Symmetric antitriangular matrices and applications

Symmetric antitriangular matrices and
N. Mastronardi, R. Vandebril, and P. Van Dooren
Istituto per le Applicazioni del Calcolo “M. Picone”, CNR, sede di Bari
via Amendola 122D, I-70126 Italy
[email protected]
[email protected]
[email protected]
Indefinite symmetric matrices occur in many applications, such as optimization, least squares problems, partial differential equations and variational problems. In these applications one is often interested in computing
a factorization of the indefinite matrix that puts into evidence the inertia of
the matrix or possibly provides an estimate of its eigenvalues. In this talk
we present a new matrix decomposition that provides this information for
any symmetric indefinite matrix by transforming it to a block antitriangular
form using orthogonal similarity transformations. We discuss several of the
properties of the decomposition and show its use in the analysis of saddle
point problems. Moreover, it will be shown that such a decomposition, implemented in a recursive way, can be efficiently used for solving structured
KKT linear systems arising in Model Predictive Control.
[1] N. Mastronardi, P. Van Dooren, The anti–triangular factorization of
symmetric matrices, SIAM Journal on Matrix Analysis and Applications, to
[2] C. Kirches, H. Bock, J.P. Schl¨oder, S. Sager, A factorization with
update procedures for a KKT matrix arising in direct optimal control, Mathematical Programming Computation, 3(4), pp. 319348, 2011.