Appl Math Optim 41:227–236 (2000)
2000 Springer-Verlag New York Inc.
Recurrence Relations for Chebyshev-Type Methods
J. A. Ezquerro and M. A. Hern´andez
Department of Mathematics and Computation, University of La Rioja,
C/ Luis de Ulloa s/n, 26004 Logro˜no, Spain
Abstract. The convergence of new second-order iterative methods are analyzed in
Banach spaces by introducing a system of recurrence relations. A system of a priori
error bounds for that method is also provided. The methods are deﬁned by using
a constant bilinear operator A, instead of the second Fr´echet derivative appearing
in the deﬁning formula of the Chebyshev method. Numerical evidence that the
methods introduced here accelerate the classical Newton iteration for a suitable A
Key Words. Nonlinear equations in Banach spaces, Second-order methods, New-
ton’s method, A priori error bounds.
AMS Classiﬁcation. 47H17, 65J15.
Let X and Y be Banach spaces and consider a nonlinear operator F: ⊆ X → Y, which
is Fr´echet differentiable on an open convex domain . Assume that F
∈ L(Y, X)
exists at some x
∈ , where L(Y, X) is the set of bounded linear operators from Y into
X. The most famous iterative process to approximate a solution x
∈ of an equation
F(x) = 0 (1)
is the Newton method:
), n ≥ 0. (2)
The research reported herein was sponsored in part by the University of La Rioja (Grants API-98/A25
and API-98/B12) and the DGES (Grant PB96-0120-C03-02).