naginterfaces.library.zeros.quadratic_​real

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

quadratic_real determines the roots of a quadratic equation with real coefficients.

For full information please refer to the NAG Library document for c02aj

https://www.nag.com/numeric/nl/nagdoc_29.3/flhtml/c02/c02ajf.html

Parameters

afloat

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

bfloat

Must contain $b$, the coefficient of $z$.

cfloat

Must contain $c$, 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: $a=⟨value⟩$, $c=⟨value⟩$ and $b=⟨value⟩$.

(errno $4$)

On entry, $c=0.0$ and the root $-b/a$ overflows: $c=⟨value⟩$, $b=⟨value⟩$ and $a=⟨value⟩$.

(errno $5$)

On entry, $b$ is so large that $b^2$ is indistinguishable from $(b^2-4\times a\times c)$ and the root $-b/a$ overflows: $b=⟨value⟩$, $a=⟨value⟩$ and $c=⟨value⟩$.

Notes

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

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

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

It is based on the function QDRTC 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