Control-Theoretic Forward Error Analysis of Iterative Numerical Algorithms

It has been known for at least five decades that control theory can be used to study iterative algorithms. However, little work can be found in the control systems literature on numerical algorithms, especially on the study of finite precision effects. In this technical note, we consider numerical iterative algorithms in finite precision as dynamical systems and study the effects of finite precision using control theory. By using the control tools of input-to-state stability and results from the study of quantization in control systems, we present new systematic ways to find bounds on the forward error for iterative algorithms. The advantages of the proposed schemes are shown by applying them to find bounds for the classical iterative methods for solving a system of linear equations.