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Multidimensional rates of convergence
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applications Banach algebra Banach space bounded linear operators closed graph theorem complete metric space consistent approximation constant convergence of type convergent sequences corresponding estimate function Definition Let divided difference eigenvalue element equation f(x error estimates exact R-order exists f is Frechet formula Frechet derivative Frechet differentiable function f given Hence Hilbert space implies induction theorem inequality invertible iterative algorithm iterative procedure Lemma Let Let us denote Let us prove Lipschitz continuous mapping mathematical induction method of nondiscrete metric space X,d natural number Newton's method non-negative nondiscrete mathematical induction nonvoid norm numbers satisfying observe obtain positive functional positive numbers Proposition 1.9 Proposition Let Ptak Q-order of convergence rate of convergence real numbers relation result s(rQ secant method sequence xn)n>n star-algebra subset superlinearly Suppose Theorem Let type p,m x e Z(r xn+1