By Nicholas J. Higham
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Extra resources for Accuracy and Stability of Numerical Algorithms, Second Edition
I=l Computing s~ from this formula requires two passes through the data, one to compute x and the other to accumulate the sum of squares. A two-pass computation is undesirable for large data sets or when the sample variance is to be computed as the data is generated. 5) This formula is very poor in the presence of rounding errors because it computes the sample variance as the difference of two positive numbers, and therefore can suffer severe cancellation that leaves the computed answer dominated by roundoff.
Note that this definition is specific to problems where rounding errors are the dominant form of errors. The term stability has different meanings in other areas of numerical analysis. 2. Mixed forward-backward error for y = computed. = f(x). 6. Conditioning The relationship between forward and backward error for a problem is governed by the conditioning of the problem, that is, the sensitivity of the solution to perturbations in the data. :1x). :1X)2, 2. OE(O,I), and we can bound or estimate the right-hand side.
For illustration, consider the system Ax = b, where A is the inverse of the 5 x 5 Hilbert matrix and bi = (-l)ii. ) We solved the system in varying precisions with unit roundoffs u = 2- t , t = 15: 40, corresponding to about 4 to 12 decimal places of accuracy. ) The algorithm used was Gaussian elimination (without pivoting), which is perfectly stable for this symmetric positive definite matrix. 3 shows t against the relative errors Ilx - xlloo/llxll oo and the relative residuals lib - Axlloo/(IIAlloo Ilxll oo ).
Accuracy and Stability of Numerical Algorithms, Second Edition by Nicholas J. Higham