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

By Nicholas J. Higham

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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. This expansion leads to the notion of condition number. :1x, the relative change in the output for a given relative change in the input, and it is called the (relative) condition number of f. :1x. As an example, consider the function f(x) = logx.

Accuracy and precision are the same for the scalar computation c = a * b, but accuracy can be much worse than precision in the solution of a linear system of equations, for example. It is important to realize that accuracy is not limited by precision, at least in theory. This may seem surprising, and may even appear to contradict many of the results in this book. 9. ) In all our error analyses there is an implicit assumption that the given arithmetic is not being used to simulate arithmetic of a higher precision.

For example, if x » y ~ z > 0 then the cancellation in the evaluation of x + (y - z) is harmless. 8. 3) nor the accuracy of the computed roots can be taken for granted. The easiest issue to deal with is the choice of formula for computing the roots. 3) suffers massive cancellation. jb 2 - 4ac), is inexact, so the subtraction brings into prominence the earlier rounding errors. jb2 -4ac) 2a and the other from the equation X1X2 = cia. Unfortunately, there is a more pernicious source of cancellation: the subtraction b2 - 4ac.

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