The bisection eigenvalue method for unitary Hessenberg matrices via their quasiseparable structure

If $N_0$ is a normal matrix, then the Hermitian matrices $\\frac{1}{2}(N_0+N_0^*)$ and $\\frac{i}{2}(N_0^*-N_0)$have same eigenvectors as $N_0$. Their eigenvalues are real part imaginary of $N_0$, respectively.If unitary, only each its sign ofthe needed to completely determine eigenvalue, sincethe sum squares these two parts known be equal $1$.Since unitary upper Hessenberg matrix $U$ has quasiseparable structure order oneand we express $A=\frac{1}{2}(U+U^*)$ , can findthe and, when needed, corresponding eigenvector $x$, by using techniquesthat have been established in paper Eidelman Haimovici [Oper. Theory Adv. Appl., 271 (2018), pp. 181–200].We describe here fast procedure, which takes $1.7\%$ bisection method time, find signof part.For instance, worst case only, build one rowof multiply it of$A$, main procedure.This occurs for our algorithm amongthe $4$ numbers $\pm\cos t\pm i \sin t$ there exactly $2$ andthey opposite, so that distinguish between $\lambda,-\lambda$and $\overline\lambda,-\overline\lambda$.The performance developedalgorithm illustrated series numerical tests. The more accurate many times faster(when executed Matlab) than forgeneral two, because action generators,which small previous cited paper, replaced scalars, most them numbers.