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19
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 56 (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 20 (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.
Computation of GaussKronrod Quadrature Rules with NonPositive Weights
 Math. Comp
, 1999
"... Recently Laurie presented a fast algorithm for the computation of (2n + 1)point GaussKronrod quadrature rules with real nodes and positive weights. We describe modifications of this algorithm that allow the computation of GaussKronrod quadrature rules with complex conjugate nodes and weights or w ..."
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Cited by 20 (3 self)
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Recently Laurie presented a fast algorithm for the computation of (2n + 1)point GaussKronrod quadrature rules with real nodes and positive weights. We describe modifications of this algorithm that allow the computation of GaussKronrod quadrature rules with complex conjugate nodes and weights or with real nodes and positive and negative weights.
Gausstype quadrature rules for rational functions
 in Numerical Integration IV
"... Abstract. When integrating functions that have poles outside the interval of integration, but are regular otherwise, it is suggested that the quadrature rule in question ought to integrate exactly not only polynomials (if any), but also suitable rational functions. The latter are to be chosen so as ..."
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Cited by 13 (4 self)
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Abstract. When integrating functions that have poles outside the interval of integration, but are regular otherwise, it is suggested that the quadrature rule in question ought to integrate exactly not only polynomials (if any), but also suitable rational functions. The latter are to be chosen so as to match the most important poles of the integrand. We describe two methods for generating such quadrature rules numerically and report on computational experience with them.
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 4 (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.
The use of rational functions in numerical quadrature
 J. Comput. Appl. Math. 133(12
"... The use of rational functions in numerical quadrature ..."
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The use of rational functions in numerical quadrature
Stochastic Investigation of Flows About Airfoils at Transonic Speeds
, 2010
"... In this study, a deterministic compressible solver is coupled to a nonintrusive stochastic spectral projection method to propagate several aerodynamic uncertainties through a transonic steady flow around a NACA0012 airfoil. The stochastic model is solved in a generalized polynomial chaos framework. ..."
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In this study, a deterministic compressible solver is coupled to a nonintrusive stochastic spectral projection method to propagate several aerodynamic uncertainties through a transonic steady flow around a NACA0012 airfoil. The stochastic model is solved in a generalized polynomial chaos framework. This approach combines the advantage of not modifying the existing deterministic solver while maintaining accurate representations of the stochastic solution and its statistics. The major difficulty of this work is to deal with deterministic transonic flows for which aerodynamics nonlinearities are reported in the uncertain probabilistic space. The efficiency of the present methodology are evaluated for the propagation of random disturbances associated with the angle of attack and the freestream Mach number. An error analysis is carried out in order to determine appropriate physical and stochastic discretization levels. Different stochastic flow regimes are analyzed in details by means of various postprocessing procedures, including error bars, probabilistic density function of the aerodynamic field, and Sobol’s coefficients.
ON GENERALIZED AVERAGED GAUSSIAN FORMULAS
, 2007
"... We present a simple numerical method for constructing the optimal (generalized) averaged Gaussian quadrature formulas which are the optimal stratified extensions of Gauss quadrature formulas. These extensions exist in many cases in which real positive Kronrod formulas do not exist. For the Jacobi ..."
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We present a simple numerical method for constructing the optimal (generalized) averaged Gaussian quadrature formulas which are the optimal stratified extensions of Gauss quadrature formulas. These extensions exist in many cases in which real positive Kronrod formulas do not exist. For the Jacobi weight functions w(x) ≡ w (α,β) (x)=(1 − x) α (1 + x) β (α, β> −1) we give a necessary and sufficient condition on the parameters α and β such that the optimal averaged Gaussian quadrature formulas are internal.
Error estimates for . . . Kronrod Extensions
, 2009
"... We study the kernel Kn,s(z) of the remainder term Rn,s ( f) of Gauss–Turán–Kronrod quadrature rules with respect to one of the generalized Chebyshev weight functions for analytic functions. The location on the elliptic contours where the modulus of the kernel attains its maximum value is investigat ..."
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We study the kernel Kn,s(z) of the remainder term Rn,s ( f) of Gauss–Turán–Kronrod quadrature rules with respect to one of the generalized Chebyshev weight functions for analytic functions. The location on the elliptic contours where the modulus of the kernel attains its maximum value is investigated. This leads to effective L∞error bounds of Gauss–Turán–Kronrod quadratures. Following Kronrod, using the modulus of the difference of Gauss–Turán quadratures and their Kronrod extensions, we derive new error estimates for Gauss–Turán quadratures and compare them with the effective L1error bounds derived in