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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))/([f^'(x_n)]^2-f(x_n)f^('')(x_n))

(Scavo and Thoo 1995).


See also

Newton's Method

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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. http://citeseer.nj.nec.com/93609.html.

Referenced on Wolfram|Alpha

Schröder's Method

Cite this as:

Weisstein, Eric W. "Schröder's Method." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SchroedersMethod.html

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