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Re: CoefficientList
*To*: mathgroup at smc.vnet.net
*Subject*: [mg74536] Re: CoefficientList
*From*: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
*Date*: Sat, 24 Mar 2007 05:24:33 -0500 (EST)
*Organization*: The Open University, Milton Keynes, UK
*References*: <eu1pgs$ev0$1@smc.vnet.net>
Luke wrote:
> I'm having a little trouble understanding how CoefficientList works
> for multivarate polynomials. In the Mathematica book, there is this
> example:
> t = (1 + x)^3 (1 - y - x)^2
>
> Expand[t]
> 1 + x - 2x^2 - 2x^3 + x^4 + x^5 - 2y - 4xy + 4x^3y + 2x^4y + y^2 +
> 3xy^2 + 3x^2y^2 + x^3y^2
> CoefficientList[t,{x,y}]
> {{1, -2, 1}, {1, -4, 3}, {-2, 0, 3}, {-2, 4, 1}, {1, 2, 0}, {1, 0, 0}}
>
> I am confused as to what each entry of the output of the
> CoefficientList corresponds to. The Handbook says:
> For multivariate polynomials, CoefficientList gives an array of the
> coefficients for each power of each variable.
>
> So what exactly do the entries of the first item, {1,-2,1}, correspond
> to? Is it the 0 order terms? Why three entries then? Is
> corresponding to the 1, the x, and the -2y? What is the order of each
> of these lists? Maybe I'm just being dense, but it isn't immediately
> obvious to me how this is structured, and the Handbook is extremely
> terse in its description.
>
> Any help would be greatly appreciated.
>
> Thanks,
> ~Luke
>
>
Hi Luke,
Say we have a multivariate polynomial in x and y with highest powers n
and m, respectively. The function CoefficientList returns a rectangular
array where the rows correspond to the increasing powers of x (0 to n,
from top to bottom) and the column columns correspond to the increasing
powers of y (0 to m, from left to right). Each entry displays the
corresponding coefficient. For instance,
In[1]:=
p = a + b*x^3 + c*x^2*y + d*x*y^2 + e*y^3;
Exponent[p, {x, y}]
CoefficientList[p, {x, y}]
TableForm[%, TableHeadings -> {{x^0, x^1, x^2, x^3},
{y^0, y^1, y^2, y^3}}]
Out[2]=
{3, 3}
Out[3]=
{{a, 0, 0, e}, {0, 0, d, 0}, {0, c, 0, 0}, {b, 0, 0, 0}}
Out[4]=
Out[8]//TableForm=
2 3
1 y y y
1 a 0 0 e
x 0 0 d 0
2
x 0 c 0 0
3
x b 0 0 0
HTH,
Jean-Marc
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