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Newton-Cotes formulas

The main objective is to numerically compute an integral. In order to do so, we shall introduce Lagrange interpolation polynomials, present the notions of quadrature rules and of exact quadrature rules on polynomial spaces. Finally, we shall define Newton-Cotes formulas and the particular cases of composite formulas for rectangles, trapezes and Simpson’s formula.

Let be a continuous function. The aim is to compute the following integral:

We start by partitioning the interval into intervals with

Therefore:

We then have to determine the following integral:

It is easy to switch to the interval via the following substitution

or

therefore

where

{{{Quadrature rule}}} Let be a continuous function on . The application

is said to be a quadrature rule. The points defined by

are called points of integration of the quadrature and

are the weights of the quadrature rule . The weights and the points of integration are chosen such that:

{{{Exact quadrature rule}}} Let be the set of polynomials of degree at most equal to q. The quadrature rule is said to be exact in if

Alternatively stated, if

{{{Exact quadrature rule: Lagrange basis}}} {{Theorem.}} Let be the Lagrange-basis of relatively to the points

We have the following equivalence:

{{{Newton-Cotes formula}}} Newton-Cotes's method is obtained from the above mentioned methods for which the points are equidistant

where . This being so:

We can now approach by the composite formula :

where

Following are some classical Newton formulas as well as the degree of . |Degree p|Designation| |0|Method of rectangles| |1|Method of trapezes| |2|Simpson's method| |3|3/8 Simpson's method| |4|Boole's method| {{{Method of rectangles}}} The method of rectangles is a one-point method. In this case, and . Lagrange basis associated to is

Consequently:

The quadrature rule is defined by:

We thus have the composite rectangle formula:

{{{Method of trapezes}}} The method of trapezes is a two-point method. In this case, and . Lagrange basis associated to is

Consequently:

The quadrature rule is defined by:

We thus have the composite trapezes formula:

{{{Simpson's method}}} Simpson's method is a three-point method. In this case, and . Lagrange basis associated to is

Consequently:

The quadrature rule is defined by:

We thus have Simpson's composite formula: