Dear Simon Plouffe, I send you the number e = exp(1) to 869,894,101 decimal digits. A world record again, but why? My main reason for doing this was to collect some experience with the two-dimensional special linear group over integers. More about this group will appear soon in my dissertation ("Powerful Tests for Primality"). Moreover, this calculation shows that there exists a method to compute exp(1) faster than the direct formula infty ----- \ 1 exp(1) = ) --- / k! ----- k=0 by using the binary splitting method which was described in [4]. I have used the formula n/2 [ a_n b_n ] [ 3 1 ] ------- [ (4k-1)(16k+4)+1 4k-1 ] [ ] = [ ] | | [ ] [ c_n d_n ] [ 2 1 ] | | [ 16k+4 1 ] k=1 e = lim (4 a_n/c_n - 3). n->infty I should note that this formula is not new, it can simply be transferred in the notation of continued fractions and can be found for example in [5]. By using this formula in the matrix representation and taking the advantage of fast matrix multiplication (e.g. see [5]) in combination with the binay splitting method ([3],[4]), we have the effect of a fast algorithm to calculate exp(1). Furthermore, for all numbers which have a simple continued fraction, this method seems to be faster than the binary splitting of the series formula. Using n = 55,000,000, I have calculated 869,894,101 decimal digits of exp(1). Originally, I intended to calculate 1 billion digits but I made an error in estimating the number of iterations n. No matter! To do this record, I have taken advantage of my arithmetic Piologie [8]. A new version will appear soon. So if there are people interested in the possibilities of Piologie, they are invited to contact me via e-mail. An open question is whether the formulas of H. J. Brothers and J. A. Knox [1], [2] are faster than this way? Best regards, Sebastian Wedeniwski (wedeniws@de.ibm.com) Timings ======= The computation (with no line of assembler) was done on July 31, 1999 and it took 79 hours and 30 minutes on 16 workstations of type IBM Power2 SC 135 MHz, 2 GB RAM, GNU C++ 2.8.0, AIX 4.1.5. I have verified all 200 million digits of my previous record which had used the direct formula. Acknowledgements ================ This computational record was possible with the machine support of IBM Deutschland Entwicklung GmbH, Boeblingen. Therefore, many thanks are going to my managers Dr. Michael von Carlowitz, Reinhold Krause and Joerg Thielges who have made a series of computations possible and have been helpful coordinators. Thanks also to Henry Koplien for his very cooperative and administrative aid. And last but not least, many thanks are going to my ex-manager Dr. Oskar von Dungern who set these computations on foot. Technical Details ================= The computation of exp(1) was splitted up in 96 processes on 16 processors. During the computation it takes 11.5 GB hard disc splitted up on 31 folders. - partial computations: (0) in 29073 sec (1) in 30067 sec (2) in 30249 sec (3) in 30866 sec (4) in 31146 sec (5) in 31795 sec (6) in 32113 sec (7) in 32510 sec (8) in 32528 sec (9) in 31813 sec (10) in 32159 sec (11) in 32448 sec (12) in 32773 sec (13) in 32703 sec (14) in 32849 sec (15) in 30250 sec - combinations (0)- (1) in 7039 sec (2)- (3) in 7058 sec (4)- (5) in 7068 sec (6)- (7) in 7092 sec (8)- (9) in 7067 sec (10)-(11) in 7042 sec (12)-(13) in 7054 sec (14)-(15) in 4035 sec (0)- (3) in 14939 sec (4)- (7) in 14943 sec (8)-(11) in 14905 sec (12)-(15) in 8500 sec (0)- (7) in 30830 sec (8)-(15) in 17539 sec (0)-(15) in 9170+9420+9139+9372 sec - divisions (splitted in 65 processes): 23568 sec + 9088 sec + 23458 sec + 8972 sec + 23713 sec + 9059 sec + 23800 sec + 9089 sec + 23636 sec + 9093 sec + 23649 sec + 9156 sec + 23523 sec + 9084 sec + 23676 sec + 9166 sec + 23617 sec + 9104 sec + 23436 sec + 9208 sec + 23587 sec + 9226 sec + 23528 sec + 9124 sec + 23557 sec + 9258 sec + 23622 sec + 9177 sec + 23547 sec + 9234 sec + 23664 sec + 9194 sec + 23649 sec + 9242 sec + 23524 sec + 9270 sec + 23847 sec + 9255 sec + 23546 sec + 9287 sec + 23556 sec + 9367 sec + 23625 sec + 9367 sec + 23483 sec + 9335 sec + 23596 sec + 9343 sec + 23507 sec + 9394 sec + 23490 sec + 9450 sec + 23445 sec + 9495 sec + 23433 sec + 9535 sec + 23421 sec + 9369 sec + 23359 sec + 9423 sec + 23398 sec + 9533 sec + 23565 sec + 9554 sec + 258 sec References ========== [1] H. J. Brothers, J. A. Knox: New Closed-Form Approximations to the Logarithmic Constant e, The Mathematical Intelligencer 20 (1998), 25-29. [2] H. J. Brothers, J. A. Knox: Novel Series-based Approximations to e, The College Mathematics Journal 30 (1999), 209-215. [3] R. P. Brent: Fast Multiple-Precision Evaluation of Elementary Functions, Journal of the Association for Computing Machinery 23 (1976), 242-251. [4] B. Haible, T. Papanikolaou: Fast multiprecision evaluation of series of rational numbers, Technical Report TI-97-7, Darmstadt University of Technology, April 1997. [5] J. Minkus: A Continued Fraction, The American Mathematical Monthly 103 (1996), 605-606. [6] C. D. Olds: The Simple Continued Fraction Exansion of e, The American Mathematical Monthly 77 (1970), 968-974. [7] S. Wedeniwski: Generic Matrices and the Implementation of Fast Matrix Operations, Diploma thesis, Universitaet Tuebingen. [8] S. Wedeniwski: Piologie - Eine exakte arithmetische Bibliothek in C++, Technical Report WSI 96-35, Universitaet Tuebingen, available by anonymous ftp from "ftp://ftp.informatik.uni-tuebingen.de/pub/CA/software/Piologie/" or "ftp://ftp-ca.informatik.uni-tuebingen.de/pub/math/Piologie/". =============================================================================== 2.7182818284590452353602874713526624977572470936999595749669676277240766303535\ 475945713821785251664274274663919320030599218174135966290435729003342952605956\ 307381323286279434907632338298807531952510190115738341879307021540891499348841\ 675092447614606680822648001684774118537423454424371075390777449920695517027618\ 386062613313845830007520449338265602976067371132007093287091274437470472306969\ 772093101416928368190255151086574637721112523897844250569536967707854499699679\ 468644549059879316368892300987931277361782154249992295763514822082698951936680\ 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