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
<mike_liz....@tiscali.co.uk>:
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" <rauldmil...@gmail.com> 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/Category: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 <dongu...@gmail.com> wrote:
How about rounding to a rational of some precision like pi rounded to
22r7 ?
On Mar 9, 2016 8:52 AM, "Kip Murray" <thekipmur...@gmail.com> 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 <rogerhui.can...@gmail.com>
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 <mike_liz....@tiscali.co.uk
<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
On Tue, Mar 8, 2016 at 7:28 PM, Kip Murray <thekipmur...@gmail.com
<javascript:;>>
wrote:
That's great! It's still a nice puzzle to write your own. --Kip
On Tuesday, March 8, 2016, Raul Miller <rauldmil...@gmail.com
<javascript:;>> wrote:
2 x: 6r4
3 2
--
Raul
On Tue, Mar 8, 2016 at 10:04 PM, Kip Murray <
thekipmur...@gmail.com
<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
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