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![Science News cover](images/highlights/46sh_bailey.jpg) |
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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.)
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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 developedwhat 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 1a 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 Unlimited2001
and Beyond, Bjorn Engquist and Wilfried Schmid, eds., (Springer, 2001),
pp. 51-66.
http://www.nersc.gov/~dhbailey
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