This self-contained treatment offers a contemporary and systematic development of the theory and application of Newton methods, which are undoubtedly the most effective tools for solving equations appearing in computational sciences. Its focal point resides in an exhaustive analysis of the convergence properties of several Newton variants used in connection to specific real life problems originated from astrophysics, engineering, mathematical economics and other applied areas. What distinguishes this book from others is the fact that the weak convergence conditions inaugurated here allow for a wider applicability of Newton methods; finer error bounds on the distances involved, and a more precise information on the location of the solution. These factors make this book ideal for researchers, practitioners and students.
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approximating a locally approximating a solution Assume there exist Banach Lemma Banach space bounded linear operator Cauchy sequence closed set continuous contraction mapping convergence radius D C X define function Denote divided difference equation F equation F(x error bounds hold F(xk F(xn fc+i fixed point following error bounds Frechet Frechet-differentiable operator defined function F Furthermore given Hence hypotheses of Theorem induction integer invertible operators iteration tn Kantorovich Kantorovich theorem Lemma on invertible Let F Lipschitz continuous locally unique solution Moreover the following Newton-like method Newton's method non-decreasing non-negative parameters open convex subset operator F positive zero problem of approximating relaxed monotone Remark replaced satisfying Secant method semilocal convergence theorem sequence xn simple zero sn+1 solution of equation solution x tk+1 tk+2 tk+i tn+2 tn+i x0 G D xfc+i xn+1 xn+i xoll