NAG Library Routine Document

C02ALF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

C02ALF determines the roots of a quartic equation with real coefficients.

2 Specification

SUBROUTINE C02ALF (

E, A, B, C, D, ZEROR, ZEROI, ERREST, IFAIL)

INTEGER	IFAIL
REAL (KIND=nag_wp)	E, A, B, C, D, ZEROR(4), ZEROI(4), ERREST(4)

3 Description

C02ALF attempts to find the roots of the quartic equation

e z^{4} + a z^{3} + b z^{2} + c z + d = 0,

where

e

a

b

c

and

d

are real coefficients with

e \neq 0

. The roots are located by finding the eigenvalues of the associated

4

4

(upper Hessenberg) companion matrix

H

given by

H = (\begin{array}{cccr} 0 & 0 & 0 & - d / e \\ 1 & 0 & 0 & - c / e \\ 0 & 1 & 0 & - b / e \\ 0 & 0 & 1 & - a / e \end{array}) .

The eigenvalues are obtained by a call to F08PEF (DHSEQR). Further details can be found in Section 8.

To obtain the roots of a cubic equation, C02AKF can be used.

4 References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5 Parameters

1: E – REAL (KIND=nag_wp)Input: On entry: $e$ , the coefficient of $z^{4}$ .
Constraint: $E \neq 0.0$ .
2: A – REAL (KIND=nag_wp)Input: On entry: $a$ , the coefficient of $z^{3}$ .
3: B – REAL (KIND=nag_wp)Input: On entry: $b$ , the coefficient of $z^{2}$ .
4: C – REAL (KIND=nag_wp)Input: On entry: $c$ , the coefficient of $z$ .
5: D – REAL (KIND=nag_wp)Input: On entry: $d$ , the constant coefficient.
6: ZEROR( $4$ ) – REAL (KIND=nag_wp) arrayOutput
7: ZEROI( $4$ ) – REAL (KIND=nag_wp) arrayOutput: On exit: $ZEROR (i)$ and $ZEROI (i)$ contain the real and imaginary parts, respectively, of the $i$ th root.
8: ERREST( $4$ ) – REAL (KIND=nag_wp) arrayOutput: On exit: $ERREST (i)$ contains an approximate error estimate for the $i$ th root.
9: IFAIL – INTEGERInput/Output: On entry: IFAIL must be set to $0$ , $- 1 or 1$ . 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 $- 1 or 1$ 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 $- 1 or 1$ 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

- 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,

E = 0.0

$IFAIL = 2$: The companion matrix $H$ cannot be formed without overflow.

$IFAIL = 3$: The iterative procedure used to determine the eigenvalues has failed to converge.

7 Accuracy

IFAIL = 0

on exit, then the

i

th computed root should have approximately

|\log_{10} (ERREST (i))|

correct significant digits.

8 Further Comments

The method used by the routine consists of the following steps, which are performed by routines from LAPACK in Chapter F08.

(a)	Form matrix $H$ .
(b)	Apply a diagonal similarity transformation to $H$ (to give $H^{'}$ ).
(c)	Calculate the eigenvalues and Schur factorization of $H^{'}$ .
(d)	Calculate the left and right eigenvectors of $H^{'}$ .
(e)	Estimate reciprocal condition numbers for all the eigenvalues of $H^{'}$ .
(f)	Calculate approximate error estimates for all the eigenvalues of $H^{'}$ (using the $1$ -norm).

9 Example

This example finds the roots of the quartic equation

z^{4} + 2 z^{3} + 6 z^{2} - 8 z - 40 = 0 .

NAG Library Routine DocumentC02ALF