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56
On the ADI Method for Sylvester Equations
"... This paper is concerned with the numerical solution of large scale Sylvester equations AX − XB = C, Lyapunov equations as a special case in particular included, with C having very small rank. For stable Lyapunov equations, Penzl (2000) and Li and White (2002) demonstrated that the so called Cholesky ..."
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Cited by 15 (13 self)
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This paper is concerned with the numerical solution of large scale Sylvester equations AX − XB = C, Lyapunov equations as a special case in particular included, with C having very small rank. For stable Lyapunov equations, Penzl (2000) and Li and White (2002) demonstrated that the so called Cholesky factor ADI method with decent shift parameters can be very effective. In this paper we present a generalization of the Cholesky factor ADI method for Sylvester equations. We also demonstrate that often much more accurate solutions than ADI solutions can be obtained by performing Galerkin projection via the column space and row space of the computed approximate solutions.
A spectral approach to lower bounds with applications to geometric searching
 SIAM J. Comput
, 1998
"... Abstract. We establish a nonlinear lower bound for halfplane range searching over a group. Specifically, we show that summing up the weights of n (weighted) points within n halfplanes requires Ω(n log n) additions and subtractions. This is the first nontrivial lower bound for range searching over a ..."
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Cited by 13 (3 self)
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Abstract. We establish a nonlinear lower bound for halfplane range searching over a group. Specifically, we show that summing up the weights of n (weighted) points within n halfplanes requires Ω(n log n) additions and subtractions. This is the first nontrivial lower bound for range searching over a group. By contrast, range searching over a semigroup (which forbids subtractions) is almost completely understood. Our proof has two parts. First, we develop a general, entropybased method for relating the linear circuit complexity of a linear map A to the spectrum of A ⊤ A. In the second part of the proof, we design a “highspectrum ” geometric set system for halfplane range searching and, using techniques from discrepancy theory, we estimate the median eigenvalue of its associated map. Interestingly, the method also shows that using up to a linear number of help gates cannot help; these are gates that can compute any bivariate function.
On the core of a conepreserving map
 Trans. Amer. Math. Soc
, 1994
"... ABSTRACT. This is the third of a sequence of papers in an attempt to study the PerronFrobenius theory of a nonnegative matrix and its generalizations from the conetheoretic viewpoint. Our main object of interest here is the core of a conepreserving map. If A is an n x n real matrix which leaves i ..."
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Cited by 13 (3 self)
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ABSTRACT. This is the third of a sequence of papers in an attempt to study the PerronFrobenius theory of a nonnegative matrix and its generalizations from the conetheoretic viewpoint. Our main object of interest here is the core of a conepreserving map. If A is an n x n real matrix which leaves invariant a proper cone K in IR n, then by the core of A relative to K, denoted by coreK(A), we mean the convex cone. nb:1 Ai K. It is shown that when coreK(A) is polyhedral, which is the case whenever K is, then coreK(A) is generated by the distinguished eigenvectors of positive powers of A. The important concept of a distinguished Ainvariant face is introduced, which corresponds to the concept of a distinguished class in the nonnegative matrix case. We prove a significant theorem which describes a onetoone correspondence between the distinguished Ainvariant faces of K and the cycles of the permutation induced by A on the extreme rays of coreK (A), provided that the latter cone is nonzero, simplicial. By an interplay between conetheoretic and graphtheoretic ideas, the extreme rays of the core of a nonnegative matrix are fully described. Characterizations of Kirreducibility or Kprimitivity of A are also found in terms of coreK (A). Several equivalent conditions are also given on a matrix with an invariant proper cone so that its spectral radius is an eigenvalue of index one. An equivalent condition in terms of the peripheral spectrum is also found on a real matrix A with the PerronSchaefer condition for which there exists a proper invariant cone K such that coreK (A) is polyhedral, simplicial, or a single ray. A method of producing a large class of invariant proper cones for a matrix with the PerronSchaefer condition is also offered. 1.
Evaluation of the Reliable Data Rates Supported by MultipleAntenna Coded Wireless Links for QAM Transmissions
, 2001
"... In this paper, we present some novel results about the reliable informationrate supported by pointtopoint multipleantenna Rayleighfaded wireless links for coded transmissions that employ twodimensional (QAM or PSK) data constellations. After deriving the symmetric capacity of these links, we p ..."
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Cited by 12 (2 self)
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In this paper, we present some novel results about the reliable informationrate supported by pointtopoint multipleantenna Rayleighfaded wireless links for coded transmissions that employ twodimensional (QAM or PSK) data constellations. After deriving the symmetric capacity of these links, we present fastcomputable analytical upper and lower bounds that are asymptotically exact both for high and low SNRs, and give rise to a reliable evaluation of the link capacity when perfect channel state information (CSI) is available at the receiver. Furthermore, asymptotically exact simple upper bounds are also presented for a tight evaluation of the outage probability.
Rényi's Divergence and Entropy Rates for Finite Alphabet Markov Sources
"... In this work, we examine the existence and the computation of the Renyi divergence rate, lim n!1 1 n D (p (n) kq (n) ), between two timeinvariant nitealphabet Markov sources of arbitrary order and arbitrary initial distributions described by the probability distributions p (n) and q (n) ..."
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Cited by 11 (3 self)
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In this work, we examine the existence and the computation of the Renyi divergence rate, lim n!1 1 n D (p (n) kq (n) ), between two timeinvariant nitealphabet Markov sources of arbitrary order and arbitrary initial distributions described by the probability distributions p (n) and q (n) , respectively. This yields a generalization of a result of Nemetz where he assumed that the initial probabilities under p (n) and q (n) are strictly positive. The main tools used to obtain the Renyi divergence rate are the theory of nonnegative matrices and PerronFrobenius theory. We also provide numerical examples and investigate the limits of the Renyi divergence rate as ! 1 and as # 0. Similarly, we provide a formula for the Renyi entropy rate lim n!1 1 n H (p (n) ) of Markov sources and examine its limits as ! 1 and as # 0. Finally, we briey provide an application to source coding. Index Terms: Timeinvariant Markov sources, Renyi's divergence and entropy r...
