ON AN APPLICATION OF A VARIANT OF THE CLOSED GRAPH THEOREM AND THE SECANT METHOD

The method of nondiscrete mathematical induction is used to find error bounds for the Secant method. We assume only that the operator has Holder continuous derivatives. In the case the Frechet­ derivative of the operator satisfies a Lipschitz condition our results reduce to the ones obtained by F. Potra (Num. Math. 1982).

AN ERROR ANALYSIS OF STIRLING'S METHOD IN BANACH SPACES

The method of nondiscrete mathematical induction is applied to Stirling's method. The method yields a very simple proof of the convergence and error estimates which are generally better than those given in the literature.

On Newton's method and nondiscrete mathematical induction

The method of nondiscrete mathematical induction is used to find sharp error bounds for Newton's method. We assume only that the operator has Hölder continuous derivatives. In the case when the Fréchet-derivative of the operator satisfies a Lipschitz condition, our results reduce to the ones obtained by Ptak and Potra in 1972.