Some Stability Theorems for Ordinary Difference Equations

LaSalle [5], [6], [7] and others have developed a generalization of the “second method” of Liapunov which utilizes certain invariance properties of solutions of ordinary differential equations. Invariance properties of solutions of ordinary difference equations are utilized here to develop stability theorems similar to those in LaSalle [5]. As illustrations of the application of these theorems, a region of convergence is derived for the Newton-Raphson and secant iteration methods. A modification of one of these theorems is given and applied to study the effect of roundoff errors in the Newton-Raphson and Gauss-Seidel iteration methods.