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SIAM J. on Matrix Analysis and Applications

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2004

Volume 26, Issue 1, pp. 1-294


Backward Perturbation Analysis of the Periodic Discrete-Time Algebraic Riccati Equation

Ji-Guang Sun

SIAM. J. Matrix Anal. & Appl. 26, pp. 1-19 (19 pages)

Online Publication Date: July 31, 2006

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Normwise backward errors and residual bounds for an approximate Hermitian positive semidefinite solution set to the periodic discrete-time algebraic Riccati equation are obtained. The results are illustrated by using simple numerical examples.

A Preconditioner for Generalized Saddle Point Problems

Michele Benzi and Gene H. Golub

SIAM. J. Matrix Anal. & Appl. 26, pp. 20-41 (22 pages) | Cited 45 times

Online Publication Date: July 31, 2006

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In this paper we consider the solution of linear systems of saddle point type by preconditioned Krylov subspace methods. A preconditioning strategy based on the symmetric\slash skew-symmetric splitting of the coefficient matrix is proposed, and some useful properties of the preconditioned matrix are established. The potential of this approach is illustrated by numerical experiments with matrices from various application areas.

Jacobi's Algorithm on Compact Lie Algebras

M. Kleinsteuber, U. Helmke, and K. Huper

SIAM. J. Matrix Anal. & Appl. 26, pp. 42-69 (28 pages) | Cited 2 times

Online Publication Date: July 31, 2006

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A generalization of the cyclic Jacobi algorithm is proposed that works in an arbitrary compact Lie algebra. This allows, in particular, a unified treatment of Jacobi algorithms on different classes of matrices, e.g., skew-symmetric or skew-Hermitian Hamiltonian matrices. Wildberger has established global, linear convergence of the algorithm for the classical Jacobi method on compact Lie algebras. Here we prove local quadratic convergence for general cyclic Jacobi schemes.

Cubically Convergent Iterations for Invariant Subspace Computation

P. A. Absil, R. Sepulchre, P. Van Dooren, and R. Mahony

SIAM. J. Matrix Anal. & Appl. 26, pp. 70-96 (27 pages) | Cited 5 times

Online Publication Date: July 31, 2006

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We propose a Newton-like iteration that evolves on the set of fixed dimensional subspaces of $\rr^n$ and converges locally cubically to the invariant subspaces of a symmetric matrix. This iteration is compared in terms of numerical cost and global behavior with three other methods that display the same property of cubic convergence. Moreover, we consider heuristics that greatly improve the global behavior of the iterations.

Splitting a Matrix of Laurent Polynomials with Symmetry and itsApplication to Symmetric Framelet Filter Banks

Bin Han and Qun Mo

SIAM. J. Matrix Anal. & Appl. 26, pp. 97-124 (28 pages) | Cited 10 times

Online Publication Date: July 31, 2006

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Let M be a $2\times 2$ matrix of Laurent polynomials with real coefficients and symmetry. In this paper, we obtain a necessary and sufficient condition for the existence of four Laurent polynomials (or finite-impulse-response filters) u1, u2, v1, v2 with real coefficients and symmetry such that
$$ \left[ \begin{matrix} u_1(z) &v_1(z)\\ u_2(z) &v_2(z) \end{matrix}\right] \left[ \begin{matrix} u_1(1/z) &u_2(1/z)\\ v_1(1/z) &v_2(1/z)\end{matrix}\right]=M(z) \qquad \forall\; z\in \CC \bs \{0 \} $$
and [Su1](z)[Sv2](z)=[Su2](z)[Sv1](z), where [Sp](z)=p(z)/p(1/z) for a nonzero Laurent polynomial p. Our criterion can be easily checked and a step-by-step algorithm will be given to construct the symmetric filters u1, u2, v1, v2. As an application of this result to symmetric framelet filter banks, we present a necessary and sufficient condition for the construction of a symmetric multiresolution analysis tight wavelet frame with two compactly supported generators derived from a given symmetric refinable function. Once such a necessary and sufficient condition is satisfied, an algorithm will be used to construct a symmetric framelet filter bank with two high-pass filters which is of interest in applications such as signal denoising and image processing. As an illustration of our results and algorithms in this paper, we give several examples of symmetric framelet filter banks with two high-pass filters which have good vanishing moments and are derived from various symmetric low-pass filters including some B-spline filters.

