Chebyshev method

A method for obtaining a class of iteration algorithms (cf. Iteration algorithm) for finding a simple real root of an equation

(1)

where is a sufficiently smooth function.

The basis of the method lies in the formal representation of the inverse function of via the Taylor formula. If is a sufficiently close approximation to a root of equation (1), if and if , then

(2)

where the coefficients are defined recursively from the identity using the Taylor coefficients of the function . Putting in (2), one obtains the relation

(3)

Taking a certain number of terms on the right-hand side of (3) gives a formula for the iteration algorithm; with two terms, for example, one obtains Newton's method, while with three terms one obtains an iteration method of the form

(4)

The rate of convergence of the to increases with the number of terms of (3) taken into consideration (cf. [2]). The method can be extended to functional equations (cf. [3]).

References

[1a]	P.L. Chebyshev, , Collected works , 5 , Moscow-Leningrad (1951) pp. 7–25 (In Russian)
[1b]	P.L. Chebyshev, , Collected works , 5 , Moscow-Leningrad (1951) pp. 173–176 (In Russian)
[2]	I.S. Berezin, N.P. Zhidkov, "Computing methods" , Pergamon (1973) (Translated from Russian)
[3]	M.I. Nechepurenko, Uspekhi Mat. Nauk , 9 : 2 (1954) pp. 163–170

Comments

This method is also called Chebyshev's root-finding method. A related approach is based on inverse interpolation, cf. [a1].

References

[a1]	A. Ralston, "A first course in numerical analysis" , McGraw-Hill (1965)
[a2]	P.J. Davis, "Interpolation and approximation" , Dover, reprint (1975) pp. 108–126
[a3]	F.B. Hildebrand, "Introduction to numerical analysis" , McGraw-Hill (1974)