## High-Precision Computation and Mathematical Physics

### Cached

### Download Links

Citations: | 7 - 3 self |

### BibTeX

@MISC{Bailey_high-precisioncomputation,

author = {David H. Bailey and Jonathan M. Borwein},

title = {High-Precision Computation and Mathematical Physics},

year = {}

}

### OpenURL

### Abstract

At the present time, IEEE 64-bit floating-point arithmetic is sufficiently accurate for most sci-entific applications. However, for a rapidly growing body of important scientific computing ap-plications, a higher level of numeric precision is required. Such calculations are facilitated by high-precision software packages that include high-level language translation modules to min-imize the conversion effort. This paper presents a survey of recent applications of these tech-niques and provides some analysis of their numerical requirements. These applications include supernova simulations, climate modeling, planetary orbit calculations, Coulomb n-body atomic systems, scattering amplitudes of quarks, gluons and bosons, nonlinear oscillator theory, Ising theory, quantum field theory and experimental mathematics. We conclude that high-precision arithmetic facilities are now an indispensable component of a modern large-scale scientific com-puting environment.

### Citations

828 | Deterministic non-periodic flow - Lorenz |

358 | Solving ordinary differential equations. I: Nonstiff problems - Hairer, Nørsett, et al. - 1987 |

309 | An introduction to Numerical Analysis - Atkinson - 1989 |

255 | The Art of Computer Programming, Seminumerical Algorithms Volume 2, third edition - Knuth - 1998 |

105 | A Floating-point Technique for Extending the Available Precision - Dekker - 1971 |

104 | Quadrature and interpolation formulas for tensor products of certain classes of functions - Smolyak - 1963 |

101 | On the rapid computation of various polylogarithmic constants
- Bailey, Borwein, et al.
- 1997
(Show Context)
Citation Context ...al digits beginning at the n-th digit, without needing to calculate any of the first n − 1 digits, using a simple scheme that requires very little memory and no multiple-precision arithmetic software =-=[4]-=-[17, pg. 135-143]. Since 1996, numerous other formulas of this type have been found, using the PSLQ-based computational approach, and then subsequently proven [17, pg. 147–149]. In an unexpected turn ... |

92 | Mathematics by experiment: plausible reasoning - Borwein, Bailey - 2003 |

66 | Sparse grids - Zenger - 1991 |

64 | Accurate Sum and Dot Product - Ogita, Rump, et al. - 1955 |

51 | On the random character of fundamental constant expansions
- Bailey, Crandall
- 2001
(Show Context)
Citation Context ...f events, it has been found that these computer-discovered formulas have implications for the age-old question of whether (and why) the digits of constants such as π and log2 are statistically random =-=[11]-=-[17, pg. 163–174]. This same line of investigation has further led to a formal proof of normality (statistical randomness in a specific sense) for an uncountably infinite class of explicit real number... |

41 | Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization - Bauschke, Combettes, et al. |

40 | Parallel integer relation detection: Techniques and applications
- Bailey, Broadhurst
- 2000
(Show Context)
Citation Context ...the utilization of modern computing technology as an active agent of exploration in mathematical research [17][5]. One of the key techniques used here is the PSLQ integer relation detection algorithm =-=[10]-=-. An integer relation detection scheme is a numerical algorithm which, given an n-long vector (xi) of real numbers (presented as a vector of high-precision floating-point values), attempts to recover ... |

39 | Mathematics by Experiment: Plausible Reasoning in the 21 st
- Borwein, Bailey
- 2004
(Show Context)
Citation Context ...ven to be an essential tool for the emerging discipline of “experimental mathematics,” namely the utilization of modern computing technology as an active agent of exploration in mathematical research =-=[17]-=-[5]. One of the key techniques used here is the PSLQ integer relation detection algorithm [10]. An integer relation detection scheme is a numerical algorithm which, given an n-long vector (xi) of real... |

34 |
An IMT-type double exponential formula for numerical integration
- MORI
- 1978
(Show Context)
Citation Context ...o 1000 digits. In our studies, we have used either Gaussian quadrature (in cases where the function is well behaved in a closed interval) or the “tanh-sinh” quadrature scheme due to Takahasi and Mori =-=[29]-=- (in cases where the function has an infinite derivative or blow-up singularity at one or both endpoints). For many integrand functions, these schemes exhibit “quadratic” or “exponential” convergence ... |

