[sage-devel] Re: multivariate polynomial coefficients
Thanks for the discussion about this topic. I send this mail to re-iterate
and summarize. It seems there are two things that you might want:
1) Get the coefficient of a specific monomial in the multivariate polynomial
ring.
2) Get the coefficient of the polynomial in a tower of (two) polynomial
extensions
I suggest that MPolynomial.coefficient method perform function #1 above. It
currently does some strange things over the base rings I've tested. It does
vary on base ring due to implementation differences.
I suggest that there be a new method which performs function #2 and I suggest
it be called polynomial_coefficient, but I really don't like that name. I
almost prefer coefficient_polynomial since it will sort beside coefficient
in documentation and help people realize that if they are not happy
with coefficient that they have an alternative. Of course, each method's
documentation should probably mention the other.
I'm wondering if we could have a vote on preferred syntax. I'm not going to
describe the parameters because if they are not clear enough from context, it
probably isn't a good parameter choice :) :
sage: P.v,w,x,y,z=ZZ[]
sage: f=(1-v)*(1-2*w)*(1-3*x); f
-6*v*w*x + 2*v*w + 3*v*x + 6*w*x - v - 2*w - 3*x + 1
##
# Alternative Number 1
##
sage: f.polynomial_coefficient({w:0,v:1})
3*x - 1
##
# Alternative Number 2
##
sage: f.polynomial_coefficient([x,y,z],v)
3*x - 1
##
# Alternative Number 3
# Overloading of coefficient to handle both functionalities
##
sage: f.coefficient(v,base=ZZ['x,y,z'])
3*x - 1
sage: f.coefficient(v) # here's an example of functionality #1 above
-1
Those are listed in the order of my preference for the interface. It feels to
me that Alternative Number 3 would not be much fun to implement (and
involve coercions that some view as suspect). I basically
implemented Alternative Number 1 as a function in my own program and it
feels like a straightforward implementation just iterating through the
polydict. I'd love to have some other voters.
NOTE: It is NOT sufficient to simply provide a monomial expression to
the polynomial_coefficient method because it doesn't provide for the case
when you explicitly want an exponent to be 0 -- this point seems to me to
have caused a great deal of confusion in the current implementation.
--
Joel
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[sage-devel] Re: multivariate polynomial coefficients
2007/10/12, Joel B. Mohler [EMAIL PROTECTED]: sage: P.x,y,z=ZZ[] sage: f=x*y^2*z^3+y^2*z sage: f.coefficient(y,2,z,3) # I want the coefficient of y^2*z^3 # Bang That doesn't seem very nice to me. Good point: Dictionary it is then. (Incidentally, there downs seem a non-obvious way to do this in Maple). didier -- Joel --~--~-~--~~~---~--~~ To post to this group, send email to sage-devel@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://sage.scipy.org/sage/ and http://modular.math.washington.edu/sage/ -~--~~~~--~~--~--~---
[sage-devel] Re: multivariate polynomial coefficients
Another *possible* way of sage behaving is
sage: P.x,y=ZZ[]
sage: f=x*y^2+x*y+y+x+1
sage: f.coefficient(y^2)
0
sage: f.coefficient(y^1)
1
sage: f.coefficient(y^0)
1
Now, I'm not sure if this is better or not, but I thought maybe I should
point it out. The biggest problem with this is that if you do want to extract
the whole polynomial coefficient, you will have some issues. I'm guessing
the proposal for dictionary will resolve this nicely.
Soroosh
On Fri, Oct 12, 2007 at 10:36:29AM -0400, Joel B. Mohler wrote:
This e-mail is too long. Here's the bottom line: I suggest that the
coefficient method on a multivariate polynomial ring take a dictionary
indicating the variables and degrees that you want to restrict your attention
to.
It seems that the multivariate polynomial coefficient function is a bit
inflexible (and inconsistent). I'm looking for some insight about how to
think about the following things.
sage: P.x,y=ZZ[]
sage: f=x*y^2+x*y+y+x+1
sage: f.coefficient(y^2)
x
sage: f.coefficient(y^1)
x + 1
sage: f.coefficient(y^0)
1
I realize that y^0 == 1 so that the last line is returning the constant
coefficient (and the implication that y is special to me the user is totally
unseen by the coefficient method). But, the logic seems a bit inconsistent.
