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Real Roots of cubics
*To*: mathgroup at yoda.physics.unc.edu
*Subject*: Real Roots of cubics
*From*: paul at earwax.pd.uwa.edu.au (Paul Abbott)
*Date*: Mon, 14 Mar 1994 17:10:51 +0800
>>Actually, I'd suggest that WRI's version of the cubic formula
>>is defective and provides unnecessary "I" values. This bug
>>was corrected in Macsyma, as I recall, about 20 years ago.
If you need them, the real roots of a cubic can be expressed in explicitly
form using trigs and arctrigs. The following is a slightly modified
version of a column that appeared in The Mathematica Journal (Vol 2, Issue
3, 1992):
Consider the polynomial
poly = 11 x^3 - 20 x^2 - 10 x + 22
2 3
22 - 10 x - 20 x + 11 x
Its roots:
sol = Solve[poly == 0, x];
are all real as is easily determined numerically:
N[sol] // Chop
{{x -> 1.61319}, {x -> -1.01567}, {x -> 1.22065}}
but this is certainly not obvious from the format of the exact symbolic
solution. For example,
First[sol]
1/3
20 730 2
{x -> -- + --------------------------------- +
33 1/3
33 (-36074 + 66 I Sqrt[58479])
1/3
(-36074 + 66 I Sqrt[58479])
------------------------------}
1/3
33 2
How can this be simplified to a real expression? From the general theory of
cubic equations [see Numerical Recipes in C, 2nd Ed (1992), 183-5; American
Journal of Physics 52 (3), 269 (1984)], a cubic of the form
x^3 + a x^2 + b x + c, has three real roots when
q[a_,b_] := (a^2 - 3b)/9
and
r[a_,b_,c_] := (2 a^3 - 9a b + 27 c)/54
are real and r^2 < q^3. Defining
theta[r_,q_] := ArcCos[r/q^(3/2)]
the three roots are
roots[a_,b_,c_,n_] :=
-a/3 - 2 Sqrt[q[a,b]] *
Cos[(theta[r[a,b,c],q[a,b]] + (2-n) 2 Pi)/3]
where n = 1, 2 and 3.
>From the coefficients of the polynomial:
{c,b,a} = Drop[CoefficientList[poly,x]/Coefficient[poly,x^3],-1]
10 20
{2, -(--), -(--)}
11 11
it is seen that r and q are positive real:
r[a,b,c]
18037
-----
35937
q[a,b]
730
----
1089
and also that
r[a,b,c]^2 - q[a,b]^3
19493
-(------)
395307
is negative. Hence, the three roots can be written exactly as:
Table[roots[a,b,c,n],{n,3}]
18037
2 Pi + ArcCos[-------------]
730 Sqrt[730]
2 Sqrt[730] Cos[----------------------------]
20 3
{-- - ---------------------------------------------,
33 33
18037
ArcCos[-------------]
730 Sqrt[730]
2 Sqrt[730] Cos[---------------------]
20 3
-- - --------------------------------------,
33 33
18037
-2 Pi + ArcCos[-------------]
730 Sqrt[730]
2 Sqrt[730] Cos[-----------------------------]
20 3
-- - ----------------------------------------------}
33 33
This expression is simpler, more elegant and, because it does not involve
complex numbers, much more suitable for subsequent algebraic manipulation. As
a check it is seen that the numerical values do tally:
N[%]
{1.22065, -1.01567, 1.61319}
Paul Abbott
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| Department of Physics | Interests: Computer Algebra, |
| University of Western Australia | Computational Physics, |
| Nedlands 6009 | Exact atomic research |
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