Newton Cotes integration

In Newton-Cotes integration, we place the support points uniformly, 

To improve the accuracy of the integral estimate, we can either move to a higher interpolation order, or more conveniently, subdivide the integration domain into a number of subdomains, 

An efficient means to improve accuracy is to estimate the error in each interval separately, and adaptively add new support points only to those intervals where rapid changes in /(x) yield the highest errors. We can further increase the accuracy by computing several approximate values of the definite integral with different values of h = b - a)/N, and extrapolating the results to the limit A 0. In MATLAB, adaptive refinement is applied with Simpson s rule in quad. 

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

The closed Newton-Cotes integral rules are given by ... 

The above method has been obtained by the simplest Open Newton-Cotes integral rule. 

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

Table 8.7-1 The coefficients for Newton-Cotes integration of equidistant points.

Note that the trapezoidal rule is the first-order member of this method. In the above example, fourth-order Newton-Cotes integration for /x 0.9 yields 71=1.00, tr= 26.67, gtr= 24.00, ar= 1.72, and Np = 24.00. In fact, almost equally accurate results can already be obtained with a second-order Newton-Cotes fit. 

A more accurate and systematic way of evaluating integrals is to perform the integration numerically. In the next two sections, we derive Newton-Cotes integration formulas for equally spaced intervals and Gauss quadrature for unequally spaced points. 

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). 

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

We present closed Newton-Cotes differential methods as multilayer S5unplectic integrators. 

We apply the closed Newton-Cotes methods on the Hamiltonian system (1) and we obtain as a result that the Hamiltonian energy of the system remains almost constant as the integration proceeds. 

Closed Newton-Cotes differential methods were produced from the integral rules. For Table 1 we have the following differential methods ... 

Closed Newton-Cotes can be expressed as symplectic integrators... 

J. C. Chiou and S. D. Wu, Open Newton-Cotes differential methods as multilayer symplectic integrators, J. Chem. Phys., 1997, 107, 6894-6897. 

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

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

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 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... 

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