Iterative Processes and Point Estimation Theory

This chapter is devoted to estimations of zero-finding methods from data at one point in the light of Smale’s point estimation theory [165]. One of the crucial problems in solving equations of the form f(z) = 0 is the construction of such initial conditions, which provide the guaranteed convergence of the considered numerical algorithm. These initial conditions involve an initial approximation z⁽⁰⁾ to a zero ζ of f and they should be established in such a way that the sequence {z^(m)} _m =1,2,… of approximations, generated by the implemented algorithm which starts from z⁽⁰⁾, tends to the zero of f. The construction of initial conditions and the choice of initial approximations ensuring the guaranteed convergence are very difficult problems that cannot be solved in a satisfactory way in general, even in the case of simple functions, such as algebraic polynomials.

In Sect. 2.1, we present some historical data and Smale's point estimation theory applied to Newton's method. More generalized problems are discussed in Sect. 2.2, where the whole class of quadratically convergent methods is treated, and in Sect. 2.3, where Smale's work is applied to the third-order methods. Improvements of Smale's result related to Newton's method, carried out by X. Wang and Han [181] and D. Wang and Zhao [178], are the subject of Sect. 2.4. Their approach is applied to the convergence analysis of the Durand–Kerner's method for the simultaneous determination of all zeros of a polynomial (Sect. 2.5).