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2000

Volume 21, Issue 2, pp. 357-702


A Note on Relative Perturbation Bounds

Tristan Londré and Noah H. Rhee

SIAM. J. Matrix Anal. & Appl. 21, pp. 357-361 (5 pages) | Cited 1 time

Online Publication Date: July 31, 2006

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In this paper we provide an actual bound for the distance between the original and the perturbed right singular vector subspaces of a general matrix with full column rank. We also provide actual relative componentwise bounds for perturbed eigenvectors of a positive definite matrix.

Bounds on the Extreme Eigenvalues of Real Symmetric Toeplitz Matrices

A. Melman

SIAM. J. Matrix Anal. & Appl. 21, pp. 362-378 (17 pages) | Cited 2 times

Online Publication Date: July 31, 2006

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We derive upper and lower bounds on the smallest and largest eigenvalues, respectively, of real symmetric Toeplitz matrices. The bounds are first obtained for positive-definite matrices and then extended to the general real symmetric case. They are computed as the roots of rational and polynomial approximations to spectral, or secular, equations for the symmetric and antisymmetric parts of the spectrum; this leads to separate bounds on the even and odd eigenvalues. We also present numerical results.

Lidskii's Theorem via Nonsmooth Analysis

A. S. Lewis

SIAM. J. Matrix Anal. & Appl. 21, pp. 379-381 (3 pages) | Cited 2 times

Online Publication Date: July 31, 2006

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Lidskii's theorem on eigenvalue perturbation is proved via a nonsmooth mean value theorem.

Second Logarithmic Derivative of a Complex Matrix in the Chebyshev Norm

L. Kohaupt

SIAM. J. Matrix Anal. & Appl. 21, pp. 382-389 (8 pages) | Cited 4 times

Online Publication Date: July 31, 2006

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The second logarithmic derivative $\mu^{(2)}_{\infty}[A]$ of a complex n x n matrix A in the Chebyshev norm is defined as the second right derivative of $\| \Phi(t) \|_{\infty} \,=\, \| e^{A t} \|_{\infty}$ at $t=0$, where $\| \cdot \|_{\infty}$ denotes the operator norm corresponding to the norm $\| \cdot \|_{\infty}$ in $\CC^n$. The obtained formula is illustrated by a numerical example. The result may be of interest in applications such as stability theory.

Nonsingularity of the Difference of Two Oblique Projectors

Jürgen Gross and Götz Trenkler

SIAM. J. Matrix Anal. & Appl. 21, pp. 390-395 (6 pages) | Cited 13 times

Online Publication Date: July 31, 2006

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For two real projectors P and Q of the same order it is shown that the difference P - Q is nonsingular if and only if the column spaces of P and Q are complementary and the row spaces of P and Q are complementary. Moreover, it is demonstrated that nonsingularity of P - Q implies nonsingularity of P + Q,I - QP, and P + Q - QP.

Transformation of Families of Matrices to Normal Forms and its Application to Stability Theory

Alexei A. Mailybaev

SIAM. J. Matrix Anal. & Appl. 21, pp. 396-417 (22 pages) | Cited 6 times

Online Publication Date: July 31, 2006

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Families of matrices smoothly depending on a vector of parameters are considered. Arnold [Russian Math. Surveys, 26 (1971), pp. 29--43] and Galin [Uspekhi Mat. Nauk, 27 (1972), pp. 241--242] have found and listed normal forms of families of complex and real matrices (miniversal deformations), to which any family of matrices can be transformed in the vicinity of a point in the parameter space by a change of basis, smoothly dependent on a vector of parameters, and by a smooth change of parameters. In this paper a constructive method of determining functions describing a change of basis and a change of parameters, transforming an arbitrary family to the miniversal deformation, is suggested. Derivatives of these functions with respect to parameters are determined from a recurrent procedure using derivatives of the functions of lower orders and derivatives of the family of matrices. Then the functions are found as Taylor series. Examples are given. The suggested method allows using efficiently miniversal deformations for investigation of different properties of matrix families. This is shown in the paper where tangent cones (linear approximations) to the stability domain at the singular boundary points are found.