I'd suggest that this next line work:
sage: f.coefficient({y:0})
x + 1
Does anyone have any objection? It actually seems like a dictionary of
variables whose exponents you want to control would make a much better
interface to this. In any case, we can maintain both functionalities by
checking the type of the parameter.
And another:
sage: P.x,y=QQ[] # DIFFERENT BASE RING THAN LAST TIME
sage: f=x*y^2+x*y+y+x+1
sage: f.coefficient(y^0) # DIFFERENT OUTPUT
x*y^2 + x*y + x + y + 1
What in the world is going on in that last line? (Hmm, I think that it is
getting a 1 so it is thinking that no variables are restricted -- that's
actually consistent, but seems mighty strange.)
Does anyone else agree? Or, have a better suggestion?
--
Joel
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[sage-devel] Re: multivariate polynomial coefficients
On Friday 12 October 2007 11:41, didier deshommes wrote: But: coeff(f,y,0); x + 1 returns the right answer Actually I like Maple's notation better here over the dictionary notation you proposed: it is as intuitive and I have to type less curly braces to get the same result :) Ok, that's a good reason, but what if I was working in something like this?: sage: P.x,y,z=ZZ[] sage: f=x*y^2*z^3+y^2*z sage: f.coefficient(y,2,z,3) # I want the coefficient of y^2*z^3 # Bang That doesn't seem very nice to me. -- Joel --~--~-~--~~~---~--~~ To post to this group, send email to sage-devel@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://sage.scipy.org/sage/ and http://modular.math.washington.edu/sage/ -~--~~~~--~~--~--~---
[sage-devel] Re: multivariate polynomial coefficients
On Oct 12, 7:36 am, Joel B. Mohler [EMAIL PROTECTED] wrote:
This e-mail is too long. Here's the bottom line: I suggest that the
coefficient method on a multivariate polynomial ring take a dictionary
indicating the variables and degrees that you want to restrict your attention
to.
Sounds like a great idea.
Does anyone have any objection? It actually seems like a dictionary of
variables whose exponents you want to control would make a much better
interface to this. In any case, we can maintain both functionalities by
checking the type of the parameter.
I would actually slightly prefer to remove the previous interface. It
seems very hard to use; worse, it seems like it would be easy to
accidentally write buggy code using the old interface.
(I actually have code that finds f.coefficient({y:0}) by starting with
f and subtracting out all larger coefficients of y.)
Carl
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[sage-devel] Re: multivariate polynomial coefficients
On Friday 12 October 2007 13:36, Mike Hansen wrote:
If you're doing a dictionary anyway, doesn't it make more sense to use
**kwargs? For example,
sage: P.x,y=ZZ[]
sage: f=x*y^2+x*y+y+x+1
sage: f.coefficient(y=2)
x
sage: f.coefficient(y=1)
x + 1
sage: f.coefficient(x=1, y=2)
1
It takes a little bit to get used to the semantics of it, but the
syntax is much more natural / Pythonic.
Hmm, possibly. kwargs feels scary to me with variables since the 'x' in the
kwargs parameter list is a totally different 'x' than the one in P.gen(0).
They just happened to be named the same in some different scope. I think
that allowing both methods might be ok.
Another more concrete reason is that I want to do something like this:
P=PolynomialRing(ZZ,'x',n)
d={}
d[P.gen(1)] = 1
d[P.gen(2)] = 2
poly.coefficient(d)
That is, I want to fill out my dictionary very dynamically. I suppose there
is probably a way to do this with kwargs, but it feels like a reach into some
dark corner syntactically.
