Naslov
On deflation process and solving the quadratic eigenvalue problems
(On deflation process and solving the quadratic eigenvalue problems)

Autori
Šain Glibić , Ivana

Vrsta, podvrsta i kategorija rada
Sažeci sa skupova, neobjavljeni rad, znanstveni

Skup
Advances in NLA: Celebrating the birth of James H. Wilkinson

Mjesto i datum
Manchester, Ujedinjeno Kraljevstvo, 29.05.2019. - 30.05.2019

Vrsta sudjelovanja
Poster

Vrsta recenzije
Nije recenziran

Ključne riječi
eigenvalues ; quadratic eigenvalue problem ; QR factorization ; QZ method ; backward error ; infinite eigenvalues

Sažetak
Quadratic eigenvalue problem appears in a variety of applications, e.g. vibration analysis of mechanical systems, acoustics, computational fluid dynamics, total least squares. The standard approach for solving quadratic eigenvalue problem is to linearize it and then use QZ algorithm for the corresponding generalized eigenvalue problem. This procedure is prone to numerical difficulties in presence of zero and infinite eigenvalues. In 2011., Hammarling, Munro and Tisseur proposed quadeig algorithm that substantially alleviated these problems by careful preprocessing. Before calling the QZ algorithm, quadeig deploys parameter scaling to equilibrate the norms of the coefficient matrices, and attempts deflating the zero and the infinite eigenvalues from the linearized problem. Deflation process relies on rank determination of leading matrix coefficient M, and constant coefficient K, and quadeig uses the (rank revealing) QR factorization with column pivoting. Using the orthogonal equivalence transformation on the linearization, n-rank(M) infinite and n-rank(K) zero eigenvalues are removed from the linearized pencil. The remaining eigenvalues are computed using the QZ algorithm. We analyse numerical properties of quadeig in more detail and propose several enhancements. New backward error analysis further clarifies numerical properties and the success of quadeig, but it also leads to modifications that lead to even more robust procedure. In addition to the theoretical considerations, we provide numerical examples to illustrate the improved accuracy of the new approach, implemented as a Matlab toolbox and as a LAPACK-style FORTRAN package. Finally, we show how our approach extends to certain higher order polynomial eigenvalue problems.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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