Comparison of Newton–Cotes and Gaussian methods of quadrature

Applied Mathematics and Computation, 2005

This paper discusses on numerical improvement of the Newton-Cotes integration rules, which are in forms of: Z b¼aþnh a f ðxÞ dx ' X n k¼0 B ðnÞ k f ða þ khÞ: It is known that the precision degree of above formula is n + 1 for even n's and is n for odd n's. However, if its bounds are considered as two additional variables (i.e. a and h in fact) we reach a nonlinear system that numerically improves the precision degree of above integration formula up to degree n + 2. In this way, some numerical examples are given to show the numerical superiority of our approach with respect to usual Newton-Cotes integration formulas.

Experimental Mathematics, 2005

The authors have implemented three numerical quadrature schemes, using the new Arbitrary Precision (ARPREC) software package, with the objective of seeking a completely "automatic" arbitrary precision quadrature facility, namely one that does not rely on a priori information of the function to be integrated. Such a facility is required, for example, to permit the experimental identification of definite integrals based on their numerical values. The performance and accuracy of these three quadrature schemes are compared using a suite of 15 integrals, ranging from continuous, well-behaved functions on finite intervals to functions with vertical derivatives and integrable singularities at endpoints, as well as several integrals on an infinite interval.

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...

In this paper, three quadrature rules for numerical integration are compared: - Boole's rule (Newton-Cotes formula of 4-th order); - Gauss-Legendre; - Tanh-Sinh (Double exponential formula). These rules are applied to the same function, and with the same number of nodes. Calcpad mathematical and engineering platform are used for the calculations. Then, the obtained approximation errors are compared.

The numerical integration of polynomial functions is one of the most interesting processes for numerical calculus and analyses, and represents thus, a compulsory step especially in finite elements analyses. Via the Gauss quadrature, the users concluded a great inconvenience that is processing at certain points which not required the based in finite element method points for deducting the form polynomials constants. In this paper, the same accuracy and efficiency as the Gauss quadrature extends for the numerical integration of the polynomial functions, but as such at the same points and nods have chosen for the determination of the form polynomials. Not just to profit from the values of the polynomials at those points and nods, but also from their first derivatives, the chosen points positions are arbitrary and the resulted deducted formulas are therefore different, as will be presented bellow and implemented.

Applied Mathematics and Computation, 2007

In MATLAB environment, a new quadrature routine based on Gaussian quadrature rule has been developed. Its performance is evaluated for improper integrals, rapidly oscillating functions and other types of functions requiring a large number of evaluations. This performance is compared against the other quadrature routines written for MATLAB in terms of capability, accuracy and computation time. It is found that our routine rates quite favourably.

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.

Applied Mathematics and Computation, 2005

One of the less-known integration methods is the weighted Newton-Cotes of closed type quadrature rule, which is denoted by:

Mathematical Theory and Modeling, 2019

In this research paper, a new family of numerical integration of closed newton cotes is introduced which uses the mean of arithmetic and geometric means at derivative value for the Evaluation of Definite Integral. These quadrature methods are shown to be more efficient than the existing quadrature rules. The error terms are obtained by using the concept of precision. Finally, the accuracy of proposed method is verified with numerical examples and the results are compared with existing methods numerically and graphically. Keywords – Numerical Integration, Closed Newton-cotes formula, Definite integral, Arithmetic mean, Geometric mean, Numerical examples. DOI : 10.7176/MTM/9-5-06 Publication date : May 31 st 2019

This study explored the piecewise approach of the closed Newton-Cotes quadrature formulas (Trapezoidal, Simpson's 1/3, and 3/8 rules) and how well they work with different kinds of functions in terms of convergence and accuracy. MATHEMATICA software was used to approximate the integrals and determine their errors, allowing for a comparison of convergence and accuracy. Simpson's 1/3 and 3/8 rules consistently outperformed the trapezoidal rule, demonstrating faster convergence and greater accuracy across a wide range of functions. However, as tolerance levels increased to a considerable magnitude, Simpson's 3/8 rule emerged as the most robust among the three methods. We recommend investigating various domains to substantiate the findings of this study including a comprehensive error analysis that includes truncation error, round-off error, and error bounds to provide a more detailed understanding of the sources and magnitude of errors and to include higher-dimensional integrals to provide valuable insights into the robustness of these methods.