--
Joel
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[sage-devel] Re: multivariate polynomial coefficients
If you're doing a dictionary anyway, doesn't it make more sense to use **kwargs? For example, sage: P.x,y=ZZ[] sage: f=x*y^2+x*y+y+x+1 sage: f.coefficient(y=2) x sage: f.coefficient(y=1) x + 1 sage: f.coefficient(x=1, y=2) 1 It takes a little bit to get used to the semantics of it, but the syntax is much more natural / Pythonic. --Mike On 10/12/07, didier deshommes [EMAIL PROTECTED] wrote: 2007/10/12, Joel B. Mohler [EMAIL PROTECTED]: sage: P.x,y,z=ZZ[] sage: f=x*y^2*z^3+y^2*z sage: f.coefficient(y,2,z,3) # I want the coefficient of y^2*z^3 # Bang That doesn't seem very nice to me. Good point: Dictionary it is then. (Incidentally, there downs seem a non-obvious way to do this in Maple). didier -- Joel --~--~-~--~~~---~--~~ To post to this group, send email to sage-devel@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://sage.scipy.org/sage/ and http://modular.math.washington.edu/sage/ -~--~~~~--~~--~--~---
[sage-devel] Re: multivariate polynomial coefficients
On Fri, 12 Oct 2007, Joel B. Mohler wrote:
On Friday 12 October 2007 13:36, Mike Hansen wrote:
If you're doing a dictionary anyway, doesn't it make more sense to use
**kwargs? For example,
sage: P.x,y=ZZ[]
sage: f=x*y^2+x*y+y+x+1
sage: f.coefficient(y=2)
x
sage: f.coefficient(y=1)
x + 1
sage: f.coefficient(x=1, y=2)
1
It takes a little bit to get used to the semantics of it, but the
syntax is much more natural / Pythonic.
Hmm, possibly. kwargs feels scary to me with variables since the 'x' in the
kwargs parameter list is a totally different 'x' than the one in P.gen(0).
They just happened to be named the same in some different scope. I think
that allowing both methods might be ok.
... I don't understand what you mean by this. When you construct a polynomial
ring, each variable gets a name, and that name is (AFAIK) immutable.
If an argument doesn't correspond to an entry in R.variable_names(), then an
exception should be raised.
Another more concrete reason is that I want to do something like this:
P=PolynomialRing(ZZ,'x',n)
d={}
d[P.gen(1)] = 1
d[P.gen(2)] = 2
poly.coefficient(d)
That is, I want to fill out my dictionary very dynamically. I suppose there
is probably a way to do this with kwargs, but it feels like a reach into some
dark corner syntactically.
--
Joel
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[sage-devel] Re: multivariate polynomial coefficients
Both the poly.coefficients({x:1,y:2}) and poly.coefficients(x=1,y=2)
seem confusing to me (the latter one downright scary. Exponents and
variable names have no business being on opposite sides of an equality
sign). In mathematical terms, what you want to do is view the
polynomial ring k[x,y,z] as the ring k[z][x,y] and ask for the
coefficient of the monomial x*y^2.
Wouldn't it be clearer to have a call like
poly.polynomial_coefficient({z},x*y^2) ?
i.e., give the sum of the terms in poly that are of the form
z^e*x*y^2 ? In this setting,
(x^2*y^2+z*x*y^2+x*y).polynomial_coefficient({z,y},x*y^2) == y+z
which may be useful. polynomial_coefficient seems an ugly name to me,
but this interface is a lot easier to explain and much more explicit
about what the routine does.
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[sage-devel] Re: multivariate polynomial coefficients
2007/10/12, Joel B. Mohler [EMAIL PROTECTED]:
This e-mail is too long. Here's the bottom line: I suggest that the
coefficient method on a multivariate polynomial ring take a dictionary
indicating the variables and degrees that you want to restrict your attention
to.
It seems that the multivariate polynomial coefficient function is a bit
inflexible (and inconsistent). I'm looking for some insight about how to
think about the following things.
sage: P.x,y=ZZ[]
sage: f=x*y^2+x*y+y+x+1
sage: f.coefficient(y^2)
x
sage: f.coefficient(y^1)
x + 1
sage: f.coefficient(y^0)
1
I realize that y^0 == 1 so that the last line is returning the constant
coefficient (and the implication that y is special to me the user is totally
unseen by the coefficient method). But, the logic seems a bit inconsistent.
I'd suggest that this next line work:
sage: f.coefficient({y:0})
x + 1
+1.
