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D. S. McNeil

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Feb 9, 2011, 12:02:10 AM2/9/11
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Hello!

First time poster, so brief introduction: planetary astronomer, been
using Sage for years for both science and fun. I'm also one of the
Editors-in-Chief of the OEIS, and have been experimenting with using
Sage for bulk sequence processing and verification. I find writing in
Sage very competitive, but there are a handful of things I'd like to
fix.

For one behaviour which startled me, I thought I followed the right
procedure: asked why it was done that way on IRC, then filed a bug
report on trac. I found the current behaviour irritating enough to
consider posting to sage-devel to complain, but one page recommended
against shouting here as a way of drawing attention to your pet bug,
so I figured before I suggest work for others it's important to show
I'm willing to do some myself. Accordingly I've spent time answering
questions on ask.sagemath.org and worked my way up to fourth, just
passing was. I don't think I'll be catching those above me (hi @niles,
@kcrisman!), as I keep having to vote them up..

Anyway, now it's time to start burning that karma. :^)


The easiest to fix of my pet peeves, each of which I'm willing to
submit patches for if there's someone willing to help navigate.

(1) gcd is broken. http://trac.sagemath.org/sage_trac/ticket/10459

gcd(2/1,4) returns 1 "for simplicity" (!), because 2/1 is a rational.
This is shockingly silly. It's especially so given the existence of
both .content and lcm! I could almost understand raising an
exception, but there's no reason not to use something which reduces to
the integer case. (Using // isn't a good option for me, because that
means surrendering a great sanity check for sequences which should be
made up of integers but aren't.) There's a pending request to
implement a more general gcd, but that's a different problem: I just
want the common case of rationals to do something sensible, and it
could be done in just a few lines given the existing code.

(2) No kwarg constraints in Partitions/Compositions should be mutually
exclusive. (I think there's a ticket for this but I can't find it
now.)

Partitions(15,length=5, parts_in=[1,3,4]) doesn't work, so you have to
do an explicit filter. Optimizing it is a separate issue, but when
there's a trivial way to implement the functionality (simply test that
partitions satisfy the specified criteria before yielding them, in
cases where we don't currently embed the constraints in the
construction process itself), then ISTM we should do so now rather
than wait for a more efficient implementation to appear.
Functionality+speed > functionality, but functionality >>> no
functionality.

Currently, partitions_restricted's docstring says not to use it but to
use RestrictedPartitions instead, which in turn says not to use
RestrictedPartitions but to use Partitions with the "parts_in"
keyword, which in its turn doesn't work.

(3) primes(10, infinity) should work.

Since we're just calling next_prime, there's no reason to require the
upper limit to be an integer. This bites me more than you'd think,
because lots of (often uninteresting, but very error-prone) OEIS
sequences are of the form "function-of (next prime > x s.t.
something-or-other)" and I like functional programming styles.
Wouldn't mind a proof=False option which calls next_probable_prime
instead, either.

(4) Probably this has already been discussed to death and presumably
vetoed for sound reasons, but I don't see the point of complaining
about whitespace issues at the end of a program after every
non-whitespace character when Python doesn't.


Doug

--
Department of Earth Sciences
University of Hong Kong

William Stein

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Feb 9, 2011, 12:52:51 AM2/9/11
to sage-...@googlegroups.com
On Tue, Feb 8, 2011 at 9:02 PM, D. S. McNeil <dsm...@gmail.com> wrote:
> Hello!
>
> First time poster, so brief introduction: planetary astronomer, been
> using Sage for years for both science and fun.  I'm also one of the
> Editors-in-Chief of the OEIS, and have been experimenting with using
> Sage for bulk sequence processing and verification.  I find writing in
> Sage very competitive, but there are a handful of things I'd like to
> fix.
>
> For one behaviour which startled me, I thought I followed the right
> procedure: asked why it was done that way on IRC, then filed a bug
> report on trac.  I found the current behaviour irritating enough to
> consider posting to sage-devel to complain, but one page recommended
> against shouting here as a way of drawing attention to your pet bug,
> so I figured before I suggest work for others it's important to show
> I'm willing to do some myself.  Accordingly I've spent time answering
> questions on ask.sagemath.org and worked my way up to fourth, just
> passing was. I don't think I'll be catching those above me (hi @niles,
> @kcrisman!), as I keep having to vote them up..

Dang, I need to post to http://ask.sagemath.org more!

>
> Anyway, now it's time to start burning that karma. :^)
>
>
> The easiest to fix of my pet peeves, each of which I'm willing to
> submit patches for if there's someone willing to help navigate.
>
> (1) gcd is broken.    http://trac.sagemath.org/sage_trac/ticket/10459
>
> gcd(2/1,4) returns 1 "for simplicity" (!), because 2/1 is a rational.
> This is shockingly silly.  It's especially so given the existence of
> both .content and lcm!  I could almost understand raising an
> exception, but there's no reason not to use something which reduces to
> the integer case.  (Using // isn't a good option for me, because that
> means surrendering a great sanity check for sequences which should be
> made up of integers but aren't.)  There's a pending request to
> implement a more general gcd, but that's a different problem: I just
> want the common case of rationals to do something sensible, and it
> could be done in just a few lines given the existing code.

I'm personally OK either way with this.

>
> (2) No kwarg constraints in Partitions/Compositions should be mutually
> exclusive.  (I think there's a ticket for this but I can't find it
> now.)
>
> Partitions(15,length=5, parts_in=[1,3,4]) doesn't work, so you have to
> do an explicit filter.  Optimizing it is a separate issue, but when
> there's a trivial way to implement the functionality (simply test that
> partitions satisfy the specified criteria before yielding them, in
> cases where we don't currently embed the constraints in the
> construction process itself), then ISTM we should do so now rather
> than wait for a more efficient implementation to appear.
> Functionality+speed > functionality, but functionality >>> no
> functionality.
>
> Currently, partitions_restricted's docstring says not to use it but to
> use RestrictedPartitions instead, which in turn says not to use
> RestrictedPartitions but to use Partitions with the "parts_in"
> keyword, which in its turn doesn't work.

You should probably email the sage-combinat list about this:

http://groups.google.com/group/sage-combinat-devel/

>
> (3) primes(10, infinity) should work.
>
> Since we're just calling next_prime, there's no reason to require the
> upper limit to be an integer.  This bites me more than you'd think,
> because lots of (often uninteresting, but very error-prone) OEIS
> sequences are of the form "function-of (next prime > x s.t.
> something-or-other)" and I like functional programming styles.
> Wouldn't mind a proof=False option which calls next_probable_prime
> instead, either.

+1 to both suggestions. I don't see any reason for somebody not to
implement both of these. Great suggestions.

> (4) Probably this has already been discussed to death and presumably
> vetoed for sound reasons, but I don't see the point of complaining
> about whitespace issues at the end of a program after every
> non-whitespace character when Python doesn't.

I don't remember any discussion about this at all. It sounds like you
might have found a bug in the preparser. Could you post an example to
nail down exactly what you're talking about?

-- William

>
>
> Doug
>
> --
> Department of Earth Sciences
> University of Hong Kong
>

> --
> To post to this group, send an email to sage-...@googlegroups.com
> To unsubscribe from this group, send an email to sage-devel+...@googlegroups.com
> For more options, visit this group at http://groups.google.com/group/sage-devel
> URL: http://www.sagemath.org
>

--
William Stein
Professor of Mathematics
University of Washington
http://wstein.org

D. S. McNeil

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Feb 9, 2011, 3:46:52 AM2/9/11
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[..]

> I'm personally OK either way with this.

IMO a*b = gcd(a,b)*lcm(a,b) should be maintained wherever possible.
There are pari codes whose direct Sage equivalent silently breaks for
this reason, and I can't bring myself to admit it to my pari-speaking
friends. :^)

> Could you post an example [re: my whitespace issues --ed] to nail down exactly what you're talking about?

sage: s = 'for i in range(3):\n' + ' '*4 + 'print i\n'
sage: # add extra space, such as can often happen in practice
sage: # when writing code, and I can't see it, because, well,
sage: # it's whitespace..
sage: s = s + '\n' + ' '*4 + '\n'
sage:
sage: fname = 'whitespace_pedantic.sage'
sage: with open(fname,'w') as fp:
....: fp.write(s)
....:
sage: # Python is happy
sage: execfile(fname)
0
1
2
sage: # Sage is not
sage: load(fname)
------------------------------------------------------------
File "<string>", line 5

^
SyntaxError: invalid syntax

David Roe

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Feb 9, 2011, 4:18:42 AM2/9/11
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[..]
> I'm personally OK either way with this.

IMO a*b = gcd(a,b)*lcm(a,b) should be maintained wherever possible.
There are pari codes whose direct Sage equivalent silently breaks for
this reason, and I can't bring myself to admit it to my pari-speaking
friends. :^)

+1.

koffie

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Feb 9, 2011, 4:53:48 AM2/9/11
to sage-devel
Hej Doug,

Nice list of bugs. I was wondering, might you be interested in
becoming a sage developer to? It's really not that hard, and it sounds
to me that if you knew how to edit the source code of sage you would
be able to fix some of these bugs yourself, and become one of the
volunteers who makes sage better. And the advantage is by being a
developer you are the first one to be able to use the fixed or
improved code!
If you are willing to learn how to contribute code to sage, I'm also
willing to aid you in guiding you trough the developement process. The
feature request in number 3 sounds like one which is really doable. If
you are interested I could teach you how to create and submit patches
to trac while fixing this issue.
Just send me an e-mail.

