Iteration

Iteration, meaning the repeated application of a process or function, appears in a surprisingly wide range of applications. Discrete dynamical systems, in which the time variable has been “quantized” into individual units (seconds, days, years, etc.) are modeled by iterative systems. Most numerical solution algorithms, for both linear and nonlinear systems, are based on iterative procedures. Starting with an initial guess, the successive iterates lead to closer and closer approximations to the true solution. For linear systems of equations, there are several iterative solution algorithms that can, in favorable situations, be employed as efficient alternatives to Gaussian Elimination. Iterative methods are particularly effective for solving the very large, sparse systems arising in the numerical solution of both ordinary and partial differential equations. All practical methods for computing eigenvalues and eigenvectors rely on some form of iteration. A detailed historical development of iterative methods for solving linear systems and eigenvalue problems can be found in the recent survey paper [84]. Probabilistic iterative models known as Markov chains govern basic stochastic processes and appear in genetics, population biology, scheduling, internet search, financial markets, and many more.