C02 Chapter Contents
C02 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentC02AJF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

C02AJF determines the roots of a quadratic equation with real coefficients.

## 2  Specification

 SUBROUTINE C02AJF ( A, B, C, ZSM, ZLG, IFAIL)
 INTEGER IFAIL REAL (KIND=nag_wp) A, B, C, ZSM(2), ZLG(2)

## 3  Description

C02AJF 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
 $z=-b±b2-4ac 2a .$
It is based on the routine QDRTC 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

## 5  Parameters

1:     $\mathrm{A}$ – REAL (KIND=nag_wp)Input
On entry: must contain $a$, the coefficient of ${z}^{2}$.
2:     $\mathrm{B}$ – REAL (KIND=nag_wp)Input
On entry: must contain $b$, the coefficient of $z$.
3:     $\mathrm{C}$ – REAL (KIND=nag_wp)Input
On entry: must contain $c$, the constant coefficient.
4:     $\mathrm{ZSM}\left(2\right)$ – REAL (KIND=nag_wp) arrayOutput
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.
5:     $\mathrm{ZLG}\left(2\right)$ – REAL (KIND=nag_wp) arrayOutput
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.
6:     $\mathrm{IFAIL}$ – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ 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).

## 6  Error Indicators and Warnings

If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
${\mathbf{IFAIL}}=1$
On entry, ${\mathbf{A}}=0.0$. In this case, ${\mathbf{ZSM}}\left(1\right)$ contains the root $-c/b$ and ${\mathbf{ZSM}}\left(2\right)$ contains zero.
${\mathbf{IFAIL}}=2$
On entry, ${\mathbf{A}}=0.0$ and ${\mathbf{B}}=0.0$. In this case, ${\mathbf{ZSM}}\left(1\right)$ contains the largest machine representable number (see X02ALF) and ${\mathbf{ZSM}}\left(2\right)$ contains zero.
${\mathbf{IFAIL}}=3$
On entry, ${\mathbf{A}}=0.0$ and the root $-c/b$ overflows. In this case, ${\mathbf{ZSM}}\left(1\right)$ contains the largest machine representable number (see X02ALF) and ${\mathbf{ZSM}}\left(2\right)$ contains zero.
${\mathbf{IFAIL}}=4$
On entry, ${\mathbf{C}}=0.0$ and the root $-b/a$ overflows. In this case, both ${\mathbf{ZSM}}\left(1\right)$ and ${\mathbf{ZSM}}\left(2\right)$ contain zero.
${\mathbf{IFAIL}}=5$
On entry, $b$ is so large that ${b}^{2}$ is indistinguishable from ${b}^{2}-4ac$ and the root $-b/a$ overflows. In this case, ${\mathbf{ZSM}}\left(1\right)$ contains the root $-c/b$ and ${\mathbf{ZSM}}\left(2\right)$ contains zero.
${\mathbf{IFAIL}}=-99$
See Section 3.8 in the Essential Introduction for further information.
${\mathbf{IFAIL}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.7 in the Essential Introduction for further information.
${\mathbf{IFAIL}}=-999$
Dynamic memory allocation failed.
See Section 3.6 in the Essential Introduction for further information.
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.

## 7  Accuracy

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.

Not applicable.

None.

## 10  Example

This example finds the roots of the quadratic equation ${z}^{2}+3z-10=0$.

### 10.1  Program Text

Program Text (c02ajfe.f90)

### 10.2  Program Data

Program Data (c02ajfe.d)

### 10.3  Program Results

Program Results (c02ajfe.r)