better than a quartic?

From: Carlo (Carlo_02_at_libero.it)
Date: 07/03/04 

Date: 3 Jul 2004 07:17:07 -0700

I have to solve this set of equations:

t5 = t1 + (t1-t2)^2/( (t1-t3)*(Pa-1) )

t1 = t5 + (t5-t4)^1/( (t5-t3)*(Pb-1) )

t1 and t5 are unknown. t2, t3, t4, Pa, Pb are known.
In practice, these properties are also true:
0 < t2 < t3 < t4
0 < Pa < 1
0 < Pb < 1

With some manipulation, I can turn the equations above into a quartic,
and solve it numerically (I know there is also a symbolic solutions to
a quartic, however, the coefficients of this quartic are already a
mess, never mind the symbolic quartic solution...). Once I get t1 or
t5 this way, I can find the other easily by applying one of the above.

Unfortunately, with the numbers I have to play with, the quartic is
relatively a mess. For instance, when I have

t2=3
t3=6
t4=12
Pa=0.95
Pb=0.95

I have to solve:

-0.855 X^4 + 42.435 X^3 - 784.4625 X^2 + 6400.89 X - 19453.77

In other cases, the constant coefficient goes up to 10^11 while the
X^4 coefficient remains below 1.

In practice, I've tried a few algorithms (straight from the net :-) to
solve the quartic, (e.g. Descartes, Ferrari, Neumark, Yacoub &
Fraidenraich, Christianson), and it seems like the results I'm getting
are not very accurate. Programs like MuPad get the solutions "right",
but I need something that I can integrate in a small program I'm
writing, and I can't have a dependency on third-party components.

My questions are:

- is there a better way to solve the original equations, without
turning them into a quartic? I need the 4 solutions, having only one
won't do!

- if the quartic is the "best" way, is there some algorithm that works
well in cases like mine, where the coefficients (usually) have
different orders of magnitude?

Thanks a lot!

Carlo

Carlo_02 /at/ libero.it

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