• Journal of applied mathematics & informatics
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• 제6권2호
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• pp.393-406
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• 1999

Affine invariant sufficient conditions are given for two local convergence theorems involving inexact Newton-like methods. The first uses conditions on the first Frechet-derivative whereas the second theorem employs hypotheses on the second. Radius of con-vergence as well as rate of convergence results are derived. Results involving superlinear convergence and known to be true for inexact Newton methods are extended here. Moreover we show that under hypotheses on the second Frechet-derivation our radius of convergence results are derived. Results involving superlinear convergence and known to be true or inexact Newton methods are extended here. Moreover we show that under hypotheses on the second Frechet-derivative our radius of conver-gence is larger than the corresponding one in [10]. This allows a wider choice for the initial guess. A numerical example is also pro-vided to show that our radius of convergence is larger then the one in [10].

• Journal of applied mathematics & informatics
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• 제5권2호
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• pp.433-448
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• 1998

In this study we use inexact Newton-like methods to find solutions of nonlinear operator equations on Banach spaces with a convergence structure. Our technique involves the introduction of a generalized norm as an operator from a linear space into a par-tially ordered Banach space. In this way the metric properties of the examined problem can be analyzed more precisely. Moreover this approach allows us to derive from the same theorem on the one hand semi-local results of kantorovich-type and on the other hand 2global results based on monotonicity considerations. By imposing very general Lipschitz-like conditions on the operators involved on the other hand by choosing our operators appropriately we can find sharper error bounds on the distances involved than before. Furthermore we show that special cases of our results reduce to the corresponding ones already in the literature. Finally our results are used to solve integral equations that cannot be solved with existing methods.

• Journal of applied mathematics & informatics
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• 제8권2호
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• pp.345-360
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• 2001

Affine invariant sufficient conditions are given for two local convergence theorems involving inexact Newton-like methods. The first uses conditions on the first Frechet-derivative whereas the second theorem employs hypotheses on the mth(m≥2 an integer). Radius of convergence as well as rate of convergence results are derived. Results involving superlinear convergence and known to be true for inexact Newton methods are extended here. Moreover, we show that under hypotheses on the mth Frechet-derivative our radius of convergence can sometimes be larger than the corresponding one in [10]. This allows a wider choice for the initial guess. A numerical example is also provided to show that our radius of convergence is larger than the one in [10].

• Journal of applied mathematics & informatics
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• 제8권3호
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• pp.743-753
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• 2001

We provide local convergence results in affine form for inexact Newton-like as well as quasi-Newton iterative methods in a Banach space setting. We use hypotheses on the second or on the first and mth Frechet-derivative (m≥2 an integer) of the operator involved. Our results allow a wider choice of starting points since our radius of convergence can be larger than the corresponding one given in earlier results using hypotheses on the first-Frechet-derivative only. A numerical example is provided to illustrate this fact. Our results apply when the method is, for example, a difference Newton-like or update-Newton method. Furthermore, our results have direct applications to the solution of autonomous differential equations.

• Journal of applied mathematics & informatics
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• 제7권3호
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• pp.765-772
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• 2000

In this study we are concerned with the problem of approximating a solution of a nonlinear equation in Banach space using Newton-like methods. Due to rounding errors the sequence of iterates generated on a computer differs from the sequence produced in theory. Using Lipschitz-type hypotheses on the second Frechet-derivative instead of the first one, we provide sufficient convergence conditions for the inexact Newton-like method that is actually generated on the computer. Moreover, we show that the ratio of convergence improves under our conditions. Furthermore, we provide a wider choice of initial guesses than before. Finally, a numerical example is provided to show that our results compare favorably with earlier ones.

• Journal of applied mathematics & informatics
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• 제7권1호
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• pp.139-159
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• 2000

In this manuscript we study perturbed Newton-like methods for the solution of nonlinear operator equations in a Banach space and their discretized versions in connection with the mesh independence principle. This principle asserts that the behavior of the discretized process is asymptotically the same as that for the original iteration and consequently, the number of steps required by the two processes to converge to within a given tolerance is essentially the same. So far this result has been proved by others using Newton's method for certain classes of boundary value problems and even more generally by considering a Lipschitz uniform discretization. In some of our earlierpapers we extend these results to include Newton-like methods under more general conditions. However, all previous results assume that the iterates can be computed exactly. This is mot true in general. That in why we use perturbed Newton-like methods and even more general conditions. Our results, on the one hand, extend, and on the other hand, make more practical and applicable all previous results.