The principal goal in the exposition is to characterize the relationship between the dynamics of classical iterative methods and that of certain differential systems. The author reviews various kinds of numerical algorithms, especially those related to linear algebra problems, such as linear and nonlinear equations, matrix factorizations, and eigenvalue problems, and explore the possibility of recasting them as dynamical systems. A few of these ideas have already been reported in an earlier review by the author [SIAM Rev. 30, No. 3, 375‒387 (1988; Zbl 0657.65075)]. Various aspects of the recent development and application in this direction are discussed in the present paper. Several differential equations whose solutions evolve in submanifolds of matrices are cast in fairy frameworks. It is shown that, in some cases, there are remarkable connections between smooth flows and discrete numerical algorithms and, in other cases, the flow approaches seems advantageous in tracking very difficult open problems. The interplay between dynamical systems and computational methods is not only of theoretical interest, but also has important consequences.
Reviewer: Liu Xinguo (Qingdao)