nag_quartic_roots (c02alc) (PDF version)
c02 Chapter Contents
c02 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_quartic_roots (c02alc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_quartic_roots (c02alc) determines the roots of a quartic equation with real coefficients.

2  Specification

#include <nag.h>
#include <nagc02.h>
void  nag_quartic_roots (double e, double a, double b, double c, double d, double zeror[], double zeroi[], double errest[], NagError *fail)

3  Description

nag_quartic_roots (c02alc) attempts to find the roots of the quartic equation
ez4 + az3 + bz2 + cz + d = 0 ,
where e , a , b , c  and d  are real coefficients with e0 . The roots are located by finding the eigenvalues of the associated 4 by 4 (upper Hessenberg) companion matrix H  given by
H = -d / e -c / e -b / e -a / e .
Further details can be found in Section 8.
To obtain the roots of a cubic equation, nag_cubic_roots (c02akc) can be used.

4  References

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

5  Arguments

1:     edoubleInput
On entry: e , the coefficient of z 4 .
Constraint: e0.0 .
2:     adoubleInput
On entry: a , the coefficient of z 3 .
3:     bdoubleInput
On entry: b , the coefficient of z 2 .
4:     cdoubleInput
On entry: c , the coefficient of z .
5:     ddoubleInput
On entry: d , the constant coefficient.
6:     zeror[4]doubleOutput
7:     zeroi[4]doubleOutput
On exit: zeror[i-1]  and zeroi[i-1]  contain the real and imaginary parts, respectively, of the i th root.
8:     errest[4]doubleOutput
On exit: errest[i-1]  contains an approximate error estimate for the i th root.
9:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_C02_NOT_CONV
The iterative procedure used to determine the eigenvalues has failed to converge.
NE_C02_OVERFLOW
The companion matrix H  cannot be formed without overflow.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_REAL
On entry, e=0.0 .
Constraint: e0.0 .

7  Accuracy

If fail=NE_NOERROR  on exit, then the i th computed root should have approximately log 10 errest[i-1]  correct significant digits.

8  Further Comments

The method used by the function consists of the following steps, which are performed by functions from LAPACK.
(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

To find the roots of the quartic equation
z 4 + 2 z 3 + 6 z 2 - 8 z - 40 = 0 .

9.1  Program Text

Program Text (c02alce.c)

9.2  Program Data

Program Data (c02alce.d)

9.3  Program Results

Program Results (c02alce.r)


nag_quartic_roots (c02alc) (PDF version)
c02 Chapter Contents
c02 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012