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Rutishauser's Rule


Let m and m+h be two consecutive critical indices of f and let F be (m+h)-normal. If the polynomials p^~_k^((n)) are defined by

p^~_0^((n))(u)=1
(1)
p^~_(k+1)^((n))(u)=up^~_k^((n+1))(u)-q_(m+k+1)^((n))p^~_k^((n))(u)
(2)

for n=0, 1, ... and k=0, ..., h-1, then, under the hypothesis below, there exists an infinite set N of positive integers such that

 lim_(n->infty; n in N)p^~_h^((n))(u)=p^~_h(u),
(3)

where

 p^~_h(u)=(u-u_(m+1))(u-u_(m+2))...(u-u_(m+h)).
(4)

By hypothesis, if m=0, the polynomials p^~_k^((n)) are identical to the Hadamard polynomials p_k^((n)), and if m>0, the algorithm for constructing the p^~_k^((n)) is applied to the qd scheme suitably bounded by columns e_m^((n)) and e_(m+h)^((n)) (Henrici 1988, pp. 642-643).


See also

Critical Index

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References

Henrici, P. Applied and Computational Complex Analysis, Vol. 1: Power Series-Integration-Conformal Mapping-Location of Zeros. New York: Wiley, pp. 642-643, 1988.

Cite this as:

Weisstein, Eric W. "Rutishauser's Rule." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RutishausersRule.html