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**The strength of nonstationary iteration.**
*(English)*
Zbl 0543.65025

The paper deals with the iterative solution of a nonlinear operator equation \(F=0\) where F is an analytical multivariate or abstract function having only simple zeros. A very general definition of information and iteration without memory is introduced. The definiton of globally convergent iterations is discussed and the constant of global convergence is defined. It is proved that, for the class \({\mathcal F}_ 1\) of all analytic operators having simple zeros, the constant of global convergence is not larger than 1/2 for any iteration. Furthermore it is proved that any globally convergent iteration is a ”one-point” iteration.

A globally convergent nonstationary interpolatory iteration \(I_ 0\) for operators \(F\in {\mathcal F}_ 1\) is constructed where the constant of global convergence is not less than 1/3. The i-th step of this iteration requires the computation of \(F(x_ 0),F'(x_ 0),...,F^{(i-1)}(x_ o)\) and the solution of a polynomial equation of degree i-1. The generalization of \(I_ 0\) to iterations with memory is derived with the constant of global convergence not less than 1/4.

A globally convergent nonstationary interpolatory iteration \(I_ 0\) for operators \(F\in {\mathcal F}_ 1\) is constructed where the constant of global convergence is not less than 1/3. The i-th step of this iteration requires the computation of \(F(x_ 0),F'(x_ 0),...,F^{(i-1)}(x_ o)\) and the solution of a polynomial equation of degree i-1. The generalization of \(I_ 0\) to iterations with memory is derived with the constant of global convergence not less than 1/4.

Reviewer: J.Hřebíček

### MSC:

65H10 | Numerical computation of solutions to systems of equations |

### Keywords:

nonstationary iteration; global convergence; analytical multivariate function; one-point iteration; simple zeros; globally convergent iterations
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\textit{G. W. Wasilkowski}, Aequationes Math. 24, 243--260 (1983; Zbl 0543.65025)

### References:

[1] | Ortega, J. M. andReinboldt, W. C.,Iterative solution of nonlinear equations in several variables. Academic Press, New York, 1970. |

[2] | Rall, L. B.,Computational solution of nonlinear operator equations. John Wiley and Sons, New York, 1969. · Zbl 0175.15804 |

[3] | Traub, J. F.,Iterative methods for the solution of equations. Prentice Hall, Englewood Cliffs, N.J., 1964. · Zbl 0121.11204 |

[4] | Traub, J. F. andWoźniakowski, H.,Optimal radius of convergence of interpolatory iterations for operator equations. Department of Computer Science Report, Carnegie-Mellon University, 1976. See also Aequationes Math.21 (1980), 159–172. · Zbl 0447.65027 |

[5] | Traub, J. F. andWoźniakowski, H.,General theory of optimal error algorithms and analytic complexity, part B: Iterative information model. Department of Computer Science Report, Carnegie-Mellon University, 1978. See alsoA General Theory of Optimal Algorithms, Academic Press, New York, 1980. |

[6] | Wasilkowski, G. W.,Can any stationary iteration using linear information be globally convergent? Department of Computer Science Report, Carnegie-Mellon University, 1978. See also J. Assoc. Comput. Mach.27 (1981), 263–269. · Zbl 0439.65045 |

[7] | Wasilkowski, G. W.,Any iteration for polynomial equations using linear information has infinite complexity. Department of Computer Science Report, Carnegie-Mellon University, 1979. To appear in J. Theoret. Comput. Sci. · Zbl 0496.65024 |

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