Kind regards,
Maarten Derickx

Niles

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Feb 9, 2011, 8:23:05 AM2/9/11
to sage-devel
Hi @DSM!

On Feb 9, 4:53 am, koffie <m.derickx.stud...@gmail.com> wrote:
> Hej Doug,
>
> Nice list of bugs. I was wondering, might you be interested in
> becoming a sage developer too?

+1

It's *really* not that hard :) I'd be happy to help you get started
too, and in particular you could let me know when you need a reviewer
for #3 or #2.

The Developer's Guide is a great place to start -- I would make two
modifications for someone who's fairly comfortable with writing code
already: First, start with Mercurial queues right away:

http://www.sagemath.org/doc/developer/walk_through.html#being-more-efficient-mercurial-queues

The description makes it sound too hard to be worthwhile for a new
contributor, but I think they're more straightforward than making
clones of the entire sage library. Just remember to start a new patch
('hg qnew <patchname>') *before* you start making changes -- it's
easier that way:

http://www.sagemath.org/doc/developer/walk_through.html#creating-your-own-patch-with-queues

Second, I wouldn't bother with installing a separate "development
version" of Sage. Mercurial queues let you revert all of your changes
in bulk, without losing them, and the kinds of things you'll be
starting with are relatively straightforward, so it's unlikely that
you'll render your Sage install unusable and irreparable. If you do,
*then* install a second version of Sage :)

Final note: to use Mercurial queues, you'll need the following in
"~/.hgrc" (update for your information, but an email address is not
required). This is explained earlier on the page "Walking Through the
Development Process", but not entirely in the section "Getting Started
with Mercurial Queues".

[ui]
username = Carl Friedrich Gauss <cfg...@uni-goettingen.de>

[extensions]
# Enable the Mercurial queue extension.
hgext.mq =

enjoy!
Niles

p.s. I hope I'm not duplicating or contradicting too much of what
Maarten might have already told you -- if so, just ignore my
comments :)

luisfe

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Feb 9, 2011, 9:57:36 AM2/9/11
to sage-devel


On Feb 9, 9:46 am, "D. S. McNeil" <dsm...@gmail.com> wrote:
> >> (1) gcd is broken. http://trac.sagemath.org/sage_trac/ticket/10459
> [..]
> > I'm personally OK either way with this.
>
> IMO a*b = gcd(a,b)*lcm(a,b) should be maintained wherever possible.
> There are pari codes whose direct Sage equivalent silently breaks for
> this reason, and I can't bring myself to admit it to my pari-speaking
> friends. :^)

+1 to this. I am interested on this ticket. But I have zero time to
spend developing sage right now.

There is also something wrong with lcm for rationals

sage: a = 2/3 # rational
sage: b = 1 # integer
sage: gcd(a,b)
1
sage: lcm(a,b)
...
TypeError
sage: b = QQ(1)
sage: lcm(a,b)
2
sage: a.lcm(b)
2
sage: a._lcm(b)
1

So, it seems that Rational.lcm has bugs. Please someone correct me if
I am mistaken, but Rational.lcm should not expect that the other term
is itself a rational. For this assumption we have Rational._lcm

Also, I am little surprised about the definition of lcm here. I would
expect, maybe naively, that the lcm of 1/3 and 1/2 is 1/6. That is:
lcm(numerators) / lcm(denominators)
Although the current behavior is well documented. Thinking a little
bit the lcm seems to be defined as the smallest 'integer' multiple of
1/3 and 1/2. The generator of the intersection of the ZZ-modules
ZZ[1/3] and ZZ[1/2]. So it makes sense...

Willem Jan Palenstijn

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Feb 9, 2011, 10:05:40 AM2/9/11
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On Wed, Feb 09, 2011 at 04:46:52PM +0800, D. S. McNeil wrote:
> > Could you post an example [re: my whitespace issues --ed] to nail down
> > exactly what you're talking about?
>
> sage: s = 'for i in range(3):\n' + ' '*4 + 'print i\n'
> sage: # add extra space, such as can often happen in practice
> sage: # when writing code, and I can't see it, because, well,
> sage: # it's whitespace..
> sage: s = s + '\n' + ' '*4 + '\n'
> sage:
> sage: fname = 'whitespace_pedantic.sage'
> sage: with open(fname,'w') as fp:
> ....: fp.write(s)
> ....:
> sage: # Python is happy
> sage: execfile(fname)
> 0
> 1
> 2
> sage: # Sage is not
> sage: load(fname)
> ------------------------------------------------------------
> File "<string>", line 5
>
> ^
> SyntaxError: invalid syntax

I think this should be fixed by trac ticket #9363 (merged in
sage-4.6.2.alpha1).

-Willem Jan

rjf

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Feb 10, 2011, 1:18:01 AM2/10/11
to sage-devel
You say,
> gcd(2/1,4) returns 1 "for simplicity" (!), because 2/1 is a rational.
> This is shockingly silly.

I don't know exactly how this came up, but if 2/1 is in a different
domain (rational)
from 2, (integer), then gcd should probably be 1, since any non-
zero
rational number divides any other, and one commonly uses the positive
"unit" 1 for
such a case. You could argue that since you can coerce 2/1, you
should.

That's sometimes OK, but not always.

Really, the issue is much broader. for example, do you also want to
treat the complex number
1+0*i the same as 1? do you want to treat the floating point number
1.0 the same as 1?

What about 1X1 matrices?

Is 1^0 the same as 1^0.0 or 1.0^0 or 1.0^0.0? Do you perhaps wish
to consider/dismiss
the existence of number systems with signed zeros (IEEE floating-point
standard) on the
grounds that -0 = +0, [true, for numerical comparison] and therefore
there should be
only a single zero?

While I don't know the exact formulation of this GCD problem, the
issue of
implicit coercion is one of the troubling sore spots in a system
design, and should not
be decided by counting up casual +1 votes.

I think the Axiom people might have thought more about it than others.

daly

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Feb 10, 2011, 1:46:53 AM2/10/11
to sage-...@googlegroups.com, axiom-d...@nongnu.org

It is a question of domains. In Axiom you can specify the domains.

2/1 is a Fraction(Integer) aka rational
4 is an Integer
(2/1)::Integer => 2 where 2 is an Integer.
4::Fraction(Integer) is a Fraction(Integer)

So there are several cases:
gcd((2/1),4::Fraction(Integer)) => 1 of type Fraction(Integer)
gcd((2/1)::Integer,4)) => 2 of type PositiveInteger
gcd(2/1,4) => 1 of type Fraction(Integer)
gcd(2,4) => 2 of type PositiveInteger
gcd(2,4::Fraction(Integer)) => 1 of type Fraction(Integer)

Tim Daly

William Stein

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Feb 10, 2011, 1:58:15 AM2/10/11
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Thanks. The above is _precisely_ what Sage currently does:


sage: a = gcd(2/1,QQ(4)); print a, type(a)
1 <type 'sage.rings.rational.Rational'>
sage: a = gcd(ZZ(2/1), 4); print a, type(a)
2 <type 'sage.rings.integer.Integer'>
sage: a = gcd(2/1, 4); print a, type(a)
1 <type 'sage.rings.rational.Rational'>
sage: a = gcd(2, 4); print a, type(a)
2 <type 'sage.rings.integer.Integer'>
sage: a = gcd(2, QQ(4)); print a, type(a)
1 <type 'sage.rings.rational.Rational'>

I personally see no *harm* in allowing gcd(a,b) to be a different
choice of generator for the ideal (a,b), which is all that the OP is
requesting. PARI does this, and it is definitely very useful there.
Always returning 1 (or 0) in the rationals isn't very useful.

-- William

Simon King

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Feb 10, 2011, 2:01:43 AM2/10/11
to sage-devel
Hi all!

On 9 Feb., 15:57, luisfe <lftab...@yahoo.es> wrote:
> On Feb 9, 9:46 am, "D. S. McNeil" <dsm...@gmail.com> wrote:
>
> > >> (1) gcd is broken.    http://trac.sagemath.org/sage_trac/ticket/10459
> > [..]
> > > I'm personally OK either way with this.
>
> > IMO a*b = gcd(a,b)*lcm(a,b) should be maintained wherever possible.

What does "Wherever possible" mean in an ordered ring, R? In
particular, is it possible for R=QQ? Or even for R=ZZ?

If I am not mistaken, lcm(x,y) could be defined as
inf {x*m | m in R, x*m>0, there is n in R with x*m==y*n} (1)
At least, that definition matches the situation in ZZ (note that
lcm(-2,-2) = 2).
Or is there any reason to have a minimum rather than an infimum?

On the other hand, what is the notion of a "multiple" of x? Is it
simply x*m for any m in R? Or is it rather x*m for any abs(m)>=1 in R?
In that case, definition (1) above became
inf {x*m | m in R, abs(m)>=1, x*m>0, there is n in R with abs(n)>=1
and x*m==y*n} (2)
Again, it's one definition that matches the situation in ZZ.

Now, consider R=QQ.
According to definition (1), the least common multiple of 1/1 and 1/1
is: Zero! 'Cause any positive rational is a multiple of 1/1, and the
infimum is zero.
But according to definition (2), the least common multiple of 1/1 and
1/1 is: 1/1.

