The apparent randomness of pi's digits is represented by a random-walk landscape to illustrate the September 1, 2001 Science News cover story describing David Bailey and Richard Crandall's research into the normality of certain mathematical constants. (Illustration: David V. Chudnovsky and Gregory V. Chudnovsky. Copyright ©2001 Science Service. Reprinted with permission.)

David Bailey and Xiaoye Sherry Li, NERSC, Lawrence Berkeley National Laboratory

Research Objectives
This project seeks to develop several easy-to-use, preferably Web-based tools for experimental mathematics, such as high-precision arithmetic constant evaluations, definite integral evaluations, integer relation detection, and others. The numerical code underlying these calculations has already been developed—what remains is to package these tools so that someone other than highly trained numerical analysts can use them.

Computational Approach
The approach is first to gather together a number of tools that appear to have promise as tools for experimental mathematics. In some cases, some additional development or polishing is needed. Once this is done, they will be placed in a common repository with detailed instructions and examples of usage. A few key items will be provided to users by means of an easy-to-use Web interface. This interface may require developing processes for moving heavier computation to other platforms, including parallel platforms (in order to provide excellent real-time, interactive performance).

Accomplishments
In a previous project, we took a major step toward answering the age-old question of whether the digits of p and other mathematical constants are "normal," which means that their digits are random in a certain statistical sense. Our results indicate that the normality of certain constants is a consequence of a plausible conjecture in the field of chaotic dynamics, which states that sequences of a particular kind are uniformly distributed between 0 and 1—a conjecture we refer to as "Hypothesis A." We have thus translated a heretofore unapproachable problem to a more tractable question in the field of chaotic processes. Previous work on the PSLQ integer relation algorithm was selected as one of ten "Algorithms of the Century" by the publication "Computing in Science and Engineering."

A high-quality software package was completed that provides double-double (128-bit) and quad-double (256-bit) floating-point arithmetic. This package includes translation modules for both Fortran-90 and C/C++, which greatly reduce the programming effort to use these routines. In most cases it is only necessary to change a few type declarations to utilize these facilities. We used the quad-double software to perform a large simulation of a vortex roll-up phenomenon, running on the IBM SP and Cary T3E systems. Using this software, this calculation ran approximately five times faster than with arbitrary-precision software, thus saving thousands of CPU-hours of run time. The resulting calculations confirm that an instability occurs in these situations that had not been observed in previous studies.

Significance
Although high performance computer technology is now a mainstay in many fields of scientific research, and much of modern computer technology has its roots in pure/applied mathematics, the field of mathematics has not yet benefited much. Several valuable software tools have been developed, including several that were developed at NERSC. But for the most part they remain research codes, typically written in Fortran-90 or requiring parallel platforms, which places them out of range for most mathematicians. This work seeks to bridge this gap, making these tools available to average mathematicians for the first time.

Publications
David H. Bailey and Richard E. Crandall, "On the random character of fundamental constant expansions," Experimental Mathematics 10, 175 (2001).

Yozo Hida, Xiaoye S. Li, and David H. Bailey, "Algorithms for quad-double precision floating-point arithmetic," in Proceedings of ARITH-15 (2001).

David H. Bailey and Jonathan M. Borwein, "Experimental mathematics: Recent developments and future outlook," in Mathematics Unlimited—2001 and Beyond, Bjorn Engquist and Wilfried Schmid, eds., (Springer, 2001), pp. 51-66.

http://www.nersc.gov/~dhbailey