Thanks; of course, and together with Ben Gorte's
suggestion delivers the wanted result:
] v=. 5+i.4
5 6 7 8
}: ,v ,.($v) # 1
5 1 6 1 7 1 8
-M
At 2016-03-11 13:12, you wrote:
Would this be helpful? ,v,.($v)#1 0 1 1 1 2 1
3 1 Linda -----Original Message----- From:
Programming
[mailto:[email protected]]
On Behalf Of Martin Kreuzer Sent: Friday,
March 11, 2016 7:30 AM To:
[email protected] Subject: Re:
[Jprogramming] Numerator and Denominator btw
(and as expected) D.E. Knuth, The Art of
Computer Programming, 2nd, vol 2, pages 339 ff
uses this convention as a shorthand to describe
a regular continued fraction: / x1, x2, ...,xn /
like in these examples 8/29 = / 3, 1, 1, 1, 2 /
e = 2 + / 1, 2, 1, 1, 4, 1, 1, 6, 1,... / -M At
2016-03-10 22:01, you wrote: >[ a0 ; a1 a2 a3
... ] ? Usage seems to vary - the ";" appears to
be >optional, but if used, it >can be thought
of as the integer part: a0 + 1 >% (a1 + 1 % (
a2 + 1 % (a3 +... ) ) ) The notation may be
extended to >use a comma (",") to indicate
repeated groups, so, dropping the ";" >rt2 = [1
2 2 2 2... ] == [1 ,2] rt3 = [1 1 2 >1 2 1 2 ...
] == [1,1 2] and here's an extension I hadn't
seen before: >e = [2 1 2 1 1 4 1 1 6 1 ...] ==
[2, 1 2n+2 1] I'll have a look at >Abramowitz &
Stegun; so far I've looked
at >http://perl.plover.com/classes/cftalk/
and >https://hal.inria.fr/file/index/docid/75792/
filename/RR-0760.pdf >I expect Knuth has
something on it too. Sorry for going a bit
off >topic, Cheers, Mike On >10/03/2016 19:35,
Kip Murray wrote: > Your question is too hard
for me. >If I wanted to look into it, I would >
start with paragraph >3.10 Theorems on Continued
Fractions on page 10 of the > Handbook
of >Mathematical Functions edited by Abramowitz
and Stegun. The > matrix >theorem shown there
may help. > > What does your notation [ a0 ; a1
a2 >a3 ... ] mean? > > --Kip > > On Thursday,
March 10, 2016, Mike
Day ><[email protected]> wrote: > >> So
how do you add [a0; a1 a2 >a3 ...] and [b0; b1
b2 b3 b4] >> to yield [c0;c1 c2 c3 c4...]
using >just their encodings as cfs? >> >> For
example, 3r11 >+ 1r2 = 17r22 ; >> or, as
c.f. [1;5 2] + [0;2] >= [1;1 2 7] >> How would
you do the latter directly? >> >> Mike >> >> >On
10/03/2016 17:18, Kip Murray wrote: >> >>> J
Continued Fractions >>> >>>> Let us consider the
continued fraction >>> >>> 1 + 2 % 3 + 4 % 5
+ >6 % 7 +
>>>> >>> This representss the
infinite list >>> >>> 1 , >(1+2%3), (1 + 2%3 + 4
% 5),
>>> >>> I follow the conventionn
that >each termm after the first ends with a >>>
denominator. >>> >>> We can >calculate 1 + 2 %
3 >+ 4 % 5 using. +`% / 1 2 3 4 5 >>> >>>
Try >it! >>> >>> The verb cf below calculates
terms >of our continued fraction >>> >>> cf
=: [: >+`%/ 1 + [: i. 1 + 2 *
] >>> cf >0 >>> 1 >>> cf >1 >>>
1.666666667 >>> cf >2 >>>
1.526315789 >>> cf >3 >>>
1.543046358 >>> cf"0 i. >4 >>> 1
1.666666667 1.526315789 >1.543046358 >>> >>>
--Kip Murray >>> >>> On Thursday, March 10,
2016, >Mike Day <[email protected]>
wrote: >>> >>> Thanks, Bo. >>>> I >was going to
reply saying "yes, but what about doing
arithmetic with >>>>> cf?" >>>> >>>> But I see
there are methods for addition etc with >such
representations. >>>> They're >>>> fairly
formidable, and I don't >quite see yet how to
control the >>>> precision >>>> of
such >>>> >operations. Unlike p-adics, they're
