MACM 316 Lecture Notes - Lecture 10: Intermediate Value Theorem, Richardson Extrapolation, Machine Epsilon
15 Oct 2018
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Given a function f (x), it is often not possible to evaluate its deriva- tive or integral exactly. Sometimes we may not even know what f (x) is! In this section, we"ll develop numerical methods for approximat- ing integrals and derivatives of a function. If we have a polynomial p (x) that interpolates a function f (x), the derivative or integral of p (x) will approximate the derivative or in- tegral of f (x). Note: the degree of p (x) is kept low to avoid runge"s phenomenon. We consider forming the interpolating polynomial of f (x), p (x), with the lagrange form. We then approximate the derivative of f (x) with the derivative of p (x). Suppose that x0, x1 [a, b] and f c 2[a, b]. We can construct a linear lagrange interpolant as then. Typically, we only care about the values of f at the interpolation points. For example, when x = x0, we get.
