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w1s_div_diff - 1 Problem Derivatives re-visited In...

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1 Problem. Derivatives re-visited In engineering and scientific problems data are often “noisy” – even if the tabulated function f ( x ) is supposed to be nice and smooth, errors in measurement and computation might affect the result. Often the errors in measuring f are independently random and roughly of the same magnitude for each x . If that is the case and you have some additional information about function f (e.g., f might be known to be linear, or quadratic, or exponential, etc.), then you can recover “the most likely” true values of the smooth f from your noisy measurements. Here we address a somewhat different problem: detecting an isolated error in the tabulation of a smooth function. We will assume that all f ( x i ) ’s are listed correctly, except for one x j for which f ( x j ) was computed or measured with much less precision. Our task (to find x j among the benign x i ’s) is not as easy as it sounds. Table 1 on the next page contains values of a smooth function measured at 11 equidistant points on the interval [2 , 3] . Ten of these measurements were carried out very accurately; the eleventh value was contaminated with a relatively large error (on the order of 10 - 3 ). Since the function values range from 1 . 4 to 1 . 8 , you are unlikely to see that error visually on the graph of f ( x ) , but it can still hurt you later when you attempt to use f ( x j ) .
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