#
A Matrix-Free Preconditioner for Sparse Symmetric Positive Definite Systems
and Least-Squares Problems

###
Technical Report ERGO-11-010

**
S. Bellavia,
J. Gondzio and
B. Morini
**

**
Abstract
**

We analyze and discuss matrix-free and limited-memory preconditioners (LMP)
for sparse symmetric positive definite systems and normal equations
of large and sparse least-squares problems. The preconditioners are
based on a partial Cholesky factorization and can be coupled with
a deflation strategy. The construction of the preconditioners requires
only matrix-vector products, is breakdown-free, and does not need
to form the matrix. The memory requirements of the preconditioners
are defined in advance and they do not depend on the number of nonzero
entries in the matrix.
When eigenvalue deflation is used, the preconditioners turn out
to be suitable for solving sequences of slowly changing systems
or linear systems with different right-hand sides. Numerical
results are provided, including the case of sequences arising
in nonnegative linear least-squares problems solved
by interior point methods.

**
Key words:
**
sparse symmetric positive definite systems and least-squares problems,
preconditioners, matrix-free, memory constraints, Cholesky factorization,
deflation, interior point methods.

**
Text
**

PDF ERGO-11-010.pdf.

**
History:
**

Written: July 13, 2011, revised March 22, 2012 and September 6, 2012.

Published:
*SIAM Journal on Scientific Computing* 35 (2013) No 1, pp. A192-A211.