Roy Vincent L. Canseco, MSEE Jan 2014
An interpolating function fits the data points exactly. Quadrature rules are based on polynomial interpolation. The integral of the interpolant approximates the integral of the function of interest.
An n-point quadrature formula is of the form
It is approximated by a quadrature rule of the form
We integrate a polynomial interpolant to come up with a quadrature rule.
The Lagrange polynomial for datapoints (t[i], y[i]) is of the form
We integrate that.
Comparing the above with the quadrature rule and taking into account that y[i] in the data set is equal to the value of function f at t[i] , we determine that the weights function w(t[i]) for this case must be.