In case it's not clear from the J code, I'm evaluating this continued
fraction:
e = 3 + _1
--------------
4 + _2
----------
5 + _3
-------
6 + _4
----
7 + ...
On Thu, Jul 25, 2019 at 11:32 AM Devon McCormick <[email protected]> wrote:
> 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
>
>
--
Devon McCormick, CFA
Quantitative Consultant
----------------------------------------------------------------------
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