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Want a general method to extract cases resulting from Reduce

  • To: mathgroup at smc.vnet.net
  • Subject: [mg88433] Want a general method to extract cases resulting from Reduce
  • From: dontdont at gmail.com
  • Date: Mon, 5 May 2008 06:13:54 -0400 (EDT)

Someone recently posted asking "How to remove unneeded constraints."
I can see a general use for something perhaps related to this.

Reduce often gives back fairly complicated nested boolean structures
of
constraints.  Consider the results from the following concrete
example:

Reduce[{a x1^2 + b x1 + c == y1, a x2^2 + b x2 + c == y2, a( -b/
(2a))^2+
   b ( -b/(2a)) + c == y3, y1 < y3, y2 < y3, x1 < x2,
Element[Alternatives[
   x1, x2, x3, y1, y2, y3, a, b, c], Reals]}, {a, b, c}]

or even worse

Reduce[{a x1^2 + b x1 + c == y1, a x2^2 + b x2 + c == y2, a( -b/
(2a))^2+
   b ( -b/(2a)) + c == y3, y1 < y3, y2 < y3, x1 < x2,
Element[Alternatives[
   x1, x2, x3, y1, y2, y3, a, b, c], Reals]}, {a, b, c},
Backsubstitution->True]

For examples like these I then spend time carefully matching up () and
trying to determine how many layers and which of the nested
constraints
apply to each of the alternatives.  I now realize I could often use a
general
purpose solution to inspect results like these.  For example,
(all the conditions except the last one are just the conditions I gave
Reduce,
the y1<y2 I then manually provide after inspecting the result from
Reduce)

Assuming[y1<y3&&y2<y3&&x1<x2&&Element[Alternatives[x1, x2, x3,
   y1, y2, y3, a, b, c], Reals]&&y1<y2, reduceExtract[%]]

to get from the first example

a == (y1 + y2 - 2*y3)/(x1 - x2)^2 - 2*Sqrt[(y1*y2 - y1*y3 - y2*y3 +
y3^2)/(x1 - x2)^4] ||
a == (y1 + y2 - 2*y3)/(x1 - x2)^2 + 2*Sqrt[(y1*y2 - y1*y3 - y2*y3 +
y3^2)/(x1 - x2)^4]

and in general, to have reduceExtract do the equivalent of unification
with all the
nested conditions and give me back all the alternatives that match but
without
the constraints that I have assumed to be in force.  (It is not
necessary that
Assuming[] be used, but that would be an understandable way of reading
this)

LogicalExpand will at least flatten out the nested conditionals and
this might be
useful in creating reduceExtract, but by itself it makes the result
from Reduce
even larger and still leaves me with manually searching through each
alternative,
checking which constraints apply, and then doing cut and paste to try
to get
what I am after without making a mistake.

reduceExtract would also have to deal with things like
   And[Element[x3, Reals], Element[Alternatives[x2, y3], Reals]]
that are generated by Reduce, even though constraints on x3,x2,y3 were
given
in exactly the same form to Reduce.

I've searched old postings and searched help, trying to see if someone
had
described a way of doing this but I haven't found any hits.

Can anyone come up with something that will do this general
reduceExtract?

On a different and much smaller note, it would be nice if Reduce had
noticed
that it had expr^(4n) of non-zero real expressions in the denominators
of args to
to Sqrt and pulled those outside the Sqrt as expr^(2n).  Then Together
would have
happily made the resullting expressions smaller.  Reduce knows enough
to do
this, it just doesn't do it.  It does not appear that leaving those
inside reduces
the complexity measure of the expressions.  But if I could have
reduceExtract
then I'd happily live with manually cutting and pasting to fix this
detail.

All this is still under 5.2, but my reading of the documentation says
this still
applies to 6.0.

Thank you


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