Exactly,
e=. &.>
u=. ('('"_ , 'u '"_ , ] , ')'"_) :('('"_ , [ , ' u '"_ , ] , ')'"_)
YS=. <;._1 ' Y0 Y1 .. YN'
u e/\ YS
┌──┬─────────┬────────────────┬───────────────────────┐
│Y0│(Y0 u Y1)│(Y0 u (Y1 u ..))│(Y0 u (Y1 u (.. u YN)))│
└──┴─────────┴────────────────┴───────────────────────┘
u e scan YS
┌──┬─────────┬────────────────┬───────────────────────┐
│Y0│(Y0 u Y1)│((Y0 u Y1) u ..)│(((Y0 u Y1) u ..) u YN)│
└──┴─────────┴────────────────┴───────────────────────┘
On Tue, Jan 10, 2023 at 3:25 PM Raul Miller <rauldmil...@gmail.com> wrote:
> scan=. ((~/)\.)(&.|.)
> -scan p:i.5
> 2 _1 _6 _13 _24
> -/\ p:i.5
> 2 _1 4 _3 8
>
> --
> Raul
>
> On Tue, Jan 10, 2023 at 3:18 PM Jose Mario Quintana
> <jose.mario.quint...@gmail.com> wrote:
> >
> > Indeed,
> >
> > scan=. ((~/)\.)(&.|.)
> >
> > %&:{./"_1 +/ .* /\ (1 0,:~,&1)"0 ]10$1
> > 1 2 1.5 1.66667 1.6 1.625 1.61538 1.61905 1.61765 1.61818
> > %&:{./"_1 +/ .*scan (1 0,:~,&1)"0 ]10$1
> > 1 2 1.5 1.66667 1.6 1.625 1.61538 1.61905 1.61765 1.61818
> >
> > stp=. ] (([ ((<;._1 '|Sentence|Space|Time|Space * Time') , (,
> */&.:>@:(1
> > 2&{))@:(] ; 7!:2@:] ; 6!:2)&>) (10{a.) -.&a:@:(<;._2@,~) ]) [ (0 0 $
> > 13!:8^:((0 e. ])`(12"_)))@:(2 -:/\ ])@:(".&.>)@:((10{a.) -.&a:@:(<;._2@
> ,~)
> > ]) ::(0 0&$@(1!:2&2)@:('Mismatch!'"_))) ".@:('0( : 0)'"_)
> >
> > stp 11
> > %&:{./"_1 +/ .*/\ (1 0,:~,&1)"0 ]1000$1
> > %&:{./"_1 +/ .*scan (1 0,:~,&1)"0 ]1000$1
> > )
> >
> ┌─────────────────────────────────────────────┬──────┬───────────┬────────────┐
> > │Sentence │Space │Time │Space *
> > Time│
> >
> ├─────────────────────────────────────────────┼──────┼───────────┼────────────┤
> > │ %&:{./"_1 +/ .*/\ (1 0,:~,&1)"0 ]1000$1│79552 │0.059218 │4710.91
> > │
> >
> ├─────────────────────────────────────────────┼──────┼───────────┼────────────┤
> > │ %&:{./"_1 +/ .*scan (1 0,:~,&1)"0 ]1000$1│206656│0.000644564│133.203
> > │
> >
> └─────────────────────────────────────────────┴──────┴───────────┴────────────┘
> >
> > 2,1}.,1 1&,"0]2*1+i.3
> > 2 1 2 1 1 4 1 1 6
> >
> > %&:{./"_1 +/ .*/\ (1 0,:~,&1)"0 ]2,1}.,1 1&,"0]2*1+i.3
> > 2 3 2.66667 2.75 2.71429 2.71875 2.71795 2.71831 2.71828
> > %&:{./"_1 +/ .*scan (1 0,:~,&1)"0 ]2,1}.,1 1&,"0]2*1+i.3
> > 2 3 2.66667 2.75 2.71429 2.71875 2.71795 2.71831 2.71828
> >
> >
> > stp 11
> > %&:{./"_1 +/ .*/\ (1 0,:~,&1)"0 ]2,1}.,1 1&,"0]2*1+i.30
> > %&:{./"_1 +/ .*scan (1 0,:~,&1)"0 ]2,1}.,1 1&,"0]2*1+i.30
> > )
> >
> ┌────────────────────────────────────────────────────────────┬─────┬───────────┬────────────┐
> > │Sentence │Space│Time
> > │Space * Time│
> >
> ├────────────────────────────────────────────────────────────┼─────┼───────────┼────────────┤
> > │ %&:{./"_1 +/ .*/\ (1 0,:~,&1)"0 ]2,1}.,1
> > 1&,"0]2*1+i.30│19840│0.00190225 │37.7407 │
> >
> ├────────────────────────────────────────────────────────────┼─────┼───────────┼────────────┤
> > │ %&:{./"_1 +/ .*scan (1 0,:~,&1)"0 ]2,1}.,1
> > 1&,"0]2*1+i.30│30336│0.000149318│4.52972 │
> >
> └────────────────────────────────────────────────────────────┴─────┴───────────┴────────────┘
> >
> > NB. 300 instead of 30 is too many...
> >
> > On Mon, Jan 9, 2023 at 11:21 AM Henry Rich <henryhr...@gmail.com> wrote:
> >
> > > f/\ is quadratic but f/\. shouldn't be.
