Issues on Polarity for Quadratic Hypersurfaces and a framework for Conjugate Direction Methods

Giovedi' 7 Marzo 2013 - Giovanni Fasano


Giovedi' 7 Marzo 2013 alle ore 15:00 in aula 1BC45, Giovanni Fasano (Department of Management, University of Venice "Ca' Foscari") terra' un seminario dal titolo "Issues on Polarity for Quadratic Hypersurfaces and a framework for Conjugate Direction Methods".

We first recast some results from the theory of polarity in homogeneous coordinates, used to show several geometric properties of the Conjugate Gradient Method (CG), for the solution of positive definite linear systems. We also describe how our analysis might suggest the basis for possible extensions and generalizations of the CG. In the second part of the paper we introduce a parameter dependent class of Krylov-based methods, namely CD, for the solution of symmetric linear systems. The parameter dependent framework of algorithms in the latter class, draws its inspiration from the relation between polarity and conjugacy for quadratic hypersurfaces. Moreover, we give evidence that in our proposal we generate sequences of conjugate directions, extending some properties of the standard CG, in order to preserve the conjugacy. For specific values of the parameters in our framework we obtain schemes equivalent to both the CG and the scaled-CG. We also prove the finite convergence of the algorithms in CD, and we provide some error analysis. Finally, preconditioning is introduced for CD, and we show that standard error bounds for the preconditioned CG also hold for the preconditioned CD.

Rif. int. F. Rinaldi

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