
#include <nagmk26.h>

$$u{z}^{3}+r{z}^{2}+sz+t=0\text{,}$$ 
$$H=\left(\begin{array}{ccr}0& 0& t/u\\ 1& 0& s/u\\ 0& 1& r/u\end{array}\right)\text{.}$$ 
(a)  Form matrix $H$. 
(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). 
$${z}^{3}\left(23i\right){z}^{2}+\left(5+14i\right)z\left(40+5i\right)=0\text{.}$$ 