Nyström-Clenshaw-Curtis quadrature for integral equations with discontinuous kernels

Kang, Sheon-Young; Koltracht, I.; Rawitscher, George

doi:10.1090/s0025-5718-02-01431-x

Cited by 30 publications

(25 citation statements)

References 23 publications

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“…The QUAD-PACK library (Piessens et al, 1983), for instance, and also well-known commercial libraries, such as NAG (http://www.nag.co.uk) and IMSL (http://www.vni.com), make use of the CC quadrature to numerically integrate functions with singularities, oscillatory functions or functions with weights. Further, the CC quadrature has been used to solve integral equations (Sloan and Smith, 1982;Piessens, 2000;Kang et al, 2002). In particular, if f is Riemann-integrable, it is possible to show that the numerical solution, obtained using the CC quadrature, converges to the exact solution of the integral equation as n → ∞ (Sloan and Smith, 1978).…”

Section: Section 2 Average Run Lengths For the Csewma Control Chartmentioning

confidence: 99%

Evaluation of the run-length distribution for a combined Shewhart-EWMA control chart

Capizzi

Masarotto

2009

Stat Comput

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A simple algorithm is introduced for computing the run length distribution of a monitoring scheme combining a Shewhart chart with an Exponentially Weighted Moving Average control chart. The algorithm is based on the numerical approximation of the integral equations and integral recurrence relations related to the run-length distribution. In particular, a modified Clenshaw-Curtis quadrature rule is applied for handling discontinuities in the integrand function due to the simultaneous use of the two control schemes. The proposed algorithm, implemented in R and publicy available, compares favourably with the Markov chain approach originally used to approximate the run length properties of the combined Shewhart-EWMA.

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Section: Section 2 Average Run Lengths For the Csewma Control Chartmentioning

confidence: 99%

Evaluation of the run-length distribution for a combined Shewhart-EWMA control chart

Capizzi

Masarotto

2009

Stat Comput

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“…If the function in the integrand of (28) is represented by its Chebyshev expansion (19), the integration in (28) can be done analytically (Curtis-Clenshaw integration [18]). This solves the problem because the coefficients C j in (29) must be expressible in terms of c j but these relations are not particularly simple.…”

Section: Anti-derivativementioning

confidence: 99%

“…The possibility that the configuration space non-relativistic scattering problem might be formulated in terms of an inhomogeneous Volterra equation of the second kind was first noted by Drukarev [25] more than half a century ago. Although his method soon received the necessary mathematical background [26], the Volterra equation approach went into oblivion to be revived only recently [27][28][29].…”

Section: Configuration Spacementioning

confidence: 99%

Semi-spectral Chebyshev method in quantum mechanics

Deloff¹

2007

Annals of Physics

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Traditionally, finite differences and finite element methods have been by many regarded as the basic tools for obtaining numerical solutions in a variety of quantum mechanical problems emerging in atomic, nuclear and particle physics, astrophysics, quantum chemistry, etc. In recent years, however, an alternative technique based on the semi-spectral methods has focused considerable attention. The purpose of this work is first to provide the necessary tools and subsequently examine the efficiency of this method in quantum mechanical applications. Restricting our interest to timeindependent two-body problems, we obtained the continuous and discrete spectrum solutions of the underlying Schrö dinger or Lippmann-Schwinger equations in both, the coordinate and momentum space. In all of the numerically studied examples we had no difficulty in achieving the machine accuracy and the semi-spectral method showed exponential convergence combined with excellent numerical stability.

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“…Semiseparable matrices appear in several types of applications, e.g., the field of integral equations [24,27,28], boundary value problems [26,24,30,39], in the theory of Gauss-Markov processes [29], time varying linear systems [10,23], in statistics [25], acoustic and electromagnetic scattering theory [9] and rational interpolation [40].…”

Section: Introductionmentioning

confidence: 99%

A note on the representation and definition of semiseparable matrices

Vandebril

Barel

Mastronardi

2005

Numerical Linear Algebra App

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In this paper the definition of semiseparable matrices is investigated. Properties of the frequently used definition and the corresponding representation by generators are deduced. Corresponding to the class of tridiagonal matrices another definition of semiseparable matrices is introduced preserving the nice properties dual to the class of tridiagonal matrices. Several theorems and properties are included showing the viability of this alternative definition.Because of the alternative definition, the standard representation of semiseparable matrices is not satisfying anymore. The concept of a representation is explicitely formulated and a new kind of representation corresponding to the alternative definition is given. It is proved that this representation keeps all the interesting properties of the generator representation.As an example of the effectivity of the new representation, we design on O(n) algorithm for the multiplication of a semiseparable matrix given by the new representation, with a vector. Because of the alternative definition, the standard representation of semiseparable matrices is not satisfying anymore. The concept of a representation is explicitely formulated and a new kind of representation corresponding to the alternative definition is given. It is proved that this representation keeps all the interesting properties of the generator representation.As an example of the effectivity of the new representation, we design on O(n) algorithm for the multiplication of a semiseparable matrix given by the new representation, with a vector.

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Nyström-Clenshaw-Curtis quadrature for integral equations with discontinuous kernels

Cited by 30 publications

References 23 publications

Evaluation of the run-length distribution for a combined Shewhart-EWMA control chart

Evaluation of the run-length distribution for a combined Shewhart-EWMA control chart

Semi-spectral Chebyshev method in quantum mechanics

A note on the representation and definition of semiseparable matrices

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