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18
Structured Inverse Eigenvalue Problems
 ACTA NUMERICA
, 2002
"... this paper. More should be said about these constraints in order to define an IEP. First we recall one condition under which two geometric entities intersect transversally. Loosely speaking, we may assume that the structural constraint and the spectral constraint define, respectively, smooth manifo ..."
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Cited by 44 (13 self)
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this paper. More should be said about these constraints in order to define an IEP. First we recall one condition under which two geometric entities intersect transversally. Loosely speaking, we may assume that the structural constraint and the spectral constraint define, respectively, smooth manifolds in the space of matrices of a fixed size. If the sum of the dimensions of these two manifolds exceeds the dimension of the ambient space, then under some mild conditions one can argue that the two manifolds must intersect and the IEP must have a solution. A more challenging situation is when the sum of dimensions emerging from both structural and spectral constraints does not add up to the transversal property. In that case, it is much harder to tell whether or not an IEP is solvable. Secondly we note that in a complicated physical system it is not always possible to know the entire spectrum. On the other hand, especially in structural design, it is often demanded that certain eigenvectors should also satisfy some specific conditions. The spectral constraints involved in an IEP, therefore, may consist of complete or only partial information on eigenvalues or eigenvectors. We further observe that in practice it may occur that one of the two constraints in an IEP should be enforced more critically than the other due, say, to the physical realizability. Without the realizability, the physical system simply cannot be built. There are also situations when one constraint could be more relaxed than the other due, say, to the physical uncertainty. The uncertainty arises when there is simply no accurate way to measure the spectrum or there is no reasonable means to obtain the entire information. When the two constraints cannot be satisfied simultaneously, the IEP could be formulat...
The interplay between classical analysis and (numerical) linear algebra  a tribute to Gene H. Golub
 ELECTRONIC TRANSACTIONS ON NUMERICAL ANALYSIS
, 2002
"... Much of the work of Golub and his collaborators uses techniques of linear algebra to deal with problems in analysis, or employs tools from analysis to solve problems arising in linear algebra. Instances are described of such interdisciplinary work, taken from quadrature theory, orthogonal polynomia ..."
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Cited by 17 (2 self)
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Much of the work of Golub and his collaborators uses techniques of linear algebra to deal with problems in analysis, or employs tools from analysis to solve problems arising in linear algebra. Instances are described of such interdisciplinary work, taken from quadrature theory, orthogonal polynomials, and least squares problems on the one hand, and error analysis for linear algebraic systems, elementwise bounds for the inverse of matrices, and eigenvalue estimates on the other hand.
Krylov subspace spectral methods for variablecoefficient initialboundary value problems
 ETNA
"... Abstract. This paper presents an alternative approach to the solution of diffusion problems in the variablecoefficient case that leads to a new numerical method, called a Krylov subspace spectral method. The basic idea behind the method is to use Gaussian quadrature in the spectral domain to compute ..."
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Cited by 7 (7 self)
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Abstract. This paper presents an alternative approach to the solution of diffusion problems in the variablecoefficient case that leads to a new numerical method, called a Krylov subspace spectral method. The basic idea behind the method is to use Gaussian quadrature in the spectral domain to compute components of the solution, rather than in the spatial domain as in traditional spectral methods. For each component, a different approximation of the solution operator by a restriction to a lowdimensional Krylov subspace is employed, and each approximation is optimal in some sense for computing the corresponding component. This strategy allows accurate resolution of all desired frequency components without having to resort to smoothing techniques to ensure stability. (1.1)
Orthogonal Polynomials and Quadrature
 Elec. Trans. Numer. Anal
"... Various concepts of orthogonality on the real line are reviewed that arise in connection with quadrature rules. Orthogonality relative to a positive measure and Gausstype quadrature rules are classical. More recent types of orthogonality include orthogonality relative to a signvariable measure, wh ..."
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Cited by 3 (2 self)
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Various concepts of orthogonality on the real line are reviewed that arise in connection with quadrature rules. Orthogonality relative to a positive measure and Gausstype quadrature rules are classical. More recent types of orthogonality include orthogonality relative to a signvariable measure, which arises in connection with GaussKronrod quadrature, and power (or implicit) orthogonality encountered in Tur'antype quadratures. Relevant questions of numerical computation are also considered. 1.
An Efficient Algorithm for Calculating the Heat Capacity of a Largescale Molecular System
, 2001
"... We present an ecient algorithm for computing the heat capacity of a largescale molecular system. The new algorithm is based on a special Gaussian quadrature whose abscissas and weights are obtained by a simple Lanczos iteration. Our numerical results have indicated that this new computational schem ..."
