High-precision arithmetic in mathematical physics

Mathematics

Volumn 3, Issue 2, 2015, Pages 337-367

  Bailey, David H ^a   Borwein, Jonathan M ^b  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b UNIVERSITY OF NEWCASTLE   (Australia)

Author keywords

High precision arithmetic; Ising integrals; Numerical integration; Poisson equation; PSLQ algorithm

Indexed keywords

EID: 85013655797     PISSN: None     EISSN: 22277390     Source Type: Journal    
DOI: 10.3390/math3020337     Document Type: Article

References (99)

1. Stodden, V.; Borwein, J. M.; Bailey, D. H. Publishing Standards for Computational Science: 'Setting the Default to Reproducible.' SIAM News 2013, 46, 4-6
2. Stodden, V. ; Bailey, D. H.; Borwein, J. M.; LeVeque, R. J.; Rider, W.; Stein, W. Setting the default to reproducible: Reproducibility in computational and experimental mathematics, 2013. Available online: http://www.davidhbailey.com/dhbpapers/icerm-report.pdf (accessed on 30 April 2015)
3. He, Y.; Ding, C.; Using accurate arithmetics to improve numerical reproducibility and stability in parallel applications. J. Supercomput. 2001, 18, 259-277
4. Robey, R. W.; Robey, J. M.; Aulwes, R. In search of numerical consistency in parallel programming. Parallel Comput. 2011, 37, 217-219
5. Bailey, D. H.; Barrio, R.; Borwein, J. M. High-precision computation: Mathematical physics and dynamics. Appl. Math. Computat. 2012, 218, 10106-10121
6. Computer Science Curriculum 2008: An interim revision of CS 2001. Association for Computing Machinery and IEEE Computer Society: New York, NY, USA 2008. Available online: http://www.acm.org/education/curricula/ComputerScience2008.pdf (accessed on 30 April 2015)
7. Rubio-Gonzalez, C.; Nguyen, C.; Nguyen, H. D.; Demmel, J.; Kahan, W.; Sen, K.; Bailey, D. H.; Iancu, C. Precimonious: Tuning assistant for floating-point precision. Proc. of SC13, Salt Lake City, UT, November 2013. Available online: http://www.davidhbailey.com/dhbpapers/precimonious.pdf (accessed on 30 April 2015)
8. Press, W. H.; Eukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes: The Art of Scientific Computing, 3rd ed.; Cambridge University Press: Cambridge, UK, 2007
9. LINPACK. Available online: http://www.netlib.org/linpack (accessed on 30 April 2015)
10. LAPACK. Available online: http://www.netlib.org/lapack (accessed on 30 April 2015)
11. Demmel, J.; Koev, P. The accurate and efficient solution of a totally positive generalized Vandermonde linear system. SIAM J. Matrix Anal. Appl. 2005, 27, 145-152
12. Knuth, D. E. The Art of Computer Programming: Seminumerical Algorithms 3rd ed.; Addison-Wesley: New York, NY, USA, 1998
13. Shewchuk, J. R. Adaptive precision floating-point arithmetic and fast robust geometric predicates. Discr. and Comp. Geometry, 1997, 18, 305-363
14. Hida, Y.; Li, X. S.; Bailey, D. H. Algorithms for Quad-Double Precision Floating Point Arithmetic. Proc. of the 15th IEEE Symposium on Computer Arithmetic, IEEE Computer Society, Vail, CO, USA, June 2001. Available online: http://crd-legacy.lbl.gov/~dhbailey/mpdist (accessed on 30 April 2015)
15. Borwein, J. M.; Bailey, D. H. Mathematics by Experiment: Plausible Reasoning in the 21st Century, 2nd ed.; A.K. Peters: Natick, MA, USA, 2008
16. Brent, R. P.; Zimmermann, P. Modern Computer Arithmetic; Cambridge Univ. Press: Cambridge, UK, 2010
17. Bailey, D. H.; Hida, Y.; Li, Y.; Thompson, B. ARPREC: An arbitrary precision computation package, 2002. Available online: http://www.davidhbailey.com/dhbpapers/arprec.pdf (accessed on 30 April 2015). Software available online: http://crd-legacy.lbl.gov/~dhbailey/mpdist (accessed on 30 April 2015)
18. CLN-Class Library for Numbers. Available online: http://www.ginac.de/CLN (accessed on 30 April 2015)
19. The GNU multiple precision library. Available online: https://gmplib.org (accessed on 30 April 2015)
