• East Asian mathematical journal
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• v.25 no.2
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• pp.147-151
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• 2009

We use an improved Newton-Kantorovich theorem introduced in [2] to analyze interior point methods. Our approach requires less number of steps than before [5] to achieve a certain error tolerance for both Newton's and Modified Newton's methods.

• East Asian mathematical journal
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• v.24 no.3
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• pp.289-293
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• 2008

We recently showed in [5] a semilocal convergence theorem that guarantees convergence of Newton's method to a locally unique solution of a nonlinear equation under hypotheses weaker than those of the Newton-Kantorovich theorem [7]. Here, we first weaken Miranda's theorem [1], [9], [10], which is a generalization of the intermediate value theorem. Then, we show that operators satisfying the weakened Newton-Kantorovich conditions satisfy those of the weakened Miranda’s theorem.

• Journal of applied mathematics & informatics
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• v.16 no.1_2
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• pp.1-17
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• 2004

The famous Newton-Kantorovich hypothesis has been used for a long time as a sufficient condition for the convergence of Newton method to a solution of an equation in connection with the Lipschitz continuity of the Frechet-derivative of the operator involved. Using Lipschitz and center-Lipschitz conditions we show that the Newton-Kantorovich hypothesis is weakened. The error bounds obtained under our semilocal convergence result are finer and the information on the location of the solution more precise than the corresponding ones given by the dominating Newton-Kantorovich theorem, and under the same hypotheses/computational cost, since the evaluation of the Lipschitz also requires the evaluation of the center-Lipschitz constant. In the case of local convergence we obtain a larger convergence radius than before. This observation is important in computational mathematics and can be used in connection to projection methods and in the construction of optimum mesh independence refinement strategies.

• Journal of the Korean Mathematical Society
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• v.33 no.4
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• pp.865-878
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• 1996

We consider numerical methods for finding a solution to a nonlinear system of algebraic equations $$ (1) f(x) = 0, $$ where the function $f : R^n \to R^n$ is ain $x \in R^n$. In [10], a quasi-Gauss-Newton method is proposed and shown the computational efficiency over SQRT algorithm by numerical experiments. The convergence rate of the method has not been proved theoretically. In this paper, we show theoretically that the iterate $x_k$ obtained from the quasi-Gauss-Newton method for the problem (1) actually converges to a root by Kantorovich-type convergence analysis. We also show the rate of convergence of the method is superlinear.

• Journal of applied mathematics & informatics
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• v.6 no.2
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• pp.369-378
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• 1999

In This study we examine the applicability of Newton's method and the modified Newton's method for a, pp.oximating a lo-cally unique solution of a nonlinear equation in a Banach space. We assume that the newton-Kantorovich hypothesis for Newton's method is violated but the corresponding condition for the modified Newton method holds. Under these conditions there is no guaran-tee that Newton's method starting from the same initial guess as the modified Newton's method converges. Hence it seems that we must always use the modified Newton method under these condi-tions. However we provide a numerical example to demonstrate that in practice this may not be a good decision.

• Journal of the Korean Mathematical Society
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• v.51 no.1
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• pp.137-162
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• 2014

In this work we study the local and semilocal convergence of the relaxed Newton's method, that is Newton's method with a relaxation parameter 0 < ${\lambda}$ < 2. We give a Kantorovich-like theorem that can be applied for operators defined between two Banach spaces. In fact, we obtain the recurrent sequence that majorizes the one given by the method and we characterize its convergence by a result that involves the relaxation parameter ${\lambda}$. We use a new technique that allows us on the one hand to generalize and on the other hand to extend the applicability of the result given initially by Kantorovich for ${\lambda}=1$.

• The Pure and Applied Mathematics
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• v.15 no.4
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• pp.365-375
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• 2008

We present a new theorem for the semilocal convergence of Newton's method to a locally unique solution of an equation in a Banach space setting. Under a gamma-type condition we show that we can extend the applicability of Newton's method given in [12]. We also provide a comparative study between results using the classical Newton-Kantorovich conditions ([6], [7], [10]), and the ones using the gamma-type conditions ([12], [13]). Numerical examples are also provided.

• East Asian mathematical journal
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• v.27 no.5
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• pp.521-525
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• 2011

We use a weaker version of the celebrated Newton-Kantorovich theorem [3] reported by us in [1] to find solutions of discrete dynamical systems involving operators with chaotic behavior. Our results are obtained by extending the application of the shadowing lemma [4], and are given under the same computational cost as before [4]-[6].

• The Pure and Applied Mathematics
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• v.14 no.4
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• pp.279-282
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• 2007

Using the center instead of the Lipschitz condition we show how to provide weaker sufficient convergence conditions of the modified Newton Kantorovich method for the solution of nonlinear singular integral equations with Curleman shift (NLSIES). Finer error bounds on the distances involved and a more precise information on the location of the solution are also obtained and under the same computational cost than in [1].

• East Asian mathematical journal
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• v.23 no.2
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• pp.257-260
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• 2007

Using the center instead of the Lipschitz condition we show how to provide weaker sufficient convergence conditions of the modified Newton Kantorovich method for the solution of nonlinear singular integral equations with Curleman shift (NLSIES). Finer error bounds on the distances involved and a more precise information on the location of the solution are also obtained and under the same computational cost than in [1].