Comparison of three quadrature rules

Symmetry

In this research, some new and efficient quadrature rules are proposed involving the combination of function and its first derivative evaluations at equally spaced data points with the main focus on their computational efficiency in terms of cost and time usage. The methods are theoretically derived, and theorems on the order of accuracy, degree of precision and error terms are proved. The proposed methods are semi-open-type rules with derivatives. The order of accuracy and degree of precision of the proposed methods are higher than the classical rules for which a systematic and symmetrical ascendancy has been proved. Various numerical tests are performed to compare the performance of the proposed methods with the existing methods in terms of accuracy, precision, leading local and global truncation errors, numerical convergence rates and computational cost with average CPU usage. In addition to the classical semi-open rules, the proposed methods have also been compared with some Gau...

International Journal of Multidisciplinary Research and Development, 2015

In this paper, two mixed quadrature rules (I and II) are constructed for approximating the evaluation of numerical integration, by blending Cleanshaw-Curtis five point rule of precision five with another two quadrature rules (five point Fejer’s second rule and Boole’s rule) of same precision, provided that the new constructed rules are of precision seven. In addition, another mixed quadrature rule (III) has been formed by taking the combination of two newly formed quadrature rules (I and II). Moreover, errors of the constructed rules are analyzed and approximated.

In this paper, a new modified algorithm is derived for combined numerical integration. Numerical integration is used to evaluate definite integral which cannot be evaluated analytically. In order to obtain higher order accuracy in the solution, mostly higher order quadrature rules are used which also introduces rounding off errors. In order to mitigate such errors, it has been proposed to use a combination of lower order rules to improve the accuracy and reduce error. Several definite integrals have been approximated and the results have been compared with the existing rules and a rule proposed by (Md.Amanat Ullah, 2015). The rules from the closed quadrature family, namely, Weddle's rule, Boole's rule, Simpson's rule and Trapezoidal rule have also been used. It has been found that the new proposed modified algorithm attains improved order of accuracy in comparison of the existing rules and the rule of Md.Amanat Ullah for a fixed number of subintervals.

This article deals with the numerical integration techniques in new way. We develop a new quadrature rule for numerical integration. Apart from all existing rule which uses vertical stripes in different ways to find the area under the curve. In this study, we use small circles to approximate it. Though the approximation in this rule is very rough, however the formulae consist the mathematical constant . We also evaluate the error of this formula.

SINDH UNIVERSITY RESEARCH JOURNAL (SCIENCE SERIES), 2019

In this paper, a new modified algorithm is derived for combined numerical integration. Numerical integration is used to evaluate definite integral which cannot be evaluated analytically. In order to obtain higher order accuracy in the solution, mostly higher order quadrature rules are used which also introduces rounding off errors. In order to mitigate such errors, it has been proposed to use a combination of lower order rules to improve the accuracy and reduce error. Several definite integrals have been approximated and the results have been compared with the existing rules and a rule proposed by (Md.Amanat Ullah, 2015). The rules from the closed quadrature family, namely, Weddle's rule, Boole's rule, Simpson's rule and Trapezoidal rule have also been used. It has been found that the new proposed modified algorithm attains improved order of accuracy in comparison of the existing rules and the rule of Md.Amanat Ullah for a fixed number of subintervals.

Bulletin of Pure & Applied Sciences- Mathematics and Statistics, 2015

A mixed quadrature rule of blending Clenshaw-Curties five point rule and Gauss-Legendre 3 point rule is formed. The mixed rule has been tested and found to be more effective than that of its constituent Clenshaw-Curtis five point rule for the approximate evaluation of the integral of an analytic function over a line segment in complex plane. An asymptotic error estimate of the rule has been determined and the rule has been numerically verified.

Numerical integration plays very important role in Mathematics. There are a large number of numerical integration methods in the literature and this paper overviews on the most common one, namely the Quadrature method including the Trapezoidal, Simpson's and Weddle's rule. Different procedures are compared and tried to evaluate the more accurate values of some definite integrals. Then it is sought whether a particular method is suitable for all cases. A combined approach of different integral rules has been proposed for a definite integral to get more accurate value for all cases.

Applied Mathematics and Computation, 2004

A new quadrature formula has been proposed which uses weight functions derived from a new probabilistic approach. Unlike the complicatedly structured quadrature formulae of Gauss, Hermite and others of similar type, the proposed quadrature formula only needs the values of integrand at user-defined equidistant points in the interval of integration and weight functions are not constants. The quadrature formula has been compared empirically with two other simple fundamental methods of integration. The percentage relative gain in efficiency of the quadrature formula with respect to fundamental methods of integration has been computed for certain selected functions and with different number of node points in the interval of integration. It has been observed that the proposed quadrature formula produces significantly better results than the other simple fundamental methods of integration.

Applied Mathematics and Computation, 2005

Among all integration rules with n points, it is well-known that n-point Gauss-Legendre quadrature rule Z 1 À1 f ðxÞ dx ' X n i¼1 w i f ðx i Þ has the highest possible precision degree and is analytically exact for polynomials of degree at most 2n À 1, where nodes x i are zeros of Legendre polynomial P n ðxÞ, and w i 's are corresponding weights. In this paper we are going to estimate numerical values of nodes x i and weights w i so that the absolute error of introduced quadrature rule is less than a preassigned tolerance 0 , say 0 ¼ 10 À8 , for monomial functions f ðxÞ ¼ x j ; j ¼ 0; 1;. .. ; 2n þ 1: (Two monomials more than precision degree of Gauss-Legendre quadrature rules.) We also consider some conditions under which the new rules act, numerically, more q Research supported in part by MIM Grant no. A82-109.

The concept of mixed quadrature rules has been used for construction of such types of rules of precision 5, 7, and 9. The error associated with these rules has been analyzed and some definite real integrals have been approximately evaluated by the rules and found to yield good approximation to the exact values of the integrals otherwise obtained.