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.
Convergence under Fréchet differentiability. Convergence under twice Fréchet differentiability. Newton’s method on unbounded domains. Continuous analog of Newton’s method. Interior point techniques. Regular smoothness. ω-convergence. Semilocal convergence and convex majorants. Local convergence and convex majorants. Majorizing sequences.
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.
Convergence. Average-Lipschitz conditions.
Convergence with outer inverses. Convergence of a Moser-type Method. Convergence with slantly differentiable operator. A intermediate Newton method.
Residual control conditions. Average Lipschitz conditions. Two-step methods. Zabrejko-Zincenko-type conditions.
METHODS FOR VARIATIONAL INEQUALITIES
Subquadratic convergent method. Convergence under slant condition. Newton-Josephy method.
FAST TWO-STEP METHODS
FIXED POINT METHODS
Successive substitutions methods