Have you ever wondered how people compute numbers with many digits quickly and accurately? For example, how can they compute the googolth bit of the Erdős-Borwein constant?

This collection by the well-known computational mathematician Richard Crandall (Apple, NeXT, and Reed College) gives some of the answers. The obvious idea of using many fast computers in parallel for a long time is mentioned, but there is so much more. Before going into a production run, many algorithmic ideas have to be explored. Among these are:

• Using fast Fourier transforms (FFTs) to speed up both numeric and polynomial multiplication;
• Obtaining quickly converging series by adding or subtracting quickly converging series or series with known closed-form summations;
• Using symbolic mathematical software to search for the terms to add or subtract, or to even find new formulas;
• Continuing the function from the reals to the complexes and using complex integration; and
• Adding parameters to the function and then finding values for these parameters, which give faster evaluation techniques.

This large compendium of 18 papers contains more than the topics mentioned above. There is a section on number theory that includes discussions on how to show that a number is irrational, transcendental, normal, or normal in a particular base. There is also a section on scientific applications that makes the point that algorithmic thinking and analysis is necessary in using computers to find answers to scientific questions.

This book gives some idea of the breadth of Crandall’s work, but his purview is even broader. A recently published volume by him covers some of his more scientific and less algorithmic work [1].

As expected in a compendium, the different papers vary from undergraduate level to professional research level. The target audience is unclear, but I would recommend this book to graduate students and a few exceptional undergraduates to give them an idea of what kind of work a computational mathematician does.