not built in to Pari GP, but >this >>>> is J of
course, >>>> which has neither p-adics nor
c.f. >built-in.... >>>> >>>> Cheers, >>>>
Mike >>>> >>>> On 10/03/2016 08:00, >Bo Jacoby
wrote: >>>> >>>> The standard approach to get
rational >approximations to irrational >>>>>
numbers is that of
continued >fractions. >>>>> >>>>> Den
0:23 torsdag >den 10. marts 2016 skrev Mike Day
< >>>>> >[email protected]>: >>>>> >>>>>
I'm >not at all sure that you'll find
this >relevant, but here goes, >>>>>
anyway: >>>>> I recently solved Euler >problem
541 which concerns harmonic
numbers: >>>>> >https://projecteuler.net/problem=
541 >>>>> >>>>> I started out, in J
of >course, with rationals, but the size of
the >>>>> problem is >>>>> >against at least my
naive application of them. I _did_ manage to
solve >>>>>> it >>>>> using Pari GP, and it
helped a lot that Pari GP has a >"p-adic"
type, >>>>> >>>>> For example, a p-adic
representation of >22r7 in base 7 is >3*7^-1 + 1
+ >>>>> O(7) >>>>> So I toyed around with p-adic
arithmetic >in J, but maintaining a >>>>>
consistent >>>>> level of precision
has >somewhat escaped me. Nevertheless, I've
managed >>>>> to set up verbs >to add, subtract,
multiply, divide p-adics, and even >>>>> raise
them >>>>>> to integer powers. >>>>> >>>>> So,
for example, to express 3r5 >1r15 22r7 in base
7 >to within O(7^6) >>>>> (7
6&rat2padic)"0] >3r5 1r15 22r7 >>>>> >>>>> 7 6 0
2 4 5 2 1 >4 >>>>> >>>>> 7 6 0 1 5 3 6 0
5 >>>>> >>>>> 7 6 >_1 1 3 0 0 0
0 >>>>> >>>>> >>>>> The leading digit is the
base, the >next one is the precision, and
the >>>>> next is a scaling power, 0
by >default. The remaining values are >>>>>
coefficients >>>>> of powers >of the
base. >>>>> >>>>> >>>>> So 3r5 is
approx >>>>> >(7 #.|.2 4 5 2 1
4)%7^6 >>>>> >>>>> >0.600005 >>>>> >>>>> >>>>>
And it can >do: >>>>> >>>>> padicadd/7 6
0 2 4 5 2 1 >4,:7 6 _1 1 3 NB. "add" 3r5 and
22r7 >>>>> >>>>> >7 6 _1 1 5 4 5 2 1 >>>>> >>>>>
which agrees >with >>>>> (7
6&rat2padic)3r5+22r7 NB. >represent 3r5 +
22r7 >>>>> >>>>> 7 6 _1 1 5 4 5 >2 1 >>>>> NB.
Loses the coefficient for
7^5 >(!) >>>>> >>>>> >>>>> In Pari
GP: >>>>> >>>>> >(22:49) gp > 3/5+O(7^6) \\
express 3r5 as p-adic base 7 >>>>> %1 = 2 + >4*7
+ 5*7^2 + 2*7^3 >+ 7^4 + 4*7^5 + O(7^6) >>>>>
(22:49) gp > >22/7+O(7^6) \\ express 22r7 as
p-adic base >7 >>>>> %2 = 7^-1 + 3 +
O(7^6) >>>>> (22:50) gp >
3/5+22/7+O(7^6) \\ >add them... >>>>> %3
= >7^-1 + 5 + 4*7 + 5*7^2 + 2*7^3 + 7^4 + 4*7^5
+ >O(7^6) >>>>> (22:52) gp > \\ coefficients of
1 >5 4 5 2 1 (4) match my J
ones >>>>> >>>>> >>>>> I don't
remember >studying p-adics at ("high") school or
university... >>>>> >>>>> >>>>> >They seem
pretty powerful in some
circumstances. >>>>> >>>>> >>>>> I >tried using
my J version of p-adics to solve the Euler
problem, but >>>>>> (a) my implementation is
far too slow, and >>>>> (b) precision >remains a
problem way before reaching the target
base, >>>>> namely >137. >>>>> >>>>> >>>>> I
suspect the precision difficulty is to do
with >the way I chose to >>>>> handle
multiplication and division rather
than >inability of a 64-bit >>>>> machine to
handle rationals! (Also >addition (subtraction)
of p-adics >>>>> with different scaling
powers, >as for 3r5 +
22r7 >above.) >>>>> >Currently >>>>>
padicdivide/7 6 0 2 4 5 2 >1 4,:7 6 _1 1 3 NB.
