Square roots and Solve/Reduce
- To: mathgroup@smc.vnet.net
- Subject: [mg11330] Square roots and Solve/Reduce
- From: "Richard Anderson" <richardj.anderson@stonebow.otago.ac.nz>
- Date: Fri, 6 Mar 1998 00:40:50 -0500
Hello all,
My question is a seemingly simple one. Is there any way to tell the
functions Solve and Reduce that I am only interested in using a +ve or
-ve form of a square root?
For instance in the following situation, eigen1 and eigen 2 are the
eigenvalues of a matrix.
eigen1=(2 - t - Sqrt[4 - 4*s + t^2])/4 eigen2=(2 - t +Sqrt[4 - 4*s +
t^2])/4
Now consider that we are interested in the boundaries of -1 < eigen1 < 1
and -1 < eigen2 <1. (Incidentally we are only interested in the
boundaries between -1 < s <1, -1 < t < 1)
I do:
In[3]:=
Reduce[eigen1==1,{s,t}]
Out[3]=
s==-t
In[4]:=
Reduce[eigen1==-1,{s,t}]
Out[4]=
s==-8+3 t
In[5]:=
Reduce[eigen2==1,{s,t}]
Out[5]=
s==-t
In[6]:=
Reduce[eigen2==-1,{s,t}]
Out[6]=
s==-8+3 t
So according to Reduce, the two eigenvalues give the same boundary
conditions. However, this is clearly not the case : as becomes apparent
when plotting them e.g. with
ContourPlot[eigen1,{s,-1,1},{t,-1,1},Contours->{-1,1}]
and
ContourPlot[eigen2,{s,-1,1},{t,-1,1},Contours->{-1,1}]
What seems to be happening is that Reduce is allowing the square root to
once again be +ve or -ve, thus making the equations for the eigenvalues
synonymous! How can
I prevent this? Why does ContourPlot 'know' differently? Any help would
be much appreciated
Richard
********************************************************************************
Richard Anderson
e-mail : richardj.anderson@stonebow.otago.ac.nz
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