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SIAM J. on Matrix Analysis and Applications

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1998

Volume 19, Issue 4, pp. 847-1110


On Hyperbolic Triangularization: Stability and Pivoting

Michael Stewart and G. W. Stewart

SIAM. J. Matrix Anal. & Appl. 19, pp. 847-860 (14 pages) | Cited 9 times

Online Publication Date: July 31, 2006

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This paper treats the problem of triangularizing a matrix by hyperbolic Householder transformations. The stability of this method, which finds application in block updating and fast algorithms for Toeplitz-like matrices, has been analyzed only in special cases. Here we give a general analysis which shows that two distinct implementations of the individual transformations are relationally stable. The analysis also shows that pivoting is required for the entire triangularization algorithm to be stable.

Approximate Semidefinite Matrices in a Linear Variety

Charles R. Johnson and Pablo Tarazaga

SIAM. J. Matrix Anal. & Appl. 19, pp. 861-871 (11 pages)

Online Publication Date: July 31, 2006

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We both characterize and give a convergent algorithm for finding a matrix in a linear variety of matrices that is nearest (in the Frobenius norm) to the positive semidefinite (PSD) matrices. Our motivation is from matrix completions, and in that setting our observations take an especially useful form that we use to bound, and sometimes give closed-form formulae for, the distance from the set of completions to the PSD matrices in terms only of specified data.

Kronecker Stratification of the Space of Quadruples of Matrices

Ma I. García-Planas

SIAM. J. Matrix Anal. & Appl. 19, pp. 872-885 (14 pages) | Cited 1 time

Online Publication Date: July 31, 2006

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In this paper, we study the partition of the space of quadruples of matrices according to the set of discrete structural invariants, proving that it is a stratification and that the structural stability under this equivalence relation is a generic property in the space of quadruples of matrices.
We give an application to obtain bifurcation diagrams for some few-parameters families of quadruples of matrices.

Eigenvalue Locations of Generalized Companion Predictor Matrices

Licio H. Bezerra and Fermin S. V. Bazán

SIAM. J. Matrix Anal. & Appl. 19, pp. 886-897 (12 pages) | Cited 2 times

Online Publication Date: July 31, 2006

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Generalized predictor companion matrices arise in the linear prediction approach for the fit of a weighted sum of $n$ exponentials to a given set of data points. They are special solutions of matrix equations of the type ${\sf H}(l+p)\, {\sf S}={\sf H}(l)$, where for each $l\ge 0$ ${\sf H}(l)$ is an $M\times N$ Hankel matrix obtained from this data ($M\ge N>n$). We discuss in this paper results about the eigenvalue locations of this class of solutions by means of linear algebra techniques. An application of these results in the case that all the exponents have either negative or positive real parts is that the n exponentials can correspond to eigenvalues which are outside the unit circle depending on the choice of generalized predictor companion matrices. The other (N-n) eigenvalues of these matrices always lie inside the unit circle and approach zero when p increases. This separation can facilitate their numerical calculation.

Some Recent Results on The Linear Complementarity Problem

G. S. R. Murthy, T. Parthasarathy, and B. Sriparna

SIAM. J. Matrix Anal. & Appl. 19, pp. 898-905 (8 pages) | Cited 2 times

Online Publication Date: July 31, 2006

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In this article we present some recent results on the linear complementarity problem. It is shown that (i) within the class of column adequate matrices, a matrix is in $\Qnot$ if and only if it is completely $\Qnot$ (ii) for the class of $\Cnotf$-matrices introduced by Murthy and Parthasarathy [SIAM J. Matrix Anal. Appl., 16 (1995), pp. 1268--1286], we provide a sufficient condition under which a matrix is in $\Pnot$ and as a corollary of this result, we give an alternative proof of the result that $\Cnotf \cap \Qnot \sset \Pnot$ (iii) within the class of INS-matrices introduced by Stone [Department of Operations Research, Stanford University, Stanford, CA, 1981], a nondegenerate matrix must necessarily have the block property introduced by Murthy, Parthasarathy, and Sriparna [G. S. R. Murthy, T. Parthasarathy, and B. Sriparna, Linear Algebra Appl., 252 (1997), pp. 323--337]. Furthermore, we conjecture that if a matrix has block property, then it must be Lipschitzian. This problem is an important one from two angles: if the conjecture is true, it provides a finite test to check whether a given matrix is Lipschitzian or nondegenerate INS; and it settles an open problem posed by Stone. It is shown that the conjecture is true in the cases of 2 x 2-matrices, nonnegative and nonpositive matrices of general order.