Perhaps there are other possible ways to extend the notion of lcm from
ZZ to any ordered ring. I could imagine that different textbooks use
slightly different ways of defining lcm and gcd in ZZ, extending to QQ
differently.

So, is QQ reasonably covered by "Wherever possible"?? I doubt.
Note that currently we have
sage: gcd(-2,1)
1
sage: lcm(-2,1)
2
So, gcd(x,y)*lcm(x,y) == x*y doesn't even hold in ZZ. Why should it
hold in QQ?

Cheers,
Simon

Simon King

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Feb 10, 2011, 2:34:26 AM2/10/11
to sage-devel
PS:

On 9 Feb., 15:57, luisfe <lftab...@yahoo.es> wrote:
> There is also something wrong with lcm for rationals
>
> sage: a = 2/3 # rational
> sage: b = 1    # integer
> sage: gcd(a,b)
> 1
> sage: lcm(a,b)
> ...
> TypeError

I would agree that this is a bug. I think it would be just consequent
to first coerce a and b before computing their gcd or lcm.


> Also, I am little surprised about the definition of lcm here. I would
> expect, maybe naively, that the lcm of 1/3 and 1/2 is 1/6. That is:
> lcm(numerators) / lcm(denominators)
> Although the current behavior is well documented. Thinking a little
> bit the lcm seems to be defined as the smallest 'integer' multiple of
> 1/3 and 1/2. The generator of the intersection of the ZZ-modules
> ZZ[1/3] and ZZ[1/2]. So it makes sense...

Ah, yet another definition with yet another answer for the rationals.
So, now we have at least six different ways to extend the common
notion of lcm from ZZ to QQ:

lcm(1/4,1/6) = 0 (infimum over positive multiples)

lcm(1/4,1/6) = 1/4 (infimum over positive multiples with factor at
least one)

lcm(1/4,1/6) = 1/12 (lcm(numerators) / lcm(denominators))

lcm(1/4,1/6) = 1/2 (infimum over positive multiples with integer
factor)

lcm(1/4,1/6) = 1 (minimal positive multiple that is an integer)

lcm(1/4,1/6) = 1/2 (lcm(numerators)/gcd(denominators))

... [add further definitions here]

Tough choice!
At least, the last one has the property gcd(x,y)*lcm(x,y)==x*y,
assuming that we also define gcd(a/b,c/d) = gcd(a,c)/lcm(b,d).

I agree with Richard Fateman that in this case one shouldn't simply
decide by counting "+1" (isn't there a board for that purpose?).
But I think the problem here is not coercion but the "right" (or at
least reasonable) choice of a definition. What would textbooks say?

Cheers,
Simon

Simon King

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Feb 10, 2011, 2:41:07 AM2/10/11
to sage-devel
PPS:

On 10 Feb., 08:34, Simon King <simon.k...@uni-jena.de> wrote:
> ...
> But I think the problem here is not coercion but the "right" (or at
> least reasonable) choice of a definition. What would textbooks say?

I should add: I do believe that the choice is already made and the
choice is fine (see Tim's and William's posts above). Only, lcm(2/1,4)
should not raise an error but use coercion (similar to what gcd(2/1,4)
does).

Best regards,
Simon

D. S. McNeil

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Feb 10, 2011, 3:40:20 AM2/10/11
to sage-...@googlegroups.com
@rjf:

> I don't know exactly how this came up, but if 2/1 is in a different domain (rational) from 2, (integer), then gcd should probably be 1, since any

> non-zero rational number divides any other, and one commonly uses the positive "unit" 1 for such a case.

One also commonly uses the content, as it provides information and is
likely to be useful, whereas setting the "gcd" to 1 doesn't, really: I
could simply use 1 directly instead. In practice:

Mma and Pari both have my preferred behaviour, gcd(2/1, 4) = 2,
gcd(2/3, 4/3) = 2/3, lcm(2/1, 4) = 4, lcm(2/3, 4/3) = 4/3.

Maple gives instead gcd(2/1, 4) = 2, gcd(2/3, 4/3) = 1, lcm(2/1, 4) =
4, lcm(2/3, 4/3) = 8/9, which I'd also be okay with. Let a thousand
flowers bloom!

I don't know if Maxima has an lcm function, but at least gcd(2/1,4) =
2 and gcd(2/3, 4/3) = 2/3.

Magma barfs at rational input, which is defensible. (Maybe there's
another gcd which doesn't, not sure.) This would frustrate me, but at
least avoids errors such as the one that got me started on this
subject in the first place.

Sage, by comparison, gives 1, 1, error (or 4 if you use 4/1), 4/3,
which doesn't seem nearly as useful as either the Mma/Pari or Maple
behaviour. They choose different conventions, but both make sense to
me, and convince me that I'm not crazy. :^)

It's worth emphasizing that Sage __already gives the Pari answers__
for the cases of (rational, rational) and (integer, rational) argument
to lcm (and has a Rational.content implementation); I'd be interested
in understanding why gcd should be different.


> Really, the issue is much broader. for example, do you also want to treat the complex number
> 1+0*i the same as 1?   do you want to treat the floating point number 1.0 the same as 1?

As the use cases seem far less common, I have no issues requiring
explicit coercions for symbolic complex numbers. I certainly don't
have problems requiring explicit coercions for finite-precision types,
and have no opinions about any of the thousand other possibilities.


@Simon King: as you note, there are multiple ways to extend the
concept of gcds and lcms to the rationals. In such a situation, it
would seem that two minimal things you would like would be (1) to
reduce to the integer case for integer values, and (2) to maintain
some nice properties so that the names "gcd" and "lcm" still fit.
Given some definition satisfying (1), coercing down to integers from
rationals isn't much of a problem, because the values will be the
same. Pari, Mma, and Maple all do this in a way which makes sense,
and Sage already halfway does it (with lcm). Choosing any definition
which doesn't reduce to the integer one, as is currently done, seems
problematic to me as a design decision, given that it's far more
likely to be used that way in error than it is that someone decided to
obfuscate "1" by writing it as gcd(some rational, some integer).

> So, is QQ reasonably covered by "Wherever possible"?? I doubt.
> Note that currently we have
> sage: gcd(-2,1)
> 1
> sage: lcm(-2,1)
> 2
> So, gcd(x,y)*lcm(x,y) == x*y doesn't even hold in ZZ. Why should it hold in QQ?

I'd return different signatures, myself, but even if you don't agree,
the property can easily hold as-is for Z+ and Q+. Why is "for
positive integers and rationals" not an acceptable content for
"wherever possible"? (I'm a physicist, not a mathematician, so I'm
sometimes physics-sloppy when writing: what should I have written
instead of "wherever possible" as shorthand for something like "on the
largest region containing the regime of interest while preserving the
relationships under discussion"?)

Simon King

unread,
Feb 10, 2011, 4:42:57 AM2/10/11
to sage-devel
Hi Doug!

On 10 Feb., 09:40, "D. S. McNeil" <dsm...@gmail.com> wrote:
> @Simon King: as you note, there are multiple ways to extend the
> concept of gcds and lcms to the rationals. In such a situation, it
> would seem that two minimal things you would like would be (1) to
> reduce to the integer case for integer values, and (2) to maintain
> some nice properties so that the names "gcd" and "lcm" still fit.
> Given some definition satisfying (1), coercing down to integers from
> rationals isn't much of a problem, ...

There is no "coercing down to integers from rationals". One important
property of a coercion map is that it is a map, in contrast to a
partial map. Actually, it even is a morphism.

So, a coercion from QQ to ZZ would presumably be a morphism from QQ to
ZZ in the category of unital rings - which doesn't exist.

When I wrote "lcm(2/1,4) should not raise an error but use coercion",
I meant of course that the integer 4 should be coerced into the common
parent of 2/1 and 4, which is the rational field.

> Choosing any definition
> which doesn't reduce to the integer one, as is currently done, seems
> problematic to me as a design decision, given that it's far more
> likely to be used that way in error than it is that someone decided to
> obfuscate "1" by writing it as gcd(some rational, some integer).

No, I think you are mistaken. If you work with a,b rational or, worse,
real numbers, then it is very highly unlikely that a and b *are*
integers if they happen to seem like integers. Just think of rounding
errors. This is another reason why "coercing down to the integers"
won't make sense: It is highly unlikely that 1.0 really is the integer
number 1.

So, I think it is by far better to have a consistent notion than to
have to *guess* whether a user really means the integer 2 if s/he
write 4/2 (which in the first place is a rational, not an integer).
Bugs that are result of guesswork are the most ugly, IMHO.

Cheers,
Simon

Bruno Le Floch

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Feb 10, 2011, 6:26:57 AM2/10/11
to sage-...@googlegroups.com
Hi all,

> So, a coercion from QQ to ZZ would presumably be a morphism from QQ to
> ZZ in the category of unital rings - which doesn't exist.

Agreed.

> So, I think it is by far better to have a consistent notion than to
> have to *guess* whether a user really means the integer 2 if s/he
> write 4/2 (which in the first place is a rational, not an integer).
> Bugs that are result of guesswork are the most ugly, IMHO.

True. But in the case of Q (and more generally in the case of the
quotient field of a (principal?) ring), we can be consistent with the
ring of integers, without any guess-work.