20. Julia. Available online: http://julialang.org (accessed on 30 April 2015)
21. The GNU MPFR library. Available online: http://www.mpfr.org (accessed on 30 April 2015)
22. MPFR++. Available online: http://perso.ens-lyon.fr/nathalie.revol/software.html (accessed on 30 April 2015)
23. MPFR C++. Available online: http://www.holoborodko.com/pavel/mpfr (accessed on 30 April 2015)
24. Bailey, D. H. MPFUN2015: A thread-safe arbitrary precision computation package, 2015. Available online: http://www.davidhbailey.com/dhbpapers/mpfun2015.pdf (accessed on 30 April 2015)
25. mpmath: Python library for arbitrary-precision floating-point arithmetic. Available online: https://code.google.com/p/mpmath/ (accessed on 30 April 2015)
26. NTL: A library for doing number theory. Available online: http://www.shoup.net/ntl/ (accessed on 30 April 2015)
27. PARI/GP. Available online: http://pari.math.u-bordeaux.fr (accessed on 30 April 2015)
28. SageMath. Available online: http://sagemath.org (accessed on 30 April 2015)
29. Bailey, D. H.; Borwein, J. M.; Crandall, R. E. Integrals of the Ising class. J. Physics A: Math. Gen. 2006, 39, 12271-12302
30. Ma, D.; Saunders, M. Solving multiscale linear programs using the simplex method in quadruple precision, 2014. Available online: http://stanford.edu/group/SOL/multiscale/papers/quadLP3.pdf (accessed on 30 April 2015)
31. Ma, D.; Saunders, M. Experiments with linear and nonlinear optimization using quad precision, 2014. Available online: http://stanford.edu/group/SOL/multiscale/talks/14informsQuadMINOS.pdf (accessed on 30 April 2015)
32. Tucker, W. A rigorous ODE solver and Smale's 14th problem. Found. Comput. Math. 2002, 2, 53-117
33. Macfarlane, M. H. A high-precision study of anharmonic-oscillator spectra. Ann. Phys. 1999, 271, 159-202
34. Lake, G.; Quinn, T.; Richardson, D. C. From Sir Isaac to the Sloan survey: Calculating the structure and chaos due to gravity in the universe. Proceedings of the 8th ACM-SIAM Symposium on Discrete Algorithms, New Orleans, LA, USA, November 1997; pp. 1-10
35. Farrés, A.;l Laskar, J.; Blanes, S.; Casas, F.; Makazaga, J.; Murua, A. High precision symplectic integrators for the Solar System. Celest. Mech. Dyn. Astron. 2013, 116, 141-174
36. Frolov, A. M.; Bailey, D. H. Highly accurate evaluation of the few-body auxiliary functions and four-body integrals. J. Phys. B 2003, 36, 1857-1867
37. Yan, Z.-C.; Drake, G. W. F. Bethe logarithm and QED shift for Lithium. Phys. Rev. Lett. 2003, 81, 774-777
38. 7 ln α) to the fine structure splittings of Helium and He-Like ions. Phys. Rev. Lett. 1994, 77, 1715-1718
39. Ellis, R. K.; Giele, W. T.; Kunszt, Z.; Melnikov, K.; Zanderighi, G. One-loop amplitudes for W+3 jet production in hadron collisions, 2008. Available online: http://arXiv.org/abs/0810.2762 (accessed on 30 April 2015)
40. Berger, C. F.; Bern, Z.; Dixon, L. J.; Febres Cordero, F.; Forde, D.; Ita, H.; Kosower, D. A.; Maitre, D. An automated implementation of on-shell methods for one-loop amplitudes. Phys. Rev. D 2008, 78, 036003
41. Ossola, G.; Papadopoulos, C. G.; Pittau, R. CutTools: A program implementing the OPP reduction method to compute one-loop amplitudes. J. High-Energy Phys. 2008, 0803, 042
42. Czakon, M. Tops from light quarks: Full mass dependence at two-Loops in QCD. Phys. Lett. B 2008, 664, 307
43. Beliakov, G.; Matiyasevich, Y. A parallel algorithm for calculation of large determinants with high accuracy for GPUs and MPI clusters, 2013. Available online: http://arxiv.org/abs/1308.1536 (accessed on 30 April 2015)
44. Simó, C. Global dynamics and fast indicators. Global Analysis of Dynamical Systems, Krauskopf, B., Broer, H. W., Vegtr, G., Eds.; Taylor and Francis: New York, USA, 2001; pp. 373-389
45. Barrio, R. Performance of the Taylor series method for ODEs/DAEs. Appl. Math. and Comput. 2005, 163, 525-545
