A record has a dropped sample

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics

Error, Chance and the Machine, Ch Approximating a function: interpolation and differentiation

Problem

A record has a dropped sample, so around the point x0x_0 only the values f(x0−h)f(x_0-h), f(x0)f(x_0) and f(x0+2h)f(x_0+2h) are available. We want weights c−1,c0,c2c_{-1},c_0,c_2 in f′(x0)≈c−1f(x0−h)+c0f(x0)+c2f(x0+2h).f'(x_0)\approx c_{-1}f(x_0-h)+c_0f(x_0)+c_2f(x_0+2h). ((a)) Derive the weights by requiring the formula to be exact for 11, x−x0x-x_0 and (x−x0)2(x-x_0)^2. ((b)) Show that (x−x0)3(x-x_0)^3 is the first power the formula fails on, and find the leading error, formula minus f′(x0)f'(x_0), in terms of hh and f′′′(x0)f'''(x_0). ((c)) For f=sin⁡f=\sin, x0=1x_0=1 and h=0.1h=0.1, compare the leading error with the error computed directly from the three sine values.

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