On the Geometry of Halley's Method

According to Traub [Tra64], Halley's iteration function (I.F.) "must share with the secant I.F. the distinction of being the most frequently rediscovered I.F. in the literature." Halley's method is a close relative of Newton's method, an iterative technique depicted as a sequence of tangent lines with zeros converging to a root of a function. The usual derivation of Halley's method, however, lacks any obvious geometric interpretation. We present a derivation of Halley's method having such an interpretation, and give a brief history of Halley's work and the method that bears his name.

[1]  E. Kobald.  Notiz, betreffend die Berechnung der Wurzeln numerischer Gleichungen , 1891 .

[2]  H. Bateman,et al.  Halley's Methods for Solving Equations , 1938 .

[3]  E. Bodewig,et al.  On types of convergence and on the behavior of approximations in the neighborhood of a multiple root of an equation , 1949 .

[4]  H. J. Hamilton.  A Type of Variation on Newton's Method , 1950 .

[5]  J. K. Stewart.  Another Variation of Newton's Method , 1951 .

[6]  J. S. Frame.  The Solution of Equations by Continued Fractions , 1953 .

[7]  I. Kiss,et al.  Über eine Verallgemeinerung des Newtonschen Näherungsverfahrens , 1954 .

[8]  R. W. Snyder.  One More Correction Formula , 1955 .

[9]  Carl B. Boyer,et al.  A History of Mathematics. , 1993 .

[10]  Joseph F. Traub,et al.  On a class of iteration formulas and some historical notes , 1961, CACM.

[11]  J. F. Traub,et al.  Comparison of iterative methods for the calculation of nth roots , 1961 .

[12]  Thomas Richard McCalla,et al.  Introduction to Numerical Methods and Fortran Programming , 1967 .

[13]  V. Shamanskii.  A modification of Newton's method , 1967 .

[14]  A. Ostrowski.  Solution of equations and systems of equations , 1967 .

[15]  A. Ostrowski.  Solution of equations in Euclidean and Banach spaces , 1973 .

[16]  B. Dawson,et al.  On the global convergence of Halley's iteration formula , 1975 .

[17]  E. Hansen,et al.  A family of root finding methods , 1976 .

[18]  George H. Brown,et al.  On Halley's Variation of Newton's Method , 1977 .

[19]  D. B. Popovski,et al.  A family of one-point iteration formulae for finding roots , 1980 .

[20]  Götz Alefeld,et al.  On the Convergence of Halley's Method , 1981 .

[21]  J. Traub.  Iterative Methods for the Solution of Equations , 1982 .

[22]  W. Gander.  On Halley's Iteration Method , 1985 .

[23]  L. Feigenbaum,et al.  Brook Taylor and the method of increments , 1985 .

[24]  D. F. Bailey,et al.  A Historical Survey of Solution by Functional Iteration , 1989 .

[25]  S. Gordon,et al.  A parabolic extension of Newton's method , 1990 .

[26]  Nick Kollerstrom.  Thomas Simpson and ‘Newton's method of approximation’: an enduring myth , 1992 .

[27]  Tjalling J. Ypma,et al.  Historical Development of the Newton-Raphson Method , 1995, SIAM Rev..

[28]  Ioannis K. Argyros,et al.  On the method of tangent hyperbolas , 1996, Approximation Theory and its Applications.