Schröder's Method

Two families of equations used to find roots of nonlinear functions of a single variable. The ``B'' family is more robust and can be used in the neighborhood of degenerate multiple roots while still providing a guaranteed convergence rate. Almost all other root-finding methods can be considered as special cases of Schröder's method. Householder humorously claimed that papers on root-finding could be evaluated quickly by looking for a citation of Schröder's paper; if the reference were missing, the paper probably consisted of a rediscovery of a result due to Schröder (Stewart 1993).

One version of the ``A'' method is obtained by applying Newton's Method to $f/f'$,

$$x_{n+1}=x_n-{f(x_n)f'(x_n)\over [f'(x_n)]^2-f(x_n)f''(x_n)}$$

(Scavo and Thoo 1995).

See also Newton's Method

References

Householder, A. S. The Numerical Treatment of a Single Nonlinear Equation. New York: McGraw-Hill, 1970.

Scavo, T. R. and Thoo, J. B. ``On the Geometry of Halley's Method.'' Amer. Math. Monthly 102, 417-426, 1995.

Schröder, E. ``Über unendlich viele Algorithmen zur Auflösung der Gleichungen.'' Math. Ann. 2, 317-365, 1870.

Stewart, G. W. ``On Infinitely Many Algorithms for Solving Equations.'' English translation of Schröder's original paper. College Park, MD: University of Maryland, Institute for Advanced Computer Studies, Department of Computer Science, 1993. ftp://thales.cs.umd.edu/pub/reports/imase.ps.