If you are interested in a symbolic solution or numerical solution without
missing any of the possible roots (real or imaginary) you could check out
sympy, a CAS that offers a quartic solution. It uses appropriate
simplifications besides a general solution to the quartic. The code for this is
in the polys/polyroots.py file; that code makes some calls to other routines in
sympy so it's not totally self contained.
It can be downloaded from sympy.org. It can be sandboxed at live.sympy.org (but
that has an error in the quartic solution right now). It's a pretty quick
install.
Here it is in use:
>>> from sympy import solve, Rational, var, nsimplify
>>> var('x') # create a variable
x
>>> solve(x**4 + 3*x**3 - 2*x + x/3 + .7) # note the 0.7 float
...
... error message ending with
...
CoercionFailed: can't convert 2.1 of type RR to ZZ
So either use nsimplify to replace floats with rationals
>>> ans = solve(nsimplify(x**4 + 3*x**3 - 2*x + x/3 + .7, rational=1))
>>> ans[0].n()
-2.74495954970930
Or enter floats as Rationals from the start
>>> ans = solve(x**4 + 3*x**3 - 2*x + x/3 + Rational(7, 10))
>>> ans[0].n()
-2.74495954970930
You can use a symbol for the number that will take on a float
>>> var('a')
a
>>> ans = solve(x**4 + 3*x**3 - 2*x + x/3 + a, x) # now indicate you want to
>>> solve for x
Substitute 'a' with your desired float:
>>> [w.subs(a, .7).n() for w in ans] # .n() means "give a numerical answer"
[0.423047811447149 - 0.229393202547502*I, -2.74495954970930, 0.423047811447149
+ 0.229393202547502*I, -1.10113607318500]
The answer is there at index 1.
Best regards,
Chris
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