Convergence of Subdivision Schemes Associated with Nonnegative Masks

Rong-Qing Jia and Ding-Xuan Zhou

SIAM. J. Matrix Anal. & Appl. 21, pp. 418-430 (13 pages) | Cited 7 times

Online Publication Date: July 31, 2006

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This paper is concerned with refinement equations of the type $$ f = \sum_{\alpha\in\bbbz^s} a(\alpha) f({M\kern .12em\cdot}-\alpha), $$ where f is the unknown function defined on the s-dimensional Euclidean space $\bbbr^s$, a is a finitely supported sequence on $\bbbz^s$, and M is an s x s dilation matrix with m := |det M|. The solution of a refinement equation can be obtained by using the subdivision scheme associated with the mask. In this paper we give a characterization for the convergence of the subdivision scheme when the mask is nonnegative. Our method is to relate the problem of convergence to m column-stochastic matrices induced by the mask. In this way, the convergence of the subdivision scheme can be determined in a finite number of steps by checking whether each finite product of those column-stochastic matrices has a positive row. As a consequence of our characterization, we show that the convergence of the subdivision scheme with a nonnegative mask depends only on the location of its positive coefficients. Several examples are provided to demonstrate the power and applicability of our approach.

Any Circulant-Like Preconditioner for Multilevel Matrices Is Not Superlinear

S. Serra Capizzano and E. Tyrtyshnikov

SIAM. J. Matrix Anal. & Appl. 21, pp. 431-439 (9 pages) | Cited 24 times

Online Publication Date: July 31, 2006

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Superlinear preconditioners (those that provide a proper cluster at 1) are very important for the cg-like methods since they make these methods converge superlinearly. As is well known, for Toeplitz matrices generated by a continuous symbol, many circulant and circulant-like (related to different matrix algebras) preconditioners were proved to be superlinear. In contrast, for multilevel Toeplitz matrices there has been no proof of the superlinearity of any multilevel circulants. In this paper we show that such a proof is not possible since any multilevel circulant preconditioner is not superlinear, in the general case of multilevel Toeplitz matrices. Moreover, for matrices not necessarily Toeplitz, we present some general results proving that many popular structured preconditioners cannot be superlinear.

A Bound on the Solution to a Structured Sylvester Equation with an Application to Relative Perturbation Theory

Ren-Cang Li

SIAM. J. Matrix Anal. & Appl. 21, pp. 440-445 (6 pages) | Cited 2 times

Online Publication Date: July 31, 2006

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Assuming only that the spectra of A and B are disjoint as opposed to the more restrictive assumption previously used, we obtain a bound in all unitarily invariant norms on the solution to the structured Sylvester equation AX - XB = A1/2EB1/2. This bound is the first of its kind in all unitarily invariant norms under only the disjointedness assumption. An application of the bound to the relative perturbation theory for scaled Hermitian eigenvalue problems is given.

The Solution to a Structured Matrix Approximation Problem Using GrassmanCoordinates

Craig S. MacInnes

SIAM. J. Matrix Anal. & Appl. 21, pp. 446-453 (8 pages)

Online Publication Date: July 31, 2006

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A method for finding the best approximation of a matrix A by a full rank Hankel matrix is given. The initial problem of best approximation of one matrix by another is transformed to a problem involving best approximation of a given vector by a second vector whose elements are constrained so that its inverse image is a Hankel matrix. The map from a matrix to a vector is the invertible map between a subspace represented as the row space of the matrix A and the Grassman vector representing that subspace. The relation between the principle angles associated with a pair of subspaces and the angle between the Grassman vectors associated with the subspaces is established.

