MATH 417 Lecture Notes - Lecture 13: Hermite Interpolation, Lagrange Polynomial
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Guaranteed convergence under slight restrictions => f is continuos and in a closed and bounded interval. (cid:12254) (cid:12254) Ex: use a xed point iteration to nd a root of (x^3) - 4x -3 lying in [2,3). In particular , specify a starting point and a xed point function. Thus we must only show that f(2) > 2 and f(3) < 3. Must be twice continuously di erentiable in order for it to satisfying newto"s method. Ex: f(x) = (x^3) - 4x - 3; nd a root in [2,3] using newton"s method. Will be asked to write down a divided di erence table. (cid:12254) (cid:12254) Data from a hermite interpolation only have two or three points. Ex: lagrange interpolation: let x0 = 1, x1 = 2, x2 = 4 and f(x) = e^x. Bound |f(3) - p(3)|, and max|f(x) - p(x)|, 1.
