Abstract
In this paper we describe a new technique for generating iteration formulas — of arbitrary order — for determining a zero (assumed simple) of a functionf, assumed analytic in a region containing the zero. The 1/p Padé Approximant (p≧0) to the functiong(t)≡f(z) is formed wherez=w+t, using the Taylor series forf at the pointw, an approxination to the zero off. The value oft for which the 1/p Padé Approximant vanishes provides the basis of iteration formulas of orderp+2.
Some known iteration formulas, e.g., Newton-Raphson's, Halley's and Kiss's of order of convergence two, three and four, are directly obtained by settingp=0,1 and 2, respectively.
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References
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Nourein, AW.M. Root determination by use of Padé approximants. BIT 16, 291–297 (1976). https://doi.org/10.1007/BF01932271
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DOI: https://doi.org/10.1007/BF01932271