This book introduces advanced numerical-functional analysis to beginning computer science researchers. The reader is assumed to have had basic courses in numerical analysis, computer programming, computational linear algebra, and an introduction to real, complex, and functional analysis. Although the book is of a theoretical nature, each chapter contains several new theoretical results and important applications in engineering, in dynamic economics systems, in input-output system, in the solution of nonlinear and linear differential equations, and optimization problem.
INTRODUCTION NEWTON'S METHOD Convergence under Frechet differentiability. Convergence under twice Frechet differentiability. Newton's method on unbounded domains. Continuous analog of Newton's method. Interior point techniques. Regular smoothness. Ie-convergence. Semilocal convergence and convex majorants. Local convergence and convex majorants. Majorizing sequences. SECANT METHOD Convergence. Least squares problems. Nondiscrete induction and Secant method. Nondiscrete induction and a double step Secant method. Directional Secant Methods. Efficient three step Secant methods. STEFFENSEN'S METHOD Convergence GAUSS-NEWTON METHOD Convergence. Average-Lipschitz conditions. NEWTON-TYPE METHODS Convergence with outer inverses. Convergence of a Moser-type Method. Convergence with slantly differentiable operator. A intermediate Newton method. INEXACT METHODS Residual control conditions. Average Lipschitz conditions. Two-step methods. Zabrejko-Zincenko-type conditions. WERNER'S METHOD Convergence analysis HALLEY'S METHOD Local convergence METHODS FOR VARIATIONAL INEQUALITIES Subquadratic convergent method. Convergence under slant condition. Newton-Josephy method. FAST TWO-STEP METHODS Semilocal convergence FIXED POINT METHODS Successive substitutions methods