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

Abstract

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