A quadratically convergent iteration method for computing zeros of operators satisfying autonomous differential equations

Author:
L. B. Rall

Journal:
Math. Comp. **30** (1976), 112-114

MSC:
Primary 65H05; Secondary 47H15

MathSciNet review:
0405831

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Abstract: If the Fréchet derivative *P*' of the operator *P* in a Banach space *X* is Lipschitz continuous, satisfies an autonomous differential equation , and has the bounded inverse , then the iteration process

*f*is Lipschitz continuous and exists, then the global existence of is shown to follow if is uniformly bounded by a sufficiently small constant. The replacement of the uniform boundedness of

*P*by Lipschitz continuity gives a semilocal theorem for the existence of and the quadratic convergence of the sequence to .

**[1]**Robert G. Bartle,*Newton’s method in Banach spaces*, Proc. Amer. Math. Soc.**6**(1955), 827–831. MR**0071730**, 10.1090/S0002-9939-1955-0071730-1**[2]**J. E. Dennis Jr.,*Toward a unified convergence theory for Newton-like methods*, Nonlinear Functional Anal. and Appl. (Proc. Advanced Sem., Math. Res. Center, Univ. of Wisconsin, Madison, Wis., 1970) Academic Press, New York, 1971, pp. 425–472. MR**0278556****[3]**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****[4]**L. B. Rall,*Convergence of Stirling’s method in Banach spaces*, Aequationes Math.**12**(1975), 12–20. MR**0366030**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1976-0405831-4

Keywords:
Nonlinear operator equations,
iteration methods,
quadratic convergence,
variants of Newton's method

Article copyright:
© Copyright 1976
American Mathematical Society