46. Barrio, R. ; Blesa, F.; Lara, M. VSVO formulation of the Taylor method for the numerical solution of ODEs. Comput. Math. Appl. 2005, 50, 93-111
47. Corliss, G.; Chang, Y. F. Solving ordinary differential equations using Taylor series. ACM Trans. Math. Software 1982, 8, 114-144
48. Barrio, R. Sensitivity analysis of ODEs/DAEs using the Taylor series method. SIAM J. Sci. Comput. 2006, 27, 1929-1947
49. Dena, A.; Rodriguez, M.; Serrano, S.; Barrio, R. High-precision continuation of periodic orbits. Abstr. Appl. Anal. 2012, 716024
50. Hairer, E.; Nørsett, S.; Wanner, G. Solving ordinary differential equations. I. Nonstiff problems; 2nd ed.; Springer-Verlag, Berlin, Germany, 1993
51. Hairerm E.; Ostermann, A. Dense output for extrapolation methods. Numer. Math. 1990, 58, 419-439
52. Barrio, R.; Rodriguez, M.; Abad, A.; Blesa, F. Breaking the limits: The Taylor series method. Appl. Math. and Comput. 2011, 217, 7940-7954
53. Lorenz, E. Deterministic nonperiodic flow. J. Atmospheric Sci. 1963, 20, 130-141
54. Abad, A.; Barrio, R.; Blesa, F.; Rodriguez, M. TIDES: a Taylor series Integrator for Differential EquationS. ACM Trans. Math. Software 2012, 39, 1
55. Dormand, J. R.; Prince, P. J. A family of embedded Runge-Kutta formulae. J. Comput. Appl. Math. 1980, 6, 19-26
56. Viswanath, D. The Lindstedt-Poincaré technique as an algorithm for computing periodic orbits. SIAM Rev. 2001, 43, 478-495
57. Viswanath, D. The fractal property of the Lorenz attractor. J. Phys. D 2004, 190, 115-128
58. Viswanath, D.; Şahutǒglu, S. Complex singularities and the Lorenz attractor. SIAM Rev. 2010, 52, 294-314
59. Abad, A.; Barrio, R.; Dena, A. Computing periodic orbits with arbitrary precision. Phys. Rev. E 2011, 84, 016701
60. Calvo, F.; Pahl, E.; Wormit, M.; Schwerdtfeger, P. Evidence for low-temperature melting of Mercury owing to relativity. Angew. Chem. Intl. Ed. Engl, 2013, doi: 10.1002/anie.201302742
61. Borwein, J. M.; Bailey, D. H.; Girgensohn, R. Experimentation in Mathematics: Computational Paths to Discovery, A.K. Peters: Natick, MA, USA, 2004
62. Ferguson, H. R. P.; Bailey, D. H.; Arno, S. Analysis of PSLQ, an integer relation finding algorithm. Math. Comput. 1999, 68, 351-369
63. Bailey, D. H.; Broadhurst, D. Parallel integer relation detection: Techniques and applications. Math. Comput. 2000, 70, 1719-1736
64. Chen, J.; Stehle, D.; Villard, G. A new view on HJLS and PSLQ: Sums and projections of lattices. Proceedings of ISSAC'13, Boston, MA, USA, June 2013; pp. 149-156
65. Bailey, D. H. A compendium of BBP-type formulas. 2011. Available online: http://www.davidhbailey.com/dhbpapers/bbp-formulas.pdf (accessed on 30 April 2015)
66. 2 and Catalan's constant. Not. AMS 2013, 60, 844-854
67. Karrels, E. Computing digits of pi with CUDA, 2013. Available online: http://www.karrels.org/pi (accessed on 30 April 2015)
68. Becher, V.; Heiber, P. A.; Slaman, T. A. A computable absolutely normal Liouville number, 2014. Available online: http://math.berkeley.edu/~slaman/papers/liouville.pdf (accessed on 30 April 2015)
69. Stoneham, R. On absolute (j; ")-normality in the rational fractions with applications to normal numbers. Acta Arithmetica 1973, 22, 277-286
70. Bailey, D. H.; Borwein, J. M. Nonnormality of Stoneham constants. Ramanujan J. 2012, 29, 409-422; DOI:10.1007/s11139-012-9417-3
71. Takahasi, H.; Mori, M. Double exponential formulas for numerical integration. Pub. RIMS 1974, 9, 721-741
72. NIST Digital Library of Mathematical Functions, version 1.0.6, May 2013. Available online: http://dlmf.nist.gov (accessed on 30 April 2015)
73. NIST Handbook of Mathematical Functions; Olver, F. W. J., Lozier, D. W., Boisvert, R. F., Clark, C. W., Eds.; Cambridge University Press: New York, NY, USA, 2010
74. The Inverse Symbolic Calculator 2.0. Available online: https:/isc.carma.newcastle.edu.au (accessed on 30 April 2014)