Statistical Condition Estimation for Linear Least Squares

C. S. Kenney, A. J. Laub, and M. S. Reese

SIAM. J. Matrix Anal. & Appl. 19, pp. 906-923 (18 pages) | Cited 9 times

Online Publication Date: July 31, 2006

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Statistical condition estimation is applied to the linear least squares problem. The method obtains componentwise condition estimates via the Fréchet derivative. A rigorous statistical theory exists that determines the probability of accuracy in the estimates. The method is as computationally efficient as normwise condition estimation methods, and it is easily adapted to respect structural constraints on perturbations of the input data. Several examples illustrate the method.

Global Block-Similarity and Pole Assignment of Class Cp

Josep Ferrer and Ferran Puerta

SIAM. J. Matrix Anal. & Appl. 19, pp. 924-932 (9 pages) | Cited 2 times

Online Publication Date: July 31, 2006

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Starting from the existence of a Cp-basis for any Cp-family of subspaces having constant dimension, we construct a Brunovsky basis of class Cp for a Cp-family of pairs of matrices having constant Brunovsky type. We derive a global pole assignment theorem for such kinds of pairs. In all the cases we assume that the manifold of parameters is contractible.

Stability and Convergence of Principal Component Learning Algorithms

Wei-Yong Yan

SIAM. J. Matrix Anal. & Appl. 19, pp. 933-955 (23 pages) | Cited 2 times

Online Publication Date: July 31, 2006

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This paper is concerned with the differential equation approximating the subspace learning algorithm for extracting principal components. Two issues are fully resolved. First, all the stable equilibria are found. Second, the global convergence rate is explicitly obtained. The whole treatment is without the nonsingularity assumption on the covariance matrix.

Relative Perturbation Theory: I. Eigenvalue and Singular Value Variations

Ren-Cang Li

SIAM. J. Matrix Anal. & Appl. 19, pp. 956-982 (27 pages) | Cited 16 times

Online Publication Date: July 31, 2006

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The classical perturbation theory for Hermitian matrix eigenvalue and singular value problems provides bounds on the absolute differences between approximate eigenvalues (singular values) and the true eigenvalues (singular values) of a matrix. These bounds may be bad news for small eigenvalues (singular values), which thereby suffer worse relative uncertainty than large ones. However, there are situations where even small eigenvalues are determined to high relative accuracy by the data much more accurately than the classical perturbation theory would indicate. In this paper, we study how eigenvalues of a Hermitian matrix A change when it is perturbed to $\wtd A=D^*AD$, where D is close to a unitary matrix, and how singular values of a (nonsquare) matrix B change when it is perturbed to $\wtd B=D_1^*BD_2$, where D1 and D2 are nearly unitary. It is proved that under these kinds of perturbations small eigenvalues (singular values) suffer relative changes no worse than large eigenvalues (singular values). Many well-known perturbation theorems, including the Hoffman--Wielandt and Weyl--Lidskii theorems, are extended.

Analyses, Development, and Applications of TLS Algorithms in Frequency Domain System Identification

Rik Pintelon, Patrick Guillaume, Gerd Vandersteen, and Yves Rolain

SIAM. J. Matrix Anal. & Appl. 19, pp. 983-1004 (22 pages) | Cited 8 times

Online Publication Date: July 31, 2006

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This paper gives an overview of frequency domain total least squares (TLS) estimators for rational transfer function models of linear time-invariant multivariable systems. The statistical performance of the different approaches are analyzed through their equivalent cost functions. Both generalized and bootstrapped total least squares (GTLS and BTLS) methods require the exact knowledge of the noise covariance matrix. The paper also studies the asymptotic (the number of data points going to infinity) behavior of the GTLS and BTLS estimators when the exact noise covariance matrix is replaced by the sample noise covariance matrix obtained from a (small) number of independent data sets. Even if only two independent repeated observations are available, it is shown that the estimates are still strongly consistent without any increase in the asymptotic uncertainty.

On Construction of a Family of Smooth Nonseparable Prewavelets via Infinite Products of Triangularizable Matrices

Mohsen Maesumi

SIAM. J. Matrix Anal. & Appl. 19, pp. 1005-1026 (22 pages)

Online Publication Date: July 31, 2006

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Infinite products of matrices arise in many areas, such as the study of subdivision and interpolation schemes, Markov chains, and construction of wavelets of compact support. These products are used here to give sufficient conditions for the continuity and differentiability of a class of rectangular compactly supported nonseparable N-dimensional prewavelets or scaling functions. This paper considers the dilation equation $\phi(X)=\sum_K C_K \phi(2X-K)$, where $K\in \{0, \ldots, m\}^N$, $\phi : {\cal R}^N \to {\cal R}$, and $C_K \in {\cal R}$. First, the one-dimensional case is studied, and sufficient conditions on CK, which guarantee a continuous scaling function $\phi(X)$, are given. These conditions are based on simultaneous triangularizability of two special matrices with entries in terms of CK. Then, these results are generalized to N dimensions and applied to the particular case where CK's are obtained by binomial interpolation of their values at the corners of the N-cube, $\{0,m\}^N$. A set of inequalities, based on sums of CK's on the corners of various faces of the N-cube gives sufficient conditions for the existence of smooth solutions to the dilation equation.

