Wyniki wyszukiwania

Wyszukiwano:
w słowach kluczowych: Fréchet derivative

Sortuj według:

Ogranicz wyniki do:

The effect of rounding errors on a certain class of iterative methods

100%

Argyros I. K.

Applicationes Mathematicae

2000

tom 27

nr 3

369-375

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 mth Fréchet derivative (m ≥ 2 an integer) 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.

Local convergence of inexact Newton methods under affine invariant conditions and hypotheses on the second Fréchet derivative

86%

Argyros I. K.

Applicationes Mathematicae

1999

tom 26

nr 4

457-465

We use inexact Newton iterates to approximate a solution of a nonlinear equation in a Banach space. Solving a nonlinear equation using Newton iterates at each stage is very expensive in general. That is why we consider inexact Newton methods, where the Newton equations are solved only approximately, and in some unspecified manner. In earlier works [2], [3], natural assumptions under which the forcing sequences are uniformly less than one were given based on the second Fréchet derivative of the operator involved. This approach showed that the upper error bounds on the distances involved are smaller compared with the corresponding ones using hypotheses on the first Fréchet derivative. However, the conditions on the forcing sequences were not given in affine invariant form. The advantages of using conditions given in affine invariant form were explained in [3], [10]. Here we reproduce all the results obtained in [3] but using affine invariant conditions.

Ograniczanie wyników

2 Applicationes Mathematicae

2 Argyros I. K.

1 2000

1 1999

Wyniki wyszukiwania

The effect of rounding errors on a certain class of iterative methods

Local convergence of inexact Newton methods under affine invariant conditions and hypotheses on the second Fréchet derivative