Inexact Krylov Subspace Methods for Linear Systems

Jasper van den Eshof and Gerard L. G. Sleijpen

SIAM. J. Matrix Anal. & Appl. 26, pp. 125-153 (29 pages) | Cited 9 times

Online Publication Date: July 31, 2006

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There is a class of linear problems for which the computation of the matrix-vector product is very expensive since a time consuming method is necessary to approximate it with some prescribed relative precision. In this paper we investigate the impact of approximately computed matrix-vector products on the convergence and attainable accuracy of several Krylov subspace solvers. We will argue that the sensitivity towards perturbations is mainly determined by the underlying way the Krylov subspace is constructed and does not depend on the optimality properties of the particular method. The obtained insight is used to tune the precision of the matrix-vector product in every iteration step in such a way that an overall efficient process is obtained. Our analysis confirms the empirically found relaxation strategy of Bouras and Frayssé for the GMRES method proposed in [A Relaxation Strategy for Inexact Matrix-Vector Products for Krylov Methods, Technical Report TR/PA/00/15, CERFACS, France, 2000]. Furthermore, we give an improved version of a strategy for the conjugate gradient method of Bouras, Frayssé, and Giraud used in [A Relaxation Strategy for Inner-Outer Linear Solvers in Domain Decomposition Methods, Technical Report TR/PA/00/17, CERFACS, France, 2000].

Stability Estimates on the Jacobi and Unitary Hessenberg Inverse Eigenvalue Problems

Leonid Knizhnerman

SIAM. J. Matrix Anal. & Appl. 26, pp. 154-169 (16 pages)

Online Publication Date: July 31, 2006

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Perturbation bounds for the Jacobi inverse eigenvalue problem (JIEP), which are more realistic than the earlier ones, are proved and illustrated by numerical experiments. The technique of orthonormal polynomials and integral representation of Hankel determinants is used. The same technique is then applied to the unitary Hessenberg inverse eigenvalue problem (UHIEP).

Convexity and Elasticity of the Growth Rate in Size-Classified Population Models

S. J. Kirkland, M. Neumann, and J. Xu

SIAM. J. Matrix Anal. & Appl. 26, pp. 170-185 (16 pages)

Online Publication Date: July 31, 2006

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This paper investigates both the convexity and elasticity of the growth rate of size-classsified population models. For an irreducible population projection matrix, we discuss the convexity properties of its Perron eigenvalue under perturbation of the vital rates, extending work of Kirkland and Neumann on Leslie matrices. We also provide nonnegative attainable lower bounds on the derivatives of the elasticity of the Perron eigenvalue under perturbation of the vital rates, sharpening, in the context of population projection matrices, the main result of Kirkland, Neumann, Ormes, and Xu.

V-cycle Optimal Convergence for Certain (Multilevel) Structured Linear Systems

Antonio Aricò, M. Donatelli, and S. Serra Capizzano

SIAM. J. Matrix Anal. & Appl. 26, pp. 186-214 (29 pages) | Cited 13 times

Online Publication Date: July 31, 2006

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In this paper we are interested in the solution by multigrid strategies of multilevel linear systems whose coefficient matrices belong to the circulant, Hartley, or $\tau$ algebras or to the Toeplitz class and are generated by (the Fourier expansion of) a nonnegative multivariate polynomial f. It is well known that these matrices are banded and have eigenvalues equally distributed as f, so they are ill-conditioned whenever f takes the zero value; they can even be singular and need a low-rank correction.
We prove the V-cycle multigrid iteration to have a convergence rate independent of the dimension even in presence of ill-conditioning. If the (multilevel) coefficient matrix has partial dimension nr at level r, r=1,...,d, then the size of the algebraic system is $N(n)=\prod_{r=1}^d n_r$, O(N(n)) operations are required by our technique, and therefore the corresponding method is optimal.
Some numerical experiments concerning linear systems arising in applications, such as elliptic PDEs with mixed boundary conditions and image restoration problems, are considered and discussed.

Tridiagonal-Diagonal Reduction of Symmetric Indefinite Pairs

Françoise Tisseur

SIAM. J. Matrix Anal. & Appl. 26, pp. 215-232 (18 pages) | Cited 3 times

Online Publication Date: July 31, 2006

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We consider the reduction of a symmetric indefinite matrix pair (A,B), with B nonsingular, to tridiagonal-diagonal form by congruence transformations. This is an important reduction in solving polynomial eigenvalue problems with symmetric coefficient matrices and in frequency response computations. The pair is first reduced to symmetric-diagonal form. We describe three methods for reducing the symmetric-diagonal pair to tridiagonal-diagonal form. Two of them employ more stable versions of Brebner and Grad's pseudosymmetric Givens and pseudosymmetric Householder reductions, while the third is new and based on a combination of Householder reflectors and hyperbolic rotations. We prove an optimality condition for the transformations used in the third reduction. We present numerical experiments that compare the different approaches and show improvements over Brebner and Grad's reductions.

