## MC01VD

### Roots of a quadratic equation with real coefficients

[Specification] [Arguments] [Method] [References] [Comments] [Example]

Purpose

```  To compute the roots of a quadratic equation with real
coefficients.

```
Specification
```      SUBROUTINE MC01VD( A, B, C, Z1RE, Z1IM, Z2RE, Z2IM, INFO )
C     .. Scalar Arguments ..
INTEGER           INFO
DOUBLE PRECISION  A, B, C, Z1IM, Z1RE, Z2IM, Z2RE

```
Arguments

Input/Output Parameters

```  A       (input) DOUBLE PRECISION
The value of the coefficient of the quadratic term.

B       (input) DOUBLE PRECISION
The value of the coefficient of the linear term.

C       (input) DOUBLE PRECISION
The value of the coefficient of the constant term.

Z1RE    (output) DOUBLE PRECISION
Z1IM    (output) DOUBLE PRECISION
The real and imaginary parts, respectively, of the largest
root in magnitude.

Z2RE    (output) DOUBLE PRECISION
Z2IM    (output) DOUBLE PRECISION
The real and imaginary parts, respectively, of the
smallest root in magnitude.

```
Error Indicator
```  INFO    INTEGER
= 0:  successful exit;
= 1:  if on entry, either A = B = 0.0 or A = 0.0 and the
root -C/B overflows; in this case Z1RE, Z1IM, Z2RE
and Z2IM are unassigned;
= 2:  if on entry, A = 0.0; in this case Z1RE contains
BIG and Z1IM contains zero, where BIG is a
representable number near the overflow threshold
of the machine (see LAPACK Library Routine DLAMCH);
= 3:  if on entry, either C = 0.0 and the root -B/A
overflows or A, B and C are non-zero and the largest
real root in magnitude cannot be computed without
overflow; in this case Z1RE contains BIG and Z1IM
contains zero;
= 4:  if the roots cannot be computed without overflow; in
this case Z1RE, Z1IM, Z2RE and Z2IM are unassigned.

```
Method
```  The routine computes the roots (r1 and r2) of the real quadratic
equation
2
a * x  + b * x + c = 0

as
- b - SIGN(b) * SQRT(b * b - 4 * a * c)             c
r1 = ---------------------------------------  and r2 = ------
2 * a                            a * r1

unless a = 0, in which case

-c
r1 = --.
b

Precautions are taken to avoid overflow and underflow wherever
possible.

```
Numerical Aspects
```  The algorithm is numerically stable.

```
```  None
```
Example

Program Text

```*     MC01VD EXAMPLE PROGRAM TEXT
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     .. Parameters ..
INTEGER          NIN, NOUT
PARAMETER        ( NIN = 5, NOUT = 6 )
*     .. Local Scalars ..
DOUBLE PRECISION A, B, C, Z1IM, Z1RE, Z2IM, Z2RE
INTEGER          INFO
*     .. External Subroutines ..
EXTERNAL         MC01VD
*     .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
*     Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) A, B, C
*     Solve the quadratic equation A*x**2 + B*x + C = 0.
CALL MC01VD( A, B, C, Z1RE, Z1IM, Z2RE, Z2IM, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
WRITE ( NOUT, FMT = 99996 ) Z1RE, Z1IM, Z2RE, Z2IM
END IF
*
STOP
*
99999 FORMAT (' MC01VD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MC01VD = ',I2)
99997 FORMAT (' The roots of the quadratic equation are ')
99996 FORMAT (/' x = ',F8.4,2X,SP,F8.4,'*j',SS,/' x = ',F8.4,2X,SP,F8.4,
\$       '*j')
END
```
Program Data
``` MC01VD EXAMPLE PROGRAM DATA
0.5  -1.0  2.0
```
Program Results
``` MC01VD EXAMPLE PROGRAM RESULTS

The roots of the quadratic equation are

x =   1.0000   +1.7321*j
x =   1.0000   -1.7321*j
```