Newton-Cotes formulae

Newton-Cotes formulae for the numerical i Schrodinger equation... 

The classical fourth order closed Newton-Cotes formulae (Method I)... 

The closed Newton-Cotes formulae developed in ref. 104 (Method IV)... 

Theorem 1. For the Closed Newton-Cotes formulae studied in this paper we have ... 

The Closed Newton-Cotes Formula developed in ref. 106 (which is indicated as Method VI)... 

The closed Newton-Cotes formulae case k = 6 have better behaviour (generally) than the closed Newton-Cotes formulae case k = 4. The closed Newton-Cotes formulae case k = 4 have better behaviour (generally) than the closed Newton-Cotes formulae case k = 2. 

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

Values of y , y , y 2 and y 3 are required to caleulate y , . Milne s method uses Newton-Cotes formula for the predictor and Simpson s rule for the corrector. 

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. 

For many practical problems, the close forms of Newton-Cotes formulae diverge from the solution, while the points are increased. In other words, it is not suitable to use high orders of Newton-Cotes formulae. 

The other open and semiopen Newton-Cotes formulae are of purely historical interest the open formulae are less effective than the Gauss formulae, and both the open and semiopen formulae are harder than the close formulae in their... 

If a dose Newton-Cotes formula is iteratively applied to adjacent intervak, the extended Newton-Cotes formulae are obtained and they exploit the points shared by the adjacent intervak. 

The first column obtained with the Romberg method coincides with the one obtained with the CavaUeri-Simpson method. The other columns can be different with respect to the ones that we can obtain using the close Newton-Cotes formulae. In particular, the formulae with many T are different from the high-order Newton-Cotes and this provides a valid motive for the extrapolation with more elements. 

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. 

Gauss formulae have certain advantages with respect to Newton-Cotes formulae. 

Whereas the Newton-Cotes formulae become less efficient as the order increases and thus are rarely used beyond the Boole formula, the Gauss formulae become increasingly efficient for almost every problem. 

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. 

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