Computational Mathematicsematics

Accuracy and Stability of Numerical Algorithms, Second by Nicholas J. Higham

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By Nicholas J. Higham

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1. 71828 .... 1. The Need for Pivoting Suppose we wish to compute an LU factorization A [f1 -1]1 [1 10] [un0 = = O

The subtraction 1- c is exact, but this subtraction produces a result of the same size as the error in c. In other words, the subtraction elevates the importance of the earlier error. In this particular example it is easy to rewrite f(x) to avoid the cancellation. 5, which is correct to 10 significant figures. To gain more insight into the cancellation phenomenon consider the subtraction (in exact arithmetic) x = a-b, where a = a(l+Lla) and b = b(l+Llb). The terms Lla and Llb are relative errors or uncertainties in the data, perhaps attributable to previous computations.

This fact underlies the success of many computations, including some of those described earlier in this chapter. 75237658077857 , - - - , - - - - - - , - - - , - - - - - - , - - ' 1 - - ' 1 - - ' 1 ' x XX X ~ ,. l" ,r fl(r(x» X ~-. 6. 606 + (k - 1)r52; solid line is the "exact" r(x). 8. Here we simply give a revealing numerical example (due to W. Kahan). Define the rational function r(x) = 622 - x(751 - x(324 - x(59 - 4x))) , 112 - x(151- x(72 - x(14 - x))) which is expressed in a form corresponding to evaluation of the quartic polynomials in the numerator and denominator by Horner's rule.

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