MACM 316 Lecture Notes - Lecture 11: Lagrange Polynomial, Mean Value Theorem, Numerical Integration
15 Oct 2018
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In many problems we cannot evaluate the integral r b a f (x)dx. For example, we may only know the values of f at some number of points xi. Numerical quadrature is the process of approximating integrals by weighted sums of the form. Both the weights ai and the nodes xi may be chosen to make the approximation as accurate as possible. For a given set of weights and nodes we refer to (1) as a numerical quadrature rule. In some applications the nodes xi may be xed, but in others there may be. Main idea: interpolate f (x) with a polynomial p (x) and integrate that instead. Write x0 = a, x1 = b, h = b a and let. P (x) = be the linear interpolating polynomial written in the lagrange basis. Therefore, by the error formula for p (x), The function (x x0)(x x1) does not change sign on [x0, x1]. E(x) = where x0 x1.
