Lanczos-Conjugate Gradient method and pseudoinverse computation, in unconstrained optimization

This paper extends some theoretical properties of the Conjugate Gradient-type method FLR [Fas05], for iteratively solving indefinite linear systems of equations. The latter algorithm is a generalization of the Conjugate Gradient (CG) by Hestenes and Stiefel [HS52]. On one hand, here we carry out a complete relationship between algorithm FLR and the Lanczos process, in case of indefinite and possibly singular matrices. On the other hand we develop simple theoretical results for algorithm FLR, in order to construct an approximation of the Moore-Penrose pseudoinverse of an indefinite matrix. Our approach supplies theory for applications within nonconvex optimization.