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 .
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.
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
From my previous on Lagrange Interpolation,
Here is the Interpolatory quadrature that Andrew Perry and I wrote:
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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.
and the weight is for .
Here is a Matlab code implementing Gauss-Chebyshev Quadrature:
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