This work is concerned with the development and study of a class of limited memory preconditioners for the solution of sequences of linear systems. To this aim, we consider linear systems with the same symmetric positive definite matrix and multiple right-hand sides available in sequence. We first propose a general class of preconditioners, called Limited Memory Preconditioners (LMP), whose construction involves only a small number of linearly independent vectors and their product with the matrix to precondition. After exploring and illustrating the theoretical properties of this new class of preconditioners, we more particularly study three members of the class named spectral-LMP, quasi-Newton-LMP and Ritz-LMP, and show that the two first correspond to two well-known preconditioners (see Fisher 1998 and Morales and Nocedal 2000, respectively), while the third one appears to be a new and quite promising preconditioner, as illustrated by numerical experiments.

## Citation

Technical report 2008/16, Department of Mathematics, University of Namur, Namur, Belgium, November 2008