  For the solution of full-rank ill-posed linear systems a new approach based
on the Arnoldi algorithm is presented. Working with regularized systems, the
method theoretically reconstructs the true solution by means of the computation
of a suitable function of matrix. In this sense the method can be referred to
as an iterative refinement process. Numerical experiments arising from integral
equations and interpolation theory are presented. Finally, the method is
extended to work in connection with the standard Tikhonov regularization with a
right hand side contaminated by noise.
