A quad-double number is an unevaluated sum of four IEEE double precision numbers, capable of representing at least 212 bits of significand. We present the algorithms for various arithmetic operations (including the four basic operations and various algebraic and transcendental operations) on quad-double numbers. The performance of the algorithms, implemented in C++, is also presented. 1.

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.

The authors have implemented three numerical quadrature schemes, using the Arbitrary Precision (ARPREC) software package. The objective here is a quadrature facility that can efficiently evaluate to very high precision a large class of integrals typical of those encountered in experimental mathematics, relying on a minimum of a priori information regarding the function to be integrated. Such a facility is useful, for example, to permit the experimental identification of definite integrals based on their numerical values. The performance and accuracy of these three quadrature schemes are compared using a suite of 15 integrals, ranging from continuous, well-behaved functions on finite intervals to functions with infinite derivatives and blow-up singularities at endpoints, as well as several integrals on an infinite interval. In results using 412-digit arithmetic, we achieve at least 400-digit accuracy, using two of the programs, for all problems except one highly oscillatory function on an infinite interval. Similar results were obtained using 1012-digit arithmetic.

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.

A 1996 paper by the author, Peter Borwein and Simon Plouffe showed that any mathematical constant given by an infinite series of a certain type has the property that its n-th digit in a particular number base could be calculated directly, without needing to compute any of the first n−1 digits, by means of a simple algorithm that does not require multiple-precision arithmetic. Several such formulas were presented in that paper, including formulas for the constants π and log 2. Since then, numerous other formulas of this type have been found. This paper presents a compendium of currently known results of this sort, both formal and experimental. Many of these results were found in the process of compiling this collection and have not previously appeared in the literature. Several conjectures suggested by these results are mentioned.

Abstract not found

Abstract. We propose an experimental mathematics approach leading to the computer-driven discovery of various structural properties of general counting functions coming from enumeration of walks. 1.

We document the discovery of two generating functions for ζ(2n + 2), analogous to earlier work for ζ(2n + 1) and ζ(4n + 3), initiated by Koecher and pursued further by Borwein, Bradley and others. 1

Challenge ” of Nick Trefethen, beautifully described in [12] (see also [13]). Indeed, these ten numeric challenge problems are also listed in [15, pp. 22–26], where they are followed by the ten symbolic/numeric challenge problems that are discussed in this article. Our intent in [15] was to present ten problems that are characteristic of the sorts of problems

In numerical algebraic geometry witness sets are numerical representations of positive dimensional solution sets of polynomial systems. Considering the asymptotics of witness sets we propose certificates for algebraic curves. These certificates are the leading terms of a Puiseux series expansion of the curve starting at infinity. The vector of powers of the first term in the series is a tropism. For proper algebraic curves, we relate the computation of tropisms to the calculation of mixed volumes. With this relationship, the computation of tropisms and Puiseux series expansions could be used as a preprocessing stage prior to a more expensive witness set computation. Systems with few monomials have fewer isolated solutions and fewer data are needed to represent their positive dimensional solution sets.