I recently posted something about using a (Ramanujan-like) continued fraction for calculating e: ramaE=: ([: +`%`:3 [: , [: x: (3 + i.) ,. [: - [: >: i.)"0 ramaE 10 9864101r3628800 50j48 ": ramaE 10 NB. Format as decimal 2.718281801146384479717813051146384479717813051146 50j48 ": ramaE 20 NB. More precision... 2.718281828459045235339784490666415886146403434540 50j48 ": ramaE 50 NB. Even more precision... 2.718281828459045235360287471352662497757247093700 50j48 ": ramaE 80 NB. More than 48 digits? 2.718281828459045235360287471352662497757247093700 102j100 ": ramaE 80 NB. Yes - more than 48 digits? 2.7182818284590452353602874713526624977572470936999595749669676277240766303535475945713821785251664274
This appears to add more digits quickly as you increase the argument. You can tell how many digits are good by comparing a given run with one using a higher value: try2=. ramaE 80 90 try2 194545954561539067326581042506243811231423891946873480598031074615399345559765318585225493549256921092507319165187339201r71569457046263802294811533723186532165584657342365752577109445058227039255480148842668944867280814080000000000000000000 119434520557937... =/102j100":&>try2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 =/202j200":&>try2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 ... 0 i.~ =/202j200":&>try2 122 So "ramaE 80" gives us e to 122 digits (120 decimal digits). On Thu, Jul 25, 2019 at 11:04 AM Ulrich Vollert <[email protected]> wrote: > Hello, > > I came across ’The Google Test’ by Eugene McDonnell ( > https://www.jsoftware.com/papers/play211.htm < > https://www.jsoftware.com/papers/play211.htm>) and was wondering how to > calculate many digits of Euler’s number e. > > I found an (ancient) article "The calculation of e to many significant > digits" by A. H. J. Sale in The Computer Journal 11(2) · August 1968 ( > https://www.researchgate.net/publication/266281843_The_Calculation_of_e_to_Many_Significant_Digits > < > https://www.researchgate.net/publication/266281843_The_Calculation_of_e_to_Many_Significant_Digits>) > which uses a clever algorithm implemented in Algol 60 (!) to calculate e > digit by digit and using only integer arithmetic. > > With this algorithm in J (see below) I could calculate as many digits of > e as I want (I used 140 digits) and solved the Google Test which asks for: > > 1. First 10-digit prime found in consecutive digits of e. > > 2. The number F(5) which follows > > F(1)= 7182818284 > F(2)= 8182845904 > F(3)= 8747135266 > F(4)= 7427466391 > F(5)=__________ > > (BTW In the article of Eugene McDonell the second number F(2) contains > transposed digits 9 and 0.) > > Here is my question: > > How would you calculate many significant digits of e in J? > > Regards, > Ulrich > > > NB. *** problem 1 *** > > NB. value to check m > checkm =: 3 : 0 > r =. -: ^. 6.2831852 * y > r + y * (^. y) - 1 > ) > > NB. given the number of digits of e which are wanted, > NB. calculate the number of required terms > number_of_terms =: 3 : 0 > test =. 2.30258509 * >: y > (>: ^: (test&>: @: checkm) ^:_) 4 > ) > > NB. from JforC > LoopWithInitial =: 2 : 'u&.>/\.&.(,&(<v))&.|.&.(<"_1)' > > NB. x is coeff and y is j, carry, follwed by the coeffs so far > NB. answer are the next j and carry, the new coeff, and old coeffs > term =: 4 : 0 > j =. {. y > carry =. 1 { y > coeffs =. 2 }. y > temp =. carry + 10 * x > (<: j), ((0, j) #: temp), coeffs > ) > > NB. initialise with m > init =: 3 : 0 > term LoopWithInitial (y, 0) (<: y) $ 1 > ) > > NB. given is an output of term LoopWithInitial > NB. find next output > next =: 3 : 0 > NB. take last row of output of term LoopWithInitial > NB. the row contains: j carry coeffs > last =. {: y > NB. carry is the next digit of e, forget j > ee =: ee , ": 1 { last > term LoopWithInitial ((<: #last), 0) |. 2 }. last > ) > > NB. calculate the number of digits of e > NB. the digits are collected in a global variable ee > calc_e =: 3 : 0 > ee =: '' > next^:y init number_of_terms y > ee > ) > > digits_e =: ". 10 ]\ calc_e 140 > > solution1 =: {. (I. 1 p: digits_e) { digits_e > NB. -> 7427466391 > > NB. *** problem 2 *** > > NB. cross sum of a number > csum =: 3 : '+/ "."0 ": y' > > NB. the given numbers have all the same cross sum of 49 > NB. csum"0 (7182818284 8182845904 8747135266 7427466391) -> 49 49 49 49 > > NB. find the next number in the digits of e with a cross sum of 49 > solution2 =: 4 }. (] {~ [: I. 49 = csum"0) digits_e > NB. -> 5966290435 > > > > > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm > -- Devon McCormick, CFA Quantitative Consultant ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
