Convergence of the shifted 𝑄𝑅 algorithm for unitary Hessenberg matrices

Abstract

This paper shows that for unitary Hessenberg matrices the Q R Q\!R algorithm, with (an exceptional initial-value modification of) the Wilkinson shift, gives global convergence; moreover, the asymptotic rate of convergence is at least cubic, higher than that which can be shown to be quadratic only for Hermitian tridiagonal matrices, under no further assumption. A general mixed shift strategy with global convergence and cubic rates is also presented.