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Notes
1Suppose the number of digits in c is 15, but the computer can store 16 digits. Then c will be padded with a zero in the sixteenth place. Suppose that a package can accurately compute that 16th place, and it is not zero. Then that package’s LRE will not be calculated as 15, even though it should be. Moreover, it may be calculated as less than 15, thus not properly reflecting the package’s accuracy. In most packages, correcting for this is both extremely tedious and unnecessary. Mathematica’s extreme accuracy makes this correction necessary. Fortunately, with Mathematica’s facilities for handling numbers, it is also easy.
2Rationals (e.g., “2”), are less susceptible to rounding error than Reals (e.g., “2.0”). Consider y1 = 1/3 * 3/5 * 5/7 and y2 = 1.0/3.0 * 3.0/5.0 * 5.0/7.0, y1 being Rational and y2 being Real, y1 is automatically reduced to 1/7 before being evaluated, and incurs only one numerical error in the sole division when finally evaluated. By contrast, y2 requires the evaluation of three divisions and two multiplications, and so is contaminated by five errors.
3Numerical analysis has standard definitions of “accuracy” and “precision” (Higham, 1996, p. 7). These definitions are based on a floating point arithmetic for which precision is fixed. However, Mathematica use a significance arithmetic in which precision is not fixed, and so needs alternative definitions for these important concepts. In Mathematica, “Accuracy” is the negative of the scale of the absolute error, and “Precision” is the negative of the scale of the relative error. These are not identical to, but are consistent with standard definitions.
4Since there is more than one definition of the autocorrelation function, it is important to use the same formula as NIST: \(r = \sum\nolimits_{i = 2}^n {({x_i} - \bar x)({x_{i - 1}} - \bar x)/} \sum\nolimits_{i = 1}^n {{{({x_i} - \bar x)}^2}} \).
7Note that the latter computation is N[1-…] and not 1 — N[…] so as to delay computation of the real value to the last possible moment and maximize the opportunity for symbolic calculation; though to no avail, in this case.
8In fact, Least SignificantByteFirst is hardware dependent rather than OS dependent, which is another reason to avoid writing binaries in the present case.
10available at www.wolfram.com/solutions/statistics/palette.html
11This is a very general procedure and can be applied to problems for which standard errors do not exist. Therefore it does not produce standard errors.
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Acknowledgements
The views expressed herein axe those of the author and do not necessarily reflect those of the Commission. Thanks for useful comments and suggestions are due to R. Ostermann and two anonymous referees, as well as D. Belsley, S. Hunka, L. Knüsel, and H. Varian. Thanks are also due to Wolfram Research for its cooperation, and to its employees for assistance, especially Ian Brooks. Any remaining errors are mine.
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McCullough, B.D. The accurary of Mathematica 4 as a statistical package. Computational Statistics 15, 279–299 (2000). https://doi.org/10.1007/PL00022713
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DOI: https://doi.org/10.1007/PL00022713