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Application of the Cauchy integral approach to singular and highly oscillatory integrals

INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 2021年第10期98卷 2097-2114页

作者： Kayijuka, Idrissa Alfaqeih, Suliman Ozis, Turgut Ege Univ Dept Math Fac Sci TR-35100 Izmir Bornova Turkey

This paper presents a method that is based on the sum of line integrals for fast computation of singular and highly oscillatory integrals integral(c) (d) G(x) e(i mu(x-c)k) dx, -infinity > c > d > infinity, and integral(1)(-1) f (x)H-l(x) e(i mu x) dx, l = 1, 2, 3. Where G and f are non-oscillatory sufficiently smooth functions on the interval of integration. H-l is a product of singular factors and mu >> 1 is an oscillatory parameter. The computation of these integrals requires f and G to be analytic in a large complex region C accommodating the interval of integration. The integrals are changed into a problem of integrals on [0, infinity);which are later computed using the generalized Gauss-Laguerre rule or by the construction of Gauss rules relative to a Freud weights function e-xk with k positive. MATHEMATICA programming code, algorithms and illustrative numerical examples are provided to test the efficiency of the presented experiments.

关键词： Highly oscillatory integrals Gauss quadrature rules Numerical steepest descent method Gauss-Laguerre quadrature rule chebyshev algorithm

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Gaussian rule for integrals involving Bessel functions

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BIT NUMERICAL MATHEMATICS 2023年第4期63卷 53-53页

作者： Denich, Eleonora Novati, Paolo Univ Trieste Via A Valerio 12-1 I-34127 Trieste Italy

In this work we develop the Gaussian quadrature rule for weight functions involving powers, exponentials and Bessel functions of the first kind. Besides the computation based on the use of the standard and the modified chebyshev algorithm, here we present a very stable algorithm based on the preconditioning of the moment matrix. Numerical experiments are provided and a geophysical application is considered.

关键词： Gaussian quadrature Moment-based methods chebyshev algorithm Bessel function of the first type

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Evaluation of the use of the chebyshev algorithm with the quadrature method of moments for simulating aerosol dynamics

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JOURNAL OF AEROSOL SCIENCE 2012年 44卷 11-23页

作者： Upadhyay, Rochan R. Univ Texas Austin ICES Austin TX 78712 USA

The quadrature method of moments (QMOM) has been widely used for the simulation of the evolution of moments of the aerosol general dynamic equations. However, there are several shortcomings in a crucial component of the method, the product-difference (P-D) algorithm. The P-D algorithm is used to compute the quadrature points and weights from the moments of an unknown distribution. The algorithm does not work for all types of distributions or for even reasonably high-order quadrature. In this work, we investigate the use of the chebyshev algorithm and show that it is more robust than the P-D algorithm and can be used for a wider class of problems. The algorithm can also be used in a number of applications, where accurate computations of weighted integrals are required. We also illustrate the use of QMOM with the chebyshev algorithm to solve several problems in aerosol science that could not be solved using the P-D algorithm. (C) 2011 Elsevier Ltd. All rights reserved.

关键词： Aerosol general dynamic equation Moments Quadrature method of moments chebyshev algorithm Product-difference algorithm

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Polynomials orthogonal with respect to exponential integrals

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NUMERICAL algorithmS 2015年第1期70卷 215-226页

作者： Gautschi, Walter Purdue Univ Dept Comp Sci W Lafayette IN 47907 USA

Moment-based methods and related Matlab software are provided for generating orthogonal polynomials and associated Gaussian quadrature rules having as weight function the exponential integral E (nu) of arbitrary positive order nu supported on the positive real line or on a finite interval [0,c], c > 0. By using the symbolic capabilities of Matlab, allowing for variable-precision arithmetic, the codes provided can be used to obtain as many of the recurrence coefficients for the orthogonal polynomials as desired, to any given accuracy, by choosing d-digit arithmetic with d large enough to compensate for the underlying ill-conditioning.

关键词： Orthogonal polynomials Exponential integrals chebyshev algorithm Matlab software

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Estimates of the trace of the inverse of a symmetric matrix using the modified chebyshev algorithm

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NUMERICAL algorithmS 2009年第3期51卷 309-318页

作者： Meurant, Gerard 30 Rue Sergent Bauchat F-75012 Paris France

In this paper we study how to compute an estimate of the trace of the inverse of a symmetric matrix by using Gauss quadrature and the modified chebyshev algorithm. As auxiliary polynomials we use the shifted chebyshev polynomials. Since this can be too costly in computer storage for large matrices we also propose to compute the modified moments with a stochastic approach due to Hutchinson (Commun Stat Simul 18:1059-1076, 1989).

关键词： Trace Inverse chebyshev algorithm

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Forecast of water using improved chebyshev neural network

Shiyou Huagong Gaodeng Xuexiao Xuebao/Journal of Petrochemic...

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Shiyou Huagong Gaodeng Xuexiao Xuebao/Journal of Petrochemical Universities 2005年第1期18卷 70-72页

作者： Chen, Xue-Yan Li, Ping Yuan, Yi Shi, Xiang-Xing School of Information Engineering Liaoning University of Petroleum and Chemical Technology Fushun 113001 China

A kind of chebyshev neural network which was improved from algorithm and network structure was presented according to existing insufficiency of chebyshev neural network. Improved chebyshev neural network does not only accord with the basic characteristics of biology neural network, but also has a simple algorithm, a high speed convergence of learning process, and input value of this network can be random. It is a multi-input network structure, so it expands the identification ability and learning adaptation of the network, which has excellent characteristics in the linear and nonlinear accurate approximation. This relatively new technique of improved chebyshev neural network was proposed to modeling and forecasting the water demand in urban areas. The simulation results indicate that the improved chebyshev neural network offers an effective way to forecast domestic water demand. It has a good estimate ability and high convergent speed with the same precision compared to the general network of BP.

关键词： chebyshev algorithm Forecast Neural network Water demand

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A vector chebyshev algorithm

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NUMERICAL algorithmS 1998年第1-2期17卷 33-50页

作者： Roberts, DE Napier Univ Dept Math Edinburgh EH14 1DJ Midlothian Scotland

We consider polynomials orthogonal relative to a sequence of vectors and derive their recurrence relations within the framework of Clifford algebras. We state sufficient conditions for the existence of a system of such polynomials. The coefficients in the above relations may be computed using a cross-rule which is linked to a vector version of the quotient-difference algorithm, both of which are proved here using designants. An alternative route is to employ a vector variant of the chebyshev algorithm. This algorithm is established and an implementation presented which does not require general Clifford elements. Finally, we comment on the connection with vector Pade approximants.

关键词： Clifford algebras orthogonal polynomials quotient-difference algorithm chebyshev algorithm vector Pade approximants designants

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ON GENERATING ORTHOGONAL POLYNOMIALS

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SIAM JOURNAL ON SCIENTIFIC AND STATISTICAL COMPUTING 1982年第3期3卷 289-317页

作者： GAUTSCHI, W

We consider the problem of numerically generating the recursion coefficients of orthogonal polynomials, given an arbitrary weight distribution of either discrete, continuous, or mixed type. We discuss two classical methods, respectively due to Stieltjes and chebyshev, and modern implementations of them, placing particular emphasis on their numerical stability properties. The latter are being studied by analyzing the numerical condition of appropriate finite-dimensional maps. A number of examples are given to illustrate the strengths and weaknesses of the various methods and to test the theory developed for them.

关键词： orthogonal polynomials recurrence relations for orthogonal polynomials Stieltjes procedure discretized Stieltjes procedure chebyshev algorithm modified chebyshev algorithm condition numbers

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