From the author’s abstract: We introduce two new methods for solving large sparse nonsymmetric linear systems with several right-hand sides. These methods are the global Hessenberg and global CMRH methods. Using the global Hessenberg process, these methods are less expensive than the global FOM and global GMRES methods presented by K. Ibilou, A. Messaoudi, and H. Sadok [Appl. Numer. Math. 31, No. 1, 49-63 (1999; Zbl 0935.65024)]. Theoretical results about the new methods are given, and experimental results that show good performances of these new methods are presented.