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By Nicholas J. Higham

A remedy of the behaviour of numerical algorithms in finite precision mathematics that mixes algorithmic derivations, perturbation idea, and rounding blunders research. software program practicalities are emphasised all through, with specific connection with LAPACK and MATLAB.

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The terms omitted constitute the truncation error, and for many methods the size of this error depends on a parameter (often called h, “the stepsize”) whose appropriate value is a compromise between obtaining a small error and a fast computation. Because the emphasis of this book is on finite precision computation, with virtually no mention of truncation errors, it would be easy for the reader to gain the impression that the study of numerical methods is dominated by the study of rounding errors.

4) where the sample mean Computing from this formula requires two passes through the data, one to compute 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. 10 SOLVING LINEAR EQUATIONS 13 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.

The reason for the pronounced oscillating behaviour of the relative error (but not the residual) for the inverse Hilbert matrix is not clear. An example in which increasing the precision by several bits does not improve the accuracy is the evaluation of y = x + a sin(bx), a = 10-8, b = 224. 4 plots t versus the absolute error, for precisions u = 2-t, t = 10:40. 55×10-9, for t less than about 20 the error is dominated by the error in representing x = l/7. For 22 < t < 31 the accuracy is (exactly) constant!

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