(*) This is a white lie (see below).
Every rational number has a unique(*) form Product(p^(a_p), p prime)
for some integer powers a_p. The rational is an integer iff all a_p
are non-negative. In that case,

gcd(Product(p^(a_p)), Product(p^(b_p))) = Product(p^(min(a_p,b_p)))

and lcm is defined with max(a_p,b_p). But actually this definition
does not rely at all on the fact that the a_p and b_p are positive. So
we have a definition of the gcd and lcm for free on the quotient field
of any (principal?) ring. Then, gcd(x,y).lcm(x,y)=x.y; the notion
reduces to the one for integers, etc. This definition amounts to the
definition of lcm(x,y) as the smallest integer multiple of x which is
also an integer multiple of y.

(*) In fact, there is the issue of the sign, or more generally units
(elements that are invertible in the ring (here, ZZ) ). For this,
there has to be some arbitrariness on the sign of gcd and lcm of
negative numbers.

Note that for RDF and its colleagues, this does not apply (since they
are not the quotient field of any sensible ring), and we should stick
with the definition gcd(x,y)=1, however much x and y look like
integers.

Regards,
Bruno

Simon King

unread,
Feb 10, 2011, 8:10:04 AM2/10/11
to sage-devel
Hi Bruno

On 10 Feb., 12:26, Bruno Le Floch <blfla...@gmail.com> wrote:
> True. But in the case of Q (and more generally in the case of the
> quotient field of a (principal?) ring), we can be consistent with the
> ring of integers, without any guess-work.

Sure. This could be one of the definitions I mentioned: lcm(a/b,c/d) =
lcm(a,c)/gcd(b,d). But: Would it be a wise idea to have a totally
different definition for fields and for quotient fields?

Let me phrase it like this: There are different interpretations of the
term "consistent".

On the one hand, one could mean "consistency with respect to sub-
structures": Let S be a sub-ring of a ring R; gcd_R is consistent with
gcd_S <=> gcd_R(x,y)==gcd_S(x,y) for all x,y in S. As you have
pointed out, there is a way to define the gcd of a quotient field
consistent with the gcd of the underlying ring.

On the other hand, one could mean "consistent as algebraic notion" (or
perhaps "consistent with respect to categories"). Let me elaborate:
One could argue that there is a partial map "gcd_*" that assigns to
any object R of Rings() a function gcd_R that accepts two arguments
(namely elements of R) returning elements of R.
Now, it is possible to consider the same object R as object of
different categories. For example, we have
sage: QQ in QuotientFields()
True
sage: QQ in Fields()
True
sage: QQ in PrincipalIdealDomains()
True

So, it makes sense to call gcd_* "consistent as algebraic notion", if
gcd_R for R object of some category C is the same as gcd_R for the
same R considered as an object of a different category C'.

What you propose would be "consistent with respect to
subrings" ( gcd(QQ(m),QQ(n))==gcd(m,n) for m,n in ZZ), but it would be
"inconsistent as algebraic notion", as gcd(a/b,c/d) would depend on
whether we consider QQ as a quotient field or as a principal ideal
domain.

I strongly prefer to work with things that are consistent as algebraic
notions.

Best regards,
Simon

Nicolas M. Thiery

unread,
Feb 10, 2011, 8:38:09 AM2/10/11
to sage-...@googlegroups.com, sage-comb...@googlegroups.com
Hi Doug,

Welcome to the Sage community!

On Wed, Feb 09, 2011 at 01:02:10PM +0800, D. S. McNeil wrote:
> (2) No kwarg constraints in Partitions/Compositions should be mutually
> exclusive. (I think there's a ticket for this but I can't find it
> now.)
>
> Partitions(15,length=5, parts_in=[1,3,4]) doesn't work, so you have to
> do an explicit filter. Optimizing it is a separate issue, but when
> there's a trivial way to implement the functionality (simply test that
> partitions satisfy the specified criteria before yielding them, in
> cases where we don't currently embed the constraints in the
> construction process itself), then ISTM we should do so now rather
> than wait for a more efficient implementation to appear.
> Functionality+speed > functionality, but functionality >>> no
> functionality.

Annoying, huh? This has bothered us ever since IntegersListLex was
first implemented in MuPAD 9 years ago :-)

In principle one of the goals of the upcoming Sage Days in Acadia in
May is to reimplement IntegersListLex
(http://trac.sagemath.org/sage_trac/ticket/6538). I can't guarantee
that it will actually happen though. Volunteers are welcome to
implement a temporary fix. However one should be careful: the kwarg
options are *not* mutually exclusive as long as they are consistent
(for some loose definition of consistent), and this feature is used in
many places. So one should be careful not to slow things down in those
situations.

> Currently, partitions_restricted's docstring says not to use it but to
> use RestrictedPartitions instead, which in turn says not to use
> RestrictedPartitions but to use Partitions with the "parts_in"
> keyword, which in its turn doesn't work.

Specific examples welcome!

Best,
Nicolas
--
Nicolas M. Thi�ry "Isil" <nth...@users.sf.net>
http://Nicolas.Thiery.name/

koffie

unread,
Feb 10, 2011, 9:02:37 AM2/10/11
to sage-devel
So bruno and simon agree that lcm(1/4,1/6) = 1/2 (lcm(numerators)/
gcd(denominators)) is the most logical. It also seems to satisfy dough
wanted relation up to units. I like it because it makes sense if you
think in terms of fractional ideals. And I suggest we switch to that
convention.

When trying fun stuff with fractional ideals in sage to give nice
examples I ran in to the following worrysome results.

First I tried to define a fractional ideal over QQ but I failed
misserabely. Does anybody know how to do it without the following ugly
trick (i.e. make a field extension of QQ of degree 1)?

sage: var('x')
sage: K=QQ.extension(x,'x')
sage: K*(1/4)
Fractional ideal (1/4)
sage: QQ*(1/4)
Principal ideal (1) of Rational Field

But this is not the most terrible thing yet. Look at what happens if I
intersect the rational ideal in K by itself!

sage: (K*(1/4)).intersection(K*(1/4))
Fractional ideal (1/16)


Simon King

unread,
Feb 10, 2011, 9:19:10 AM2/10/11
to sage-devel
Hi koffie,

On 10 Feb., 15:02, koffie <m.derickx.stud...@gmail.com> wrote:
> So bruno and simon agree that lcm(1/4,1/6) = 1/2   (lcm(numerators)/
> gcd(denominators)) is the most logical.

I do not agree at all with that! "lcm(1/4,1/6)=1/2" was just an
example of one (among others) way to extend lcm from ZZ to QQ. I did
*not* say that I want this behaviour!

Actually, I clearly wrote in my preceding post that I prefer a notion
of gcd/lcd for QQ that does not depend on whether QQ is regarded as a
principal ideal domain or as the quotient field of another principal
ideal domain (namely of ZZ).

Since QQ is a field, it is a principal ideal domain, where lcm and gcd
should have something to do with ideals. So, clearly lcm(4/1,2)=1.

Cheers,
Simon

luisfe

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Feb 10, 2011, 9:36:26 AM2/10/11
to sage-devel


On Feb 10, 3:19 pm, Simon King <simon.k...@uni-jena.de> wrote:
> Hi koffie,
> Since QQ is a field, it is a principal ideal domain, where lcm and gcd
> should have something to do with ideals. So, clearly lcm(4/1,2)=1.

It would be good to know what why lcm was written as it is right now.

rjf

unread,
Feb 10, 2011, 9:51:51 AM2/10/11
to sage-devel
in maxima, gcd(1/4,1/6) is 1/12, lcm is 1/2

Since maxima immediately simplifies 2/1 to 2, there is no
distinction between gcd(2/1, ....) and gcd(2, ...)

That is not to say that INTERNALLY, everything runs through the same
gcd process.
It should be clear that notions like polynomial gcd / content / etc
have to
be very carefully defined relative to the domains of interest. The
programmers
who knew rather less algebra than necessary had to study up about
rings and fields.
But this was in 1966 or so.

luisfe

unread,
Feb 10, 2011, 11:48:08 AM2/10/11
to sage-devel
On Feb 10, 2:10 pm, Simon King <simon.k...@uni-jena.de> wrote:
> Hi Bruno
You could have both consistencies. That depends on how you define gcd
and lcm:

- Quotient fields as described by Bruno.
- Fields: zero if both elements are zero. A non-zero element
otherwise (most fields would choose 1 here).
- PID: a generator of the corresponding ideal.

This is not trivial. For instance Fields do not have a default gcd/lcm
method. I asked in sage-devel about some time ago about sensible
approaches here.
http://groups.google.com/group/sage-devel/browse_thread/thread/12524b18d2325633/7b8af907c3c45c8b?lnk=gst&q=gcd+and+lcm+for+field+elements#7b8af907c3c45c8b

Simon King

unread,
Feb 10, 2011, 12:49:50 PM2/10/11
to sage-devel
Hi Luis,

On 10 Feb., 17:48, luisfe <lftab...@yahoo.es> wrote:
> ...
> You could have both consistencies. That depends on how you define gcd
> and lcm:
>
> - Quotient fields as described by Bruno.
> - Fields:  zero if both elements are zero. A non-zero element
> otherwise (most fields would choose 1 here).
> - PID: a generator of the corresponding ideal.
>
> This is not trivial. For instance Fields do not have a default gcd/lcm
> method. I asked in sage-devel about some time ago about sensible
> approaches here.http://groups.google.com/group/sage-devel/browse_thread/thread/12524b...

Yes, on my way home, I thought that perhaps "lcm(4/1,2)=1" is not so
obvious as I first found. lcm(a,b) has to be a generator of the
intersection of the ideals generated by a and b. Of course, 1 is a
quite canonical generator for the only non-zero ideal in a field --
simply because any field has a 1.

