Re: Cubic equations

In article <1164579996.853032.128010@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
mac <manojuv@xxxxxxxxx> wrote:
Dear All,
I would like to know if it is possible to check whether two cubic
equations a1x^3+b1x^2+c1x+d1 and a2x^3+b2x^2+c2x+d2 intersect each
other (x, in the interval [1,t], x real and finite) without actually
solving them. Specifically i would like to know if (a1-a2)x^3+
(b1-b2)x^2+ (c1-c2)x + (d1-d2)>0 in the interval [1,t]. Is it possible
to make conclusion by forming some relationship between the
coefficients.

Yes. Use Sturm's theorem. See e.g.
<http://eom.springer.de/S/s090790.htm>.
In Maple, with P a polynomial in x with rational or float coefficients:

sturm(sturmseq(P,x),x,a,b);

returns the number of real roots of P in the interval (a,b].

Robert Israel israel@xxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada



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