GMRES algorithms over 35 years.

• Deep insight on both sequential and parallel GMRES for linear systems. • Rich discussion on the convergence and acceleration of GMRES. • Brief account of other problems and block algorithms. This paper is about GMRES algorithms for the solution of nonsingular linear systems. We first consider basic algorithms and study their convergence. We then focus on acceleration strategies and parallel algorithms that are useful for solving challenging systems. We also briefly discuss other problems, such as systems with multiple right-hand sides, shifted systems, and singular systems. [ABSTRACT FROM AUTHOR]