Numerical Differentiation

Let p(x) be an interpolation polynomial approximating satisfactorily a given function f(x) over a certain interval I. We may hope that the result of differentiating p(x) will also satisfactorily approximate the corresponding derivative of f(x). However, if we observe a curve representing the polynomial approximating and oscillating about the curve representing f(x), we may anticipate the fact that even though the deviation between p(x) and f(x) be small throughout the interval, still the slope of the two curves representing them may differ quite appreciably. Also it is seen that the round-off errors of alternating sign in consecutive ordinates could affect the calculation of the derivative quite strongly if those ...

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