Condensed Forms for Skew-Hamiltonian/Hamiltonian Pencils

Christian Mehl

SIAM. J. Matrix Anal. & Appl. 21, pp. 454-476 (23 pages) | Cited 6 times

Online Publication Date: July 31, 2006

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In this paper we consider real or complex skew-Hamiltonian/Hamiltonian pencils $\lambda S-H$, i.e., pencils where S is a skew-Hamiltonian and H is a Hamiltonian matrix. These pencils occur, for example, in the theory of continuous time, linear quadratic optimal control problems. We reduce these pencils to canonical and Schur-type forms under structure-preserving transformations, i.e., J-congruence-transformations $(\lambda S-H) \mapsto -JP^{\ast}J(\lambda S-H)P$, where P is nonsingular or unitary.

On Optimal Banded Preconditioners for the Five-Point Laplacian

Zhanye Tong and Ahmed Sameh

SIAM. J. Matrix Anal. & Appl. 21, pp. 477-480 (4 pages) | Cited 1 time

Online Publication Date: July 31, 2006

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We prove the following theorem, which includes a conjecture by Greenbaum and Rodrigue [[ BIT, 29 (1989), pp. 610--634] as a special case: Let Ah be the five-point Laplace matrix on a unit square for grid size h, and let Mh be any symmetric positive definite preconditioner for Ah with half bandwidth k=O(1). Then the condition number $\kappa(M_h^{-{1\over 2}}A_hM_h^{-{1\over 2}})$ satisfies $\kappa(M_h^{-{1\over 2}}A_hM_h^{-{1\over 2}}) = O(h^{-2}).

Classification of Linear Periodic Difference Equations under Periodic or Kinematic Similarity

I. Gohberg, M. A. Kaashoek, and J. Kos

SIAM. J. Matrix Anal. & Appl. 21, pp. 481-507 (27 pages) | Cited 2 times

Online Publication Date: July 31, 2006

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In this paper linear periodic systems of difference equations are classified with respect to periodic similarity and kinematic similarity. Complete sets of invariants of periodic difference equations relative to such similarity transformations are given, and corresponding canonical forms are described. Also the irreducible periodic difference equations, i.e., those that cannot be reduced by such similarities to a nontrivial direct sum, are identified.

Analysis of Iterative Line Spline Collocation Methods for Elliptic Partial Differential Equations

A. Hadjidimos, E. N. Houstis, J. R. Rice, and E. Vavalis

SIAM. J. Matrix Anal. & Appl. 21, pp. 508-521 (14 pages) | Cited 6 times

Online Publication Date: July 31, 2006

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In this paper we present the convergence analysis of iterative schemes for solving linear systems resulting from discretizing multidimensional linear second-order elliptic partial differential equations (PDEs) defined in a hyperparallelepiped $\Omega$ and subject to Dirichlet boundary conditions on some faces of $\Omega$ and Neumann on the others, using line cubic spline collocation (LCSC) methods. Specifically, we derive analytic expressions or obtain sharp bounds for the spectral radius of the corresponding Jacobi iteration matrix and from this we determine the convergence ranges and compute the optimal parameters for the extrapolated Jacobi and successive overrelaxation (SOR) methods. Experimental results are also presented.

Matrices with Low-Rank-Plus-Shift Structure: Partial SVD and Latent Semantic Indexing

Hongyuan Zha and Zhenyue Zhang

SIAM. J. Matrix Anal. & Appl. 21, pp. 522-536 (15 pages) | Cited 6 times

Online Publication Date: July 31, 2006

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We present a detailed analysis of matrices satisfying the so-called low-rank-plus-shift property in connection with the computation of their partial singular value decomposition (SVD). The application we have in mind is latent semantic indexing for information retrieval, where the term-document matrices generated from a text corpus approximately satisfy this property. The analysis is motivated by developing more efficient methods for computing and updating partial SVD of large term-document matrices and gaining deeper understanding of the behavior of the methods in the presence of noise.

Distribution of Entries in a Substochastic Matrix Having Eigenvalues Near 1

D. J. Hartfiel

SIAM. J. Matrix Anal. & Appl. 21, pp. 537-546 (10 pages)

Online Publication Date: July 31, 2006

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This paper gives a quantitative result that if A is a substochastic matrix and has r eigenvalues which are sufficiently close to 1, then A has r disjoint principal submatrices which are nearly stochastic.

