By Nicholas J. Higham
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Extra resources for Accuracy and Stability of Numerical Algorithms, Second Edition
0 17 , whose purpose is to zero the vector fj, and which are determined by ratios involving the elements of fj, bear little relation to their exact counterparts, causing Ak to differ greatly from Ak for k = 11,12, .... 10). 35a)) we conclude that IIR - R112/IIA112 is bounded by a multiple of K2(A)u. 5 must eventually dip down to the unit roundoff level. We also note that IIQ - QI12 is of order u in this example, as again we can show it must be from perturbation theory. Since Q is a product of Givens rotations, this means that even though some of the intermediate Givens rotations are very inaccurate, their product is highly accurate, so in the formation of Q, too, there is extensive cancellation of rounding errors.
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. If fj = 1 then eX (1 + 15) = 1, so 1151 ::; u, which implies that the correctly rounded value of I(x) = 1 + x/2 + x 2/6 + ...
Rounding is the act of replacing a given number by the nearest p significant digit number, with some rule for breaking ties when there are two nearest. This definition of correct significant digits is mathematically elegant and agrees with intuition most of the time. 9951. 0), but does have one and three correct significant digits! A definition of correct significant digits that does not suffer from the latter anomaly states that agrees with x to p significant digits if Ix - xl is less than half a unit in the pth significant digit of x.