[Jprogramming] Approximating e

[Bo Jacoby]

   >:@*/%>:i.20
2.71828
This speeds up the series. 




Den 7:11 tirsdag den 11. marts 2014 skrev Roger Hui <rogerhui.canada at gmail.com>:
 
> If I have 32-bit numbers, when does this information become fiction.
>> Also how can I get the best possible and correct decimal approximation
>> from these rational numbers?
>
>32-bits give you about 10 decimal digits.  (0 j. n) ": x displays x to n
>decimal places.  Where the following two outputs agree tells you how many
>digits for +`%/401#1x can be trusted.
>
>    _100]\(98$' '),0j200 ": +`%/1x, 400#1
>
>                      1.
>6180339887498948482045868343656381177203091798057628621354486227052604628189024497050372347293136948
>1774505251856655015609156645743553332885289608910521042249892203715062899055972967323443765486133617
>
>    _100]\(98$' '),0j200 ": +`%/1x, 500#1
>
>                      1.
>6180339887498948482045868343656381177203091798057628621354486227052604628189024497072072041893911374
>8475134846000035278501181299327946966700279337151010482725318315024081083474561817534485930797695298
>
>
>
>
>
>On Mon, Mar 10, 2014 at 7:07 PM, Linda Alvord <lindaalvord at verizon.net>wrote:
>
>> Here's the version which gets just the final ratio.
>>
>>      +`%/1x, 400#1
>>
>> 734544867157818093234908902110449296423351r453973694165307953197296969697410
>> 619233826
>>
>>
>>
>> 734544867157818093234908902110449296423351%453973694165307953197296969697410
>> 619233826
>> 1.61803
>>
>>    (32#'O'),' ',32#'O'
>>    OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO
>>
>> If I have 32-bit numbers, when does this information become fiction.  Also
>> how can I get the best possible and correct decimal approximation from
>> these
>> rational numbers?
>>
>> Linda
>>
>>
>>
>> ----Original Message-----
>> From: programming-bounces at forums.jsoftware.com
>> [mailto:programming-bounces at forums.jsoftware.com] On Behalf Of Linda
>> Alvord
>> Sent: Monday, March 10, 2014 9:32 PM
>> To: programming at jsoftware.com
>> Subject: Re: [Jprogramming] Approximating e
>>
>> Thanks for your hints.  I always wanted to get rational approximations for
>> the Golden Section.
>>
>>      {.|.+`%/\1x, 300#1
>> 26099748102093884802012313146549r16130531424904581415797907386349
>>
>>    32#'O'
>> OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO
>>
>>    {.|.+`%/\1x, 400#1
>>
>> 734544867157818093234908902110449296423351r453973694165307953197296969697410
>> 619233826
>>
>>
>>
>> 734544867157818093234908902110449296423351%453973694165307953197296969697410
>> 619233826
>> 1.61803
>>
>> How can I get the best possible decimal approximation (I have 32 bit
>> digits)?
>>
>> Linda
>>
>>
>>
>> -----Original Message-----
>> From: programming-bounces at forums.jsoftware.com
>> [mailto:programming-bounces at forums.jsoftware.com] On Behalf Of EelVex
>> Sent: Monday, March 10, 2014 7:12 PM
>> To: Programming forum
>> Subject: Re: [Jprogramming] Approximating e
>>
>> * Summing infinite series
>>
>>    +/%!i.100x
>>    +/^ t. i.100x  NB. Taylor coefficients
>> %+/((_1&^)%!)i.100x
>> etc
>>
>> * Taking an asymptotic
>>
>> (-^~1-%) 100x
>> ((^~%~^~@>:) - (^~%^~@<:))100x
>> etc
>>
>> * Continued fractions
>>
>>    +`%/2 1,2#>:i.100x
>>    +`%/2, 2#2+i.100x
>>    (+%)/2 1, ,(1 1,~])"0 +:>:i.100x   NB. canonical form
>>
>>
>>
>>
>> On Mon, Mar 10, 2014 at 6:38 PM, km <km at math.uh.edu> wrote:
>>
>> > The rational  2721r1001  approximates  e  to six, almost seven decimal
>> > places:
>> >
>> >     0j7 ": (^ 1) ,: 2721r1001
>> >  2.7182818
>> >  2.7182817
>> >
>> > I
 got 2721r1001 from a continued fraction.  How would you look for
>> > rational approximations to  e  ?
>> >
>> > --Kip Murray
>> >
>> > Sent from my iPad
>> > ----------------------------------------------------------------------
>> > For information about J forums see http://www.jsoftware.com/forums.htm
>
>> >
>> ----------------------------------------------------------------------
>> For information about J forums see http://www.jsoftware.com/forums.htm
>>
>>
 ----------------------------------------------------------------------
>> For information about J forums see http://www.jsoftware.com/forums.htm
>>
>> ----------------------------------------------------------------------
>> For information about J forums see http://www.jsoftware.com/forums.htm
>>
>----------------------------------------------------------------------
>For information about J forums see http://www.jsoftware.com/forums.htm
>
>
>