Abstract
We prove Kantorovich’s theorem on Newton’s method using a convergence analysis which makes clear, with respect to Newton’s method, the relationship of the majorant function and the non-linear operator under consideration. This approach enables us to drop out the assumption of existence of a second root for the majorant function, still guaranteeing Q-quadratic convergence rate and to obtain a new estimate of this rate based on a directional derivative of the derivative of the majorant function. Moreover, the majorant function does not have to be defined beyond its first root for obtaining convergence rate results.
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The research of O.P. Ferreira was supported in part by FUNAPE/UFG, CNPq Grant 475647/2006-8, CNPq Grant 302618/2005-8, PRONEX–Optimization(FAPERJ/CNPq) and IMPA.
The research of B.F. Svaiter was supported in part by CNPq Grant 301200/93-9(RN) and by PRONEX–Optimization(FAPERJ/CNPq).
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Ferreira, O.P., Svaiter, B.F. Kantorovich’s majorants principle for Newton’s method. Comput Optim Appl 42, 213–229 (2009). https://doi.org/10.1007/s10589-007-9082-4
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DOI: https://doi.org/10.1007/s10589-007-9082-4