Convergence of the unitary 𝑄𝑅 algorithm with a unimodular Wilkinson shift

Abstract

In applying the Q R Q\!R algorithm to compute the eigenvalues of a unitary Hessenberg matrix, a projected Wilkinson shift of unit modulus is proposed and proved to give global convergence with (at least) a quadratic asymptotic rate for the Q R Q\!R iteration. Experimental testing demonstrates that the unimodular shift produces more efficient numerical convergence.