Summary
The slow convergence ofNewton's method for the iterative approximation of the roots of an algebraic equation may be improved by using a formula due toLaguerre. It is shown that this formula together with certain generalizations are useful for the numerical calculation of both real and complex zeros. A method is also given for the calculation of a further root of an equation after several roots have already been found, such that errors on the determination of the previous roots will not affect the accuracy with which the present one may be approximated.
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Literatur
Vergleiche hierzu zum Beispiel:E. F. Moore Math. Tables & Aids to Comp.3, 486 (1949).
Diese beiden Formeln finden sich schon beiE. Schröder, Math. Ann.2, 317 (1870).
Vgl.E. Bodewig, Proc. Acad. Sci. Amsterdam49, 911 (1946), undJ. G. Van der Corput Proc. Acad. Sci. Amsterdam49, 922 (1946).
E. N. Laguerre, Nouv. Ann. Math. [2es.]19 (1880), oderŒuvres de Laguerre, Bd. I, S. 87 bis 103.
Vgl.E. Bodewig undJ. G. Van der Corput, a. a. O.
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Maehly, H.J. Zur iterativen Auflösung algebraischer Gleichungen. Journal of Applied Mathematics and Physics (ZAMP) 5, 260–263 (1954). https://doi.org/10.1007/BF01600333
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DOI: https://doi.org/10.1007/BF01600333