#### Volume 2, pp. 183-193, 1994.

## Convergence of infinite products of matrices and inner-outer iteration schemes

Rafael Bru, L. Elsner, and M. Neumann

### Abstract

We develop conditions under which a product
$\prod_{i=0}^{\infty}T_{i}$
of matrices chosen from a possibly infinite set of matrices
${\cal S}=\{T_j | j\in J\}$ converges. We obtain the following
conditions which are sufficient for the convergence
of the product:
There exists a vector norm such that all matrices in ${\cal S}$
are nonexpansive with respect to this norm and
there exists a subsequence $\{i_k\}_{k=0}^{\infty}$ of
the sequence of the nonnegative integers such that the corresponding
sequence of operators $\left\{ T_{i_k} \right\}_{k=0}^{\infty}$
converges to an operator which is paracontracting
with respect to this norm. We deduce the continuity of the
limit of the product of matrices as a function of
the sequences $\{i_k\}_{k=0}^{\infty}$. But more importantly, we apply our
results to
the question of the convergence of inner–outer iteration schemes for solving
**singular** consistent linear systems of equations, where the outer splitting is
regular and the inner splitting is weak regular.

Full Text (PDF) [186 KB]

### Key words

iterative methods, infinite products, contractions.

### AMS subject classifications

65F10.

### ETNA articles which cite this article

Vol. 3 (1995), pp. 24-38 Rafael Bru, Violeta Migallón, José Penadés, and Daniel B. Szyld: Parallel, synchronous and asynchronous two-stage multisplitting methods |

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