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We present an ecient algorithm for computing the heat capacity of a largescale molecular system. The new algorithm is based on a special Gaussian quadrature whose abscissas and weights are obtained by a simple Lanczos iteration. Our numerical results have indicated that this new computational scheme is quite accurate. We have also shown that this method is at least a hundred times faster than the earlier approach that is based on estimating the density of states and integrating with a simple quadrature formula. 1 Introduction The heat capacity of a large molecular system is dened by C v = kB Z 1 0 h!c=kB T 2 e h!c=kB T 1 e h!c=kB T 2 g(!)d!; (1) where kB is Boltzmann's constant, c is the speed of light, h is Planck's constant, and ! is the fundamental vibrational frequency in cm 1 [11]. The function g(!), often known as the density of states, describes the distribution of the vibrational frequencies of the particle. To be specic, g(!)d! gives the number of vib...
1 Level Crossing Rate and Average Fade Duration of Dual Selection Combining with Cochannel Interference and Nakagami Fading
, 908
"... Abstract — This letter provides closedform expressions for the outage probability, the average level crossing rate (LCR) and the average fade duration (AFD) of a dual diversity selection combining (SC) system exposed to the combined influence of the cochannel interference (CCI) and the thermal nois ..."
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Abstract — This letter provides closedform expressions for the outage probability, the average level crossing rate (LCR) and the average fade duration (AFD) of a dual diversity selection combining (SC) system exposed to the combined influence of the cochannel interference (CCI) and the thermal noise (AWGN) in Nakagami fading channel. The branch selection is based on the desired signal power SC algorithm with all input signals assumed to be independent, while the powers of the desired signals in all diversity branches are mutually equal but distinct from the power of the interference signals. The analytical results reduce to known solutions in the cases of an interferencelimited system in Rayleigh fading and an AWGNlimited system in Nakagami fading. The average LCR is determined by an original approach that does not require explicit knowledge of the joint PDF of the envelope and its time derivative, which also paves the way for similar analysis of other diversity systems. Index Terms — Level crossing rate, average fade duration, cochannel interference, selection combining, Nakagami fading. I.
On Robust Matrix Completion with Prescribed Eigenvalues
, 2002
"... Matrix completion with prescribed eigenvalues is a special kind of inverse eigenvalue problems. Thus far, only a handful of specific cases concerning its existence and construction have been studied in the literature. The general problem where the prescribed entries are at arbitrary locations with a ..."
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Matrix completion with prescribed eigenvalues is a special kind of inverse eigenvalue problems. Thus far, only a handful of specific cases concerning its existence and construction have been studied in the literature. The general problem where the prescribed entries are at arbitrary locations with arbitrary cardinalities proves to be challenging both theoretically and computationally. This paper investigates some continuation techniques by recasting the completion problem as an optimization of the distance between the isospectral matrices with the prescribed eigenvalues and the affine matrices with the prescribed entries. The approach not only offers an avenue for solving the completion problem in its most general setting but also makes it possible to seek a robust solution that is least sensitive to perturbation.
ISSN 10689613. ORTHOGONAL POLYNOMIALS AND QUADRATURE ∗
"... Abstract. Various concepts of orthogonality on the real line are reviewed that arise in connection with quadrature rules. Orthogonality relative to a positive measure and Gausstype quadrature rules are classical. More recent types of orthogonality include orthogonality relative to a signvariable m ..."
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Abstract. Various concepts of orthogonality on the real line are reviewed that arise in connection with quadrature rules. Orthogonality relative to a positive measure and Gausstype quadrature rules are classical. More recent types of orthogonality include orthogonality relative to a signvariable measure, which arises in connection with GaussKronrod quadrature, and power (or implicit) orthogonality encountered in Turántype quadratures. Relevant questions of numerical computation are also considered. Key words. orthogonal polynomials, GaussLobatto, GaussKronrod, and GaussTurán rules, computation of Gausstype quadrature rules.
DOI: 10.1017/S0962492902000014 Printed in the United Kingdom Structured inverse eigenvalue problems
"... An inverse eigenvalue problem concerns the reconstruction of a structured matrix from prescribed spectral data. Such an inverse problem arises in many applications where parameters of a certain physical system are to be determined from the knowledge or expectation of its dynamical behaviour. Spectr ..."
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An inverse eigenvalue problem concerns the reconstruction of a structured matrix from prescribed spectral data. Such an inverse problem arises in many applications where parameters of a certain physical system are to be determined from the knowledge or expectation of its dynamical behaviour. Spectral information is entailed because the dynamical behaviour is often governed by the underlying natural frequencies and normal modes. Structural stipulation is designated because the physical system is often subject to some feasibility constraints. The spectral data involved may consist of complete or only partial information on eigenvalues or eigenvectors. The structure embodied by the matrices can take many forms. The objective of an inverse eigenvalue problem is to construct a matrix that maintains both the specific structure as well as the given spectral property. In this expository paper the emphasis is to provide an overview of the vast scope of this intriguing problem, treating some of its many applications, its mathematical properties, and a variety of numerical