75. Panzer, E. Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals, 2014. Available online: http://arxiv.org/abs/1403.3385 (accessed on 30 April 2015)
76. Bailey, D. H.; Borwein, J. M. Hand-to-hand combat with thousand-digit integrals. J. Comput. Sci. 2012, 3, 77-86
77. Borwein, J. M.; Straub, A.; Wan, J. Three-step and four-step random walk integrals. Exp. Math. 2013, 22, 1-14
78. Lucas, S. K.; Stone, H. A. Evaluating infinite integrals involving Bessel functions of arbitrary order. J. Comp. Appl. Math. 1995, 64, 217-231
79. Sidi, A. The numerical evaluation of very oscillatory infinite integrals by extrapolation. Math. Comput. 1982, 38 517-529
80. Sidi, A.; A user-friendly extrapolation method for oscillatory infinite integrals. Math. Comput. 1988, 51, 249-266
81. Sidi, A. A user-friendly extrapolation method for computing infinite range integrals of products of oscillatory functions. IMA J. Numer. Anal. 2012, 32, 602-631
82. Bailey, D. H.; Borwein, J. M.; Broadhurst, D. M.; Glasser, L. Elliptic integral representation of Bessel moments. J. Phys. A: Math. Theor. 2008, 41, 5203-5231
83. Wan, J. Moments of products of elliptic integrals. Adv. Appl. Math 2012, 48, 121-141
84. Rogers, M.; Wan, J. G.; Zucker, I. J. Moments of elliptic integrals and critical L-values. 2013. Available online: http://arxiv.org/abs/1303.2259 (accessed on 30 April 2015)
85. Bailey, D. H.; Borwein, J. M.; Crandall, R. E.; Zucker, J. Lattice sums arising from the Poisson equation. J. Physics A: Math. Theor. 2013, 46, 115201
86. Borwein, D.; Borwein, J. M.; Straub, A. On lattice sums and Wigner limits. J. Math. Anal. Appl. 2014, 414, 489-513
87. Borwein, J. M.; Glasser, L.; McPhedran, R.; Wan, J. G.; Zucker, I. J. Lattice Sums: Then Now Cambridge University Press: Cambridge, UK, 2013
88. Borwein, D.; Borwein, J. M.; Shail, R. Analysis of certain lattice sums. J. Math. Anal. Appl. 1989, 143, 126-137
89. Lewin, M.; Lieb, E. H. Improved Lieb-Oxford exchange-correlation inequality with gradient correction. 2014. Available online: http://arxiv.org/abs/1408.3358 (accessed 30 on April 2015)
90. Bailey, D. H.; Borwein, J. M.; Crandall, R. E. Computation and theory of extended Mordell-Tornheim-Witten sums. Math. Comput. 2015. Available online: http://www.davidhbailey.com/dhbpapers/BBC.pdf (accessed on 30 April 2015)
91. Williams, S. W.; Bailey, D. H. Parallel computer architecture. Performance Tuning of Scientific Applications, Bailey, D. H., Lucas, R. F., Williams, S. W., Eds; CRC Press: Boca Raton, FL, USA, 2011; pp. 11-33
92. Top 500 list. Nov. 2014. Available online: http://top500.org (accessed on 30 April 2015)
93. Khanna, G. High-precision numerical simulations on a CUDA GPU: Kerr black hole tails. J. Sci. Comput. 2013, 56, 366-380
94. Mukunoki, D.; Takahashi, D. Implementation and evaluation of quadruple precision BLAS functions on GPUs. Appl. Parallel Sci. Comput. 2012, 7133, 249-259
95. Chevillard, S.; Mezzarobba, M. Multiple precision evaluation of the Airy Ai function with reduced cancellation. Proceedings of the 21st IEEE Symposium on Computer Arithmetic, IEEE Computer Society: Austin, TX, USA, June 2013
96. Crandall, R. E. Unified algorithms for polylogarithm, L-series and zeta variants. Algorithmic Reflections: Selected Works; PSIPress: Portland, OR, USA, 2012
97. Johansson, F. Fast and rigorous computation of special functions to high precision; PhD thesis, RSC Institute: Linz, Austria, 2014
98. Muller, J.-M.; Brisebarre, N.; de Dinechin, F.; Jeannerod, C.-P.; Lefevre, V.; Melquiond, G.; Revol, N.; Stehle, D.; Torres, S. Handbook of Floating-Point Arithmetic; Birkhauser: Boston, MA, USA, 2010
99. Corden, M. J.; Kreitzer, D. Consistency of floating-point results using the Intel compiler, or why doesn't my application always give the same answer. Intel Corporation, 2012. Available online: http://software.intel.com/sites/default/files/article/164389/fp-consistency-102511.pdf (accessed on 30 April 2015)