Convergence of GMRES for Tridiagonal Toeplitz Matrices

J. Liesen and Z. Strakos

SIAM. J. Matrix Anal. & Appl. 26, pp. 233-251 (19 pages) | Cited 6 times

Online Publication Date: July 31, 2006

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We analyze the residuals of GMRES [Y. Saad and M. H. Schultz, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 856--859], when the method is applied totridiagonal Toeplitz matrices. We first derive formulas for the residuals as well as their norms when GMRES is applied to scaled Jordan blocks. This problem has been studied previously by Ipsen [BIT, 40 (2000), pp. 524--535] and Eiermann and Ernst [Private communication, 2002], but we formulate and prove our results in a different way. We then extend the (lower) bidiagonal Jordan blocks to tridiagonal Toeplitz matrices and study extensions of our bidiagonal analysis to the tridiagonal case. Intuitively, when a scaled Jordan block is extended to a tridiagonal Toeplitz matrix by a superdiagonal of small modulus (compared to the modulus of the subdiagonal), the GMRES residual norms for both matrices and the same initial residual should be close to each other. We confirm and quantify this intuitive statement. We also demonstrate principal difficulties of any GMRES convergence analysis which is based on eigenvector expansion of the initial residual when the eigenvector matrix is ill-conditioned. Such analyses are complicated by a cancellation of possibly huge components due to close eigenvectors, which can prevent achieving well-justified conclusions.

Normwise Scaling of Second Order Polynomial Matrices

Hung-Yuan Fan, Wen-Wei Lin, and Paul Van Dooren

SIAM. J. Matrix Anal. & Appl. 26, pp. 252-256 (5 pages) | Cited 12 times

Online Publication Date: July 31, 2006

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We propose a minimax scaling procedure for second order polynomial matrices that aims to minimize the backward errors incurred in solving a particular linearized generalized eigenvalue problem. We give numerical examples to illustrate that it can significantly improve the backward errors of the computed eigenvalue-eigenvector pairs.

Minimal Spectrally Arbitrary Sign Patterns

T. Britz, J. J. McDonald, D. D. Olesky, and P. van den Driessche

SIAM. J. Matrix Anal. & Appl. 26, pp. 257-271 (15 pages) | Cited 10 times

Online Publication Date: July 31, 2006

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An $n\times n$ sign pattern $\mathcal{A}$ is spectrally arbitrary if given any self-conjugate spectrum there exists a matrix realization of $\mathcal{A}$ with that spectrum. If replacing any nonzero entry of $\mathcal{A}$ by zero destroys this property, then $\mathcal{A}$ is a minimal spectrally arbitrary sign pattern. Several families of sign patterns are presented that, for all $n\geq 3$, each contain an $n\times n$ minimal spectrally arbitrary sign pattern. These are the first families proven to have this property, and they improve previously known results. Furthermore, all $3\times 3$ minimal spectrally arbitrary sign patterns are determined, it is proved that any irreducible $n\times n$ spectrally arbitrary sign pattern must have at least $2n-1$ nonzero entries, and it is conjectured that the minimum number of nonzero entries is $2n$.

A Dual Approach to Semidefinite Least-Squares Problems

Jérôme Malick

SIAM. J. Matrix Anal. & Appl. 26, pp. 272-284 (13 pages) | Cited 26 times

Online Publication Date: July 31, 2006

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In this paper, we study the projection onto the intersection of an affine subspace and a convex set and provide a particular treatment for the cone of positive semidefinite matrices. Among applications of this problem is the calibration of covariance matrices. We propose a Lagrangian dualization of this least-squares problem, which leads us to a convex differentiable dual problem. We propose to solve the latter problem with a quasi-Newton algorithm. We assess this approach with numerical experiments which show that fairly large problems can be solved efficiently.

On Some Inverse Eigenvalue Problems with Toeplitz-Related Structure

Fasma Diele, Teresa Laudadio, and Nicola Mastronardi

SIAM. J. Matrix Anal. & Appl. 26, pp. 285-294 (10 pages) | Cited 2 times

Online Publication Date: July 31, 2006

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Some inverse eigenvalue problems for matrices with Toeplitz-related structure are considered in this paper. In particular, the solutions of the inverse eigenvalue problems for Toeplitz-plus-Hankel matrices and for Toeplitz matrices having all double eigenvalues are characterized, respectively, in close form. Being centrosymmetric itself, the Toeplitz-plus-Hankel solution can be used as an initial value in a continuation method to solve the more difficult inverse eigenvalue problem for symmetric Toeplitz matrices. Numerical testing results show a clear advantage of such an application.
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