Invio il seguente avviso di seminario
Dario
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Gorno: Martedi 17 ottobre
Ora: 11:00
Aula: Seminari, Dipartimento di Matematica
Speaker: Leonardo Robol, ISTI --- CNR
Titolo: Backward error analysis for polynomial rootfinders.
Abstract: The most common methods to compute roots of polynomials is to
rephrase the problem in linear algebra terms: one constructs a companion
matrix (or pencil) that has the roots of the polynomial as eigenvalues.
However, even when the eigenvalue method used is backward stable, it's
non-trivial to map the backward error back on the polynomial. This
problem has been studied extensively in the past and it has been shown
that relying on the QR method does not provide a normwise
backward-stable rootfinder. For this reason, one needs to rely on QZ +
appropriate scaling.
However, companion matrices are unitary plus rank 1, and we claim that
relying on fast QR method that exploits this structure can lead to a
backward-stable method. To this end, one needs to ensure that the errors
on the unitary and rank 1 parts satisfy different bounds.
We show how to perform this analysis of "mixed perturbation
backward-error", as well as a method that satisfies these constrains.
Time permitting, we discuss some more details that one should consider.
We show, for example, that normwise backward stable operations (i.e.,
unitary rotations in our case) do not necessarily imply backward stable
operations on the unitary and low-rank factors, unless some particular
identities involving sines and cosines are preserved.
This is joint work with Jared Aurentz, Thomas Mach, Raf Vandebril, and
David Watkins.
