Xi - d, i = 1: n, which does not change s~), but a good estimate is not always available and there are alternative one-pass algorithms that will always produce an acceptably accurate answer. 6b) after which s~ = Qnl(n -1). Note that the only subtractions in these recurrences are relatively harmless ones that involve the data Xi. 6) produces the exact answer. 7). The problem of computing the sample variance illustrates well how mathematically equivalent formulae can have different numerical stability properties.
1. Backward and forward errors for y computed. = f(x). 3. If the backward error is no larger than these uncertainties then the computed solution can hardly be criticized-it may be the solution we are seeking, for all we know. The second attraction of backward error analysis is that it reduces the question of bounding or estimating the forward error to perturbation theory, which for many problems is well understood (and only has to be developed once, for the given problem, and not for each method).
00000005 to the significant digits shown. 00000000 x 10-8 = 1. 3245 47 66. 00000006. 00000005. We see that Algorithm 2 obtains very inaccurate values of eX - 1 and log eX, but the ratio of the two quantities it computes is very accurate. Conclusion: errors cancel in the division in Algorithm 2. A short error analysis explains this striking cancellation of errors. We assume that the exp and log functions are both computed with a relative error not exceeding the unit roundoff u. The algorithm first computes fj = eX (1 + 15), 1151 ::; u.