But any other non-zero element is fine as well, in a field. So, after
all, defining lcm(a/b,c/d)=lcm(a,c)/gcd(b,d) for fraction fields of
principal ideal domains makes more sense than I originally thought.
And with gcd(a/b,c/d)=gcd(a,c)/lcm(b,d), we would indeed have
gcd(x,y)*lcm(x,y)=x*y.

According to Richard Fateman, that definition seems to be be used in
Maxima ("in maxima, gcd(1/4,1/6) is 1/12, lcm is 1/2"). But
according to Tim Daly, Axiom returns 1 as lcm of any two rationals.
So, should Sage stay on the side of Axiom or switch to the side of
Maxima?

Cheers,
Simon

William Stein

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Feb 10, 2011, 12:55:59 PM2/10/11
to sage-...@googlegroups.com

It should switch to the side of Maxima/Pari/Mathematica/etc. in this.

-- William

William Stein

unread,
Feb 10, 2011, 1:01:58 PM2/10/11
to sage-...@googlegroups.com, Alex Ghitza, Andrey Novoseltsev
On Thu, Feb 10, 2011 at 9:55 AM, William Stein <wst...@gmail.com> wrote:
> [... gcd stuff ...]

It seems like nobody explained how the current gcd definition got
included. It's from a patch to rational.pyx from Alex Ghitza (who I
cc'd) that did this:

- d = self.denom()*other.denom()
- self_d = self.numer()*other.denom()
- other_d = other.numer()*self.denom()
- return self_d.gcd(other_d) / d
+ if self == 0 and other == 0:
+ return Rational(0)
+ else:
+ return Rational(1)


This was from trac 3214 "uniformise the behaviour of gcd for rational numbers":

http://trac.sagemath.org/sage_trac/ticket/3214

which was reported by Andrey Novoseltsev.

So if Andrey or Alex cared so much, they may want to pipe up.

This thread is at least:

http://groups.google.com/group/sage-devel/browse_thread/thread/cd05585cf395b3a0/160c549d6bdc8867#160c549d6bdc8867

William

John Cremona

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Feb 10, 2011, 1:15:01 PM2/10/11
to sage-...@googlegroups.com
I have not taken the time to read this whole thread, but here goes anyway:

The distinction is between ideals of Q (which are of course only (0)
and (1)) and sub-Z-modules of Q, a.k.a. fractional ideals (since in
the generalization to number fields K we (ab)use the terminology
"ideal of K" to mean "fractional ideal of K" which is the same as
"OK_submodule of K (of maximal rank)".

Every f.g. Z-submodule of Q is cyclic, and instead of the current
behaviour of gcd(x,y) for rationals (which is to give the generator of
the Q-ideal generated by x and y) the old -- and perhaps desired --
behaviour is to give the generator of the Z-module generated by x and
y. The latter is, of course, unique up to sign. It's the same as
Simon's generator of the sum of the Z-submodules generated by x and by
y. And then lcm(x,y) is the genrator of their intersection.

This way, both gcd(x,y) and lcm(x,y) are positive rationals (or 0 but
not when x and y are nonzero). And we have

gcd(x,y)*lcm(x,y) = abs(x*y)

which I think is a better convention for when x and/or y are negative
than deciding to make one of the gcd and the lcm negative.

John

> --
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Andrey Novoseltsev

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Feb 10, 2011, 1:42:18 PM2/10/11
to sage-devel
>    http://groups.google.com/group/sage-devel/browse_thread/thread/cd0558...
>
> William

Well, I used to use gcd for obtaining the primitive integral vector
with a specified rational direction. My concern on Trac 3214 was that
gcd(a1, ..., ak) depended on the order of arguments and I wanted it to
be fixed. The eventual solution was to agree that gcd as the "greatest
common divisor" does not really make much sense for fields, but
instead of raising an exception it can just return 1. This meant that
I cannot use it for my original purpose (not a big deal - it is easy
to do more directly), but I think that it was quite a sensible
decision:
1) I don't recall seeing gcd of rational numbers in any book or paper
2) there is clearly no natural extension of this notion from ZZ to QQ
3) the name itself indeed is very strange applied to fields.

So I personally think that any kind of gcd/lcm combinations of
numerators/denominators of rational numbers should have some other
more appropriate names, since making up some conventions for gcd is
potentially dangerous and may make code using it harder to understand
if a reader thinks of gcd differently than the original author...

Thank you,
Andrey

David Roe

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Feb 10, 2011, 2:48:44 PM2/10/11
to sage-...@googlegroups.com
Well, I used to use gcd for obtaining the primitive integral vector
with a specified rational direction. My concern on Trac 3214 was that
gcd(a1, ..., ak) depended on the order of arguments and I wanted it to
be fixed. The eventual solution was to agree that gcd as the "greatest
common divisor" does not really make much sense for fields, but
instead of raising an exception it can just return 1. This meant that
I cannot use it for my original purpose (not a big deal - it is easy
to do more directly), but I think that it was quite a sensible
decision:
1) I don't recall seeing gcd of rational numbers in any book or paper
2) there is clearly no natural extension of this notion from ZZ to QQ
3) the name itself indeed is very strange applied to fields.

So I personally think that any kind of gcd/lcm combinations of
numerators/denominators of rational numbers should have some other
more appropriate names, since making up some conventions for gcd is
potentially dangerous and may make code using it harder to understand
if a reader thinks of gcd differently than the original author...

I agree that it's a little bit awkward, but I think that we should go with the maxima/pari/mathematica convention.
* The issue the OP brought up, where gcd silently gave nonsensical issues when one of the "integers" accidentally got divided by 1 (or you used / instead of // in a case where you know one is divisible by the other) is compelling.
* The gcd should be a generator of the fractional ideal generated by the inputs; the lcm of the intersection as long as that fractional ideal is principal (if we're in a non-PID or fraction field of a non-PID I think we should raise an error).  There's an ambiguity about units; we should make a consistent choice between the PID and its fraction field.  In the case of integers and rationals, this means we choose the results to be positive and have the relation
gcd(x,y)*lcm(x,y) = abs(x,y)
as John pointed out.
David

Bruno Le Floch

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Feb 10, 2011, 7:37:06 PM2/10/11
to sage-...@googlegroups.com
>> Let me phrase it like this: There are different interpretations of the
>> term "consistent".

@Simon: You are right to distinguish the two kinds of consistencies.
And I can understand that sometimes it is preferable to have the
algebraic consistency. I tend to care about elements of the objects
more than the objects of the category (i.e. individual rational
numbers rather than the field/PID/quotient field QQ), and thus I tend
towards subring consistency.


> You could have both consistencies. That depends on how you define gcd
> and lcm:
>
> - Quotient fields as described by Bruno.
> - Fields: zero if both elements are zero. A non-zero element
> otherwise (most fields would choose 1 here).
> - PID: a generator of the corresponding ideal.

I don't see how this brings in both consistencies. Algebraic
consistency requires gcd and lcm on QQ to have different outputs
depending on whether QQ is seen a Field, a PID, a Quotient Field... Is
there a clear way for the user to indicate "which QQ" he wants?

Or we could have (I don't really know how this is done ;-) )

lcm(10/21, 14/15, type="PID") = 1
lcm(10/21, 14/15, type="Field") = 1
lcm(10/21, 15/14, type="quotient-of-ZZ") = 30/7

I doubt that the "field" version is useful at all: the lcm is
basically always 1 (except when one of the arguments is zero). lcm and
gcd should only be defined for PIDs, where they are interesting (or
for factorization rings? I can't remember my undergrad).

> http://groups.google.com/group/sage-devel/browse_thread/thread/12524b18d2325633/7b8af907c3c45c8b?lnk=gst&q=gcd+and+lcm+for+field+elements#7b8af907c3c45c8b

Interesting read, thanks.

Simon King

unread,
Feb 11, 2011, 1:57:45 AM2/11/11
to sage-devel
Hi Bruno!

On 11 Feb., 01:37, Bruno Le Floch <blfla...@gmail.com> wrote:
> > You could have both consistencies. That depends on how you define gcd
> > and lcm:
>
> > - Quotient fields as described by Bruno.
> > - Fields:  zero if both elements are zero. A non-zero element
> > otherwise (most fields would choose 1 here).
> > - PID: a generator of the corresponding ideal.
>
> I don't see how this brings in both consistencies. Algebraic
> consistency requires gcd and lcm on QQ to have different outputs
> depending on whether QQ is seen a Field, a PID, a Quotient Field... Is
> there a clear way for the user to indicate "which QQ" he wants?

I am sorry for the FUD that I spread in my earlier posts.

Meanwhile people convinced me that it is indeed possible to have both
consistencies. The point is: In a PID, "the" lcm of a and b is only
defined up to a unit - it is only required that lcm(a,b) generates the
intersection of the ideals generated by a and b. My mistake was: 1 is
certainly "the" canonical choice of a generator of the ideal <a>\cap
<b> (a,b non-zero); but that does not mean that it is the best return
value of lcm(a,b)!

So, if lcm(a,b) for a,b non-zero returns *any* non-zero element, then
"consistency from the category point of view" is granted -
lcm(1/2,1/4) = 42 is not wrong in QQ.

But that freedom means: We can *in addition* achieve consistency with
respect to sub-structures, namely by seeing QQ as a quotient field.

