naginterfaces.library.zeros.cubic_​real

naginterfaces.library.zeros.cubic_real(u, r, s, t)

cubic_real determines the roots of a cubic equation with real coefficients.

For full information please refer to the NAG Library document for c02ak

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/c02/c02akf.html

Parameters

ufloat

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

rfloat

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

sfloat

$s$, the coefficient of $z$.

tfloat

$t$, the constant coefficient.

Returns

zerocomplex, ndarray, shape $\left(3\right)$

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

errestfloat, ndarray, shape $\left(3\right)$

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

Raises

NagValueError

(errno $1$)

On entry, $u=0.0$.

Constraint: $u\ne 0.0$.

(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.

cubic_real attempts to find the roots of the cubic equation

$$u{z}^3+r{z}^2+sz+t=0\text{,}$$

where $u$, $r$, $s$ and $t$ are real coefficients with $u\ne 0$. The roots are located by finding the eigenvalues of the associated $3\times 3$ (upper Hessenberg) companion matrix $H$ given by

$$\begin{array}{c}H=\left(\begin{array}{ccc}0& 0& -t/u\\ 1& 0& -s/u\\ 0& 1& -r/u\end{array}\right)\text{.}\end{array}$$

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

To obtain the roots of a quartic equation, quartic_real() can be used.

References

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