On Inverse Quadratic Eigenvalue Problems With Partially Prescribed Eigenstructure
 SIAM J. MATRIX ANAL. APPL
, 2004
"... The inverse eigenvalue problem of constructing real and symmetric square matrices M,C and K of size n n for the quadratic pencil Q(#) = # M + #C +K so that Q(#) has a prescribed subset of eigenvalues and eigenvectors is considered. This paper consists of two parts addressing two related but di ..."
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Cited by 10 (5 self)
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The inverse eigenvalue problem of constructing real and symmetric square matrices M,C and K of size n n for the quadratic pencil Q(#) = # M + #C +K so that Q(#) has a prescribed subset of eigenvalues and eigenvectors is considered. This paper consists of two parts addressing two related but di#erent problems. The first part deals with the inverse problem where M and K are required to be positive definite and semidefinite, respectively. It is shown via construction that the inverse problem is solvable for any k given complex conjugately closed pairs of distinct eigenvalues and linearly independent eigenvectors, provided k n. The construction also allows additional optimization conditions to be built into the solution so as to better refine the approximate pencil. The eigenstructure of the resulting Q(#) is completely analyzed. The second part deals with the inverse problem where M is a fixed positivedefinite matrix (and hence may be assumed to be the identity matrix In ). It is shown via construction that the monic quadratic pencil Q(#) = In +#C+K with n+1 arbitrarily assigned complex conjugately closed pairs of distinct eigenvalues and column eigenvectors which span the space C always exists. Su#cient conditions under which this quadratic inverse eigenvalue problem is uniquely solvable are specified.
Computing reliability and message delay for cooperative wireless distributed sensor networks subject to random failures
 IEEE Transactions on Reliability
"... Abstract—One of the most compelling technological advances of this decade has been the advent of deploying wireless networks of heterogeneous smart sensor nodes for complex information gathering tasks. A wireless distributed sensor network (DSN) is a selforganizing, adhoc network of a large number ..."
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Cited by 9 (0 self)
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Abstract—One of the most compelling technological advances of this decade has been the advent of deploying wireless networks of heterogeneous smart sensor nodes for complex information gathering tasks. A wireless distributed sensor network (DSN) is a selforganizing, adhoc network of a large number of cooperative intelligent sensor nodes. Due to the limited power of sensor nodes, energyefficient DSN are essentially multihop networks. The selforganizing capabilities, and the cooperative operation of DSN allow for forming reliable clusters of sensors deployed near, or at, the sites of target phenomena. Reliable monitoring of a phenomenon (or event detection) depends on the collective data provided by the target cluster of sensors, and not on any individual node. The failure of one or more nodes may not cause the operational data sources to be disconnected from the data sinks (command nodes or end user stations). However, it may increase
Solving rational eigenvalue problems via linearization
, 2008
"... Abstract. Rational eigenvalue problem is an emerging class of nonlinear eigenvalue problems arising from a variety of physical applications. In this paper, we propose a linearizationbased method to solve the rational eigenvalue problem. The proposed method converts the rational eigenvalue problem i ..."
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Cited by 9 (0 self)
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Abstract. Rational eigenvalue problem is an emerging class of nonlinear eigenvalue problems arising from a variety of physical applications. In this paper, we propose a linearizationbased method to solve the rational eigenvalue problem. The proposed method converts the rational eigenvalue problem into a wellstudied linear eigenvalue problem, and meanwhile, exploits and preserves the structure and properties of the original rational eigenvalue problem. For example, the lowrank property leads to a trimmed linearization. We show that solving a class of rational eigenvalue problems is just as convenient and efficient as solving linear eigenvalue problems. Key words. Rational eigenvalue problem, linearization, nonlinear eigenvalue problem AMS subject classifications. 65F15, 65F50, 15A18
Functions of matrices
 Society for Industrial and Applied Mathematics (SIAM
, 2008
"... Reports available from: And by contacting: ..."
Definite matrix polynomials and their linearization by definite pencils
 Manchester Institute for Mathematical Sciences, The University of Manchester
, 2008
"... Abstract. Hyperbolic matrix polynomials are an important class of Hermitian matrix polynomials that contain overdamped quadratics as a special case. They share with definite pencils the spectral property that their eigenvalues are real and semisimple. We extend the definition of hyperbolic matrix po ..."
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Cited by 8 (7 self)
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Abstract. Hyperbolic matrix polynomials are an important class of Hermitian matrix polynomials that contain overdamped quadratics as a special case. They share with definite pencils the spectral property that their eigenvalues are real and semisimple. We extend the definition of hyperbolic matrix polynomial in a way that relaxes the requirement of definiteness of the leading coefficient matrix, yielding what we call definite polynomials. We show that this class of polynomials has an elegant characterization in terms of definiteness intervals on the extended real line, and that it includes definite pencils as a special case. A fundamental question is whether a definite matrix polynomial P can be linearized in a structurepreserving way. We show that the answer to this question is affirmative: P is definite if and only if it has a definite linearization in H(P), a certain vector space of Hermitian pencils; and for definite P we give a complete characterization of all the linearizations in H(P) that are definite. For the important special case of quadratics, we show how a definite quadratic polynomial can be transformed into a definite linearization with a positive definite leading coefficient matrix—a form that is particularly attractive numerically.