This book shows the importance of studying semilocal convergence in iterative methods through
Newton's method and addresses the most important aspects of the Kantorovich's theory including
implicated studies. Kantorovich's theory for Newton's method used techniques of functional
analysis to prove the semilocal convergence of the method by means of the well-known majorant
principle. To gain a deeper understanding of these techniques the authors return to the
beginning and present a deep-detailed approach of Kantorovich's theory for Newton's method
where they include old results for a historical perspective and for comparisons with new
results refine old results and prove their most relevant results where alternative
approaches leading to new sufficient semilocal convergence criteria for Newton's method are
given. The book contains many numerical examples involving nonlinear integral equations two
boundary value problems and systems of nonlinear equations related to numerous physical
phenomena. The book is addressed to researchers in computational sciences in general and in
approximation of solutions of nonlinear problems in particular.
