Mechanische Quadraturen mit positiven Cotesschen Zahlen

Fejér, Leopold

doi:10.1007/bf01474575

Cited by 116 publications

(63 citation statements)

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“…(1) k under the assumption that f (−1) = f (1) = 0.) Quadrature (3.21) is usually known as Fejér's first quadrature [4,5]. The weight expressions (3.18), (3.20) deviate from the standard ones, but we prefer them because they underline the weights' positivity.…”

Section: K the Zeros Of The Chebyshev Polynomial T K If The Weightmentioning

confidence: 99%

Fast memory efficient evaluation of spherical polynomials at scattered points

Ivanov

Petrushev

2014

Adv Comput Math

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Abstract. A method for fast evaluation of spherical polynomials (band-limited functions) at many scattered points on the unit 2-d sphere is presented. The method relies on the sub-exponential localization of the father needlet kernels and their compatibility with spherical harmonics. It is fast, local, memory efficient, numerically stable and with guaranteed (prescribed) accuracy. The speed is independent of the band limit and depends logarithmically on the precision. The method can be also applied for approximation on the sphere, verification of spherical polynomials and for fast generation of surfaces in computeraided geometric design. It naturally generalizes to higher dimensions.

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Section: K the Zeros Of The Chebyshev Polynomial T K If The Weightmentioning

confidence: 99%

Fast memory efficient evaluation of spherical polynomials at scattered points

Ivanov

Petrushev

2014

Adv Comput Math

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“…The general-purpose Fejer's quadrature [13] can be used for the discretization in (13). Gautschi [12] showed that the performance of discretization can be increased through multi-component discretization.…”

Section: Polynomial Chaos For Arbitrary Distribution and Its Associatmentioning

confidence: 99%

Robust PID controller design for processes with stochastic parametric uncertainties

Duong

Lee

2012

Journal of Process Control

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a b s t r a c tThe stability and performance of a system can be inferred from the evolution of statistical characteristics of the system's states. Wiener's polynomial chaos can provide an efficient framework for the statistical analysis of dynamical systems, computationally far superior to Monte Carlo simulations. This work proposes a new method of robust PID controller design based on polynomial chaos for processes with stochastic parametric uncertainties. The proposed method can greatly reduce computation time and can also efficiently handle both nominal and robust performance against stochastic uncertainties by solving a simple optimization problem. Simulation comparison with other methods demonstrated the effectiveness of the proposed design method.

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“…Applying this procedure to the nodes (2.1) directly yields the ClenshawCurtis rules. Fejér's second rule [4] is obtained by omitting the nodes x 0 = 1 and x n = −1 and using the interpolating polynomial of degree n − 2. This may also be achieved by keeping the boundary points as nodes, but preassigning the corresponding weights as w 0 = w n = 0.…”

Section: The Rules By Fejér and Clenshaw-curtismentioning

confidence: 99%

“…We will adopt this unconventional approach in order to obtain a unified treatment of the two rules. Fejér's first rule [4] is obtained by using the well-known Chebyshev points as nodes, i.e., x k from (2.1) with k = 1 2 , 3 2 , . .…”

Section: The Rules By Fejér and Clenshaw-curtismentioning

confidence: 99%

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Fast Construction of the Fejér and Clenshaw–Curtis Quadrature Rules

Waldvogel

2006

Bit Numer Math

133

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Abstract.We present an elegant algorithm for stably and quickly generating the weights of Fejér's quadrature rules and of the Clenshaw-Curtis rule. The weights for an arbitrary number of nodes are obtained as the discrete Fourier transform of an explicitly defined vector of rational or algebraic numbers. Since these rules have the capability of forming nested families, some of them have gained renewed interest in connection with quadrature over multi-dimensional regions.

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Mechanische Quadraturen mit positiven Cotesschen Zahlen

Cited by 116 publications

References 0 publications

Fast memory efficient evaluation of spherical polynomials at scattered points

Fast memory efficient evaluation of spherical polynomials at scattered points

Robust PID controller design for processes with stochastic parametric uncertainties

Fast Construction of the Fejér and Clenshaw–Curtis Quadrature Rules

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