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1Purpose

c02ahf determines the roots of a quadratic equation with complex coefficients.

2Specification

Fortran Interface
 Subroutine c02ahf ( ar, ai, br, bi, cr, ci, zsm, zlg,
 Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: ar, ai, br, bi, cr, ci Real (Kind=nag_wp), Intent (Out) :: zsm(2), zlg(2)
#include <nag.h>
 void c02ahf_ (const double *ar, const double *ai, const double *br, const double *bi, const double *cr, const double *ci, double zsm[], double zlg[], Integer *ifail)
The routine may be called by the names c02ahf or nagf_zeros_quadratic_complex.

3Description

c02ahf 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
 $z=-b±b2-4ac 2a .$
It is based on the routine CQDRTC from Smith (1967).
Note:  it is not necessary to scale the coefficients prior to calling the routine.
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

5Arguments

1: $\mathbf{ar}$Real (Kind=nag_wp) Input
2: $\mathbf{ai}$Real (Kind=nag_wp) Input
On entry: ar and ai must contain the real and imaginary parts respectively of $a$, the coefficient of ${z}^{2}$.
3: $\mathbf{br}$Real (Kind=nag_wp) Input
4: $\mathbf{bi}$Real (Kind=nag_wp) Input
On entry: br and bi must contain the real and imaginary parts respectively of $b$, the coefficient of $z$.
5: $\mathbf{cr}$Real (Kind=nag_wp) Input
6: $\mathbf{ci}$Real (Kind=nag_wp) Input
On entry: cr and ci must contain the real and imaginary parts respectively of $c$, the constant coefficient.
7: $\mathbf{zsm}\left(2\right)$Real (Kind=nag_wp) array Output
On exit: the real and imaginary parts of the smallest root in magnitude are stored in ${\mathbf{zsm}}\left(1\right)$ and ${\mathbf{zsm}}\left(2\right)$ respectively.
8: $\mathbf{zlg}\left(2\right)$Real (Kind=nag_wp) array Output
On exit: the real and imaginary parts of the largest root in magnitude are stored in ${\mathbf{zlg}}\left(1\right)$ and ${\mathbf{zlg}}\left(2\right)$ respectively.
9: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
Note: if ${\mathbf{ifail}}>{\mathbf{0}}$ on exit, then ${\mathbf{zlg}}\left(1\right)$ contains the largest machine representable number (see x02alf) and ${\mathbf{zlg}}\left(2\right)$ contains zero.
${\mathbf{ifail}}=1$
On entry, $\left({\mathbf{ar}},{\mathbf{ai}}\right)=\left(0,0\right)$.
${\mathbf{ifail}}=2$
On entry, $\left({\mathbf{ar}},{\mathbf{ai}}\right)=\left(0,0\right)$ and $\left({\mathbf{br}},{\mathbf{bi}}\right)=\left(0,0\right)$.
${\mathbf{ifail}}=3$
On entry, $\left({\mathbf{ar}},{\mathbf{ai}}\right)=\left(0,0\right)$ and the root $-\left({\mathbf{cr}},{\mathbf{ci}}\right)/\left({\mathbf{br}},{\mathbf{bi}}\right)$ overflows: ${\mathbf{ar}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{cr}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{br}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{ai}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{ci}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{bi}}=⟨\mathit{\text{value}}⟩$.
${\mathbf{ifail}}=4$
On entry, $\left({\mathbf{cr}},{\mathbf{ci}}\right)=\left(0,0\right)$ and the root $-\left({\mathbf{br}},{\mathbf{bi}}\right)/\left({\mathbf{ar}},{\mathbf{ai}}\right)$ overflows: ${\mathbf{cr}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{br}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{ar}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{ci}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{bi}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{ai}}=⟨\mathit{\text{value}}⟩$.
${\mathbf{ifail}}=5$
On entry, $B$ is so large that ${B}^{2}$ is indistinguishable from $\left({B}^{2}-4×A×C\right)$ and the root $-\left({\mathbf{br}},{\mathbf{bi}}\right)/\left({\mathbf{ar}},{\mathbf{ai}}\right)$ overflows: $B=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(|{\mathbf{br}}|,|{\mathbf{bi}}|\right)=⟨\mathit{\text{value}}⟩$, $A=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(|{\mathbf{ar}}|,|{\mathbf{ai}}|\right)=⟨\mathit{\text{value}}⟩$, $C=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(|{\mathbf{cr}}|,|{\mathbf{ci}}|\right)=⟨\mathit{\text{value}}⟩$, ${\mathbf{br}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{bi}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{ar}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{ai}}=⟨\mathit{\text{value}}⟩$.
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7Accuracy

If ${\mathbf{ifail}}={\mathbf{0}}$ on exit, then the computed roots should be accurate to within a small multiple of the machine precision except when underflow (or overflow) occurs, in which case the true roots are within a small multiple of the underflow (or overflow) threshold of the machine.

8Parallelism and Performance

c02ahf is not threaded in any implementation.

None.

10Example

This example finds the roots of the quadratic equation ${z}^{2}-\left(3.0-1.0i\right)z+\left(8.0+1.0i\right)=0$.

10.1Program Text

Program Text (c02ahfe.f90)

10.2Program Data

Program Data (c02ahfe.d)

10.3Program Results

Program Results (c02ahfe.r)