scholarly journals Revisited Optimal Error Bounds for Interpolatory Integration Rules

2016 ◽  
Vol 2016 ◽  
pp. 1-8 ◽  
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.

scholarly journals On Corrected Quadrature Rules and Optimal Error Bounds

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
François Dubeau
Keyword(s):  
Error Bounds ◽  
Truncation Error ◽  
Quadrature Rules ◽  
Optimal Error ◽  
First Derivative ◽  
Weight Parameter ◽  
Best Parameter ◽  
Weighted Values

We present an analysis of corrected quadrature rules based on the method of undetermined coefficients and its associated degree of accuracy. The correcting terms use weighted values of the first derivative of the function at the endpoint of the subinterval in such a way that the composite rules contain only two new values. Using Taylor’s expansions and Peano’s kernels we obtain best truncation error bounds which depend on the regularity of the function and the weight parameter. We can minimize the bounds with respect to the parameter, and we can find the best parameter value to increase the order of the error bounds or, equivalently, the degree of accuracy of the rule.

ESTIMATES FOR THE ERROR TERM IN A UNIFORM ASYMPTOTIC EXPANSION OF THE JACOBI POLYNOMIALS

2003 ◽  
Vol 01 (02) ◽  
pp. 213-241 ◽  
Author(s):  
R. WONG ◽  
Y.-Q. ZHAO
Keyword(s):  
Asymptotic Expansion ◽  
Explicit Expression ◽  
Canonical Form ◽  
Error Term ◽  
Jacobi Polynomials ◽  
Recursive Formula ◽  
Contour Integral ◽  
Integration By Parts ◽  
Uniform Asymptotic Expansion ◽  
Repeated Use

There are now several ways to derive an asymptotic expansion for [Formula: see text], as n → ∞, which holds uniformly for [Formula: see text]. One of these starts with a contour integral, involves a transformation which takes this integral into a canonical form, and makes repeated use of an integration-by-parts technique. There are two advantages to this approach: (i) it provides a recursive formula for calculating the coefficients in the expansion, and (ii) it leads to an explicit expression for the error term. In this paper, we point out that the estimate for the error term given previously is not sufficient for the expansion to be regarded as genuinely uniform for θ near the origin, when one takes into account the behavior of the coefficients near θ = 0. Our purpose here is to use an alternative method to estimate the remainder. First, we show that the coefficients in the expansion are bounded for [Formula: see text]. Next, we give an estimate for the error term which is of the same order as the first neglected term.

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

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 

scholarly journals Optimal error bounds for two-grid schemes applied to the Navier–Stokes equations

2012 ◽  
Vol 218 (13) ◽  
pp. 7034-7051 ◽  
Author(s):  
Javier de Frutos ◽  
Bosco García-Archilla ◽  
Julia Novo
Keyword(s):  
Error Bounds ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Optimal Error ◽  
Navier Stokes Equations

scholarly journals Optimal Error Bounds for Convergents of a Family of Continued Fractions

1996 ◽  
Vol 197 (3) ◽  
pp. 767-773 ◽  
Author(s):  
Yair Shapira ◽  
Avram Sidi ◽  
Moshe Israeli
Keyword(s):  
Error Bounds ◽  
Continued Fractions ◽  
Optimal Error

Norm-Preserving Dilations and Their Applications to Optimal Error Bounds

1982 ◽  
Vol 19 (3) ◽  
pp. 445-469 ◽  
Author(s):  
Chandler Davis ◽  
W. M. Kahan ◽  
H. F. Weinberger
Keyword(s):  
Error Bounds ◽  
Optimal Error

scholarly journals Errors of Gauss-Radau and Gauss-Lobatto quadratures with double end point

2019 ◽  
Vol 13 (2) ◽  
pp. 463-477
Author(s):  
Aleksandar Pejcev ◽  
Ljubica Mihic
Keyword(s):  
Explicit Expression ◽  
Error Bounds ◽  
Analytic Functions ◽  
Remainder Term ◽  
Quadrature Rules ◽  
Quadrature Formulas ◽  
End Points ◽  
End Point ◽  
Appl Math

Starting from the explicit expression of the corresponding kernels, derived by Gautschi and Li (W. Gautschi, S. Li: The remainder term for analytic functions of Gauss-Lobatto and Gauss-Radau quadrature rules with multiple end points, J. Comput. Appl. Math. 33 (1990) 315{329), we determine the exact dimensions of the minimal ellipses on which the modulus of the kernel starts to behave in the described way. The effective error bounds for Gauss- Radau and Gauss-Lobatto quadrature formulas with double end point(s) are derived. The comparisons are made with the actual errors.

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