1: $e$ – doubleInput: On entry: $e$ , the coefficient of $z^{4}$ .

Constraint: $e \neq 0.0$ .
2: $a$ – doubleInput: On entry: $a$ , the coefficient of $z^{3}$ .
3: $b$ – doubleInput: On entry: $b$ , the coefficient of $z^{2}$ .
4: $c$ – doubleInput: On entry: $c$ , the coefficient of $z$ .
5: $d$ – doubleInput: 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: $fail$ – NagError *Input/Output: The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

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: $e \neq 0.0$ .

7

Accuracy

fail = NE_NOERROR

on exit, then the

i

th computed root should have approximately

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

correct significant digits.

nag_quartic_roots (c02alc) 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 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).

To find the roots of the quartic equation

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