RE: Re: FindRoot
- To: mathgroup@smc.vnet.net
- Subject: [mg11594] RE: [mg11506] Re: FindRoot
- From: Ersek_Ted%PAX1A@mr.nawcad.navy.mil
- Date: Tue, 17 Mar 1998 10:43:22 -0500
Daniel Lichtblau gave a detailed analysis: Part of his analysis is
given below.
Once in a while members of the mathgroup show how GroebnerBasis can be
used to solve real world problems (as below). Has anyone written a
tutorial on GroebnerBasis and PolynomialReduce that assumes no
advanced background in polynomial algebra?
If not it would be a good topic for a future article.
Also:
Often times we are referred to an article in an the Mathematica Journal.
I have looked for it in several University libraries, and have yet to
find one that carries it. Does anyone how where I could find an old
issue? In some cases I may be willing to purchase a copy.
Ted Ersek
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|
|Now I compute a Groebner basis that in effect triangulates the system
of |polynomials. I first do this with a nonzero modulus, so as to keep
|coefficient growth from murdering the computation (it turns out this
is |not a problem, but often it is). We will see that we do not have
seven |independent polynomials, only six.
|
|
|
|In[29]:= Timing[gb = GroebnerBasis[p2, vars, Modulus->Prime[1111]];]
|Out[29]= {0.27 Second, Null}
|
|
|Note that the first polynomial is not in x7 alone, but has x6 as well.
|Indeed, fortuitiously, it can be written as a polynomial in x6*x7.
|Before preceding further we will recompute this in characteristic
zero. |
|In[31]:= Timing[gb = GroebnerBasis[p2, vars];] Out[31]= {0.72 Second,
|Null}
|
|
|In[33]:= InputForm[pol1 =
| Last[PolynomialReduce[gb[[1]], x6*x7 - z, {x6,x7,z}]];] |