Abstract
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 lambda B-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 compution 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.
Anno
2021
Autori IAC
Tipo pubblicazione
Altri Autori
Mastronardi n., Van Dooren P.
Editore
Academic Press,
Rivista
IMA journal of numerical analysis