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----- Message d'origine ----- <br>
De : Sam Mathew <br>
Date : 08/11/2010 05:10:
<blockquote
cite="mid:AANLkTinhEq2xv5B=sQdrwyTLDH=o=4EF7hg81AtAnHnY@mail.gmail.com"
type="cite">Thanks Samuel for that solution. Just that I'm not
able to figure out the explanation for the last part:<br>
<br>
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Is it possible to also perform such transformations if a
function is of two<br>
variables x and y, and the coordinate axes undergo
different transformations<br>
(translation, rotation, etc.).<br>
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<u>It could be, whether the polynomial could be applied to a<br>
matrix (it can) in a matrix way. But it is not the case :<br>
horner() works on a matrix only in a element-wise way.</u><br>
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<br>
You are right, it is a bit obscure, and somewhat useless. <br>
I was just wondering about a way to implement<br>
bi-variate polynomials with matrices.<br>
// Let<br>
x=grand(4,1,"uin",0,9)<br>
// be the coefficients of p(x)<br>
x = <br>
2. <br>
2. <br>
4. <br>
5. <br>
// and <br>
y=grand(1,3,"uin",0,9)<br>
// be the coefficients of q(y)<br>
y = <br>
4. 1. 9. <br>
// Then, the coefficients of P(x,y)=p(x).q(y) are in the matrix x*y<br>
P=x*y<br>
P =<br>
8. 2. 18. <br>
8. 2. 18. <br>
16. 4. 36. <br>
20. 5. 45. <br>
// where P(i,j) is the coeff of the x^(i-1).y^(j-1) term<br>
<br>
From there, it should be possible to write a library of matrix-based
<br>
functions implementing operations between bi-variates polynomials,<br>
or even N-variates polynomials (through hypermatrices).<br>
<br>
Then, the evaluation of such polynomials could not be done with any<br>
horner() algo, that does not fit for that, even if it could be
called in<br>
a matrix-oriented way.<br>
<br>
For instance, for x=2 and y=3, the following would be required:<br>
<br>
[X,Y] = ndgrid(x^(0:size(P,1)-1),y^(0:size(P,2)-1))<br>
Result = sum(P.*X.*Y) // evaluates P(x=2,y=3)<br>
<br>
Other functions would be required for multiplying by-variate
polynomials<br>
with each others, etc<br>
<br>
Besides all this stuff, it could be worthwhile to implement a
horner() <br>
release applicable in a matrix-wise way, instead of a element-wise
one.<br>
But this has nothing to do with multivariate polynomials.<br>
<br>
Samuel<br>
<br>
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