> > >
> > > Henry Rich
> > >
> > > On 1/9/2023 11:20 AM, Marshall Lochbaum wrote:
> > > > It's not just Fibonacci numbers. This is equivalent to a general
> method
> > > > for computing continued fractions convergents in linear time, using
> > > > matrix multiplication. There's a nice explanation of why it works in
> > > > here:
> > > >
> > > > https://perl.plover.com/classes/cftalk/INFO/gosper.txt
> > > >
> > > > And a J version, with phi's continued fraction 1 1 1... :
> > > >
> > > > %&:{./"_1 +/ .*/\ (1 0,:~,&1)"0 ]10$1
> > > > 1 2 1.5 1.66667 1.6 1.625 1.61538 1.61905 1.61765 1.61818
> > > >
> > > > Here's the continued fraction sequence for e:
> > > >
> > > > 2,1}.,1 1&,"0]2*1+i.3
> > > > 2 1 2 1 1 4 1 1 6
> > > >
> > > > %&:{./"_1 +/ .*/\ (1 0,:~,&1)"0 ]2,1}.,1 1&,"0]2*1+i.3
> > > > 2 3 2.66667 2.75 2.71429 2.71875 2.71795 2.71831 2.71828
> > > >
> > > > My timings show J taking quadratic time for the scan, so this
> > > > formulation is pretty slow but shows the principle.
> > > >
> > > > Marshall
> > > >
> > > > On Sun, Jan 08, 2023 at 07:05:46PM -0800, Elijah Stone wrote:
> > > >> Can we generate phi, the golden ratio, in parallel?
> > > >>
> > > >> Of course we can! Follows is an exposition of a classic method for
> > > doing
> > > >> it, which may be of interest; I have not seen it satisfactorily
> > > described
> > > >> elsewhere.
> > > >>
> > > >> The classic method for generating phi in j uses a continued
> fraction:
> > > >>
> > > >> (+%)/10#1
> > > >> 1.61818
> > > >>
> > > >> (It can be equivalently spelt (1+%)^:n]1.)
> > > >>
> > > >> Using rational numbers clarifies slightly:
> > > >>
> > > >> (+%)/10#1x
> > > >> 89r55
> > > >>
> > > >> Unsurprisingly, it's generating ratios of successive fibonacci
> > > numbers. Can
> > > >> it be parallelised? The operation used for reduction is not
> > > associative:
> > > >>
> > > >> ((1 +% 1) +% 1) -: (1 +% (1 +% 1))
> > > >> 0
> > > >>
> > > >> So it's not obvious how we would parallelise it. Taking a step
> back, it
> > > >> performs repeated division, where we don't know the divisor a
> priori,
> > > >> instead generating it recursively, so attacking it thus seems
> hopeless.
> > > >>
> > > >> Here's another method, based more directly on fibonacci numbers,
> which
> > > is
> > > >> more promising:
> > > >>
> > > >> (u=. {:,+/)^:9]1 1
> > > >> 55 89
> > > >>
> > > >> Here we generate fibonacci numbers via a recurrence relation, using
> two
> > > >> successive terms to generate the next.
> > > >>
> > > >> At first glance, this seems just as hopelessly sequential as the
> first
> > > >> solution, but the use of ^: is a tell. u^:n is the same as u@:u@
> :u@:
> > > ... n
> > > >> times. And _@:_ is associative! So if we can somehow express the
> > > operation
> > > >> of u as 'data', and ditto the composition of any number of us, then
> we
> > > can
> > > >> create a big array of n copies of u, and then reduce @: over it (in
> > > >> parallel), and finally apply the reduced u^:n to our y.
> > > >>
> > > >> Of course, this all depends on finding an efficient way of
> representing
> > > >> compositions of u. If we can't do that, then we'll waste a bunch of
> > > >> parallel work constructing compositions, only to _still_ need linear
> > > >> sequential work at the end, once we apply our composed u.
> > > >>
> > > >> Let's look at the operation of u _symbolically_:
> > > >>
> > > >> u a,b is b,a+b
> > > >> u u a,b is (a+b),(a+2*b)
> > > >> u u u a,b is (a+2*b),((2*a)+(3*b))
> > > >>
> > > >> In other words, u gives two results, each of which is a linear
> > > combination
> > > >> of its two inputs. And we have a convenient way of representing
> such
> > > >> functions: matrices! Hence obtains an alternate implementation of
> u,
> > > as a
> > > >> matrix product:
> > > >>
> > > >> u2=. (0 1,:1 1)&(+/ . *)
> > > >> u2^:9]1 1 NB.same result as u
> > > >> 55 89
> > > >>
> > > >> Hence, we can generate an efficient representation for u^:n in
> parallel,
> > > >> with only logarithmic span, for matrix multiplication is isomorphic
> to
> > > >> composition (and it is associative). The resultant function always
> > > takes
> > > >> the form of a 2x2 matrix, so we have only a constant amount of
> > > additional
> > > >> work to do at the end.
> > > >>
> > > >> (+/ .*/9#,:0 1,:1 1) +/ .* 1 1
> > > >> 55 89
> > > >>
> > > >> The astute may notice that this is wasted parallelism. We use a
> > > parallel
> > > >> reduction to find +/ .*/n#,:0 1,:1 1 in logarithmic time, but this
> is in
> > > >> fact just the nth power of matrix 0 1,:1 1, which can be calculated
> > > >> _sequentially_ in logarithmic time, by repeated squaring. But this
> > > method
> > > >> has a key advantage over that: if we want the entire fibonacci
> > > sequence, we
> > > >> can generate it, still with logarithmic span and linear work, by
> simply
> > > >> replacing our reduction with a scan:
> > > >>
> > > >> (+/ .*/\9#,:0 1,:1 1) +/ .* 1 1
> > > >> 1 2
> > > >> 2 3
> > > >> 3 5
> > > >> 5 8
> > > >> 8 13
> > > >> 13 21
> > > >> 21 34
> > > >> 34 55
> > > >>
> ----------------------------------------------------------------------
> > > >> 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
>
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