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Numerical stability of the Halley-iteration for the solution of a system of nonlinear equations

Author: Annie A. M. Cuyt
Journal: Math. Comp. 38 (1982), 171-179
MSC: Primary 65H10; Secondary 65G05, 65J15
MathSciNet review: 637295
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Abstract: Let $ F:{{\mathbf{R}}^q} \to {{\mathbf{R}}^q}$ and $ {x^ \ast }$ a simple root in $ {{\mathbf{R}}^q}$ of the system of nonlinear equations $ F(x) = 0$.

Abstract Padé approximants (APA) and abstract Rational approximants (ARA) for the operator F have been introduced in [2] and [3]. The adjective ``abstract'' refers to the use of abstract polynomials [5] for the construction of the rational operators.

The APA and ARA have been used for the solution of a system of nonlinear equations in [4]. Of particular interest was the following third order iterative procedure:

$\displaystyle {x_{i + 1}} = {x_i} + \frac{{a_i^2}}{{{a_i} + \frac{1}{2}F_i^{'- 1}F_i^{''}a_i^2}},$

with $ F_i'$ the 1st Fréchet-derivative of F in $ {x_1},{a_i} = - F_i^{'- 1}{F_i}$ the Newton-correction where $ {F_i} = F({x_i}),F_i^{''}$ the 2nd Fréchet-derivative of F in $ {x_i}$ where $ F_i^{''}a_i^2$ is the bilinear operator $ F_i^{''}$ evaluated in $ ({a_i},{a_i})$, and componentwise multiplication and division in $ {{\mathbf{R}}^q}$. For $ q = 1$ this technique is known as the Halley-iteration [6, p. 91]. In this paper the numerical stability [7] of the Halley-iteration for the case $ q \geqslant 1$ is investigated and illustrated by a numerical example.

References [Enhancements On Off] (What's this?)

  • [1] Robert G. Bartle, The elements of real analysis, 2nd ed., John Wiley & Sons, New York-London-Sydney, 1976. MR 0393369
  • [2] Annie A. M. Cuyt, Abstract Padé-approximants in operator theory, Padé approximation and its applications (Proc. Conf., Univ. Antwerp, Antwerp, 1979) Lecture Notes in Math., vol. 765, Springer, Berlin, 1979, pp. 61–87. MR 561445
  • [3] Annie A. M. Cuyt, On the properties of abstract rational (1-point) approximants, J. Operator Theory 6 (1981), no. 2, 195–216. MR 643691
  • [4] A. Cuyt & P. Van der Cruyssen, Abstract Padé Approximants for the Solution of a System of Nonlinear Equations, Report 80-17, University of Antwerp, 1980.
  • [5] Louis B. Rall, Computational solution of nonlinear operator equations, With an appendix by Ramon E. Moore, John Wiley & Sons, Inc., New York-London-Sydney, 1969. MR 0240944
  • [6] J. F. Traub, Iterative methods for the solution of equations, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0169356
  • [7] H. Woźniakowski, Numerical stability for solving nonlinear equations, Numer. Math. 27 (1976/77), no. 4, 373–390. MR 0443323,
  • [8] D. Young, A Survey of Numerical Mathematics. I, Addison-Wesley, Reading, Mass., 1972.

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Article copyright: © Copyright 1982 American Mathematical Society

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