c02 Chapter Contents
c02 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_cubic_roots (c02akc)

## 1  Purpose

nag_cubic_roots (c02akc) determines the roots of a cubic equation with real coefficients.

## 2  Specification

 #include #include
 void nag_cubic_roots (double u, double r, double s, double t, double zeror[], double zeroi[], double errest[], NagError *fail)

## 3  Description

nag_cubic_roots (c02akc) attempts to find the roots of the cubic equation
 $uz3 + rz2 + sz + t = 0 ,$
where $u,r,s$ and $t$ are real coefficients with $u\ne 0$. The roots are located by finding the eigenvalues of the associated 3 by 3 (upper Hessenberg) companion matrix2 $H$ given by
 $H = 0 0 -t / u 1 0 -s / u 0 1 -r / u .$
Further details can be found in Section 8.
To obtain the roots of a quadratic equation, nag_quartic_roots (c02alc) 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:     udoubleInput
On entry: $u$, the coefficient of ${z}^{3}$.
Constraint: ${\mathbf{u}}\ne 0.0$.
2:     rdoubleInput
On entry: $r$, the coefficient of ${z}^{2}$.
3:     sdoubleInput
On entry: $s$, the coefficient of $z$.
4:     tdoubleInput
On entry: $t$, the constant coefficient.
5:     zeror[$3$]doubleOutput
6:     zeroi[$3$]doubleOutput
On exit: ${\mathbf{zeror}}\left[i-1\right]$ and ${\mathbf{zeroi}}\left[i-1\right]$ contain the real and imaginary parts, respectively, of the $i$th root.
7:     errest[$3$]doubleOutput
On exit: ${\mathbf{errest}}\left[i-1\right]$ contains an approximate error estimate for the $i$th root.
8:     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, ${\mathbf{u}}=0.0$.
Constraint: ${\mathbf{u}}\ne 0.0$.

## 7  Accuracy

If ${\mathbf{fail}}\mathbf{.}\mathbf{code}=\mathrm{NE_NOERROR}$ on exit, then the $i$th computed root should have approximately $\left|{\mathrm{log}}_{10}\left({\mathbf{errest}}\left[i-1\right]\right)\right|$ correct significant digits.

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

## 9  Example

To find the roots of the cubic equation
 $z 3 + 3 z 2 + 9 z - 13 = 0 .$

### 9.1  Program Text

Program Text (c02akce.c)

### 9.2  Program Data

Program Data (c02akce.d)

### 9.3  Program Results

Program Results (c02akce.r)