Speaker: David H Bailey, Lawrence Berkeley National Lab (retired) and U.C. Davis
When: Thursday, November 5th, 2015 at 3:10pm
Where: 1131 Kemper Hall
Host: Cindy Rubio Gonzalez
In two previous studies of lattice sums arising from the Poisson equation of mathematical physics, we found that the lattice sum phi_2(x,y) is given by 1 / pi * log(A), where A is algebraic (i.e., the root of a polynomial with integer coefficients). After computing high-precision numerical values of these sums and applying an integer relation algorithm, we obtained the explicit minimal polynomials for the algebraic numbers A in a set of specific cases. Based on these computations, Jason Kimberley conjectured a number-theoretic formula for the degree of A in the case x = y = 1/s for some integers.
The earlier work was hampered by the enormous cost and complexity of these computations. In this study, we describe three major improvements: (a) a new thread-safe, high-level arbitrary precision package (MPFUN-MPFR), which is approximately 3X faster than the package used earlier; (b) a new three-level multipair PSLQ integer relation scheme, which is approximately 5X faster than the scheme used earlier; and (c) a parallel implementation on a 16-core system. These enhancements resulted in a combined speedup of approximately 156X over the software used earlier (based on one typical case). As a result, we have been able to substantially extend our previous results, confirming that Kimberley’s formula holds for all integers s up to 50 (except for s = 41, 43, 47, 49, which are still too costly to test). As far as we are aware, these computations, employing up to 51,000-digit precision, with polynomial degrees up to 324 and integer coefficients up to 10^145, constitute the largest successful integer relation computations performed to date.
David H. Bailey (recently retired from Lawrence Berkeley National Lab; Research Fellow at U.C. Davis) is a senior figure in high-performance computing and computational mathematics, with five books and over 200 published papers in the field. In high-performance computing, he is best known as lead author for the NAS Parallel Benchmarks, which was just awarded the 2015 “Test of Time Award” by the Supercomputing Conference. In computational mathematics, he is best known for studies using very high-precision computations to discover new identities of mathematics and mathematical physics. He has received the Chauvenet and Merten Hesse Prizes from the Mathematical Association of America, the Sidney Fernbach Award from the IEEE Computer Society, and the Gordon Bell Prize from the Association of Computing Machinery (the only person to have received major awards from all three of these societies).
1131 Kemper Hall