Re: Up-Values with GroebnerBasis

• To: mathgroup at smc.vnet.net
• Subject: [mg19436] Re: [mg19424] Up-Values with GroebnerBasis
• From: "Clifford J. Nelson" <cnelson9 at gte.net>
• Date: Wed, 25 Aug 1999 01:25:10 -0400
• References: <7ptaqh\$lj4@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```
"Ersek, Ted R" wrote:

>  Cliff Nelson wrote:
> ------------------------
> The Solve function in Mathematica version 3.0.1 uses GroebnerBasis but
> GroebnerBasis does not use up-values with which I define my own kind of
> numbers. So, how can I use Solve to quickly find solutions to equations
> with coefficients which have up-values for addition, subtraction,
> multiplication, division, and powers defined by my own rules for B
> numbers(a list of numbers wrapped by the head B[{a,b,...}]) in version
> 3.0.1? It works fine in version 2.1, 2.2 and 2.2.2.
>
> ------------------------
>
> Are you sure the kernel isn't ignoring your up-values for addition,
> subtraction, and
> multiplication?  In Mathematica Version 3 and 4 the kernel uses built-in
> rules for Plus, Times before user defined rules (including your upvalues).
> I recommend you start by looking into that.
>
> As to how you can use Solve to quickly find solutions:
> - I don't really understand your problem.
> - I only have a vague understanding of GroebnerBasis.
>
> But if you send in an example you will probably get a solution.  Keep in
> mind the probability of a good solution is inversely proportional to the
>
> -------------------------
> Regards,
> Ted Ersek
>
> For Mathematica tips, tricks see
> http://www.dot.net.au/~elisha/ersek/Tricks.html

( http://forum.swarthmore.edu/epigone/geometry-research/brydilyum ) is where
you can find the links to mathsource for the package with the up-values from
RBFields.m. This is long. I hope someone can get it to work. Maybe Method ->
n in the Solve function or something like that? The answers to the two Solve
statements should be the same. Thank you for your time.

Cliff Nelson

In[1]:=
<<RBFields.m

Out[1]=
{"E"}

In[2]:=
GroebnerBasis[{x 2,x y 3}]

Out[2]=
{x}

In[3]:=
GroebnerBasis[{x B[{2,2,-4}],x y B[{3,3,-6}]}]

Out[3]=
{x*y*B[{3, 3, -6}], x*B[{2, 2, -4}]}

In[4]:=
m= {{x,y},{xx,yy}}

Out[4]=
{{x, y}, {xx, yy}}

In[5]:=
divv[x_] := (Plus @@({v,1} First[x]))/(Plus @@ ( {v,1} Last[x]))

In[6]:=
eq = Range[5]

Out[6]=
{1, 2, 3, 4, 5}

In[7]:=
exeq = ( divv/@  NestList [m.#&,m,4]) - eq;

In[8]:=
Solve[exeq=={0,0,0,0,0},{v,x,xx,y,yy}]

>From In[8]:=
Solve::svars: Equations may not give solutions for all "solve" variables.

Out[8]=
{{v -> 0, y -> yy, x -> yy, xx -> 0}}

In[9]:=
beq = N[ {B[{1,1,-2}],B[{2,2,-4}],B[{3,3,-6}],B[{4,4,-8}],B[{5,5,-10}]}]

Out[9]=
{B[{1., 1., -2.}], B[{2., 2., -4.}], B[{3., 3., -6.}], B[{4., 4., -8.}],
B[{5., 5., -10.}]}

In[10]:=
exbnumeq = ( divv /@  NestList [m.#&,m,4]) - beq;

In[11]:=
Solve[exbnumeq=={0,0,0,0,0},{v,x,xx,y,yy}]

Out[11]=
\$Aborted

```

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