On QZ Steps with Perfect Shifts and Computing the Index of a Differential Algebraic Equation

In this paper we revisit the problem of performing a QZ step with a so-called perfect shift, which is an exact eigenvalue of a given regular pencil lambdaB-A in unreduced Hessenberg triangular form. In exact arithmetic, the QZ step moves that eigenvalue to the bottom of the pencil, while the rest of the pencil is maintained in Hessenberg triangular form, which then yields a deflation of the given eigenvalue. But in finite precision the QZ step gets blurred and precludes the deflation of the given eigenvalue. In this paper we show that when we first compute the corresponding eigenvector to sufficient accuracy, then the QZ step can be constructed using this eigenvector, so that the deflation is also obtained in finite precision. An important application of this technique is the computation of the index of a system of differential algebraic equations, since an exact deflation of the infinite eigenvalues is needed to impose correctly the algebraic constraints of such differential equations.