Mathematical Analysis

Download Accuracy and Stability of Numerical Algorithms by Nicholas J. Higham PDF

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

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18. Designing Stable Algorithms There is no simple recipe for designing numerically stable algorithms. ) it is not good news for computational scientists in general. 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. 18 DESIGNING STABLE ALGORITHMS 27 1. Try to avoid subtracting quantities contaminated by error (though such subtractions may be unavoidable).

We solved the system in varying precisions with unit roundoffs u = 2-t, t = 15:40, corresponding to about 4 to 12 decimal places of accuracy. ) The algorithm used was Gaussian elimination (without pivoting), which is perfectly stable for this symmetric positive definite matrix. 3 gives corresponding results for A = P5 + 51, where P5 is the Pascal matrix of order 5. 55 x 105 for the shifted Pascal matrix. In both cases the general trend is that increasing the precision decreases the residual and relative error, but the behaviour is not monotonic.

1 gives results computed in single precision (u w 6 x 10-8). The approximations are poor, degrading as n approaches the reciprocal of the machine precision. For n a power of 10, 1/n has a nonterminating binary expansion. When 1 + 1/n is formed for n a large power of 10, only a few significant digits from 1/n are retained in the sum. 1). 1. 5. Strassen's method for fast matrix multiplication provides another example of the unpredictable relation between the number of arithmetic operations and the error.

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