A Numerical Method for the Inverse Stochastic Spectrum Problem

Moody T. Chu and Quanlin Guo

SIAM. J. Matrix Anal. & Appl. 19, pp. 1027-1039 (13 pages) | Cited 5 times

Online Publication Date: July 31, 2006

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The inverse stochastic spectrum problem involves the construction of a stochastic matrix with a prescribed spectrum. The problem could be solved by first constructing a nonnegative matrix with the same prescribed spectrum. A differential equation aimed to bring forth the steepest descent flow in reducing the distance between isospectral matrices and nonnegative matrices, represented in terms of some general coordinates, is described. The flow is further characterized by an analytic singular value decomposition to maintain the numerical stability and to monitor the proximity to singularity. This flow approach can be used to design Markov chains with specified structure. Applications are demonstrated by numerical examples.

On the Singularity of LCM Matrices

Bo-Ying Wang

SIAM. J. Matrix Anal. & Appl. 19, pp. 1040-1044 (5 pages) | Cited 1 time

Online Publication Date: July 31, 2006

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Let S=x1,x2,...,xn be a set of distinct positive integers. The set S is called gcd-closed if it contains the greatest common divisor (xi,xj) of xi and xj for $1\le i,j\le n.$ The matrix [S] is called the least common multiple (LCM) matrix on S if its i,j entry is the least common multiple [xi,xj] of xi and xj. Bourque and Ligh conjectured that the LCM matrix on a gcd-closed set is invertible [Linear Algebra Appl., 174 (1992), pp. 65--74]. The aim of this note is to show that this conjecture holds if $n\le 7$, but it does not hold in general when $n\ge 8$.

A Truncated RQ Iteration for Large Scale Eigenvalue Calculations

D. C. Sorensen and C. Yang

SIAM. J. Matrix Anal. & Appl. 19, pp. 1045-1073 (29 pages) | Cited 5 times

Online Publication Date: July 31, 2006

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We introduce a new Krylov subspace iteration for large scale eigenvalue problems that is able to accelerate the convergence through an inexact (iterative) solution to a shift-invert equation. The method also takes full advantage of exact solutions when they can be obtained with sparse direct method. We call this new iteration the truncated RQ (TRQ) iteration. It is based upon a recursion that develops in the leading k columns of the implicitly shifted RQ iteration for dense matrices. Inverse-iteration-like convergence to a partial Schur decomposition occurs in the leading k columns of the updated basis vectors and Hessenberg matrices. The TRQ iteration is competitive with the rational Krylov method of Ruhe when the shift-invert equations can be solved directly and with the Jacobi--Davidson method of Sleijpen and Van der Vorst when these equations are solved inexactly with a preconditioned iterative method. The TRQ iteration is related to both of these but is derived directly from the RQ iteration and thus inherits the convergence properties of that method. Existing RQ deflation strategies may be employed directly in the TRQ iteration.

Bulge Exchanges in Algorithms of QR Type

David S. Watkins

SIAM. J. Matrix Anal. & Appl. 19, pp. 1074-1096 (23 pages) | Cited 2 times

Online Publication Date: July 31, 2006

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The QR algorithm and its variants are among the most popular methods for calculating eigenvalues of matrices. Typical implementations chase bulges from top to bottom of an upper Hessenberg matrix. It is also possible to chase bulges from bottom to top. There are some situations in which it may be advantageous to chase bulges in both directions at once, in which case one needs a procedure for passing bulges through each other without mixing up the information that the bulges convey. This paper derives a procedure for passing bulges of arbitrary degree through each other. Experiments with a Fortran 90 program show that the procedure works well in practice for bulges of degree two.

A Modified Cholesky Algorithm Based on a Symmetric Indefinite Factorization

Sheung Hun Cheng and Nicholas J. Higham

SIAM. J. Matrix Anal. & Appl. 19, pp. 1097-1110 (14 pages) | Cited 3 times

Online Publication Date: July 31, 2006

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Given a symmetric and not necessarily positive definite matrix A, a modified Cholesky algorithm computes a Cholesky factorization P(A+E)PT = RT R, where P is a permutation matrix and E is a perturbation chosen to make A+E positive definite. The aims include producing a small-normed E and making A+E reasonably well conditioned. Modified Cholesky factorizations are widely used in optimization. We propose a new modified Cholesky algorithm based on a symmetric indefinite factorization computed using a new pivoting strategy of Ashcraft, Grimes, and Lewis. We analyze the effectiveness of the algorithm, both in theory and practice, showing that the algorithm is competitive with the existing algorithms of Gill, Murray, and Wright and Schnabel and Eskow. Attractive features of the new algorithm include easy-to-interpret inequalities that explain the extent to which it satisfies its design goals, and the fact that it can be implemented in terms of existing software.
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