33 | ARPREC: An Arbitrary Precision Computation Package,” 2002, http://crd.lbl.gov/~dhbailey/dhbpapers/arprec.pdf . The corresponding software is available at http://crd.lbl.gov/~dhbailey/mpdist - Bailey, Hida, et al. |

32 | Splitting of separatrices for the Chirikov standard map - Lazutkin - 2003 |

31 | Numerical recipes: The art of scientific computing. 3rd. 2007: Cambridge - Press, Teukolsky, et al. |

30 | Computational aspects of three-term recurrence relations - Gautschi - 1967 |

29 | Solving ordinary differential equations using Taylor series - Corliss, Chang - 1982 |

23 | Random generators and normal numbers
- BAILEY, CRANDALL
(Show Context)
Citation Context ...plicit real numbers. The simplest example of this class is the constant α2,3 = ∞ 1 ∑ n=1 3n23n , which is provably 2-normal: every string of m binary digits appears, in the limit, with frequency 2 −m =-=[12]-=-[17, pg. 174–178]. 3.9 Ising Integrals Several recent applications of high-precision computation have attempted to recognize definite integrals (typically arising in mathematical physics applications)... |

22 | Integrals of the Ising class
- Bailey, Borwein, et al.
(Show Context)
Citation Context ...fixed power of h. In a recent study, the present authors together with Richard Crandall applied tanh-sinh quadrature, implemented using the ARPREC package, to study the following classes of integrals =-=[7]-=-. The Dn integrals arise in the Ising theory of mathematical physics, and the Cn have tight connections to quantum field theory. Cn = 4 � ∞ � ∞ ··· n! 0 0 Dn = 4 � ∞ � ∞ ··· n! 0 0 � 1 En = 2 0 ··· � ... |

22 | Finding best approximation pairs relative to two closed convex sets in Hilbert space - Bauschke, Combettes, et al. - 2004 |

20 | A Comparison of Three High-Precision Quadrature Schemes - Bailey, Li |

20 | Experimentation in Mathematics: Computational Paths to Discovery, A K Peters - Borwein, Bailey, et al. - 2004 |

20 | Reversing symmetries in dynamical systems - Lamb - 1992 |

18 |
Hypergeometric forms for Ising-class integrals
- Bailey, Borwein, et al.
(Show Context)
Citation Context ... 180 digits beyond the level that could reasonably be ascribed to numerical round-off error; thus we are quite confident in this result even though we do not have a formal proof. In a follow-on study =-=[9]-=-, we examined the following generalization of the Cn integrals: Cn,k = 4 � ∞ � ∞ ··· n! 0 0 1 � n ∑ j=1 (u j + 1/u j) � du1 k+1 u1 ··· dun . un Here we made the initially surprising discovery—now prov... |

17 |
CutTools: A program implementing the OPP reduction method to compute one-loop amplitudes
- Ossola, Papadopoulos, et al.
- 2008
(Show Context)
Citation Context ...they find that their average evaluation time is not significantly increased [16]. Two other recent examples of employing high-precision arithmetic in fundamental physics calculations of this type are =-=[27]-=- and [20]. 3.7 Nonlinear Oscillator Theory Quinn, Rand, and Strogatz recently described a nonlinear oscillator system by means of the formula � N � 0 = 2 1 − s2 (1 − 2(i − 1)/(N − 1)) 2 � 1 − � . 1 − ... |

17 | The arithmetic-geometric mean and fast computation of elementary functions - Borwein, Borwein - 1984 |

16 | Parallel Integer Relation Detection - Bailey, Broadhurst |

16 | The accurate and efficient solution of a totally positive generalized Vandermonde linear system - Demmel, Koev |

15 | Experimental mathematics: examples, methods and implications
- Bailey, Borwein
(Show Context)
Citation Context ...to be an essential tool for the emerging discipline of “experimental mathematics,” namely the utilization of modern computing technology as an active agent of exploration in mathematical research [17]=-=[5]-=-. One of the key techniques used here is the PSLQ integer relation detection algorithm [10]. An integer relation detection scheme is a numerical algorithm which, given an n-long vector (xi) of real nu... |

