Rational QZ steps with perfect shifts

Abstract In this paper we analyze the stability of problem performing a rational QZ step with shift that is an eigenvalue given regular pencil $$H-\lambda K$$ H - λ K in unreduced Hessenberg–Hessenberg form. exact arithmetic, backward moves to top pencil, while rest maintained form, which then yields deflation shift. But finite-precision gets “blurred” and precludes at pencil. show when first compute corresponding eigenvector sufficient accuracy, can be constructed using eigenvector, so also obtained finite-precision.