Interestingly enough, Maple has the same limitation when you pass the
exponent to it directly:
coeff(f,y^0);
Error, invalid input: coeff received 1, which is not valid for its 2nd
argument, x
But:
coeff(f,y,0);
x + 1
returns the right answer
Actually I like Maple's notation better here over the dictionary
notation you proposed: it is as intuitive and I have to type less
curly braces to get the same result :)
didier
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[sage-devel] Re: multivariate polynomial coefficients
On Oct 12, 12:27 pm, [EMAIL PROTECTED] wrote: On Fri, 12 Oct 2007, Joel B. Mohler wrote: Hmm, possibly. kwargs feels scary to me with variables since the 'x' in the kwargs parameter list is a totally different 'x' than the one in P.gen(0). They just happened to be named the same in some different scope. I think that allowing both methods might be ok. ... I don't understand what you mean by this. When you construct a polynomial ring, each variable gets a name, and that name is (AFAIK) immutable. I think I understand. It just feels cleaner if variables are variables, and strings are strings, and the two only meet during parsing and printing (and not necessarily even during parsing). For me, this is partly an aesthetic issue, but there are practical consequences as well. If some ways of dealing with polynomials require variable names as strings, then there's no way of dealing with this: sage: QQ['x']['x']['x', 'x'] Multivariate Polynomial Ring in x, x over Univariate Polynomial Ring in x over Univariate Polynomial Ring in x over Rational Field sage: QQ['x']['x']['x', 'x'].random_element() ((-4*x^2 + x + 2)*x^2 + (2*x^2 + 8*x - 1/2)*x + 1/6*x^2 + 2*x + 1/4)*x + ((1/7*x^2 - 2*x - 1)*x^2 + (x^2 + x + 1/10)*x + 1/6*x^2 + 2*x + 1)*x + (1/2*x^2 + x + 1)*x^2 + (-2*x^2 + 9*x - 1)*x + 1/2*x^2 + x + 3/2 That's an absurd example, of course, but there are more important issues as well. Suppose you have an algorithm that takes a polynomial in an arbitrary multivariate ring, and your algorithm needs to lift the polynomial to a ring with one more variable, do some computations there, and then drop back to the original ring. What to name the new variable? If names are unimportant, and the identity of the variable object is all that matters, then it's easy...just always name the new variable z. However, if names are vital, then it's important to make sure the name doesn't clash with any names used in the current polynomial ring (or its base rings, recursively). However, given all that, I think separating variables and their names is a lost cause in SAGE. Polynomials must have distinct, sensible variable names, or else they cannot be shared with Pari, or with any subprocess-based interface. There's already code all through SAGE that maps back and forth between variables and their names with abandon, so adding one more place won't hurt much. There is one more issue; I wonder if cutting down on the number of places we need to map from strings to variables would make things faster? Carl --~--~-~--~~~---~--~~ To post to this group, send email to sage-devel@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://sage.scipy.org/sage/ and http://modular.math.washington.edu/sage/ -~--~~~~--~~--~--~---
[sage-devel] Re: multivariate polynomial coefficients
On Friday 12 October 2007 15:52, Nils Bruin wrote:
Both the poly.coefficients({x:1,y:2}) and poly.coefficients(x=1,y=2)
seem confusing to me (the latter one downright scary. Exponents and
variable names have no business being on opposite sides of an equality
sign).
Yes, I agree, mathematically it appears risky.
In mathematical terms, what you want to do is view the
polynomial ring k[x,y,z] as the ring k[z][x,y] and ask for the
coefficient of the monomial x*y^2.
Wouldn't it be clearer to have a call like
poly.polynomial_coefficient({z},x*y^2) ?
I think this confuses me. What is the {z} indicating?
Suppose I had a poly ring with 19 variables and one of them was named y.
How would I get the coefficient for y^0 in your syntax? (That is, the
constant term in k[y][...].) This is the sticking point in what is currently
implemented.
--
Joel
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[sage-devel] Re: multivariate polynomial coefficients
On Oct 12, 1:25 pm, Joel B. Mohler [EMAIL PROTECTED] wrote:
[...]
Suppose I had a poly ring with 19 variables and one of them was named y.
How would I get the coefficient for y^0 in your syntax? (That is, the
constant term in k[y][...].) This is the sticking point in what is currently
implemented.
No, that would be the constant term in k[...][y]. The syntax would
then be:
poly.coeff({ a for a in poly.parent().generators() if a != y }, 1)
(i.e., all variables except y are allowed)
except that python doesn't like set comprehension (why not? is there
an unordered list type with quick membership test?), so we might want
to make it a list instead:
poly.coeff([ a for a in poly.parent().generators() if a != y ], 1)
If that is the most common usage scenario you have in mind, then this
notation might be a tad wasteful. Would surgery on poly.dict() solve
your problems?
Incidentally, poly.coefficient does have documentation. The behaviour
you describe is a bug relative to that description. The alternative
behaviour that you describe, nor mine, is in line with the definition
given there either.
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