Quadrature is the numerical evaluation of definit inegrals in the following form:

This can approximated by in this form:

There are many forms of quadratures available. Each of these methods are in the same form above. The only difference is the choice of called the *quadrature nodes, *and weights .

**Interpolatory Quadrature:**

*Previous* subject area had been about polynomial interpolation. The integral of the polynomial intepolant of the maybe be used to derive the quadrature formula of . Lets this polynomial integral be denoted as from this point on.

through points.

=

*Example using Lagrange form:*

The polynomial interpolation in the Lagrange form is: . The integral of a given function which is interpolated in Lagrange form can be approximated by:

Interchanging the order of summation and integration, we obtain:

=

The quadrature formula is then:

the weight is now . The most commonly used and the simplest quadrature formula are those with equally spaced nodes and its knows as the Newton-Cotes formulas. Here is an example of 2-point closed Newton-Cotes formula using the Lagrange form above:

set , and . The interpolatory quadrature is equivalent to , where

and

From my previous on Lagrange Interpolation,

and

Here is the Interpolatory quadrature that Andrew Perry and I wrote:

INSERT CODE HERE!

INSERT NUMERICAL RESULT HERE!

This form can be extended for point of any Polynomial base interpolation:

=

The above equation can be used for any of the Rienmann Sum rules.

**Gauss-Chebyshev Quadrature Rule:**

The Newton-Cotes formulas are based on interpolating polynomials. However, interpolation at equidistant points experiences the Runge Phenomenon even for well-behaved function. For explicitly given , other choices of quadrature should be considered, such as the Gaussian quadrature.

where

and the weight is for .

Here is a Matlab code implementing Gauss-Chebyshev Quadrature:

INSERT CODE HERE

INSERT NUMERICAL RESULT