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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 conception, and rounding blunders research. software program practicalities are emphasised all through, with specific connection with LAPACK and MATLAB.

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This analysis shows that subtractive cancellation causes relative errors or uncertainties already present in and b to be magnified. In other words, subtractive cancellation brings earlier errors into prominence. It is important to realize that cancellation is not always a bad thing. There are several reasons. First, the numbers being subtracted may be error free, as when they are from initial data that is known exactly. 3). The second reason is that cancellation may be a symptom of intrinsic ill conditioning of a problem, and may therefore be unavoidable.

One way to obtain a scale-independent quantity is to divide by || A|| ||y||, yielding the relative residual The importance of the relative residual is explained by the following resuit, which was probably first proved by Wilkinson (see the Notes and References). 1. With the notation above, and for the 2-norm, Proof. 7) is attainable. 1 says that p(y) measures how much A (but not b) must be perturbed in order for y to be the exact solution to the perturbed system, that is, p(y) equals a normwise relative backward error.

Precision Versus Accuracy The terms accuracy and precision are often confused or used interchangeably, but it is worth making a distinction between them. Accuracy refers to the absolute or relative error of an approximate quantity. 1)). 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.

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