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Third order iterative methods without using second Frechet derivative

Summary: Third order iterative methods without using
second Fr´echet derivative
Great quantity of general problems may be reduced to Ţnding zeros. The roots
of a nonlinear equation cannot in general be expressed in closed form. Thus, in
order to solve nonlinear equations, we have to use approximate methods. One
of the most important techniques to study these equations is the use of itera-
tive processes, starting from an initial approximation x0, called pivot, successive
approaches (until some predetermined convergence criterion is satisŢed) xi are
computed , i = 1, 2, . . . , with the help of certain iteration function : X X,
xi+1 := (xi), i = 0, 1, 2 . . . (1)
Certainly Newton's method is the most useful iteration for this purpose. The
advance of computational techniques has allowed the development of some more
complicated iterative methods in order to obtain greater order of convergence as
Chebyshev and Halley methods. In these methods we have to evaluate Ţrst and
overall second derivatives. These difficulties are usually harder than the advantage
because of the order of these methods. So, two order iterative methods are widely
In this paper, we present a modiŢcation of classical third order iterative me-
thods. The main advantage of these methods is they do not need evaluate any
second derivative, but having the same properties of convergence than the classical


Source: Amat, Sergio - Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena


Collections: Mathematics