Norms of Large Toeplitz Band Matrices

A. Böttcher, S. Grudsky, A. Kozak, and B. Silbermann

SIAM. J. Matrix Anal. & Appl. 21, pp. 547-561 (15 pages)

Online Publication Date: July 31, 2006

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Let T(bjk) be a finite collection of infinite Toeplitz band matrices, let Tn(bjk) denote their (n+1) x (n+1) truncations, and put \[ A_n=\sum_j\prod_k T_n(b_{jk}),\quad M_p=\lim_{n\to\infty}\|A_n\|_p, \] where $\|\cdot\|_p$ stands for the operator norm associated with the $l^p$-norm $(1\le p<\infty)$ on Cn+1. We establish tight two-sided estimates for the difference Mp-||An||p. It is well known that if T(b) is a Hermitian Toeplitz band matrix and An=Tn(b), then M2-||An||2 goes to zero with polynomial speed. We show that such a slow convergence rate is, in a sense, an exceptional case, and we prove that in the generic case Mp - ||An||p approaches zero with exponential speed. Our results yield good error estimates when computing the norms of certain infinite matrices via their large truncations and, conversely, when determining the norms of certain large matrices by having recourse to known norms of infinite matrices.

Accurate Singular Value Decompositions of Structured Matrices

James Demmel

SIAM. J. Matrix Anal. & Appl. 21, pp. 562-580 (19 pages) | Cited 12 times

Online Publication Date: July 31, 2006

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We present new O(n3) algorithms to compute very accurate singular value decompositions of Cauchy matrices, Vandermonde matrices, and related "unit-displacement-rank" matrices. These algorithms compute all the singular values with guaranteed relative accuracy, independent of their dynamic range. In contrast, previous O(n3) algorithms can potentially lose all relative accuracy in the tiniest singular values.

Interlacing Inequalities and Cartan Subspaces of Classical Real Simple Lie Algebras

Tin-Yau Tam

SIAM. J. Matrix Anal. & Appl. 21, pp. 581-592 (12 pages) | Cited 1 time

Online Publication Date: July 31, 2006

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The well-known interlacing inequalities are examined in the framework of classical real simple Lie algebras.

More Results on Eigenvector Saddlepoints and Eigenpolynomials

Mark A. Mendlovitz

SIAM. J. Matrix Anal. & Appl. 21, pp. 593-612 (20 pages)

Online Publication Date: July 31, 2006

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This paper extends the author's earlier work which proved that, for a certain class of Hermitian eigenproblems $AP=\lambda BP$ (including, most notably, the Toeplitz eigenproblem), the eigenvectors can be represented as saddlepoints of a special form of the Rayleigh quotient. Each saddlepoint solves a min-max/max-min optimization problem whose optimal value is the eigenvalue. The zeros of the eigenpolynomial factors are located with respect to the unit circle. These results were proved, in part, by using an inertia theorem for Stein equations. For another class of Hermitian eigenproblems $AP=\lambda BP,$ an analogous set of results are derived here using an inertia theorem for Lyapunov equations. In this case, the zeros of the eigenpolynomial factors are located with respect to the imaginary axis, and sufficient conditions for the eigenpolynomial factors to have only extended imaginary zeros are established. The highlight of the article is an important theorem that yields a general factorization of saddlepoint eigenpolynomials and parameterizes all eigenvector saddlepoint representations for an m-dimensional eigenspace. Several key extensions to the theory are also presented.

Iterative Regularization and MINRES

Misha Kilmer and G. W. Stewart

SIAM. J. Matrix Anal. & Appl. 21, pp. 613-628 (16 pages) | Cited 8 times

Online Publication Date: July 31, 2006

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In this paper we present three theorems which give insight into the regularizing properties of MINRES. While our theory does not completely characterize the regularizing behavior of the algorithm, it provides a partial explanation of the observed behavior of the method. Unlike traditional attempts to explain the regularizing properties of Krylov subspace methods, our approach focuses on convergence properties of the residual rather than on convergence analysis of the harmonic Ritz values. The import of our analysis is illustrated by two examples. In particular, our theoretical and numerical results support the following important observation: in some circumstances the dimension of the optimal Krylov subspace can be much smaller than the number of the components of the truncated spectral solution that must be computed to attain comparable accuracy.

