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14
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 42 (14 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 12 (1 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 5 (5 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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Cited by 1 (0 self)
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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...
The Use of Rational Functions in Numerical Quadrature
"... Quadrature problems involving functions that have poles outside the interval of integration can profitably be solved by methods that are exact not only for polynomials of appropriate degree, but also for rational functions having the same (or the most important) poles as the function to be integrate ..."
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Quadrature problems involving functions that have poles outside the interval of integration can profitably be solved by methods that are exact not only for polynomials of appropriate degree, but also for rational functions having the same (or the most important) poles as the function to be integrated. Constructive and computational tools for accomplishing this are described and illustrated in a number of quadrature contexts. The superiority of such rational/polynomial methods is shown by an analysis of the remainder term and documented by numerical examples. Key words: Rational quadrature rules; Remainder term; Rational Fej'er quadrature; Rational Gauss, GaussKronrod, and GaussTur'an quadrature; Rational quadrature rules for Cauchy principal value integrals 1
STIELTJESTYPE POLYNOMIALS ON THE UNIT CIRCLE
, 2008
"... Stieltjestype polynomials corresponding to measures supported on the unit circle T are introduced and their asymptotic properties away from T are studied for general classes of measures. As an application, we prove the convergence of an associated sequence of interpolating rational functions to th ..."
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Stieltjestype polynomials corresponding to measures supported on the unit circle T are introduced and their asymptotic properties away from T are studied for general classes of measures. As an application, we prove the convergence of an associated sequence of interpolating rational functions to the corresponding Carathéodory function. In turn, this is used to give an estimate of the rate of convergence of certain quadrature formulae that resemble the GaussKronrod rule, provided that the integrand is analytic in a neighborhood of T.
Available online at www.sciencedirect.com Linear Algebra and its Applications 428 (2008) 2405–2409 Personal report
"... www.elsevier.com/locate/laa ..."
Article electronically published on October 27, 2008 STIELTJESTYPE POLYNOMIALS ON THE UNIT CIRCLE
"... Abstract. Stieltjestype polynomials corresponding to measures supported on the unit circle T are introduced and their asymptotic properties away from T are studied for general classes of measures. As an application, we prove the convergence of an associated sequence of interpolating rational functi ..."
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Abstract. Stieltjestype polynomials corresponding to measures supported on the unit circle T are introduced and their asymptotic properties away from T are studied for general classes of measures. As an application, we prove the convergence of an associated sequence of interpolating rational functions to the corresponding Carathéodory function. In turn, this is used to give an estimate of the rate of convergence of certain quadrature formulae that resemble the GaussKronrod rule, provided that the integrand is analytic in a neighborhood of T. 1.
1 Comparison of Consistent Integration Versus Adaptive Quadrature For Taming Aliasing Errors
, 2009
"... Most spectral/hp element methods, whether employing a modal or nodal basis representation, evaluate nonlinear differential operators in ”physical space ” at a collection of collocation or quadrature points. The number of points used is often set to what is needed to represent the original solution ..."
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Most spectral/hp element methods, whether employing a modal or nodal basis representation, evaluate nonlinear differential operators in ”physical space ” at a collection of collocation or quadrature points. The number of points used is often set to what is needed to represent the original solution over an element (or integrate the square of the function over an element), and not to what is needed to represent the square of the function. This discrepancy leads to aliasing errors, which when the fields are highly resolved have little appreciative impact and hence can be ignored. In underresolved scenarios, aliasing can pollute the solution leading to decreased accuracy and issues of stability. These errors can be eliminated by consistent integration at the price of increased computational cost. In most engineering simulations, however, the issue is not binary: not all elements within a simulation domain contain underresolved solutions nor fullresolved solutions. The location and times at which elements support underresolved solutions varies based upon the dynamics of the system. Hence an efficient mean of taming aliasing errors can be through dynamic quadrature. In this report, we present analysis that compares the computational efficiency of an adaptive