# Publication Peculiarities: Sequences of Papers

This is the third post is my sequence on publication peculiarities.

It is not unusual to see a sequence of related papers with similar titles, sometimes labelled “Part I”, “Part II” etc. Here I present two sequences of papers with intriguing titles and interesting stories behind them.

## Computing the Logarithm of a Complex Number

The language Algol 60 did not have complex numbers as a built-in data type, so it was necessary to write routines to implement complex arithmetic. The following sequence of papers appeared in Communications of the ACM in the 1960s and concerns writing an Algol 60 code to evaluate the logarithm of a complex number.

J. R. Herndon (1961). Algorithm 48: Logarithm of a complex number. Comm. ACM, 4(4), 179.

A. P. Relph (1962). Certification of Algorithm 48: Logarithm of a complex number. Comm. ACM, 5(6), 347.

M. L. Johnson and W. Sangren (1962). Remark on Algorithm 48: Logarithm of a complex number. Comm. CACM, 5(7), 391.

D. S. Collens (1964). Remark on remarks on Algorithm 48: Logarithm of a complex number. Comm. ACM, 7(8), 485.

D. S. Collens (1964). Algorithm 243: Logarithm of a complex number: Rewrite of Algorithm 48. Comm. CACM, 7(11), 660.

“Remark on remarks”, “rewrite”—what are the reasons for this sequence of papers?

The first paper, by Herndon, gives a short code (7 lines in total) that uses the arctan function to find the argument of a complex number $x+iy$ as $\arctan(y/x)$. Relph notes that the code fails when the real part is zero and that, because it adds $\pi$ to the $\arctan$, the imaginary part is on the wrong range, which should be $(-\pi,\pi]$ for the principal logarithm. Moreover, the original code incorrectly uses log (log to base 10) instead of ln (the natural logarithm).

It would appear that at this time codes were not always run and tested before publication, presumably because of the lack of an available compiler. Indeed Herndon’s paper was published in the April 1961 issues of CACM, and the first Algol 60 compilers had only become available the year before. according to this Wikipedia timeline.

Johnson and Sangren give more discussion about division by zero and obtaining the correct signs.

In his first paper, Collens notes that the Johnson and Sangren code wrongly gives $\log 0 = 0$ and has a missing minus sign in one statement.

Finally, Collens gives a rewritten algorithm that addresses the previously noted deficiencies. It appears to have been run, since some output is shown.

This sequence of papers from the early days of digital computing emphasizes that even for what might seem to be a trivial problem it is not straightforward to design correct, reliable algorithms and codes.

I am working on logarithms and other multivalued functions of matrices, for which many additional complications are present.

## Slow Manifolds

Edward Lorenz is well-known for introducing the Lorenz equations, discovering the Lorenz attractor, and describing the “butterfly effect”. His sequence of papers

E. N. Lorenz (1986). On the existence of a slow manifold. J. Atmos. Sci., 43(15), 1547–1557.

E. N. Lorenz and V. Krishnamurthy (1987). On the nonexistence of a slow manifold. J. Atmos. Sci., 44(20), 2940–2950.

E. N. Lorenz (1992). The slow manifold—What is it? J. Atmos. Sci., 49(24), 2449–2451.

seems to suggest a rather confused line of research!

However, inspection of the papers reveals the reasoning behind the choice of titles. The first paper discusses whether or not a slow manifold exists and shows that this question is nontrivial. The second paper shows that a slow manifold does not exist for one particular model. The third paper shows that the apparent contradiction to the second paper’s result by another author’s 1991 proof of the existence of a slow manifold for the same model can be explained by the use of different definitions of slow manifold.