Halley's method

Abstract: Newton’s method and its modifications have long played a central role in finding solutions of non-linear equations and systems. The work of Kantorovich has been seminal in extending and codifying Newton’s method. Kantorovich’s approach, which dominates the literature in this area, has these features: (a) weak differentiability hypotheses are made on the system, e.g., the map is C 2 on some domain in a Banach space; (b) derivative bounds are supposed to exist over the whole of this domain. In contrast, here strong hypotheses on differentiability are made; analyticity is assumed. On the other hand, we deduce consequences from data at a single point. This point of view has valuable features for computation and its theory. Theorems similar to ours could probably be deduced with the Kantorovich theory as a starting point; however, we have found it useful to start afresh.

Abstract: A one parameter family of iteration functions for finding roots is derived. The family includes the Laguerre, Halley, Ostrowski and Euler methods and, as a limiting case, Newton's method. All the methods of the family are cubically convergent for a simple root (except Newton's which is quadratically convergent). The superior behavior of Laguerre's method, when starting from a pointz for which |z| is large, is explained. It is shown that other methods of the family are superior if |z| is not large. It is also shown that a continuum of methods for the family exhibit global and monotonic convergence to roots of polynomials (and certain other functions) if all the roots are real.

Abstract: A modified Newton method for polynomials is discussed. It is assumed one has approximations for all the roots of the polynomial. Three variations are described. If the roots are simple, it is shown that under appropriate conditions, two of the variations are cubically convergent, 1. ! ,l t r o d u e t io, l,et t:.here be given film polynomial J'(:C) = aMt(' @ a,:.-.tX\"' @ \"'\" + aix-I-ao (1) ,i i,! where we assume the roots are distinct. Assume X~0), ... , ,~,, tare n distinct guesses for the roots of f(x).

Abstract: (1985). On Halley's Iteration Method. The American Mathematical Monthly: Vol. 92, No. 2, pp. 131-134.