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
Abstract: Let and a simple root in of the system of nonlinear equations .
Abstract Padé approximants (APA) and abstract Rational approximants (ARA) for the operator F have been introduced in  and . The adjective ``abstract'' refers to the use of abstract polynomials  for the construction of the rational operators.
The APA and ARA have been used for the solution of a system of nonlinear equations in . Of particular interest was the following third order iterative procedure:
-  Robert G. Bartle, The elements of real analysis, 2nd ed., John Wiley & Sons, New York-London-Sydney, 1976. MR 0393369
-  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
-  Annie A. M. Cuyt, On the properties of abstract rational (1-point) approximants, J. Operator Theory 6 (1981), no. 2, 195–216. MR 643691
-  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.
-  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
-  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
-  H. Woźniakowski, Numerical stability for solving nonlinear equations, Numer. Math. 27 (1976/77), no. 4, 373–390. MR 0443323, https://doi.org/10.1007/BF01399601
-  D. Young, A Survey of Numerical Mathematics. I, Addison-Wesley, Reading, Mass., 1972.
- R. G. Bartle, The Elements of Real Analysis, Wiley, New York, 1976. MR 0393369 (52:14179)
- Annie A. M. Cuyt, Abstract Padé Approximants in Operator Theory, Lecture Notes in Math., Vol. 765, Springer-Verlag, Berlin and New York, 1979, pp. 61-87. MR 561445 (81g:41042)
- Annie A. M. Cuyt, ``On the properties of Abstract Rational (1-point) Approximants,'' J. Operator Theory, v. 5, 1981. (To appear.) MR 643691 (83h:41020)
- 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.
- Louis B. Kall, Computational Solution of Nonlinear Operator Equations, Wiley, New York, 1969; reprinted by Krieger, Huntington, New York, 1979. MR 0240944 (39:2289)
- J. F. Traub, Iterative Methods for the Solution of Equations, Prentice-Hall, Englewood Cliffs, N. J., 1964. MR 0169356 (29:6607)
- H. Woźniakowski, ``Numerical stability for solving nonlinear equations,'' Numer. Math., v. 27, 1977, pp. 373-390. MR 0443323 (56:1693)
- D. Young, A Survey of Numerical Mathematics. I, Addison-Wesley, Reading, Mass., 1972.