1: $u$ – double Input: On entry: $u$ , the coefficient of $z^{3}$ .

Constraint: $u \neq 0.0$ .
2: $r$ – double Input: On entry: $r$ , the coefficient of $z^{2}$ .
3: $s$ – double Input: On entry: $s$ , the coefficient of $z$ .
4: $t$ – double Input: On entry: $t$ , the constant coefficient.
5: $zeror [3]$ – double Output
6: $zeroi [3]$ – double Output: On exit: $zeror [i - 1]$ and $zeroi [i - 1]$ contain the real and imaginary parts, respectively, of the $i$ th root.
7: $errest [3]$ – double Output: On exit: $errest [i - 1]$ contains an approximate error estimate for the $i$ th root.
8: $fail$ – NagError * Input/Output: The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

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, $u = 0.0$ .
Constraint: $u \neq 0.0$ .

7 Accuracy

fail . code = NE_NOERROR

on exit, then the

i

th computed root should have approximately

| \log_{10} (errest [i - 1]) |

correct significant digits.

Background information to multithreading can be found in the Multithreading documentation.

c02akc is not threaded in any implementation.

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 $H^{'}$ ).
(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).

To find the roots of the cubic equation

z^{3} + 3 z^{2} + 9 z - 13 = 0 .

c02ak: FL CL CPP AD PY MB