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Serious bug in FunctionExpand
*To*: mathgroup at smc.vnet.net
*Subject*: [mg25632] Serious bug in FunctionExpand
*From*: Jack Goldberg <jackgold at math.lsa.umich.edu>
*Date*: Mon, 16 Oct 2000 03:04:50 -0400 (EDT)
*Sender*: owner-wri-mathgroup at wolfram.com
Hi group;
I'll cut to the quick (whatever that means):
Let
p[x] = x^6-x^4+2x-10
This polynomial has two real root, x1 = 1.4774 and x2 = -1.4774.
The remaining 4 roots are complex with moderately sized imaginary parts. A
simple plot shows that p[x] > 0 whenever x is exterior to [-x1,x2] and
is < 0 in (x1,x2). From this we can conclude that
UnitStep[ p[x] ] = 1 for x exterior to (x1,x2)
UnitStep[ p[x] ] = 0 for x interior to (x1,x2)
We find
FunctionExpand[ UnitStep[p[x] ] -> 0
a terrible error from my perspective! (I will now have to rewrite a key
component of my package on Piecewise Functions.)
I have done some exploring and can offer the following tidbits:
(1). Change 10 (the constant term in p[x]) to 10. and FunctionExpand gives
the correct answer as can be verified by Ploting the output of
FunctionExpand with UnitStep[ p[x] ].
(2). FunctionExpand gives the incorrect answer for almost all (but not
all!) changes in the constant term. Try the polynomial family
Table[ p[x_,k_] := x^6-x^4+2x - k, {k,30,120}]
(3). I believe part of the problem is due to the fact that p[x] is a cubic
in x^2. I have tried many different polynomials and never had a problem
with FunctionExpand until I accidentally hit upon this one.
(4). FunctionExpand fails with the 12th degree polynomial
q[x] = x^12-x^8+2x^4-10
obtain from p[x] by replace x with x^2. BUT it works with
h[x] = x^3-x^2+2x-10
obtained from p[x] by replacing x^2 by x.
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