naginterfaces.library.zeros.quartic_​complex

naginterfaces.library.zeros.quartic_complex(e, a, b, c, d)

quartic_complex determines the roots of a quartic equation with complex coefficients.

For full information please refer to the NAG Library document for c02an

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/c02/c02anf.html

Parameters

ecomplex

$e$, the coefficient of $z^4$.

acomplex

$a$, the coefficient of $z^3$.

bcomplex

$b$, the coefficient of $z^2$.

ccomplex

$c$, the coefficient of $z$.

dcomplex

$d$, the constant coefficient.

Returns

zerocomplex, ndarray, shape $\left(4\right)$

$zero[i-1]$ contains the $i$th root.

errestfloat, ndarray, shape $\left(4\right)$

$errest[i-1]$ contains an approximate error estimate for the $i$th root.

Raises

NagValueError

(errno $1$)

On entry, $e=(0.0,0.0)$.

Constraint: $e\ne (0.0,0.0)$

(errno $2$)

The companion matrix $H$ cannot be formed without overflow.

(errno $3$)

Failure to converge in lapackeig.zhseqr.

Notes

No equivalent traditional C interface for this routine exists in the NAG Library.

quartic_complex attempts to find the roots of the quartic equation

$$e{z}^4+a{z}^3+b{z}^2+cz+d=0\text{,}$$

where $e$, $a$, $b$, $c$ and $d$ are complex coefficients with $e\ne 0$. The roots are located by finding the eigenvalues of the associated $4\times 4$ (upper Hessenberg) companion matrix $H$ given by

$$\begin{array}{c}H=\left(\begin{array}{cccc}0& 0& 0& -d/e\\ 1& 0& 0& -c/e\\ 0& 1& 0& -b/e\\ 0& 0& 1& -a/e\end{array}\right)\text{.}\end{array}$$

The eigenvalues are obtained by a call to lapackeig.zhseqr. Further details can be found in Further Comments.

To obtain the roots of a cubic equation, cubic_complex() can be used.

References

Golub, G H and Van Loan, C F, 1996, Matrix Computations, (3rd Edition), Johns Hopkins University Press, Baltimore