The Moore--Penrose Generalized Inverse for Sums of Matrices

James Allen Fill and Donniell E. Fishkind

SIAM. J. Matrix Anal. & Appl. 21, pp. 629-635 (7 pages) | Cited 16 times

Online Publication Date: July 31, 2006

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In this paper we exhibit, under suitable conditions, a neat relationship between the Moore--Penrose generalized inverse of a sum of two matrices and the Moore--Penrose generalized inverses of the individual terms. We include an application to the parallel sum of matrices.

A Note on P1- and Lipschitzian Matrices

G. S. R. Murthy, S. K. Neogy, and F. Thuijsman

SIAM. J. Matrix Anal. & Appl. 21, pp. 636-641 (6 pages)

Online Publication Date: July 31, 2006

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The linear complementarity problem (q, A) with data $A \in R^{n \times n}$ and $q \in R^n$ involves finding a nonnegative $z \in R^n$ such that $Az + q \geq 0$ and $z^t(Az + q) = 0$. Cottle and Stone introduced the class of P1-matrices and showed that if A is in P1\Q, then K(A) (the set of all q for which (q, A) has a solution) is a half-space and (q, A) has a unique solution for every q in the interior of K(A). Extending the results of Murthy, Parthasarathy, and Sriparna [ Ann. Dynamic Games, to appear], we present a number of equivalent characterizations of P1\Q. Also, we present yet another characterization of P-matrices. This widens the range of matrix classes for which a conjecture raised by Murthy, Parthasarathy, and Sriparna [SIAM J. Matrix Anal. Appl., 19 (1998), pp. 898--905] characterizing the class of Lipschitzian matrices is true.

Real Solutions of the Equation $\Phi^t(A)=\frac1nJ_n$\protect\unboldmath

Zhang Xian, Yang Zhongpeng, and Cao Chongguang

SIAM. J. Matrix Anal. & Appl. 21, pp. 642-645 (4 pages) | Cited 2 times

Online Publication Date: July 31, 2006

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For nonsingular n x n matrices $A$, $\Phi (A)=A\circ A^{-T}$ ($\circ $ denotes the Hadamard product and A-T the inverse transpose, (A-1)T, of A) arises in mathematical control theory associated with chemical engineering design problems and in a matrix theoretic problem involving the relation between the diagonal entries and eigenvalues. For any positive integer t, we show that the equation $$ \Phi ^t(A)=\frac{1}{n}J_n $$ hasat least 2t-1 and 2t mutually $\Phi $-distinct real solutions for the case n=4 and n > 4, respectively, and that these solutions can be determined by an inverted iteration. We also prove that the above equation has only one $\Phi $-distinct real solution {\scriptsize $(\begin{array}{cc} 1&1\\[-2pt] 1&-1 \end{array})$} for $t=1$ and has no real solutions for $t\ge 2$ when n=2. The above results answer the problem that has been proposed by Johnson and Shapiro in [ SIAM J. Algebraic Discrete Methods, 7 (1986), pp. 627--644]. When does the equation $\Phi (A)=A\circ A^{-T}=\frac{1}{n}J_n$ have a real solution?

