NAG Library Routine Document

c02ahf  (quadratic_complex)

 Contents

    1  Purpose
    7  Accuracy

1
Purpose

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

2
Specification

Fortran Interface
Subroutine c02ahf ( ar, ai, br, bi, cr, ci, zsm, zlg, ifail)
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)
C Header Interface
#include nagmk26.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)

3
Description

c02ahf attempts to find the roots of the quadratic equation az2+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.

4
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

5
Arguments

1:     ar – Real (Kind=nag_wp)Input
2:     ai – Real (Kind=nag_wp)Input
On entry: ar and ai must contain the real and imaginary parts respectively of a, the coefficient of z2.
3:     br – Real (Kind=nag_wp)Input
4:     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:     cr – Real (Kind=nag_wp)Input
6:     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:     zsm2 – Real (Kind=nag_wp) arrayOutput
On exit: the real and imaginary parts of the smallest root in magnitude are stored in zsm1 and zsm2 respectively.
8:     zlg2 – Real (Kind=nag_wp) arrayOutput
On exit: the real and imaginary parts of the largest root in magnitude are stored in zlg1 and zlg2 respectively.
9:     ifail – IntegerInput/Output
On entry: ifail must be set to 0, -1​ or ​1. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1​ 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 argument, the recommended value is 0. When the value -1​ or ​1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6
Error Indicators and Warnings

If on entry 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:
ifail=1
On entry, ar,ai=0,0. In this case, zsm1 and zsm2 contain the real and imaginary parts respectively of the root -c/b.
ifail=2
On entry, ar,ai=0,0 and br,bi=0,0. In this case, zsm1 contains the largest machine representable number (see x02alf) and zsm2 contains zero.
ifail=3
On entry, ar,ai=0,0 and the root -c/b overflows. In this case, zsm1 contains the largest machine representable number (see x02alf) and zsm2 contains zero.
ifail=4
On entry, cr,ci=0,0 and the root -b/a overflows. In this case, both zsm1 and zsm2 contain zero.
ifail=5
On entry, b~  is so large that b~ 2  is indistinguishable from b~ 2 - 4 a~ c~  and the root -b / a  overflows, where b~ = maxbr,bi , a~ = maxar,ai  and c~ = maxcr,ci . In this case, zsm1 and zsm2 contain the real and imaginary parts respectively of the root -c / b .
ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
ifail=-399
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
ifail=-999
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.
If ifail>0 on exit, then zlg1 contains the largest machine representable number (see x02alf) and zlg2 contains zero.

7
Accuracy

If ifail=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.

8
Parallelism and Performance

c02ahf is not threaded in any implementation.

9
Further Comments

None.

10
Example

This example finds the roots of the quadratic equation z2-3.0-1.0i z+8.0+1.0i=0.

10.1
Program Text

Program Text (c02ahfe.f90)

10.2
Program Data

Program Data (c02ahfe.d)

10.3
Program Results

Program Results (c02ahfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017