Eine stets quadratisch konvergente Modifikation des Steffensen-Verfahrens |
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| Authors: | Prof Dr H Esser |
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| Affiliation: | 1. Institut für Geometrie und Praktische Mathematik, RWTH Aachen, Templergraben 55, D-5100, Aachen, Bundesrepublik Deutschland
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| Abstract: | It is shown that the following modification of the Steffensen procedurex n+1=x n ?k s (x n )f(x n ) (fx n ,x n ?f(x n )])?1 (n=0,1,...) withk s (x)=(1?z s (x))?1,z s (x)=f(x) 2fx?f(x),x,x+f(x)]×(fx,x?f(x)])?2 is quadratically convergent to the root of the equation \(f(x) = (x - \bar x)^p g(x) = 0(p > 0,g(\bar x) \ne 0)\) . Furthermore \(\mathop {\lim }\limits_{n \to \infty } k_s (x_n ) = p\) holds. |
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