naginterfaces.library.zeros.quartic_​real

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

quartic_real determines the roots of a quartic equation with real coefficients.

For full information please refer to the NAG Library document for c02al

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/c02/c02alf.html

Parameters

efloat

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

afloat

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

bfloat

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

cfloat

$c$, the coefficient of $z$.

dfloat

$d$, 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, $e=0.0$.

Constraint: $e\ne 0.0$.

(errno $2$)

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

(errno $3$)

Failure to converge in lapackeig.dhseqr.

Notes

In the NAG Library the traditional C interface for this routine uses a different algorithmic base. Please contact NAG if you have any questions about compatibility.

quartic_real 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 real 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.dhseqr. Further details can be found in Further Comments.

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

References

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