Very nice, Raul! Much shorter than my Rube Goldberg approach. --Kip On Saturday, March 12, 2016, Raul Miller <[email protected]> wrote:
> cv=: ([: (+`%/) 1 }.,)\@|: > > I hope this helps... > > -- > Raul > > > On Sat, Mar 12, 2016 at 1:21 PM, Kip Murray <[email protected] > <javascript:;>> wrote: > > Here you go: > > > > nume =: 1 , 1 % 4x * _1 + 4 * [: *:@>:@i. <: > > > > dene =: 1 1r2 , 1 $~ _2 + ] > > > > I think I got those from Abramowitz and Stegun. > > > > (_1 , nume 6),: dene 7 > > _1 1 1r12 1r60 1r140 1r252 1r396 > > 1 1r2 1 1 1 1 1 > > > > --Kip > > > > > > On Saturday, March 12, 2016, Raul Miller <[email protected] > <javascript:;>> wrote: > > > >> How do you compute the first two rows? > >> > >> Thanks, > >> > >> -- > >> Raul > >> > >> On Saturday, March 12, 2016, Kip Murray <[email protected] > <javascript:;> > >> <javascript:;>> wrote: > >> > >> > The challenge is at the end. First a table for a finite continued > >> fraction > >> > that approximates e =: ^ 1 . > >> > --Kip Murray > >> > > >> > > >> > The table below summarizes a finite continued fraction which begins > >> > > >> > 1 > >> > 1 + ------------- > >> > 1r12 > >> > 1r2 + -------- > >> > 1r60 > >> > 1 + ------ > >> > > >> > 1 + . > >> > . > >> > . > >> > > >> > table > >> > _1 1 1r12 1r60 1r140 1r252 1r396 > >> > 1 1r2 1 1 1 1 1 > >> > 1 3 19r7 193r71 2721r1001 49171r18089 1084483r398959 > >> > > >> > > >> > You must ignore the _1 in the upper left corner. You see how the first > >> row > >> > identifies numerators and the second row numbers on the "diagonal" of > the > >> > continued fraction. > >> > > >> > > >> > The third row gives the "convergents", results of terminating the > >> continued > >> > fraction at a diagonal number. The first four convergents are > >> > > >> > 1 , (1 + 1 % 1r2) , (1 + 1 % 1r2 + 1r12 % 1) , (1 + 1 % 1r2 + 1r12 % > 1 + > >> > 1r60 % 1) > >> > > >> > > >> > The convergents of this continued fraction approximate the number e > =: > >> ^ 1 > >> > . > >> > > >> > 2 * 0.5 * {: table > >> > 1 3 2.714285714 2.718309859 2.718281718 2.718281829 2.718281828 > >> > > >> > > >> > Now, how would you write verb cv which provides the third row of the > >> table > >> > given the first two? > >> > > >> > 2 {. table > >> > _1 1 1r12 1r60 1r140 1r252 1r396 > >> > 1 1r2 1 1 1 1 1 > >> > > >> > cv 2 {. table > >> > 1 3 19r7 193r71 2721r1001 49171r18089 1084483r398959 > >> > > >> > > >> > --Kip Murray > >> > > >> > > >> > > >> > -- > >> > Sent from Gmail Mobile > >> > ---------------------------------------------------------------------- > >> > For information about J forums see > http://www.jsoftware.com/forums.htm > >> ---------------------------------------------------------------------- > >> For information about J forums see http://www.jsoftware.com/forums.htm > > > > > > > > -- > > Sent from Gmail Mobile > > ---------------------------------------------------------------------- > > For information about J forums see http://www.jsoftware.com/forums.htm > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm -- Sent from Gmail Mobile ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
