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By Nicholas J. Higham

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

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To gain more insight into the cancellation phenomenon consider the subtraction (in exact arithmetic) x — a — b, where a = a(1 + a) and b = b(l + b). The terms a and b are relative errors or uncertainties in the data, perhaps attributable to previous computations. With x = a — b we have 10 PRINCIPLES OF FINITE PRECISION COMPUTATION The relative error bound for x is large when |a — b\ |a| + |b|, that is, when there is heavy cancellation in the subtraction. This analysis shows that subtractive cancellation causes relative errors or uncertainties already present in a and b to be magnified.

This reflects instability of the algorithm rather than ill conditioning of the problem because the condition number The source of the instability is the large first multiplier, a 2 1 /a 1 1 = 107. 17. Rounding Errors Are Not Random Rounding errors, and their accumulated effect on a computation, are not random. This fact underlies the success of many computations, including some of those described earlier in this chapter. 6. 606 + (k - 1)2 -52 ; solid line is the "exact" r(x). 8. Here we simply give a revealing numerical example (due to W.

The solutions themselves are similar, both being accurate to three significant figures in each component but incorrect in the fourth significant figure. This is the accuracy we would expect from GEPP because of the rule of thumb "forward error < condition number x backward error". 9. For general n, the accuracy and stability of Cramer's rule depend on the method used to evaluate the determinants, and satisfactory bounds are not known even for the case where the determinants are evaluated by GEPP.

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