naginterfaces.library.zeros.quadratic_​complex

naginterfaces.library.zeros.quadratic_complex(a, b, c)

quadratic_complex determines the roots of a quadratic equation with complex coefficients.

For full information please refer to the NAG Library document for c02ah

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/c02/c02ahf.html

Parameters

acomplex

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

bcomplex

$b$, the coefficient of $z$.

ccomplex

$c$, the constant coefficient.

Returns

z_smallcomplex

The smallest root in magnitude.

z_largecomplex

The largest root in magnitude.

Warns

NagAlgorithmicWarning

(errno $1$)

On entry, $a=0.0$.

(errno $2$)

On entry, $a=0.0$ and $b=0.0$.

(errno $3$)

On entry, $a=0.0$ and the root $-c/b$ overflows: $Re\left(a\right)=⟨value⟩$, $Re\left(c\right)=⟨value⟩$, $Re\left(b\right)=⟨value⟩$, $Im\left(a\right)=⟨value⟩$, $Im\left(c\right)=⟨value⟩$, $Im\left(b\right)=⟨value⟩$.

(errno $4$)

On entry, $c=0.0$ and the root $-b/a$ overflows: $Re\left(c\right)=⟨value⟩$, $Re\left(b\right)=⟨value⟩$, $Re\left(a\right)=⟨value⟩$, $Im\left(c\right)=⟨value⟩$, $Im\left(b\right)=⟨value⟩$, $Im\left(a\right)=⟨value⟩$.

(errno $5$)

On entry, $B=max(\left|Re\left(b\right)\right|,\left|Im\left(b\right)\right|)=⟨value⟩$, $A=max(\left|Re\left(a\right)\right|,\left|Im\left(a\right)\right|)=⟨value⟩$ and $C=max(\left|Re\left(c\right)\right|,\left|Im\left(c\right)\right|)=⟨value⟩$. $B$ is so large that $B^2$ is indistinguishable from $(B^2-4\times A\times C)$ and the root $-b/a$ overflows. $Re\left(b\right)=⟨value⟩$, $Im\left(b\right)=⟨value⟩$, $Re\left(a\right)=⟨value⟩$ and $Im\left(a\right)=⟨value⟩$.

Notes

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

quadratic_complex attempts to find the roots of the quadratic equation $a{z}^2+bz+c=0$ (where $a$, $b$ and $c$ are complex coefficients), by carefully evaluating the ‘standard’ closed formula

$$z=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\text{.}$$

It is based on the function CQDRTC from Smith (1967).

Note: it is not necessary to scale the coefficients prior to calling the function.

References

Smith, B T, 1967, ZERPOL: a zero finding algorithm for polynomials using Laguerre’s method, Technical Report, Department of Computer Science, University of Toronto, Canada