Let {x1,x2, ···,xn} be a vector of real numbers. An integer relation algorithm is a computational scheme to find the n integers ak, if they exist, such that a1x1 +a2x2 +···+ anxn = 0. In the past few years, integer relation algorithms have been utilized to discover new results in mathematics and physics. Existing programs for this purpose require very large amounts of computer time, due in part to the requirement for multiprecision arithmetic, yet are poorly suited for parallel processing. This paper presents a new integer relation algorithm designed for parallel computer systems, but as a bonus it also gives superior results on single processor systems. Singleand multi-level implementations of this algorithm are described, together with performance results on a parallel computer system. Several applications of these programs are discussed, including some new results in mathematical number theory, quantum field theory and chaos theory.

This paper describes a new software package for performing arithmetic with an arbitrarily high level of numeric precision. It is based on the earlier MPFUN package [2], enhanced with special IEEE floating-point numerical techniques and several new functions. This package is written in C++ code for high performance and broad portability and includes both C++ and Fortran-90 translation modules, so that conventional C++ and Fortran-90 programs can utilize the package with only very minor changes. This paper includes a survey of some of the interesting applications of this package and its predecessors. 1.

This paper describes schemes for rapidly computing numerical values of definite integrals to very high accuracy (hundreds to thousands of digits) on highly parallel computer systems. Such schemes are of interest not only in computational physics and computational chemistry, but also in experimental mathematics, where high-precision numerical values of definite integrals can be used to numerically discover new identities. This paper presents performance results for 1-D and 2-D integral test suites on highly parallel computer systems. Results are also given for certain problems that derive from mathematical physics. One of these results confirms a conjecture to 20,000 digit accuracy. The performance rate for this calculation is 690 Gflop/s on 1024 CPUs of a state-of-the-art parallel system. Other results, which range in precision from 120 to 500 digits, and for 1-D, 2-D, 3-D and 4-D integrals, derive from Ising theory. The largest of these calculations required 28 hours on 256 CPUs. We believe that these are the first instances of evaluations of nontrivial 3-D and 4-D integrals to multi-hundred-digit accuracy.

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We find and prove relationships between Riemann zeta values and central binomial sums. We also investigate alternating binomial sums (also called Apéry sums). The study of non-alternating sums leads to an investigation of different types of sums which we call multiple Clausen values. The study of alternating sums leads to a tower of experimental results involving polylogarithms in the golden ratio. In the non-alternating case, there is a strong connection to polylogarithms of the sixth root of unity, encountered in the 3-loop Feynman diagrams of hep-th/9803091 and subsequently in hep-ph/9910223, hep-ph/9910224, cond-mat/9911452 and hep-th/0004010.

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IN SCIENTIFIC COMPUTATION IEEE 64-bit floating-point arithmetic is sufficient for most scientific applications, but a rapidly growing body of scientific computing applications requires a higher level of numeric precision. New software packages have yielded interesting scientific results that suggest numeric precision in scientific computations could be as important to program design as algorithms and data structures.