Numerical Recipes: Does This Paradigm Have a Future?

Abstract

li\:eEii1'6i{•ollaboration when we were both ing difficulty in integrating a second-or-) numerically, and our mutual thesis adflfat Keller, the distinguished professor of applied mathematic ~ ou using?" Keller asked us. We looked at each other b lankly•~~gni~~• 11~c's more than one numerical method? "Runge-Kut "M "Runge-Kutta! e-Kutta!" he exclaimed, banging his head with his fist. "That's a ll you physicists know!" He suggested that we look into Bulirsch-Stoer, which we did, to our eventual benefit. Eventual, but not immediate, because there is a twist to this story: Bulirsch-Stoer turned out to be a poor choice of method for our problem, while Runge-Kutta, when we eventually learned to integrate away from-not into-singularities, and to make a pre-integration change of variables in the equations, worked splendid ly. So this story illustrates not on ly our lack of knowledge about numerical methods, but also that when physicists consult professional numerical analysts, either in person or through a book or journal artic le, they not infrequently will be disappointed. This and similar early experiences firmly convinced us of the necessity for an algorithm's user (the physicist) to understand "what is inside the black box." This ideal became, later, one of the defining features of our Numerical Recipes project. It was then, and remains now, exceedingly controversial. The physicist-reader may be astonished, because physicists are all tinkerers and black-box disassemblers at heart. However, there is an opposite, and much better established, dogma from the community of numerical analysts, roughly: "Good numerical methods are sophisticated, highly tuned, and based on theorems, not tinkering. Users should interact with such algorithms through defined interfaces, and should be prevented from modifying their internals-for the users' own good." This "central dogma of the mathematical software community" informed the large scientific libraries that dominated the 1970s and 1980s, the NAG library (developed in the U.K.) and its American cousin, IMSL. It continues to be a dominant influence today in such useful computational environments as Matlab and Mathematica. We do not subscribe to this religion, but it is worth pointing out that it is the dominant religion (except perhaps among physicists and astronomers) in mathematical software. This controversy will continue, and NR's viewpoint is by no means assured of survival! An early book review of NR, by Tserles, 3 g ives an interesting and balanced perspective on the issue.

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