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Maehly's Procedure

A method for finding roots which defines

P_j(x)=(P(x))/((x-x_1)...(x-x_j)),
(1)

so the derivative is

P_j^'(x)=(P^'(x))/((x-x_1)...(x-x_j))-(P(x))/((x-x_1)...(x-x_j))sum_(i=1)^j(x-x_i)^(-1).
(2)

One step of Newton's method can then be written as

x_(k+1)=x_k-(P(x_k))/(P^'(x_k)-P(x_k)sum_(i=1)^(j)(x_k-x_i)^(-1)).
(3)

See also

Muller's Method

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References

Bauer, F. L. and Stoer, J. "Algorithm 105: Newton Maehly." J. CACM 5, 387-388, 1962.Maehly, H. J. "Zur iterativen Auflösung algebraischer Gleichungen." Z. Angew. Math. Mech. 5, 260-263, 1954.Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 371-372, 1992.

Referenced on Wolfram|Alpha

Maehly's Procedure

Cite this as:

Weisstein, Eric W. "Maehly's Procedure." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MaehlysProcedure.html

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