Newton-Cotes integration formula

A. Newton-Cotes formulae for the numerical integration of the Schrodinger equation... 

Z. Kalogiratou and T. E. Simos, Newton-Cotes formulae for long-time integration. Journal of Computational and Applied Mathematics, 2003, 158(1), 75-82. 

The Newton-Cotes formulae use a constant distance between the points within the integration interval. They can be close, open, or semiopen and they allow us to obtain the expression of the local error depending on h. 

A simple way to estimate the error with an algorithm based on the extended Newton—Cotes formulae is to compare the results by doubling the integration step. 

The Trapezoid and Simpson s rule belong to a family of integration formulas called the Newton-Cotes family. Abramowitz and Stegun (1964) provide a family of 10 Newton-Cotes formulas. They also present six additional Newton-Cotes formulas of the open type, that is, the functional values at end points (yo and y ) are not included in the integration formula. These latter formulas are particularly useful when the function values at the end points are unbounded. The first two Newton-Cotes formula of the open type are... 

In all the cases of Newton-Cotes Formulae the error does not grow as the integration proceeds. 

The three Newton-Cotes formulas of integration derived in the previous sections are summarized in Table 4.4. 

In the derivation of the Newton-Cotes formulas, the function y = f(x) is approximated by the Gregory-Newton polynomial Pjix) of degree n with remainder R x). The evaluation of the integral is performed ... 

There are three functions in MATLAB, trapz.m, quad.m, and quadS.m, that numerically evaluate the integral of a vector or a function using different Newton-Cotes formulas ... 

In the development of the Newton-Cotes formulas, we have assumed that the interval of integration could be divided into segments of equal width. This is usually possible when integrating continuous functions. However, if experimental data are to be integrated, such data may be used with a variable-width segment, it has been suggested by Chapra and Canale [4] that a combination of the trapezoidal rule with Simpson s rules may be feasible for integrating certain sets of unevenly spaced data points. 

Numerical differentiation and integration, including differentiation by backward, forward, and central finite differences Newton-Cotes formulas and the Gauss Quadrature... 

Newton-Cotes Integration Formulas (Equally Spaced Ordinates) for Functions of One Variable The definite integral la fix) dx is to be evaluated. 

Analysis of the digitized peak shapes is critical for the calculation of M2. The least-squares method was used to determine the base line for the moment calculations. Detailed description of the data analysis may be found in Ref. 1. The integrations of the integrals in Equations 1 and 4 were done by Bode s rule (Newton-Cotes four-point formula). 

The above methods have been produced by the well known Open Newton-Cotes integral formulae. 

T. E. Simos, Closed Newton-Cotes Trigonometrically-Fitted Formulae for Numerical Integration of the Schrodinger Equation, Computing Letters, 2007, 3(1), 45-57. 

T. E. Simos, High-order closed Newton-Cotes trigonometrically-fitted formulae for longtime integration. Computer Physics Communications, in press. 

One of the easiest ways to obtain useful quadrature formulas for our purpose is to use Lagrangian interpolation on an equally spaced mesh and integrate the result. This class of quadrature formulas is called the Newton-Cotes... 

An alternative closed Newton-Cotes quadrature formula of second order can be obtained by a polynomial of degree 1 which passes through the end points. This quadrature formula is called the trapezoid rule. In 2D this surface integral approximation requires the integrand values at the GCV corners. 

When we have too few points to justify linearizing the function between adjacent points (as the trapezoidal integration does) we can use an algorithm based on a higher-order polynomial, which thereby can more faithfully represent the curvature of the function between adjacent measurement points. The Newton-Cotes method does just that for equidistant points, and is a moving polynomial method with fixed coefficients, just as the Savitzky-Golay method used for smoothing and differentation discussed in sections 8.5 and 8.8. For example, the formula for the area under the curve between x, and xn, is... 

Many algorithms have been proposed to perform the numerical integration of functions. We consider only two femilies of algorithms, which are the basis for the development and implementation of an even number of general programs for the numerical integration the Newton-Cotes and the Gauss formulae. 

The second way of employing the Newton-Cotes numerical integration formula is to consider it with quadratic interpolations or with three points approximation nevertheless, for that the interval [a,b] is now more grainy divided, namely in 2n equal sub-intervals ... 

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