Symmetric antitriangular matrices and applications 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. References [1] N. Mastronardi, P. Van Dooren, The anti–triangular factorization of symmetric matrices, SIAM Journal on Matrix Analysis and Applications, to appear. [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.

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