Sur la convergence d'une classe de méthodes itératives de J.F. Traub

On the convergence of a class of iterative methods by J.F. Traub

Authors

  • I. Păvăloiu Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Romania

DOI:

https://doi.org/10.33993/jnaat21-15
Abstract views: 282

Abstract

Let \(X\) be a Banach space, \(Y\) a normed space and \(P:X\rightarrow Y\) a nonlinear operator. We study the convergence of the following method for solving the equation \(P\left( x\right) =0\)  \[ x_{n+1}=Q\left( x_{n}\right) -\left[ P^{\prime}\left( x_{n}\right) \right] ^{-1}P\left( Q(x_n)\right),\ n=0,1,..., \ x_{0}\in X \] where \(Q\) is a nonlinear operator associated to the nonlinear equation \(P\left( x\right) =0\). We show that if the successive approximations of \(Q\) converge with order \(k\geq2\), there the above sequence converge to the solution with order \(k+1\).

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References

Păvăloiu, I., Interpolation dans des espaces linéaires normés et applications. (French) Mathematica (Cluj) 12(35) (1970), 149-158, MR0299983.

Păvăloiu, I., Sur les procédés itératifs à un ordre élevé de convergence. (French) Mathematica (Cluj) 12(35) (1970), 309-324, MR0339486.

Păvăloiu, I., On iterative operators. (Romanian) Stud. Cerc. Mat. 23 (1971), 1537-1544, MR0341859.

Traub, J. F., Iterative methods for the solution of equations. Prentice-Hall Series in Automatic Computation Prentice-Hall, Inc., Englewood Cliffs, N.J. 1964 xviii+310 pp., MR0169356.

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Published

1973-02-01

How to Cite

Păvăloiu, I. (1973). Sur la convergence d’une classe de méthodes itératives de J.F. Traub: On the convergence of a class of iterative methods by J.F. Traub. Rev. Anal. Numér. Théorie Approximation, 2, 99–104. https://doi.org/10.33993/jnaat21-15

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Articles