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Computation and theory of extended MordellTornheimWitten sums
 Mathematics of Computation
, 2013
"... Abstract. We consider some fundamental generalized Mordell–Tornheim–Witten (MTW) zetafunction values along with their derivatives, and explore connections with multiplezeta values (MZVs). To achieve this, we make use of symbolic integration, high precision numerical integration, and some interestin ..."
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Cited by 5 (4 self)
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Abstract. We consider some fundamental generalized Mordell–Tornheim–Witten (MTW) zetafunction values along with their derivatives, and explore connections with multiplezeta values (MZVs). To achieve this, we make use of symbolic integration, high precision numerical integration, and some interesting combinatorics and specialfunction theory. Our original motivation was to represent unresolved constructs such as Eulerian loggamma integrals. We are able to resolve all such integrals in terms of a MTW basis. We also present, for a substantial subset of MTW values, explicit closedform expressions. In the process, we significantly extend methods for highprecision numerical computation of polylogarithms and their derivatives with respect to order.
Polynomial homotopies on multicore workstations. Accepted for publication
 in the proceedings of PASCO 2010
"... Homotopy continuation methods to solve polynomial systems scale very well on parallel machines. In this paper we examine its parallel implementation on multiprocessor multicore workstations using threads. With more cores we can speed up pleasingly parallel path tracking jobs. In addition, we can com ..."
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Cited by 3 (3 self)
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Homotopy continuation methods to solve polynomial systems scale very well on parallel machines. In this paper we examine its parallel implementation on multiprocessor multicore workstations using threads. With more cores we can speed up pleasingly parallel path tracking jobs. In addition, we can compute solutions more accurately in the same amount of time with threads, and thus achieve quality up. Focusing on polynomial evaluation and linear system solving (the key ingredients of Newton’s method) we can double the accuracy of the results with the quad doubles of QD2.3.9 in less than double the time, if we use all available eight cores on our workstation. 1
Pi Day is upon us again and we still do not know if Pi is normal
, 2013
"... The digits of π have intrigued both the public and research mathematicians from the beginning of time. This article briefly reviews the history of this venerable constant, and then describes some recent research on the question of whether π is normal, or, in other words, whether its digits are stati ..."
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The digits of π have intrigued both the public and research mathematicians from the beginning of time. This article briefly reviews the history of this venerable constant, and then describes some recent research on the question of whether π is normal, or, in other words, whether its digits are statistically random in a specific sense. 1 Pi and its day in modern popular culture The number π, unique among the pantheon of mathematical constants, captures the fascination both of the public and of professional mathematicians. Algebraic constants such as √ 2 are easier to explain and to calculate to high accuracy (e.g., using a simple Newton iteration scheme). The constant e is pervasive in physics and chemistry, and even appears in financial mathematics. Logarithms are ubiquitous in the social sciences. But none of these other constants has ever gained much traction in the popular culture. In contrast, we see π at every turn. In an early scene of Ang Lee’s 2012 movie adaptation of Yann Martel’s awardwinning book The Life of Pi, the title character Piscine (“Pi”) Molitor writes hundreds of digits of the decimal expansion of π on a blackboard to impress his teachers and schoolmates, who chant along with every digit. 1 This has even led to humorous takeoffs such as a 2013 Scott Hilburn cartoon entitled “Wife of Pi, ” which depicts a 4 figure seated next to a π figure, telling their marriage counselor “He’s irrational and he goes on and on. ” [21].
Computation and theory of extended MordellTornheimWitten sums
, 2012
"... We consider some fundamental generalized Mordell–Tornheim–Witten (MTW) zetafunction values along with their derivatives, and explore connections with multiplezeta values (MZVs). To achieve these results, we make use of symbolic integration, high precision numerical integration, and some interestin ..."
Abstract
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We consider some fundamental generalized Mordell–Tornheim–Witten (MTW) zetafunction values along with their derivatives, and explore connections with multiplezeta values (MZVs). To achieve these results, we make use of symbolic integration, high precision numerical integration, and some interesting combinatorics and specialfunction theory. Our original motivation was to represent previously unresolved constructs such as Eulerian loggamma integrals. Indeed, we are able to show that all such integrals belong to a vector space over an MTW basis, and we also present, for a substantial subset of this class, explicit closedform expressions. In the process, we significantly extend methods for highprecision numerical computation of polylogarithms and their derivatives with respect to order.
with Controlled Accuracy
, 2013
"... The present article concerns itself with the description of real numbers converter into basic positional notations (binary, denary, hexadecimal) with the controlled accuracy of fractional part of converted number formation. Here the converter functionality and the peculiarities of implementation of ..."
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The present article concerns itself with the description of real numbers converter into basic positional notations (binary, denary, hexadecimal) with the controlled accuracy of fractional part of converted number formation. Here the converter functionality and the peculiarities of implementation of the used algorithms of converting long numbers from one numerical notation into the other without making use of the processor input/output are specified. Moreover, the analysis of the program action period while converting the numbers of different exponents has been carried out.
Extending Summation Precision for Network Reduction Operations
"... Abstract—Double precision summation is at the core of numerous important algorithms such as NewtonKrylov methods and other operations involving inner products, but the effectiveness of summation is limited by the accumulation of rounding errors, which are an increasing problem with the scaling of m ..."
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Abstract—Double precision summation is at the core of numerous important algorithms such as NewtonKrylov methods and other operations involving inner products, but the effectiveness of summation is limited by the accumulation of rounding errors, which are an increasing problem with the scaling of modern HPC systems and data sets. To reduce the impact of precision loss, researchers have proposed increasedand arbitraryprecision libraries that provide reproducible error or even bounded error accumulation for large sums, but do not guarantee an exact result. Such libraries can also increase computation time significantly. We propose big integer (BigInt) expansions of double precision variables that enable arbitrarily large summations without error and provide exact and reproducible results. This is feasible with performance comparable to that of doubleprecision floating point summation, by the inclusion of simple and inexpensive logic into modern NICs to accelerate performance on largescale systems. I.
Editors: Will be set by the publisher FIRST STEPS TOWARDS MORE NUMERICAL REPRODUCIBILITY ∗
, 2013
"... Abstract. Questions whether numerical simulation is reproducible or not have been reported in several sensitive applications. Numerical reproducibility failure mainly comes from the finite precision of computer arithmetic. Results of floatingpoint computation depends on the computer arithmetic prec ..."
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Abstract. Questions whether numerical simulation is reproducible or not have been reported in several sensitive applications. Numerical reproducibility failure mainly comes from the finite precision of computer arithmetic. Results of floatingpoint computation depends on the computer arithmetic precision and on the order of arithmetic operations. Massive parallel HPC which merges, for instance, manycore CPU and GPU, clearly modifies these two parameters even from run to run on a given computing platform. How to trust such computed results? This paper presents how three classic approaches in computer arithmetic may provide some first steps towards more numerical reproducibility. 1. Numerical reproducibility: context and motivations As computing power increases towards exascale, more complex and larger scale numerical simulations are performed in various domains. Questions whether such simulated results are reproducible or not have been reported more or less recently, e.g. in energy science [1], dynamic weather forecasting [2], atomic or molecular dynamic [3,4], fluid dynamic [5]. This paper focuses on numerical nonreproducibility due to the finite precision of computer arithmetic – see [6] for other issues about “reproducible research ” in computational mathematics. The following example illustrates a typical failure of numerical reproducibility. In the energetic field, power system state simulation aims to compute at “real time ” a reliable estimate of the bus voltages for a given