c02alc attempts to find the roots of the quartic equation
where
,
,
,
and
are real coefficients with
. The roots are located by finding the eigenvalues of the associated 4 by 4 (upper Hessenberg) companion matrix
given by
Further details can be found in
Section 9.
To obtain the roots of a cubic equation,
c02akc can be used.
The method used by the function consists of the following steps, which are performed by functions from LAPACK.
-
(a)Form matrix .
-
(b)Apply a diagonal similarity transformation to (to give ).
-
(c)Calculate the eigenvalues and Schur factorization of .
-
(d)Calculate the left and right eigenvectors of .
-
(e)Estimate reciprocal condition numbers for all the eigenvalues of .
-
(f)Calculate approximate error estimates for all the eigenvalues of (using the 1-norm).
To find the roots of the quartic equation