By Nicholas J. Higham
A remedy 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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Extra info for Accuracy and Stability of Numerical Algorithms
Whenever we say a computation was “done in Fortran 90” we are referring to the use of this compiler. All the results quoted were obtained on a 486DX workstation, unless otherwise stated, but many of the experiments were repeated on a Sun SPARCstation, using the NAGWare4 FTN90 compiler [785, 1992]. 1 × 10 -16 in M ATLAB and in double precision in Fortran 90. 6]. ) 3 FTN90 is a joint trademark of Salford Software Ltd. and The Numerical Algorithms Group Ltd. 4 NAGWare is a trademark of The Numerical Algorithms Group Ltd.
Single precision corresponds to a 24-bit mantissa, so the term we are adding to s “drops off the end” of the computer word, as do all successive terms. The simplest cure for this inaccuracy is to sum in the opposite order: from smallest to largest. Unfortunately, this requires knowledge of how many terms to take before the summation begins. 6449 3406, which is correct to eight significant digits. For much more on summation, see Chapter 4. 13. Increasing the Precision When the only source of errors is rounding, a common technique for estimating the accuracy of an answer is to recompute it at a higher precision and to see how many digits of the original and the (presumably) more accurate answer agree.
10). Surprisingly, current calculators from more than one manufacturer (but not Hewlett-Packard) appear to use the one-pass formula, and they list it in their manuals. 0 by the one-pass formula (relative error 1). It might be argued that this data should be shifted by some estimate of the mean before applying the one-pass formula which does not change ), but a good estimate is not always available and there are alternative one-pass algorithms that will always produce an acceptably accurate answer.
Accuracy and Stability of Numerical Algorithms by Nicholas J. Higham