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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_zeros_cubic_real (c02ak)

## Purpose

nag_zeros_cubic_real (c02ak) determines the roots of a cubic equation with real coefficients.

## Syntax

[zeror, zeroi, errest, ifail] = c02ak(u, r, s, t)
[zeror, zeroi, errest, ifail] = nag_zeros_cubic_real(u, r, s, t)

## Description

nag_zeros_cubic_real (c02ak) attempts to find the roots of the cubic equation
 uz3 + rz2 + sz + t = 0, $uz3+rz2+sz+t=0,$
where u$u$, r$r$, s$s$ and t$t$ are real coefficients with u0$u\ne 0$. The roots are located by finding the eigenvalues of the associated 3$3$ by 3$3$ (upper Hessenberg) companion matrix H$H$ given by
H =
 0 0 − t / u 1 0 − s / u 0 1 − r / u
.
$H= 0 0 -t/u 1 0 -s/u 0 1 -r/u .$
The eigenvalues are obtained by a call to nag_lapack_dhseqr (f08pe). Further details can be found in Section [Further Comments].
To obtain the roots of a quartic equation, nag_zeros_quartic_real (c02al) can be used.

## References

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

## Parameters

### Compulsory Input Parameters

1:     u – double scalar
u$u$, the coefficient of z3${z}^{3}$.
Constraint: u0.0${\mathbf{u}}\ne 0.0$.
2:     r – double scalar
r$r$, the coefficient of z2${z}^{2}$.
3:     s – double scalar
s$s$, the coefficient of z$z$.
4:     t – double scalar
t$t$, the constant coefficient.

None.

None.

### Output Parameters

1:     zeror(3$3$) – double array
2:     zeroi(3$3$) – double array
zeror(i)${\mathbf{zeror}}\left(i\right)$ and zeroi(i)${\mathbf{zeroi}}\left(i\right)$ contain the real and imaginary parts, respectively, of the i$i$th root.
3:     errest(3$3$) – double array
errest(i)${\mathbf{errest}}\left(i\right)$ contains an approximate error estimate for the i$i$th root.
4:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:
ifail = 1${\mathbf{ifail}}=1$
 On entry, u = 0.0${\mathbf{u}}=0.0$.
ifail = 2${\mathbf{ifail}}=2$
The companion matrix H$H$ cannot be formed without overflow.
ifail = 3${\mathbf{ifail}}=3$
The iterative procedure used to determine the eigenvalues has failed to converge.

## Accuracy

If ${\mathbf{ifail}}={\mathbf{0}}$ on exit, then the i$i$th computed root should have approximately |log10(errest(i))| $|{\mathrm{log}}_{10}\left({\mathbf{errest}}\left(i\right)\right)|$ correct significant digits.

The method used by the function consists of the following steps, which are performed by functions from LAPACK in Chapter F08.
 (a) Form matrix H$H$. (b) Apply a diagonal similarity transformation to H$H$ (to give H′${H}^{\prime }$). (c) Calculate the eigenvalues and Schur factorization of H′${H}^{\prime }$. (d) Calculate the left and right eigenvectors of H′${H}^{\prime }$. (e) Estimate reciprocal condition numbers for all the eigenvalues of H′${H}^{\prime }$. (f) Calculate approximate error estimates for all the eigenvalues of H′${H}^{\prime }$ (using the 1$1$-norm).

## Example

```function nag_zeros_cubic_real_example
u = 1;
r = 3;
s = 9;
t = -13;
[zeror, zeroi, errest, ifail] = nag_zeros_cubic_real(u, r, s, t)
```
```

zeror =

1.0000
-2.0000
-2.0000

zeroi =

0
3
-3

errest =

1.0e-14 *

0.1005
0.1040
0.1040

ifail =

0

```
```function c02ak_example
u = 1;
r = 3;
s = 9;
t = -13;
[zeror, zeroi, errest, ifail] = c02ak(u, r, s, t)
```
```

zeror =

1.0000
-2.0000
-2.0000

zeroi =

0
3
-3

errest =

1.0e-14 *

0.1005
0.1040
0.1040

ifail =

0

```