In this paper we consider a geometric viewpoint to analyze the behaviour of the Conjugate Gradient (CG) method, for the solution of a symmetric linear system, when at current step a pivot breakdown possibly occurs (degenerate case). As well known this can occur when the system matrix is indefinite or singular. In the latter case the CG gets stuck, since the steplength along the current search direction cannot be computed. We show here that a simple geometric interpretation can be provided for the degenerate case, as long as some basics on projective geometry in the Euclidean space arc considered.

Polarity for Quadratic Hypersurfaces and Conjugate Gradient Method: Relation between Degenerate and Nondegenerate Cases

FASANO, Giovanni;GIOVE, Silvio;GUSSO, Riccardo
2016-01-01

Abstract

In this paper we consider a geometric viewpoint to analyze the behaviour of the Conjugate Gradient (CG) method, for the solution of a symmetric linear system, when at current step a pivot breakdown possibly occurs (degenerate case). As well known this can occur when the system matrix is indefinite or singular. In the latter case the CG gets stuck, since the steplength along the current search direction cannot be computed. We show here that a simple geometric interpretation can be provided for the degenerate case, as long as some basics on projective geometry in the Euclidean space arc considered.
Numerical Computations:Theory and Algorithms, The 2nd International Conference and Summer School (NUMTA 2016)
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10278/3677962
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