       Re: sorting polynomials by the degree of certain terms

• To: mathgroup at smc.vnet.net
• Subject: [mg49231] Re: [mg49202] sorting polynomials by the degree of certain terms
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Thu, 8 Jul 2004 02:51:07 -0400 (EDT)
• References: <200407070542.BAA25018@smc.vnet.net> <73A3931D-D019-11D8-B7F7-000A95B4967A@mimuw.edu.pl>
• Sender: owner-wri-mathgroup at wolfram.com

```I suddenly realized that I did something rather silly in the code I
sent earlier. Obviously if we are only going two make use of just two
variables then there is no need to compute the exponents of the
remaining variables at all! So here is a new version of the code, which
looks almost the same:

MyOrderQ[f_, g_, vars_] := With[{a = Plus @@ Exponent[
f, vars],
b = Plus @@ Exponent[g, vars]}, Which[a < b, True, a == b,
OrderedQ[{f,
g}], True, False]]

The point is that now vars refers just to the two variables we are
using in our ordering. Thus the examples I gave earlier now work like
this:

MyOrderQ[z^5,x y z,{x,y}]

True

In:=
MyOrderQ[z^5, x*y*z, {x, y}]

Out=
True

Sort[{x*y, x^2*y^2*z, x^4*z, z^3},
MyOrderQ[#1, #2, {x, y}] & ]

{z^3, x*y, x^4*z, x^2*y^2*z}

This approach should be a little faster, which may be significant if

Andrzej

On 7 Jul 2004, at 22:27, Andrzej Kozlowski wrote:

> On 7 Jul 2004, at 14:42, David Hessing wrote:
>
>>
>> Hi,
>>
>> I dealing with huge polynomials (around 50,000 terms). Each term may
>> contain some real coefficient, followed by any combination of 11
>> variables, each of which may be raised to some some power. However,
>> I'm interested in sorting the polynomial according to the sum of the
>> degrees of only 2 of the variables. In other words, in the sorted
>> expression, the first terms would be those where the two variables do
>> not appear (the sum of their powers is zero). The next terms would be
>> the terms where one of the two variables appeared raised to the power
>> 1. The next terms would be the terms where both variables appeared
>> with power 1, or just one of the variables appeared with power 2. And
>> so on.
>>
>> I've played with passing the expression to the Sort function, along
>> with a defined sorting function, but I can't get it to work. Any help
>> would be greatly appreciated.
>>
>> -David
>>
>
> Presumably what you have in mind is ordering monomials?
>
> You could define a monomial order function with respect to variables
> vars as follows:
>
> MyOrderQ[f_, g_, vars_] := With[{
>     a = Plus @@ Take[Exponent[f, vars], 2],
>         b = Plus @@ Take[
>           Exponent[g, vars], 2]}, Which[a < b, True, a == b,
> OrderedQ[{f,
>         g}], True, False]]
>
> What this does is compares the sums of the exponents of the first two
> variables in two monomials. The one with the  smaller sum is taken to
> be smaller. If the sums are equal then the canonical ordering is used.
>
> So, with three variables x,y,z:
>
>
> MyOrderQ[z^5,x y z,{x,y,z}]
>
>
> True
>
> This is because the sum of the exponents of x and y in z^5 is 0.
>
>
> MyOrderQ[x,y,{x,y,z}]
>
> True
>
> and
>
> MyOrderQ[y,x,{x,y,z}]
>
> False
>
> in this case the sums are the same so the canonical ordering is used
> (x comes before y)
>
> And here is how you sort several monomials:
>
>
> Sort[{x*y, x^2*y^2*z, x^4*z, z^3},
>   MyOrderQ[#1, #2, {x, y, z}] & ]
>
>
> {z^3, x*y, x^4*z, x^2*y^2*z}
>
> Here there was again a "tie" for the last place when sums of powers
> were compared so the canonical ordering was used with x^4 coming
> before x^2 y^2. Of course one could easily reverse this.
>
> (I hope I have not misunderstood you!)
>
> Also, note that you can't just "sort" a polynomial, in the sense of
> arranging the order of its terms in your own order because Matheamtica
> will rearrange it into canonical order again. So what you need to do
> is to first convert it to a list and then sort. For example, to sort
> the polynomial
>
> f = x*y + x^4*z + x^2*y^2*z + z^3;
>
> you have to do something like:
>
>
> Sort[List @@ f, MyOrderQ[#1, #2, {x, y, z}] & ]
>
>
> {z^3, x*y, x^4*z, x^2*y^2*z}
>
> But if you now apply Plus, Mathematica will at one re-arrange it in
> canonical order:
>
> Plus @@ %
>
> x*y + x^4*z + x^2*y^2*z + z^3
>
> Andrzej Kozlowski
> Chiba, Japan
> http://www.mimuw.edu.pl/~akoz/
>

```

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