Computational Mathematicsematics

Numerical methods in fluid dynamics: lectures given at the by E. F. Brezzi

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By E. F. Brezzi

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Extra resources for Numerical methods in fluid dynamics: lectures given at the 3rd 1983 session of the Centro internazionale matematico estivo

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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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