MACM 316 Lecture Notes - Lecture 13: Orthogonal Polynomials, Legendre Polynomials, Lagrange Polynomial
15 Oct 2018
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Document Summary
We saw earlier that newton-cotes formulas are not robust for large n, due to runge"s phenomenon in high-degree polynomial interpolation at equally-spaced nodes. We have discussed the use of composite rules at equally spaced nodes; however, these rules have lower order accuracy. In gaussian quadrature, we want to choose nodes xi that will give us higher order quadrature rules, in order to restore accuracy while maintaining robustness. Recall that a quadrature rules is of the form. If we can choose both the nodes x1, xn and the weights c1, , cn then we have 2n degrees of freedom. Note: we begin with the case [a, b] = [ 1, 1]. When n = 2, (1) is f (x)dx c1f (x1) + c2f (x2) (2) By linearity, if we nd c1, c2, x1, x2 such that (2) is exact for the monomials 1, x, x2, x3, then (1) will be of degree 3 as desired.
