In recent years, we have focused on the restart of the GMRES(m) method, and proposed an extension of the GMRES(m) method based on the error equations. In this paper, from the analysis on the residual polynomial of the extension of the GMRES(m) method, we propose a Look-Back GMRES(m) method under a Look-Back strategy which reconstructs the past residual polynomials of the GMRES(m) method. Some numerical experiments indicate that the Look-Back GMRES(m) method has a higher performance for efficient convergence than the GMRES(m) method.
Some protocols with secret information guarantee the secrecy of information by using probabilistic variables. The purpose of this paper aims at proving the secrecy of such protocols in an axiomatic proof system. This paper proposes such axiomatic proof system which deals with probabilistic variables. This paper picks up a simple secret sharing scheme and dining cryptographer protocol as examples, and show how the proof system proves the secrecies for these protocols.