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David
Bailey and Xiaoye Sherry Li, NERSC, Lawrence Berkeley National Laboratory
Research
Objectives
We are working to develop high-precision arithmetic software. One of the
key applications that we wish to pursue is a vortex roll-up simulation
we developed utilizing a new "quad-double" arithmetic package written
by Yozo Hida. In addition, we plan to pursue applications of the PSLQ
integer relation detection program.
Computational
Approach
This research employs advanced techniques for performing arithmetic with
more than the standard 16-digit IEEE floating-point arithmetic that is
available on most technical computers today. During the past year we have
developed "double-double" and "quad-double" software packages, which enable
ordinary C or Fortran computer programs to perform arithmetic with 32
and 64 decimal digit accuracy, respectively. In addition, we use a separate
package, written by the PI, which performs arithmetic to an arbitrarily
high level of numeric precision. Another key technique used is the PSLQ
integer relation detection algorithm developed by the PI and mathematician-sculptor
Helaman Ferguson of the Center for Computing Sciences in Maryland.
Accomplishments
During the past year five technical papers were completed based on calculations
mostly performed using NERSC systems. Other accomplishments include the
completion of two new extended precision software packages — a "double-double"
package, which provides approximately 32 decimal digit accuracy, and a
"quad-double" package, which provides approximately 64 decimal digit accuracy.
These software packages also include bindings that permit ordinary C and
Fortran programs to use these packages with only minor changes to the
source code.
In
addition, we have developed a new variant of the PSLQ integer relation
detection algorithm that is suitable for highly parallel computer systems.
Significance
Our vortex roll-up simulation research explores an unresolved question
regarding the behavior of vortices „ namely, whether they always form
a nice exponential spiral. Until now, researchers in the field have assumed
that this always happens, but our initial runs show that beyond a critical
value of a certain parameter, the exponential spiral develops chaotic
irregularities. We need to make more runs to firmly establish and better
understand this phenomenon.
The
PSLQ integer relation finding program explores relationships between constants
that arise in certain fields of mathematics and physics. For example,
PSLQ has unearthed a simple formula for calculating any binary digit of
 without calculating
the digits preceding it. In January 2000, the PSLQ algorithm was named
one of ten "algorithms of the century" by the editors of Comput-ing in
Science and Engineering. We hope to uncover some new facts of mathematics
and physics with this program.
Publications
David H. Bailey, "Integer relation detection," Computing in Science and
Engineering 2, 1 (2000).
Yozo
Hida, Xiaoye S. Li, and David H. Bailey, "Quad-double arithmetic: Algorithms,
implementation, and application," Lawrence Berkeley National Laboratory
technical report LBNL-46996 (2000). Condensed version submitted to 15th
IEEE Symposium on Computer Arithmetic.
Helaman
R. P. Ferguson, David H. Bailey, and Stephen Arno, "Analysis of PSLQ,
an integer relation finding algorithm," Mathematics of Computation 68,
90 (1999).
http://www.nersc.gov/~dhbailey
http://www.nersc.gov/news/newsroom/bailey1-20-00.html
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,
the more accurate is the solution. But severe roundoff errors occur
with small values of