The Mathematical Theory of Huygence' Principle by Baker B.B., Copson E.T.

By Baker B.B., Copson E.T.

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V) f (x) := ex2/3 - 1. For comparison, we give together with f (x) also the result of the first two terms of the power series expansion of f (x). ). However, the truncated power series gives results with increasing relative accuracy. (vi) The formula (x(x,)2 2 n n i-1:n gives the standard deviation of a sequence of data xl, ... , xn in a way as implemented in many pocket calculators. 3. The zero in the table arises because the calculator checks whether the computed argument of the square root is negative (which cannot happen in exact arithmetic) and, if so, sets or, equal to zero, but if the argument is positive, the positive result is purely due to round off.

ERROR Because f (x) = 1 for all x not a multiple of jr , there is a drastic decrease in accuracy for decreasing x due to division by the small number 1 - cos2 x, and ultimately an error occurs due to division by a computed zero. (iv) f (x) := sin x f 1 - sin2 x. 9588 ... 87... 9572 ... 91... 57... 39 The correct digits are underlined. We have f (x) = tan x for all x, and 26 The Numerical Evaluation of Expressions here the loss of accuracy is due to the fact that f (x) has a pole at x = 90°, together with the cancellation in the denominator.

We look closer at the example f (x) = x2, which always gives the range. Indeed, the square is an elementary operation, defined such that this holds. What happens with the equivalent expression x * x? (i) If f (x) := x * x, and x := [-r, r], with 0 < r < 1, then f (x) = [-r, r] [-r, r] = [-r2, r2], but the range of values off is only [f (x) I X E x} = [0, r2]. The reason for the overestimation of the range of values of f is as follows. In the multiplication x * x = O{z * X I X E X, X E x}, each z is multiplied with every element of x, but in the calculation of the range of values, x is multiplied only with itself, giving a tighter result.

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