"divide" 3r5 by >>>>> >22r7 >>>>> >>>>> 7 6 0 0
2 5 3 5 5 >>>>> Pari GP >has:(23:12) gp >
(3/5+O(7^6)) / >(22/7+O(7^6)) >>>>> %8 = 2*7 +
5*7^2 + 3*7^3 + >5*7^4 + 5*7^5 +
O(7^7) >>>>> >>>>> >>>>> The coefficients are
the same, >but Pari GP has shifted the
precision to >>>>> O(7^7)
.... >>>>> >>>>> >>>>>> So it's unfinished
business. >>>>> >>>>> >>>>> Any
interest? >>>>>> >>>>> >>>>>
Mike >>>>> >>>>> >>>>> >>>>> On 09/03/2016
21:44, Don >Guinn wrote: >>>>> >>>>> Rational
numbers have always fascinated me. I >wanted to
build a gear >>>>>> train >>>>>> for a science
fair where the >gears form a loop where the gear
train >>>>>> does >>>>>> not mesh. But >only
after many thousands of revolutions. Then a sign
on >>>>>> the >>>>>>> front asking people to
break the gears by turning the
crank. >>>>>>> >>>>>> Long ago the Hammond organ
used a gear train to >approximate the
twelfth >>>>>> root of two and powers of it to
build >the notes in the octave. Now >>>>>>
electronics do that
more >effectively. >>>>>> >>>>>> The ratio Roger
showed in a previous post >are too large to be
pleasing. >>>>>> Smaller numbers in the ratio
could >still be useful and still be >>>>>>
accurate >>>>>> enough for
some >purposes. >>>>>> >>>>>> I realize that
this request is a little vague, >but this is
just an >>>>>> extention to the original
challenge. >>>>>> >On Mar 9, 2016 9:38 AM, "Raul
Miller"
<[email protected]> >wrote: >>>>>> >>>>>>
That seems a bit >underspecified, or open-ended,
at the >moment. >>>>>> >>>>>>> For example, pi
could be >1p1 (or o.1) or pi could be any of
a >number >>>>>>> of
algorithms: >https://en.wikipedia.org/wiki/Catego
ry:Pi_algorithms > >>>>>>> >>>>>>> Meanwhile,
there's also the >precision aspect - that could
also be >>>>>>> specified in a variety
of >ways. >>>>>>> >>>>>>> In other words, you
might be asking for
something >like >this: >>>>>>>
7%~x:<.0.5+7*1p1 >>>>>>> >22r7 >>>>>>> >>>>>>>
Or you might have something different in
mind, >like perhaps >this: >>>>>>> ~.@(#~
(= ><./)@:|@:-&1p1),%/~i.100x >>>>>>> >22r7 >>>>>
>> >>>>>>> Or, significantly slower, >but
almost tolerable: >>>>>>> ~.@(#~
(= ><./)@:|@:-&1p1),%/~i.1000x >>>>>>> >355r113 >
>>>>>> >>>>>>> Or, ... any of a variety of
other mechanisms... >(and even brute
force >>>>>>> search techniques could be made to
be >significantly more efficient - a >>>>>>>
moment's thought would show >that incorporating
the rounding technique, >>>>>>> for example,
could >have increased the speed of that last
search by >>>>>>> something near >a factor of
1000). >>>>>>> >>>>>>> Anyways, if you can say a
bit more >about what you were aiming
for, >>>>>>> maybe we could do a better job >of
getting there? >>>>>>> >>>>>>>
Thanks, >>>>>>> >>>>>>> >-- >>>>>>>
Raul >>>>>>> >>>>>>> On Wed, Mar 9, >2016 at
11:18 AM, Don
Guinn ><[email protected]> >>>>>>> >wrote: >>>>>
>> >>>>>>> How about rounding to a rational of
some >precision like pi rounded to >>>>>>>> 22r7
? >>>>>>> On Mar 9, 2016 >8:52 AM, "Kip Murray"
<[email protected]> >wrote: >>>>>>>>
Thanks, Roger and Raul. Not understanding GCD
+. I >had a Rube >>>>>>>> Goldberg >>>>>>>>
solution involving Format >": and Do
". --Kip >>>>>>>> >>>>>>>>> On Wednesday, March
9, 2016, >Roger Hui
<[email protected]> >>>>>>>>> >>>>>>>>> >
wrote: >>>>>>>> den and num illustrate different
ways of computing the >same
thing, >>>>>>>> >>>>>>>>> hopefully in so doing
improves >understanding. If you want
speed >>>>>>>>>>
2&x: >>>>>>>>>> >>>>>>>>>> >is >>>>>>>> going to
be faster. >>>>>>>> >>>>>>>>> Also,
watch >this: >>>>>>>>>> 2 x: o.