Cheers,
Simon

daly

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Feb 11, 2011, 2:06:16 AM2/11/11
to sage-...@googlegroups.com
Can you suggest an algorithm to implement this?
Is there an agreed-upon answer (i.e., not 42?)

Tim Daly


Simon King

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Feb 11, 2011, 3:56:07 AM2/11/11
to sage-devel
Hi Tim,

On 11 Feb., 08:06, daly <d...@axiom-developer.org> wrote:
> ...
> Can you suggest an algorithm to implement this?
> Is there an agreed-upon answer (i.e., not 42?)

Well, I had the impression that a couple of people are in favour of
the following:
gcd(a/b,c/d) := gcd(a,c)/lcm(b,d)
lcm(a/b,c/d) := lcm(a,c)/gcd(b,d)

It seems that this definition is already used in Maxima:
sage: a=maxima(3/4)
sage: b=maxima(5/6)
sage: gcd(a,b)
1/12

Moreover, as one can easily see, the property gcd(x,y)*lcm(x,y)=x*y is
preserved by that definition. In addition, if F is the fraction field
of R and a,b are elements of R, then gcd(F(a),F(b))==gcd(a,b) and
lcm(F(a),F(b))==lcm(a,b).

So, that's the rationale.

Best regards,
Simon

David Kirkby

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Feb 11, 2011, 4:20:18 AM2/11/11
to sage-...@googlegroups.com
On 10 February 2011 14:51, rjf <fat...@gmail.com> wrote:
> in maxima, gcd(1/4,1/6)  is 1/12,  lcm is 1/2
>
> Since maxima immediately simplifies 2/1  to 2, there is no
> distinction between gcd(2/1, ....)   and gcd(2, ...)

FWIW, I just noticed that Mathematica treats 2/1 as an integer and not
as a rational.

In[1]:= Head[2]

Out[1]= Integer

In[2]:= Head[1/3]

Out[2]= Rational

In[3]:= Head[2/1]

Out[3]= Integer

Simon King

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Feb 11, 2011, 4:24:30 AM2/11/11
to sage-devel
PS:

On 11 Feb., 09:56, Simon King <simon.k...@uni-jena.de> wrote:
> Hi Tim,
>
> On 11 Feb., 08:06, daly <d...@axiom-developer.org> wrote:
>
> > ...
> > Can you suggest an algorithm to implement this?
> > Is there an agreed-upon answer (i.e., not 42?)
>
> Well, I had the impression that a couple of people are in favour of
> the following:

Or was your question: *How* to implement it?

I supposed that the functions gcd(a,b) and lcd(a,b) should first
coerce a and b into a common parent, P. So, suppose by now that a and
b both belong to P.

Currently, gcd(a,b,c...) (a,b,c,... not in ZZ or long or int) uses
sage.arith.__GCD_sequence, which then relies on the gcd-methods of the
elements.

Fraction fields can be easily detected: They belong to the category
QuotientFields(). Hence, it would be reasonable to implement gcd- and
lcm-methods for fraction field elements as ElementMethods of
QuotientFields(). Of course, the custom gcd/lcm-methods that we
currently have for the rationals (and maybe for some other fraction
fields) should be removed.

If one implements gcd/lcm as ElementMethods of QuotientFields(), then
gcd/lcm for elements of Frac(QQ[x]) would work out of the box.

Best regards,
Simon

daly

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Feb 11, 2011, 4:34:47 AM2/11/11
to sage-...@googlegroups.com, axiom-d...@nongnu.org
What does MMA do with gcd(2/1,4)


Simon King

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Feb 11, 2011, 4:49:47 AM2/11/11
to sage-devel
Hi,

On 11 Feb., 09:56, Simon King <simon.k...@uni-jena.de> wrote:
> Well, I had the impression that a couple of people are in favour of
> the following:
>  gcd(a/b,c/d) := gcd(a,c)/lcm(b,d)
>  lcm(a/b,c/d) := lcm(a,c)/gcd(b,d)

It just occurs to me that I am incredibly stupid.

The definition above wouldn't work at all, it isn't even well-defined.
Just replace gcd(1/4,1/6) by gcd(3/12,9/54). You obtain gcd(1,1)/
lcm(4,6) = 1/12, but gcd(3,9)/lcm(12,54) = 1/36.

Does anyone have a better idea? Would it be a correct definition if
one insisted on reduced fractions?

Cheers,
Simon

daly

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Feb 11, 2011, 4:55:24 AM2/11/11
to sage-...@googlegroups.com, axiom-d...@nongnu.org
That's why I was asking for an algorithm for gcd and lcm
in the subdomain. I'm not sure what answer is expected.
The unit (1) is correct but not by your definition, and
apparently not helpful for the original poster.

Tim Daly


Simon King

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Feb 11, 2011, 5:12:17 AM2/11/11
to sage-devel
Hi Tim,

I just opened #10771 for that purpose. Algorithm is proposed below.

On 11 Feb., 10:55, daly <d...@axiom-developer.org> wrote:
> On Fri, 2011-02-11 at 01:49 -0800, Simon King wrote:
> > Hi,
>
> > On 11 Feb., 09:56, Simon King <simon.k...@uni-jena.de> wrote:
> > Does anyone have a better idea? Would it be a correct definition if
> > one insisted on reduced fractions?
>
>
> That's why I was asking for an algorithm for gcd and lcm
> in the subdomain.

What do you mean by "subdomain"? Do you mean, an integral domain R as
a subdomain of Frac(R)?

> The unit (1) is correct but not by your definition, and
> apparently not helpful for the original poster.

Any unit is a correct answer from the point of view of a PID. That's
why it makes sense to use the freedom.


Here is an algorithm:

---------
ASSUMPTIONS:

R is an integral domain, F is its fraction field.
gcd and lcm are defined for R

INPUT:

x,y: Elements of F.

Since there is gcd in R, we can assume that x.numerator() and
y.denominator() are relatively prime, i.e., x=a/b and y=c/d for
a,b,c,d in R, the two fractions being reduced.

OUTPUT:

gcd(x,y) returns gcd(x.numerator(),y.numerator())/
lcm(x.denominator(),y.denominator())
lcm(x,y) returns lcm(x.numerator(),y.numerator())/
gcd(x.denominator(),y.denominator())
-----------------

PROPERTIES:

- Up to units in R, we have gcd(x,y)*lcm(x,y) = gcd(a,c)/
lcm(b,d)*lcm(a,c)/gcd(b,d) = a*c/b*d = x*y

- Moreover, again up to units in R, we have gcd(a/1,b/1) = gcd(a,b)/
lcm(1,1) = gcd(a,b) and lcm(a/1,b/1) = lcm(a,b)/gcd(1,1) = lcm(a,b)


My questions to people that get more sleep than I:
Does that algorithm makes sense, mathematically?

At least it seems to me that the gcd-definition is exactly what is
used in Maxima:

sage: P.<x> = QQ[]
sage: a = (1+x^2)*(1+2*x)^2
sage: b = 1-x^4
sage: c = (1+3*x^2)*(1+2*x)
sage: d = 1-x^5
sage: x = a/b
sage: y = c/d
sage: factor(gcd(maxima(x),maxima(y)))
(2*x+1)/((x-1)*(x+1)*(x^4+x^3+x^2+x+1))
sage: factor(gcd(x.numerator(),y.numerator())/
lcm(x.denominator(),y.denominator()))
(x - 1)^-1 * (x + 1)^-1 * (x + 1/2) * (x^4 + x^3 + x^2 + x + 1)^-1

Best regards,
Simon

luisfe

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Feb 11, 2011, 5:43:34 AM2/11/11
to sage-devel
Mathematically, if K is the fraction field of a PID R, then you should
first reduce both fractions to a common denominator r1/d and r2/d.
Then:

gcd(r1/d, r2/d) = gcd(r1,r2)/d
lcm(r1/d, r2/d) = lcm(r1, r2)/d

This approach will be independent on the representation of the
fractions (up to units in R). Although a direct translation of this
idea seems inefficient.

koffie

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Feb 11, 2011, 6:15:31 AM2/11/11
to sage-devel
First of all, sorry Simon to have misread you :(.

To awnser your question. I guess your algorithm makes sense when you
do it with fractions in the reduced form. I will do it only for prime
powers since the general case will follow from this.
Suppose we have two prime powers p^a and p^b with a,b possabilly
negative. Then gcd(p^a,p^b)=p^(min(a,b)) (*) would be sensible and up
to units not depending on any choise. It is easy to see that this
operation is associative and and commutative simply because min(a,b)
has those properties.
Now consider the cases (1) a,b>0, (2) a>0,b<0 (3) a,b<0 (and of course
the more trivial cases with a=0)
In the case (1) we get with your algorithm gcd(p^a,p^b)/
lcm(1,1)=p^(min(a,b)) which agrees with the definition (*)
In the case (2) a>0 b<0 we get gcd(p^a,1)/lcm(1,p^(-b))=1/(p^(-b))=p^b
which agrees with (*) since b<a
In the case (3) we get gcd(1,1)/lcm(p^(-a),p^(-b))=p^(-max(-a,-
b))=p^min(a,b) which also agrees with (*).

