Methods of Numerical Integration, 2nd Ed

[Wes Loewer]

Some months ago, there was an informative thread which dealt with the Romberg algorithm used on the 71B and other models (https://www.hpmuseum.org/forum/thread-14459.html). The algorithm traces its roots back to the 34c as described by the William Kahan article "Handheld Calculator Evaluates Integrals" (https://www.hpl.hp.com/hpjournal/pdfs/Is...980-08.pdf). In this article Kahan writes, "The HP-34C uses a Romberg method; for details consult reference 2" which is "P.J. Davis and P. Rabinowitz, 'Methods of Numerical Integration,' Academic Press, New York, 1975."

Curiosity prompted me to order a copy of this book, or rather a 1984 2nd edition that I found on Amazon for $2.44. As I have been reading through it over the summer, I found a few interesting tidbits.

As mentioned above, Kahan's article refers to the 1st edition of the book. The book's 2nd edition returns the favor by referencing the Kahan HP Journal article with a paragraph describing Kahan's implementation of the Romberg Method and the benefits of the x=½(3u-u³) substitution (p441). It's not often you see two sources mutually reference each other like this.

I got involved in the original thread when I incorrectly claimed that hp's implementation of the Romberg Method used midpoints rather than trapezoids. I only mention this now because the book briefly discusses using midpoints with the Romberg Method (p438).

Also in the original thread it was mentioned that using trapezoids allowed the reuse of previously calculated nodes. I had mentioned that reusing previously evaluated midpoints could in theory be accomplished by trisecting, rather than bisecting, the intervals. Not specific to Romberg, the book states, "Current practice favors subdivision by bisection. However, Hanke has produced theoretical arguments backed by some experimental evidence suggesting that trisection is preferable" (p427). The book later presents a generalized Romberg formula that allows for any number of subintervals (p437).

All in all, I'm glad I read through the book but it was by no means light reading. The Amazon description quotes from the back cover, "It offers a balanced presentation: certain sections derive from or allude to deep results of analysis, but most of the final results are expressed in a form accessible to anyone with a background in calculus." For parts of the book this was true, but other parts require a reader that has a very strong background in calculus and linear algebra.