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

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Absolute error versus precision, t = -log2 u. else z = ey % Store to inhibit extended precision evaluation. 0 in both single and double precision arithmetic. 2, in which a step function is computed in place of f(x) = x for a wide range of precisions. It is worth stressing that how precision is increased can greatly affect the results obtained. Increasing the precision without preserving important properties such as monotonicity of rounding can vitiate an otherwise reliable algorithm. Increasing the precision without maintaining a correct relationship among the precisions in different parts of an algorithm can also be harmful to the accuracy.

19. This chapter has few prerequisites and few assumptions are made about the nature of the finite precision arithmetic (for example, the base, number of digits, or mode of rounding, or even whether it is floating point arithmetic). The second chapter deals in detail with the specifics of floating point arithmetic. 12 onward are special ones chosen to illustrate particular phenomena. You may never see in practice the extremes of behaviour shown here. Let the examples show you what can happen, but do not let them destroy your confidence in finite precision arithmetic!

The best advice is to be aware of the need for numerical stability when designing an algorithm and not to concentrate solely on other issues, such as computational cost and parallelizability. A few guidelines can be given. 1. Try to avoid subtracting quantities contaminated by error (though such subtractions may be unavoidable). 2. Minimize the size of intermediate quantities relative to the final solution. The reason is that if intermediate quantities are very large then the final answer may be the result of damaging subtractive cancellation.

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