• 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.

• 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.

• 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].

• The Pure and Applied Mathematics
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• v.15 no.1
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• pp.35-45
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• 2008

In cases sufficient conditions for the semilocal convergence of Newtonlike methods are violated, we start with a modified Newton-like method (whose weaker convergence conditions hold) until we stop at a certain finite step. Then using as a starting guess the point found above we show convergence of the Newtonlike method to a locally unique solution of a nonlinear operator equation in a Banach space setting. A numerical example is also provided.

• The Pure and Applied Mathematics
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• v.16 no.4
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• pp.405-416
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• 2009

In this study we are concerned with the problem of approximating a locally unique solution of an equation in a Banach space setting using Newton's and modified Newton's methods. We provide weaker convergence conditions for both methods than before [5]-[7]. Then, we combine Newton's with the modified Newton's method to approximate locally unique solutions of operator equations. Finer error estimates, a larger convergence domain, and a more precise information on the location of the solution are obtained under the same or weaker hypotheses than before [5]-[7]. The results obtained here improve our earlier ones reported in [4]. Numerical examples are also provided.