A new n-dimensional Newton method is presented. In each step a whole n-dimensional ball is determined rather than a single new approximation point. This ball contains the desired zero of the given function. The method is globally convergent. If the given initial ball does not contain any zero, then the method stops after a finite number of steps. Depending upon the assumptions which are made, the convergence of the ball radii is linear, superlinear or quadratic.
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Karl Nickel (1981) studied this question.
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