Kind regards,
Maarten Derickx

koffie

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Feb 11, 2011, 6:25:54 AM2/11/11
to sage-devel
By the way, notice that in http://trac.sagemath.org/sage_trac/ticket/3214
they also added a function "content" which does the old gcd behaviour
(i.e. similar to what simon describes).

sage: (1/4).content(1/6)
1/12

Andrey Novoseltsev

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Feb 11, 2011, 9:59:20 AM2/11/11
to sage-devel
On Feb 11, 4:25 am, koffie <m.derickx.stud...@gmail.com> wrote:
> By the way, notice that inhttp://trac.sagemath.org/sage_trac/ticket/3214
> they also added a function "content" which does the old gcd behaviour
> (i.e. similar to what simon describes).
>
> sage: (1/4).content(1/6)
> 1/12
>

Which agrees with what I have suggested before - gcd "analog" for
fields should have some other name. After all, since any rational
number is divisible by any non-zero, how can we pick "the greatest" of
them all? For ZZ the notion of greatness agrees with the usual
understanding of greater...

How about such an approach:
1) if gcd got elements of a fraction field, e.g. (2/1, 4), it tries to
convert them to the ring of intergers and compute the gcd there (and
lcm too)
2) if it fails, raise an exception
This will take care of the original problem.

Thank you,
Andrey

Dr. David Kirkby

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Feb 11, 2011, 11:10:31 AM2/11/11
to sage-...@googlegroups.com

This is version 7.0 - the latest is 8.0

In[2]:= GCD[2/1,4]

Out[2]= 2

You can try this in Wolfram|Alpha too, which uses the latest Mathematica (8.0).
Just go to

http://www.wolframalpha.com/

and type in "GCD[2/1,4]"

A direct URL is actually:

http://www.wolframalpha.com/input/?i=GCD[2%2F1%2C4]

What is interesting is that Wolfram|Alpha shows an alternate expression too:

8/LCM[2,4]

If I evaluate that in Mathematica, I get:

In[6]:= 8/LCM[2,4]

Out[6]= 2

If you ever want to know the output from Mathematica, you can try typing it in
Wolfram|Alpha. Obviously Wolfram|Alpha does not have the full functionality of
Mathematica exposed, as otherwise nobody would buy Mathematica. But a fair
amount of it seems to be available.

Using Wolfram|Alpha is a lot easier if you know the Mathematica expression.
Using the text a human would type is a lot more tedious, though typing "whats
the greatest common divider of 2 divided by one and 4?" does actually get the
right result.

http://www.wolframalpha.com/input/?i=whats+the+greatess+common+divider+of+2+divided+by+one+and+4%3F

Dave
--
A: Because it messes up the order in which people normally read text.
Q: Why is top-posting such a bad thing?
A: Top-posting.
Q: What is the most annoying thing in e-mail?

rjf

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Feb 11, 2011, 12:07:23 PM2/11/11
to sage-devel


On Feb 11, 8:10 am, "Dr. David Kirkby" <david.kir...@onetel.net>
wrote:
> On 02/11/11 09:34 AM, daly wrote:

>
> >> FWIW, I just noticed that Mathematica treats 2/1 as an integer and not
> >> as a rational.
>

No, 2/1 is not treated as an integer, it is converted to an integer
and its
history is lost. That is to say, GCD[2/1 ...] is never really
computed because
2/1 is changed to 2, and it computes GCD[2 ...]

I think it is pointless to try to gather insight to your concern
about
algebraic categories by typing questions into the (current)
top-level commands in Mathematica. [ though anything could be re-
programmed.
e.g. you could write a whole category system and while you are at it,
write a system for doing polynomial arithmetic using arrays, etc etc.]

But to a pretty good approximation, Mathematica just messes around
with
trees. It was designed by a physicist, presumably
for physicists, and not for people who want to be so picky about
math :)



Simon King

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Feb 11, 2011, 12:56:21 PM2/11/11
to sage-devel
Hi Andrey,

On 11 Feb., 15:59, Andrey Novoseltsev <novos...@gmail.com> wrote:
> > sage: (1/4).content(1/6)
> > 1/12
>
> Which agrees with what I have suggested before - gcd "analog" for
> fields should have some other name. After all, since any rational
> number is divisible by any non-zero, how can we pick "the greatest" of
> them all? For ZZ the notion of greatness agrees with the usual
> understanding of greater...

The term "greatest" may be in the background of the original
definition for the gcd in ZZ, but it is quite common in math that the
generalisation of a notion (like "gcd(a,b) in a PID is defined to be a
generator of the principal ideal generated by a and b") has not much
to do with the original formulation (like "gcd(a,b) in ZZ is the
greatest integer that divides both a and b").

> How about such an approach:
> 1) if gcd got elements of a fraction field, e.g. (2/1, 4), it tries to
> convert them to the ring of intergers and compute the gcd there (and
> lcm too)

The suggestion is to define gcd in QQ (and, more general, in fraction
fields of a PID) such that it is not needed to try and convert 2/1 to
an integer. I gave a possible approach in a previous post.

> 2) if it fails, raise an exception

Why an exception? If the elements are in a field that is not the
fraction field of a PID, it is totally fine that gcd(a,b) returns 0 if
one of a,b is zero, and returns 1 otherwise.

I hope the whole discussion is not "painting a bike shed"...

Cheers,
Simon

D. S. McNeil

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Feb 11, 2011, 1:32:09 PM2/11/11
to sage-...@googlegroups.com
On Thu, Feb 10, 2011 at 9:38 PM, Nicolas M. Thiery
<Nicolas...@u-psud.fr> wrote:
> However one should be careful: the kwarg
> options are *not* mutually exclusive as long as they are consistent
> (for some loose definition of consistent), and this feature is used in
> many places.

My single most common use involves parts_in, though, and that's what doesn't
work: Partitions(10, min_length=2, max_length=6, parts_in=[1,2,3,5])
is a completely
consistent set of constraints, but it silently drops two of them, as
the docs warn. (I'm telling
you what you already know, of course, but this was the first case
which started to cause
me problems.) There are lots of OEIS sequences which involve
partitions restricted to
some set, so I have to write Partitions(10,
parts_in=[1,2,3,5]).filter(lambda x: 2<= len(x) <= 6)
or some such variant, which it'd be really nice to avoid.. I even
have lots of use cases where
it'd be nice to be able to use min_part and parts_in simultaneously,
cases where it's more natural
to build the parts_in list once.


Doug

--
Department of Earth Sciences
University of Hong Kong

Nils Bruin

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Feb 11, 2011, 2:25:00 PM2/11/11
to sage-devel
On Feb 11, 9:56 am, Simon King <simon.k...@uni-jena.de> wrote:

> Why an exception? If the elements are in a field that is not the
> fraction field of a PID, it is totally fine that gcd(a,b) returns 0 if
> one of a,b is zero, and returns 1 otherwise.
>
> I hope the whole discussion is not "painting a bike shed"...

I'd say: return 0 if a=b=0 and some random non-zero field element
otherwise. That will teach people to write programs that depend on
properly defined mathematical concepts rather than implementation
details. (this is not a serious proposal for this particular case but
the principle of not going out of your way to fix arbitrary choices
and to avoid depending on them being made in a particular way has
served me well).

The "positive generator of the fractional ideal" choice makes a lot of
sense to me in an interactive setting, but I'd expect that the
dependence of a program on that behaviour would almost always be an
error.

D. S. McNeil

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Feb 11, 2011, 3:42:43 PM2/11/11
to sage-...@googlegroups.com
Well, someone asked for more posts.. not sure this is what he had in mind. ;-)

Forgive my being a bear of little brain, but I've yet to grasp why
defining the default gcd rational function to be equal to 1 or (from
Simon) the lcm equal to 1 would be a _useful_ thing to do, independent
of the existence of perspectives from which it's the right
generalization. Who is going to call such a function? Who uses the
current rational gcd behaviour?

(.. I have a sneaking suspicion that the reason the rational lcm
behaviour doesn't currently match the rational gcd behaviour is
because these functions aren't getting a lot of exercise, not even by
people strongly in the gcd(2/1,4)=1 camp.)


The Pari/Mma/(Sage lcm+Maxima gcd) behaviour has pretty much
everything I want. Agrees with integer values when denominator is 1,
and so obeys least-surprise principles; is informative; preserves many
nice properties of positive integer gcd/lcm; is used in many other
places. The current Sage rational gcd behaviour surprised the heck
out of me and did so silently; returns 1 for all arguments and so is
minimally informative; doesn't preserve said nice relationships; and
doesn't match the behaviours of any of Pari, Mma, Maple, Maxima, or
Magma -- it doesn't even match Sage for lcm.

If the above doesn't speak to you in favour of the former I don't know
what else to say; we clearly have very different perspectives on
design!

If we do wind up defining gcd and/or lcm to be l, could we at least
define new short-named functions, say rgcd and rlcm, which do what
(IMHO) they should? Then I can simply explain to people "Oh, in Sage
we use 'rgcd' and 'rlcm' for gcd and lcm" and forget I brought this up
in the first place. :^)

Simon King

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Feb 11, 2011, 4:04:34 PM2/11/11
to sage-devel
Hi Nils,

On 11 Feb., 20:25, Nils Bruin <nbr...@sfu.ca> wrote:
> I'd say: return 0 if a=b=0 and some random non-zero field element
> otherwise. That will teach people to write programs that depend on
> properly defined mathematical concepts rather than implementation
> details. (this is not a serious proposal for this particular case but
> the principle of not going out of your way to fix arbitrary choices
> and to avoid depending on them being made in a particular way has
> served me well).

