Bailey D.H.High-precision floating-point arithmetic in scientific computation.2005

¡J17 ¡ J18 ¡ J19 + J20 + J21 ¡ J22 ¡ J23 + 2J25

Figure 2: Some new math identities found by high-precision computations

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PSLQ computations, that in each of ten cases with unit or zero mass, the ¯nite part the scalar 3-loop tetrahedral vacuum Feynman diagram reduces to 4-letter \words" that represent iterated integrals in an alphabetpof 7 \letters," namely the one-forms - := dx=x and !k := dx=(¸¡k ¡ x), where ¸ := (1 + ¡3)=2 is the primitive sixth root of unity, and k runs from 0 to 5. In this context, a 4-letter \word" is a 4-dimensional iterated integral, such as

There are 74 four-letter words. Only two of these are primitive terms occurring in the 3-loop Feynman diagrams: U, above, and

The remaining terms in the diagrams reduce to products of constants found in Feynman diagrams with fewer loops. These ten cases as shown in Figure 3. In these diagrams, dots indicate particles with nonzero rest mass. The formulas that have been found, using PSLQ, for the corresponding constants are given in Figure 4. Here the constant

C = Pk>0 sin(¼k=3)=k2.

In each case, these formulas were later proven to be correct. But until the \experimental" results of the computations were found, no one had any solid reason to believe that such relations might exist. Indeed, the adjective \experimental" is quite appropriate here, because the PSLQ computations played a role entirely analogous to (and just a crucial as) a conventional laboratory experiment.

13. Conclusion

We have presented here a brief survey of the rapidly expanding usage of high-precision arithmetic in modern scienti¯c computing. It is worth noting that all of these examples have arisen in the past ten years. Thus we may be witnessing the birth of a new era of scienti¯c computing, in which the numerical precision required for a computation is as important to the program design as are the algorithms and data structures.

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Figure 4: Formulas found by PSLQ for the ten cases of Figure 3

In one respect, perhaps it is not a coincidence that interest in high-precision arithmetic has arisen during the same period that a concerted attempt has been made to implement many scienti¯c computations on highly parallel and distributed systems. These systems have made possible much larger-scale runs than before, greatly magnifying numerical di±culties. Also, even on single-processor systems, memory and performance have greatly expanded (a gift of Moore's Law in semiconductor technology), enabling much larger computations than were feasible only a few years ago.

It may well be that other computations now being performed on these systems have signi¯cant numerical di±culties, but those who use these codes don't yet realize the extent of these di±culties. With the advent of relatively painless software to convert programs to use high-precision arithmetic, it is now possible to periodically run a large scienti¯c computation with double-double or higher-precision arithmetic, in order to determine how accurate its results really are.

In fact, the software tools now available for high-precision arithmetic can be used in every stage of numerical error management: (1) periodic testing of a scienti¯c code to see if it is becoming numerically sensitive; (2) determining how many digits in the results (intermediate or ¯nal) are reliable; (3) pinning down a numerical di±culty to a few lines of code; and (4) rectifying the numerical di±culties that are uncovered.

While software described in this paper and elsewhere will help scientists in the short run, in the longer run it is essential that computer vendors recognize the need to provide both hardware and software support for high-precision arithmetic. An IEEE committee has been formed for updating the IEEE-754 °oating-point arithmetic standard; 128-bit support is one of the key provisions of their draft standard. This same committee is also considering establishing standards for arbitrary-precision arithmetic. These are welcome and timely developments.

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Acknowledgements

The author acknowledges the contributions of numerous colleagues in the preparation of this article, including: Ed Baron, Ben Barrowes, Jonathan Borwein; Richard Crandall, Chris Ding, Alexei Frolov, William Kahan, Peter Nugent, Michael Strayer and Zong-Chao Yan.

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