Global convergence of shortest-residual family of conjugate gradient methods without line search

The shortest-residual family of conjugate gradient methods was first proposed by Hestenes and was studied by Pytlak, and Dai and Yuan. Recently, a no-line-search scheme in conjugate gradient methods was given by Sun and Zhang, and Chen and Sun. In this paper, we show the global convergence of two shortest-residual conjugate gradient methods (FRSR and PRPSR) without line search. In addition, computational results are presented to show that the methods with line search have similar numerical behavior to the methods without line search.