Jean Marie Linhart
Abstract
We present and compare three C functions to compute the logarithm of the cumulative standard normal distribution. The first is a new algorithm derived from Algorithm 304’s calculation of the standard normal distribution via a series or continued fraction approximation, and it is good to the accuracy of the machine. The second is based on Algorithm 715’s calculation of the standard normal distribution via rational Chebyshev approximation. This is related to, and an improvement on, the algorithm for the logarithm of the normal distribution available in the software package R. The third is a new and simple algorithm that uses the compiler’s implementation of the error function, and complement of the error function, to compute the log of the normal distribution.
Supplemental Material
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Software for Computing the Logarithm of the Normal Distribution
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Index Terms
- Algorithm 885: Computing the Logarithm of the Normal Distribution
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Reviews
Marlin W Thomas
Calculating the logarithm rapidly and accurately of points on the standard normal curve-a normal curve with mean of zero and standard deviation of one-is required for applications that involve repeated determinations of the logarithm. This paper points out the relative lack of discussion of the subject in the literature, surveys what there is of its history, presents the algorithm, describes three of its implementations along with empirical results, and provides the means of selecting one of the algorithms, based on requirements of speed and accuracy. In a brief but insightful review of the literature calculating the logarithm of the standard normal distribution, Linhart identifies a problem with accuracy in the tails of the curve and presents three candidate solutions to remedy the problem. One uses continued fractions, another uses rational Chebyshev approximations, and the third uses a compiler resident error function. Although the paper focuses on C implementations of the algorithm, a major feature is the analysis of the algorithm independent of its implementation; it makes a contribution to the literature by noting the negative effect of points at the extremes. One of the goals of the paper is to isolate ways to moderate this effect. A valuable feature of the paper is its presentation of testing results regarding error ranges. The paper provides a valuable resource for applied statisticians and for programmers interested in the implementation of statistical algorithms. Online Computing Reviews Service
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