1 >>>>>>>>>> >1285290289249
409120605684 >>>>>>>>>> x: >o.
1 >>>>>>>>>> >1285290289249r409120605684 >>>>>>>>
>> >>>>>>>>>> > 2 x:!.0 o. 1 >>>>>>>>>>
884279719003555 >281474976710656 >>>>>>>>>>
x:!.0
o. >1 >>>>>>>>>> >884279719003555r281474976710656
>>>>>>>>>> >>>>>> > >>>> >>>>>>>>>> >>>>>>>>>>
>>>>>>>>>> On Wed, >Mar 9, 2016 at 7:20 AM,
Mike Day
< >>>>>>>>>> >[email protected] >>>>>>>>
>> <javascript:;>>
wrote: >>>>>>>>>> >>>>>>>>>>> Roger, are these
(among) recommended and preferred
methods >for >>>>>>>>>> >>>>>>>>>>> recovering
num & den, or do they just show >an elegant way
of >>>>>>>>>>> avoiding 2
x: >? >>>>>>>>>>> >>>>>>>>>>> >Thanks, >>>>>>>>>>
> >>>>>>>>>>> >Mike >>>>>>>>>>> >>>>>>>>>>> >>>>
>>>>>>> On >09/03/2016 04:32, Roger
Hui >wrote: >>>>>>>>>>> >>>>>>>>>>>
x=: >%/?2$10^8x >>>>>>>>>>> >>>>>>>>>>>> >x >>>>>
>>>>>>> >69904549r40669028 >>>>>>>>>>>> >>>>>>>>
>>>> > den=: [: % 1
+. >] >>>>>>>>>>>> num=:
* >den >>>>>>>>>>>> >>>>>>>>>>>>
den >x >>>>>>>>>>>> >40669028 >>>>>>>>>>>>
num >x >>>>>>>>>>>> >69904549 >>>>>>>>>>>> >>
num >x >>>>>>>>>>>> >69904549 >>>>>>>>>>>> >>>>>>
>>>>>> >>>>>>>>>>>> > > >>>>>>>>>>> >>>>>>>>>>>>
On Tue, Mar 8, 2016 at >7:28 PM, Kip Murray
< >>>>>>>>>>>> >[email protected] >>>>>>>>>>
>> >>>>>>>>>>>>
<javascript:;>> >>>>>>>>>>>> >wrote: >>>>>>>>>>>
>>>>>>>>>>>> That's great! It's still a nice
puzzle >to write your
own. --Kip >>>>>>>>>>>> >>>>>>>>>>>> On
Tuesday, March >8, 2016, Raul Miller
<[email protected] >>>>>>>>>>>>> ><javascript
:;>> wrote: >>>>>>>>>>> 2
x: >6r4 >>>>>>>>>>> >>>>>>>>>>>> 3
2 >>>>>>>>>>>>>> >-- >>>>>>>>>>>>>> >Raul >>>>>>>
>>>>>>> >>>>>>>>>>>>>> >>>>>>>>>>>>>> > On
Tue, Mar 8, 2016 at 10:04 PM, Kip Murray
< >>>>>>>>>>>>>> >>>>>>>>>>>>>>>
[email protected] >>>>>>>>>>>>
<javascript:;> >>>>>>>>>>
<javascript:;>> >wrote: >>>>>>>>>>>> How do you
find the numerator and denominator in >lowest
terms >>>>>>>>>>>>>> of
a >>>>>>>>>>>>>> >>>>>>>>>>>>>
rational >>>>>>>>>>
fraction? For >example, >>>>>>>>>>>>>>>
nd >6r4 >>>>>>>>>>>>>>>
3 >2 >>>>>>>>>>>>>>> >>>>>>>>>>>>>>> --Kip
Murray >>>>>>>>>>>>>>> >>>>>>>>>>>>>>>> >>>>>>>>>
> > >>>>> >>>>>>>>>>>>>>> -- >>>>>>>>>>>>>>>
Sent >from
Gmail >Mobile >>>>>>>>>>>>>>> >>>>>>>>>>>>>>> >>>
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