Properties of Minimal Integration Rules. II

Abstract Finite-order weights have been introduced in recent years to describe the often occurring situation that multivariate integrands can be approximated by a sum of functions each depending only on a small subset of the variables. The aim of this paper is to demonstrate the danger of relying on this structure when designing lattice integration rules, if the true integrand has components lying outside the assumed finiteorder function space. It does this by proving, for weights of order two, the existence of 3-dimensional lattice integration rules for which the worst case error is of order O(N¯½), where N is the number of points, yet for which there exists a smooth 3- dimensional integrand for which the integration rule does not converge.

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Revisited Optimal Error Bounds for Interpolatory Integration Rules

Advances in Numerical Analysis ◽

10.1155/2016/3170595 ◽

2016 ◽

Vol 2016 ◽

pp. 1-8 ◽

Cited By ~ 1

Author(s):

François Dubeau

Keyword(s):

Error Bounds ◽

Error Term ◽

Quadrature Rules ◽

Optimal Error ◽

Integration By Parts ◽

Integration Rules

We present a unified way to obtain optimal error bounds for general interpolatory integration rules. The method is based on the Peano form of the error term when we use Taylor’s expansion. These bounds depend on the regularity of the integrand. The method of integration by parts “backwards” to obtain bounds is also discussed. The analysis includes quadrature rules with nodes outside the interval of integration. Best error bounds for composite integration rules are also obtained. Some consequences of symmetry are discussed.

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Application of an implicit and explicit integration rules

Geomechanics and Geotechnics ◽

10.1201/b10528-133 ◽

2010 ◽

pp. 855-862

Author(s):

Yunming Yang

Keyword(s):

Explicit Integration ◽

Implicit And Explicit ◽

Integration Rules

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Towards Public Services and Process Integration

Advances in Systems Analysis, Software Engineering, and High Performance Computing - Progressions and Innovations in Model-Driven Software Engineering ◽

10.4018/978-1-4666-4217-1.ch011 ◽

2013 ◽

pp. 275-287

Author(s):

Guillermo Infante Hernández ◽

Aquilino A. Juan Fuente ◽

Benjamín López Pérez ◽

Edward Rolando Núñez-Valdéz

Keyword(s):

Process Model ◽

Public Services ◽

Process Integration ◽

Domain Specific ◽

Modeling Environment ◽

Proposed Model ◽

Domain Specific Modeling ◽

Hardware Platforms ◽

Business Requirements ◽

Integration Rules

Software platforms for e-government transactions may differ in developed functionalities, languages and technologies, hardware platforms, and operating systems that support them. Those differences can be found among public organizations that share common processes, services, and regulations. This scenario hinders interoperability between these organizations. Hence, to find a technique for integrating these platforms becomes a necessity. In this chapter, a rule-based domain-specific modeling environment for public services and process integration is suggested, which consists of common identified public service elements and a set of process integration rules. This approach provides the needed integration or interoperability pursued in this domain. Furthermore a service and process model is proposed to formalize the information needed for integration of both. A set of integration rules is also presented as part of the modeling environment. This set of integration rules completes the proposed model to meet the business requirements of this domain.

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Improving the two-stage numerical integration in stability identification of oscillation with distributed delay

Advances in Mechanical Engineering ◽

10.1177/1687814018819905 ◽

2019 ◽

Vol 11 (1) ◽

pp. 168781401881990

Author(s):

Chigbogu Godwin Ozoegwu

Keyword(s):

Numerical Integration ◽

Distributed Delay ◽

Original Method ◽

Higher Order ◽

Trapezoidal Rule ◽

Point Of View ◽

Engineering Systems ◽

Two Stage ◽

Integration Rule ◽

Integration Rules

The vibration of the engineering systems with distributed delay is governed by delay integro-differential equations. Two-stage numerical integration approach was recently proposed for stability identification of such oscillators. This work improves the approach by handling the distributed delay—that is, the first-stage numerical integration—with tensor-based higher order numerical integration rules. The second-stage numerical integration of the arising methods remains the trapezoidal rule as in the original method. It is shown that local discretization error is of order [Formula: see text] irrespective of the order of the numerical integration rule used to handle the distributed delay. But [Formula: see text] is less weighted when higher order numerical integration rules are used to handle the distributed delay, suggesting higher accuracy. Results from theoretical error analyses, various numerical rate of convergence analyses, and stability computations were combined to conclude that—from application point of view—it is not necessary to increase the first-stage numerical integration rule beyond the first order (trapezoidal rule) though the best results are expected at the second order (Simpson’s 1/3 rule).

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Application of MATLAB Symbolic Maths with Variable Precision Arithmetic (VPA) to Compute Some High Order Gauss Legendre Quadrature Rules

GANIT Journal of Bangladesh Mathematical Society ◽

10.3329/ganit.v29i0.8521 ◽

1970 ◽

Vol 29 ◽

pp. 117-125

Author(s):

HT Rathod ◽

RD Sathish ◽

Md Shafiqul Islam ◽

Arun Kumar Gali

Keyword(s):

Gaussian Quadrature ◽

High Order ◽

Quadrature Rules ◽

Zeros Of Polynomials ◽

Matlab Program ◽

Present Context ◽

First Time ◽

Legendre Quadrature ◽

Weight Coefficients ◽

Integration Rules

Gauss Legendre Quadrature rules are extremely accurate and they should be considered seriously when many integrals of similar nature are to be evaluated. This paper is concerned with the derivation and computation of numerical integration rules for the three integrals: (See text for formulae) which are dependent on the zeros and the squares of the zeros of Legendre Polynomial and is quite well known in the Gaussian Quadrature theory. We have developed the necessary MATLAB programs based on symbolic maths which can compute the sampling points and the weight coefficients and are reported here upto 32 – digits accuracy and we believe that they are reported to this accuracy for the first time. The MATLAB programs appended here are based on symbolic maths. They are very sophisticated and they can compute Quadrature rules of high order, whereas one of the recent MATLAB program appearing in reference [21] can compute Gauss Legendre Quadrature rules upto order twenty, because the zeros of Legendre polynomials cannot be computed to desired accuracy by MATLAB routine roots (……..). Whereas we have used the MATLAB routine solve (……..) to find zeros of polynomials which is very efficient. This is worth noting in the present context. GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 29 (2009) 117-125 DOI: http://dx.doi.org/10.3329/ganit.v29i0.8521

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