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MathGroup Archive 1991

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Re: Question about Simplify and related

  • To: mathgroup at yoda.ncsa.uiuc.edu
  • Subject: Re: Question about Simplify and related
  • From: mbp at aurora.urich.edu (Mark B. Phillips)
  • Date: Sat, 19 Jan 91 18:07:47 EST

> 	This is a question about Simplify and related objects. Below is 
> a Mma session in which a 3x3 orthogonal matrix (u) is formed.
>
> ...
> 
> In fact I have gotten to the point below where one can show in a
> couple of minutes with a paper and pencil that uutrans is indeed the
> 3x3 unit matrix.
> 	Sould all this be so hard to do, or am I missing something 
> simple and obvious?

Part of the problem lies in that your simplification rules like si^2 +
ci^2 -> 1 really need to to be applied at different times to different
subexpressions.

For example to simplify:		c3^2 + c1^2 s3^2 + s1^2 s3^2
using only the rules
  si^2 + ci^2 -> 1
one needs to first factor
s3^2 out of the 2nd two terms:		c3^2 + s3^2( c1^2 + s1^2 )
apply s1^2 + c1^2 -> 1:			c3^2 + s3^2
and then apply c3^2 + s3^2 -> 1:	1.

Fortunately in this case the rules are simple enough to be written in
a form which will completely eliminate a variable when applied.  Using
si^2 -> 1 - ci^2 works much better because it complelely eliminates
si^2 from the expression:

In[1]:= u={{c1 c2 - c3 s1 s2,c2 s1 + c3 c1 s2,s2 s3},
      {-s2 c1 - c3 s1 c2, -s2 s1 + c3 c1 c2, c2 s3},
       {s3 s1, -s3 c1, c3}}
                
Out[1]= {{c1 c2 - c3 s1 s2, c2 s1 + c1 c3 s2, s2 s3}, 
 
>    {-(c2 c3 s1) - c1 s2, c1 c2 c3 - s1 s2, c2 s3}, {s1 s3, -(c1 s3), c3}}

In[2]:= uutrans = Expand[ u . Transpose[u] ]

Out[2]= [[ lengthy output deleted ]]

In[3]:= Expand[ uutrans /. {s1^2->1-c1^2, s2^2->1-c2^2, s3^2->1-c3^2} ]

Out[3]= {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}

There are other case, however, where the simplification rules are
not so simple, in which I have no idea how to proceed.

Mark Phillips
mbp at thales.urich.edu

Mathematics and Computer Science
University of Richmond			
Richmond, Virginia  23173
USA

phone: (804) 289-8090
fax: (804) 289-8943


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