Complexity and real computation
Complexity and real computation
Convergence of Newton's method and inverse function theorem in Banach space
Mathematics of Computation
Convergence behaviour of inexact Newton methods
Mathematics of Computation
Kantorovich's theorem on Newton's method in Riemannian Manifolds
Journal of Complexity
The Kantorovich Theorem and interior point methods
Mathematical Programming: Series A and B
Newton's method for sections on Riemannian manifolds: Generalized covariant α-theory
Journal of Complexity
A Unifying Local Convergence Result for Newton's Method in Riemannian Manifolds
Foundations of Computational Mathematics
Kantorovich's majorants principle for Newton's method
Computational Optimization and Applications
Smale's point estimate theory for Newton's method on Lie groups
Journal of Complexity
Kantorovich-type convergence criterion for inexact Newton methods
Applied Numerical Mathematics
Convergence behaviour of inexact Newton methods under weak Lipschitz condition
Journal of Computational and Applied Mathematics
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We prove that under semi-local assumptions, the inexact Newton method with a fixed relative residual error tolerance converges Q-linearly to a zero of the nonlinear operator under consideration. Using this result we show that the Newton method for minimizing a self-concordant function or to find a zero of an analytic function can be implemented with a fixed relative residual error tolerance. In the absence of errors, our analysis retrieve the classical Kantorovich Theorem on the Newton method.