite. We conjecture that this result is true if one replaces field with a division ring.     

Canonical forms for mixed symmetric-skewsymmetric quaternion matrix pencils

Canonical forms are described for pairs of quaternionic matrices, or equivalently matrix pencils, where one matrix is symmetric and the other matrix is skewsymmetric, under strict equivalence and symmetry respecting congruence. The symmetry is understood in the sense of a fixed involutory antiautomorphism of the skew field of the real quaternions; the involutory antiautomorphism is assumed to be nonstandard, i.e., other than the quaternionic conjugation. Some applications are developed, such as canonical forms for quaternionic matrices under symmetry respecting congruence, and canonical forms for matrices that are skewsymmetric with respect to a nondegenerate symmetric or skewsymmetric quaternion valued inner product.     

Spectral problem for polynomial matrix pencils. 2

For an arbitrary polynomial pencil of matrices A_i of dimensions m×n one presents an algorithm for the computation of the eigenvalues of the regular kernel of the pencil. The algorithm allows to construct a regular pencil having the same eigenvalues as the regular kernel of the initial pencil or (in the case of a dead end termination) allows to pass from the initial pencil to a pencil of smaller dimensions whose regular kernel has the same eigenvalues as the initial pencil. The problem is solved by reducing the obtained pencil to a linear one. For solving the problem in the case of a linear pencil one considers algorithms for pencils of full column rank as well as for completely singular pencils.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 111, pp. 109–116, 1981.     

Pseudospectra for matrix pencils and stability of equilibria

The concept of ε-pseudospectra for matrices, introduced by Trefethen and his coworkers, has been studied extensively since 1990. In this paper, ε-pseudospectra for matrix pencils, which are relevant in connection with generalized eigenvalue problems, are considered. Some properties as well as the practical computation of ε-pseudospectra for matrix pencils will be discussed. As an application, we demonstrate how this concept can be used for investigating the asymptotic stability of stationary solutions to time-dependent ordinary or partial differential equations; two cases, based on Burgers' equation, will be shown. This research has been supported by the Netherlands Organization for Scientific Research (N.W.O.)     

Methods for solving spectral problems for multiparameter matrix pencils

Methods for solving the partial eigenproblem for multiparameter regular pencils of real matrices, which allow one to improve given approximations of an eigenvector and the associated point of the spectrum (both finite and infinite) are suggested. Ways of extending the methods to complex matrices, polynomial matrices, and coupled multiparameter problems are indicated. Bibliography: 10 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 296, 2003, pp. 139–168.