15 | B.: The Douglas–Rachford algorithm in the absence of convexity. Fixed-Point Algorithms for - Borwein, Sims - 2011 |

14 | modular equations and pi or how to compute a billion digits of pi - M, Borwein, et al. - 1989 |

14 | P.: Searching with iterated maps - Elser, Rankenburg, et al. - 2007 |

13 | V.: Divide and concur: A general approach to constraint satisfaction - Gravel, Elser - 2008 |

13 | Exponentially small splitting of separatrices, matching in the complex plane and Borel summation. Nonlinearity 6 - Hakim, Mallick - 1993 |

13 | On recurrence relations for Sobolev orthogonal polynomials, manuscript - Evans, Littlejohn, et al. |

13 | A user-friendly extrapolation method for oscillatory infinite integrals - Sidi - 1988 |

10 | Using Accurate Arithmetics to Improve Numerical Reproducibility and Stability in Parallel Applications
- He, Ding
(Show Context)
Citation Context ... arithmetic for these loops. This single change dramatically reduced the numerical variability of the entire application, permitting computer runs to be compared for much longer run times than before =-=[25]-=-. 3.3 Planetary Orbit Calculations One central question of planetary theory is whether the solar system is stable over cosmological time frames (billions of years). Planetary orbits well known to exhi... |

10 | The Lindstedt-Poincaré technique as an algorithm for computing periodic orbits - Viswanath |

9 | A proof of a recursion for Bessel moments
- Borwein, Salvy
(Show Context)
Citation Context ...xamined the following generalization of the Cn integrals: Cn,k = 4 � ∞ � ∞ ··· n! 0 0 1 � n ∑ j=1 (u j + 1/u j) � du1 k+1 u1 ··· dun . un Here we made the initially surprising discovery—now proven in =-=[18]-=-—that there are linear relations in each of the rows of this array (considered as a doubly-infinite rectangular matrix), e.g., 0 = C3,0 − 84C3,2 + 216C3,4 9sHigh-Precision Computation and Mathematical... |

9 | Modern computer arithmetic
- Brent, Zimmermann
- 2010
(Show Context)
Citation Context ...function f(x)) cannot be reliably solved using conventional IEEE 64-bit floating-point arithmetic. 2. High-Precision Software Algorithms for performing high-precision arithmetic are fairly well known =-=[19]-=-, and software packages implementing these schemes have been available since the early days of computing. 2sHigh-Precision Computation and Mathematical Physics David H. Bailey However, many of these p... |

9 | Performance of the Taylor series method for ODEs/DAEs - Barrio |

9 | Verification methods: rigorous results using floating-point arithmetic - Rump - 2010 |

9 | The fractal property of the Lorenz attractor - Viswanath |

8 | Resolution of the Quinn–Rand–Strogatz constant of nonlinear physics, Experimental Mathematics 18
- Bailey, Borwein, et al.
- 2008
(Show Context)
Citation Context ...2074752208996... This led to a proof that the limit c exists and is the root of a Hurwitz zeta function ζ (1/2,c/2) = 0, where ζ(s,a) := ∑n≥0 1/(n+a) s . As a bonus, we obtained some asymptotic terms =-=[8]-=-. 3.8 Experimental Mathematics High-precision computations have proven to be an essential tool for the emerging discipline of “experimental mathematics,” namely the utilization of modern computing tec... |

8 |
An automated implementation of on-shell methods for one-loop amplitudes,” Phys
- Berger, Bern, et al.
(Show Context)
Citation Context ...r precision (double-double or quad-double as needed). Because only a few points have to be re-computed to higher precision, they find that their average evaluation time is not significantly increased =-=[16]-=-. Two other recent examples of employing high-precision arithmetic in fundamental physics calculations of this type are [27] and [20]. 3.7 Nonlinear Oscillator Theory Quinn, Rand, and Strogatz recentl... |

8 | VSVO formulation of the Taylor method for the numerical solution of ODEs,” Comput - Barrio, Blesa, et al. |

8 | Three-step and four-step random walk integrals,” Exp - Borwein, Straub, et al. |