Matrix Algebras and Displacement Decompositions

Carmine Di Fiore

SIAM. J. Matrix Anal. & Appl. 21, pp. 646-667 (22 pages) | Cited 2 times

Online Publication Date: July 31, 2006

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A class $\xi$ of algebras of symmetric n × n matrices, related to Toeplitz-plus-Hankel structures and including the well-known algebra $\mathcal{H}$ diagonalized by the Hartley transform, is investigated. The algebras of $\xi$ are then exploited in a general displacement decomposition of an arbitrary n × n matrix A. Any algebra of $\xi$ is a 1-space, i.e., it is spanned by n matrices having as first rows the vectors of the canonical basis. The notion of 1-space (which generalizes the previous notions of $\mathcal{L}_1$ space [Bevilacqua and Zellini, Linear and Multilinear Algebra, 25 (1989), pp. 1--25] and Hessenberg algebra [Di Fiore and Zellini, Linear Algebra Appl., 229 (1995), pp. 49--99]) finally leads to the identification in $\xi$ of three new (non-Hessenberg) matrix algebras close to $\mathcal{H}$, which are shown to be associated with fast Hartley-type transforms. These algebras are also involved in new efficient centrosymmetric Toeplitz-plus-Hankel inversion formulas.

Convexity of the Joint Numerical Range

Chi-Kwong Li and Yiu-Tung Poon

SIAM. J. Matrix Anal. & Appl. 21, pp. 668-678 (11 pages) | Cited 5 times

Online Publication Date: July 31, 2006

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Let A=(A1, . . ., Am) be an m-tuple of n × n Hermitian matrices. For $1 \le k \le n$, the $k${\rm th} joint numerical range of A is defined by $$W_k(A) = \{ ({\rm \tr}(X^*A_1X), \dots, {\rm \tr}(X^*A_mX) ): X \in {\bf C}^{n\times k}, X^*X = I_k \}.$$ We consider linearly independent families of Hermitian matrices {A1, . . . , Am} so that Wk(A) is convex. It is shown that m can reach the upper bound 2k(n-k)+1. A key idea in our study is relating the convexity of Wk(A) to the problem of constructing rank k orthogonal projections under linear constraints determined by A. The techniques are extended to study the convexity of other generalized numerical ranges and the corresponding matrix construction problems.

Conditioning of Rectangular Vandermonde Matrices with Nodes in the Unit Disk

Fermín S. V. Bazán

SIAM. J. Matrix Anal. & Appl. 21, pp. 679-693 (15 pages) | Cited 5 times

Online Publication Date: July 31, 2006

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Let WN=WN(z1,z2, . . . z1) be a rectangular Vandermonde matrix of order n × N, $N\geq n,$ with distinct nodes zj in the unit disk and $z_j^{k-1}$ as its (j,k) entry. Matrices of this type often arise in frequency estimation and system identification problems. In this paper, the conditioning of WN is analyzed and bounds for the spectral condition number $\kappa_2(W_N)$ are derived. The bounds depend on n, N, and the separation of the nodes. By analyzing the behavior of the bounds as functions of N, we conclude that these matrices may become well conditioned, provided the nodes are close to the unit circle but not extremely close to each other and provided the number of columns of WN is large enough. The asymptotic behavior of both the conditioning itself and the bounds is analyzed and the theoretical results arising from this analysis verified by numerical examples.

On a Newton-Like Method for Solving Algebraic Riccati Equations

Chun-Hua Guo and Alan J. Laub

SIAM. J. Matrix Anal. & Appl. 21, pp. 694-698 (5 pages) | Cited 10 times

Online Publication Date: July 31, 2006

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An exact line search method has been introduced by Benner and Byers [IEEE Trans. Automat. Control, 43 (1998), pp. 101--107] for solving continuous algebraic Riccati equations. The method is a modification of Newton's method. A convergence theory is established in that paper for the Newton-like method under the strong hypothesis of controllability, while the original Newton's method needs only the weaker hypothesis of stabilizability for its convergence theory. It is conjectured there that the controllability condition can be weakened to the stabilizability condition. In this article we prove that conjecture.

The p-Relative Distance is a Metric

Anders Barrlund

SIAM. J. Matrix Anal. & Appl. 21, pp. 699-702 (4 pages) | Cited 1 time

Online Publication Date: July 31, 2006

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The conjecture that the p-relative distance, $\varrho_p(\alpha,\tilde{\alpha})=|\alpha-\tilde{\alpha}| /{\sqrt[p]{|\alpha|^p+|\tilde{\alpha}|^p}}$, is a metric is proved.
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