Yes and no.

"Yes", because it is a bad idea to rely on choices that have no
mathematical justification.

"No", since it is a bad idea to *not* rely on canonical choices if one
is in a situation that allows for a canonical choice.

Here, we can argue:
If a programmer writes code for principal ideal domains, then s/he is
supposed to not rely on a particular choice of lcm/gcd -- because in a
PID F, these notions are only defined up to a unit of F.
However, if the programmer writes code for the fraction field, F, of
PIDs, R, then one certainly is in a special situation. There is a
mathematically sound way to define lcm/gcd uniquely up to a unit of R
(not of F). So, the ambiguity is less, and it would be silly to not
use the smaller ambiguity.

I think this kind of argument holds in general, not only for gcd/lcm.

Cheers,
Simon

Nicolas M. Thiery

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Feb 11, 2011, 4:44:51 PM2/11/11
to sage-...@googlegroups.com
On Sat, Feb 12, 2011 at 02:32:09AM +0800, D. S. McNeil wrote:
> My single most common use involves parts_in, though, and that's what doesn't
> work: Partitions(10, min_length=2, max_length=6, parts_in=[1,2,3,5])
> is a completely
> consistent set of constraints, but it silently drops two of them, as
> the docs warn. (I'm telling
> you what you already know, of course, but this was the first case
> which started to cause
> me problems.) There are lots of OEIS sequences which involve
> partitions restricted to
> some set, so I have to write Partitions(10,
> parts_in=[1,2,3,5]).filter(lambda x: 2<= len(x) <= 6)
> or some such variant, which it'd be really nice to avoid.. I even
> have lots of use cases where
> it'd be nice to be able to use min_part and parts_in simultaneously,
> cases where it's more natural
> to build the parts_in list once.

Ah, ok; that's a different story. Structurally, parts_in can't be
treated by the generic IntegerListsLex tool, so the rewrite of the
latter is not going to improve the situation. One instead needs to
treat it specifically, or probably best using species as they could
handle efficiently parts_in together with min_length and
max_length. Patch welcome!

Cheers,
Nicolas
--
Nicolas M. Thi�ry "Isil" <nth...@users.sf.net>
http://Nicolas.Thiery.name/

William Stein

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Feb 11, 2011, 9:20:40 PM2/11/11
to sage-...@googlegroups.com
On Friday, February 11, 2011, D. S. McNeil <dsm...@gmail.com> wrote:
> Well, someone asked for more posts.. not sure this is what he had in mind.  ;-)
>

That was me. I think this has been a great discussion.

I vote for changing the defn of sage rational gcd to match the
"Pari/Mma/(Sage lcm+Maxima gcd) " convention. Since +1 isn't having
the desired effect, I vote with my BDFL powers instead.

Somebody post a patch.

- william


>
> Doug
>
> --
> Department of Earth Sciences
> University of Hong Kong
>

> --
> To post to this group, send an email to sage-...@googlegroups.com
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> URL: http://www.sagemath.org
>

--
William Stein
Professor of Mathematics
University of Washington
http://wstein.org

luisfe

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Feb 12, 2011, 5:04:20 AM2/12/11
to sage-devel


On 12 feb, 03:20, William Stein <wst...@gmail.com> wrote:
> On Friday, February 11, 2011, D. S. McNeil <dsm...@gmail.com> wrote:
>
> I vote for changing the defn of sage rational gcd to match the
> "Pari/Mma/(Sage lcm+Maxima gcd) " convention.   Since +1 isn't having
> the desired effect, I vote with my BDFL powers instead.
>
> Somebody post a patch.
>
>    - william
>

I hope I have misunderstood this post. It sound quite unfortunate.

Simon King

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Feb 14, 2011, 6:40:47 AM2/14/11
to sage-devel, sag...@googlegroups.com
Hi William and all,

Note another severe oddity:

sage: gcd(int(3),3/1)
3
sage: gcd(3,3/1)
1

On 12 Feb., 03:20, William Stein <wst...@gmail.com> wrote:
> I vote for changing the defn of sage rational gcd to match the
> "Pari/Mma/(Sage lcm+Maxima gcd) " convention.   Since +1 isn't having
> the desired effect, I vote with my BDFL powers instead.
>
> Somebody post a patch.

I am about to post a patch -- see #10771.

There is one problem, though. There is precisely one fraction field
that has its own gcd/lcm implemented: The rational field. There is a
rationale provided in the documentation why this particular choice of
a gcd was made.

Question: Shall I remove the custom gcd/lcm for QQ and replace it by
something that restricts to the usual gcd/lcm on ZZ? Or is that likely
to break stuff? I could imagine that number theory people have a
certain preference for a particular choice of a gcd in QQ. So, I am
cross-posting to sage-nt.

Cheers,
Simon

Simon King

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Feb 14, 2011, 6:59:02 AM2/14/11
to sage-devel, sag...@googlegroups.com
On 14 Feb., 12:40, Simon King <simon.k...@uni-jena.de> wrote:
> Question: Shall I remove the custom gcd/lcm for QQ and replace it by
> something that restricts to the usual gcd/lcm on ZZ? Or is that likely
> to break stuff? I could imagine that number theory people have a
> certain preference for a particular choice of a gcd in QQ. So, I am
> cross-posting to sage-nt.

Oops, I have to slightly correct myself:

* The rationale was given for lcm, not for gcd.

* Fortunately, the current lcm does exactly what we want for the
fraction field of a PID:
sage: lcm(1/3,1/6)
1/3
So, there is no need to change lcm for the rationals.

* The gcd of the rationals should be changed: Currently, it returns
either 1 or 0, which is not interesting.

So, my patch for #10771 will provide lcm and gcd for fraction fields
of PID (so that it works, e.g., in Frac(QQ['x'])), will remove or
rename the current rational gcd (it shall in future be inherited from
the category), and will preserve the current rational lcm (for
efficiency).

Moreover, since in future lcm and gcd for fraction fields behave
nicely wrt. the base ring, there shall be coercion before computing
gcd/lcd.

Best regards,
Simon

Simon King

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Feb 14, 2011, 8:14:16 AM2/14/11
to sage-devel
Hi all!

On 12 Feb., 03:20, William Stein <wst...@gmail.com> wrote:
> I vote for changing the defn of sage rational gcd to match the
> "Pari/Mma/(Sage lcm+Maxima gcd) " convention.   Since +1 isn't having
> the desired effect, I vote with my BDFL powers instead.
>
> Somebody post a patch.

Done: #10771 is ready for review.

Cheers,
Simon

David Roe

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Feb 17, 2011, 5:12:48 PM2/17/11
to sag...@googlegroups.com, sage-devel
I just ran across a MathOverflow thread discussing gcds of algebraic integers living in rings with class number bigger than 1.  We may not have the tools in Sage to implement it yet, but if someone's interested in working on gcd and lcm in number fields, you should take a look.
David

Simon

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Robert Bradshaw

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Feb 17, 2011, 7:05:39 PM2/17/11
to sage-...@googlegroups.com
On Wed, Feb 9, 2011 at 10:46 PM, daly <da...@axiom-developer.org> wrote:
> On Wed, 2011-02-09 at 22:18 -0800, rjf wrote:
>> You say,
>> > gcd(2/1,4) returns 1 "for simplicity" (!), because 2/1 is a rational.
>> > This is shockingly silly.
>>
>> I don't know exactly how this came up, but if 2/1 is in a different
>> domain (rational)
>> from 2, (integer),  then gcd should probably be 1,  since any non-
>> zero
>> rational number divides any other, and one commonly uses the positive
>> "unit" 1 for
>> such a case. You could argue that since you can coerce 2/1, you
>> should.
>>
>> That's sometimes OK, but not always.
>>
>> Really, the issue is much broader. for example, do you also want to
>> treat the complex number
>> 1+0*i the same as 1?   do you want to treat the floating point number
>> 1.0 the same as 1?
>>
>> What about 1X1 matrices?
>>
>> Is 1^0  the same as 1^0.0  or 1.0^0  or 1.0^0.0?  Do you perhaps wish
>> to consider/dismiss
>> the existence of number systems with signed zeros (IEEE floating-point
>> standard) on the
>> grounds that -0 = +0,  [true, for numerical comparison] and therefore
>> there should be
>> only a single zero?
>>
>> While I don't know the exact formulation of this GCD problem, the
>> issue of
>> implicit coercion is one of the troubling sore spots in a system
>> design, and should not
>>  be decided by counting up casual +1 votes.
>>
>> I think the Axiom people might have thought more about it than others.
>>
>
> It is a question of domains. In Axiom you can specify the domains.

You can in Sage too. The question is whether 1 is the right answer for
the domain of rational numbers. (That's what it currently does, just
like Axiom, but it's not very useful.)

> 2/1 is a Fraction(Integer) aka rational
> 4 is an Integer
> (2/1)::Integer => 2 where 2 is an Integer.
> 4::Fraction(Integer) is a Fraction(Integer)
>
> So there are several cases:
> gcd((2/1),4::Fraction(Integer)) => 1 of type Fraction(Integer)
> gcd((2/1)::Integer,4))          => 2 of type PositiveInteger
> gcd(2/1,4)                      => 1 of type Fraction(Integer)
> gcd(2,4)                        => 2 of type PositiveInteger
> gcd(2,4::Fraction(Integer))     => 1 of type Fraction(Integer)
>
> Tim Daly
>
>
>
> --
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