# Local convergence of inexact Newton methods under affine invariant conditions and hypotheses on the second Fréchet derivative

• Volume: 26, Issue: 4, page 457-465
• ISSN: 1233-7234

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## Abstract

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We use inexact Newton iterates to approximate a solution of a nonlinear equation in a Banach space. Solving a nonlinear equation using Newton iterates at each stage is very expensive in general. That is why we consider inexact Newton methods, where the Newton equations are solved only approximately, and in some unspecified manner. In earlier works [2], [3], natural assumptions under which the forcing sequences are uniformly less than one were given based on the second Fréchet derivative of the operator involved. This approach showed that the upper error bounds on the distances involved are smaller compared with the corresponding ones using hypotheses on the first Fréchet derivative. However, the conditions on the forcing sequences were not given in affine invariant form. The advantages of using conditions given in affine invariant form were explained in [3], [10]. Here we reproduce all the results obtained in [3] but using affine invariant conditions.

## How to cite

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Argyros, Ioannis. "Local convergence of inexact Newton methods under affine invariant conditions and hypotheses on the second Fréchet derivative." Applicationes Mathematicae 26.4 (1999): 457-465. <http://eudml.org/doc/219251>.

@article{Argyros1999,
abstract = {We use inexact Newton iterates to approximate a solution of a nonlinear equation in a Banach space. Solving a nonlinear equation using Newton iterates at each stage is very expensive in general. That is why we consider inexact Newton methods, where the Newton equations are solved only approximately, and in some unspecified manner. In earlier works [2], [3], natural assumptions under which the forcing sequences are uniformly less than one were given based on the second Fréchet derivative of the operator involved. This approach showed that the upper error bounds on the distances involved are smaller compared with the corresponding ones using hypotheses on the first Fréchet derivative. However, the conditions on the forcing sequences were not given in affine invariant form. The advantages of using conditions given in affine invariant form were explained in [3], [10]. Here we reproduce all the results obtained in [3] but using affine invariant conditions.},
author = {Argyros, Ioannis},
journal = {Applicationes Mathematicae},
keywords = {superlinear; Fréchet derivative; weak convergence; inexact Newton method; strong; forcing sequence; Banach space; nonlinear operator equations; inexact Newton methods; second Fréchet derivative; error bounds; affine invariant conditions},
language = {eng},
number = {4},
pages = {457-465},
title = {Local convergence of inexact Newton methods under affine invariant conditions and hypotheses on the second Fréchet derivative},
url = {http://eudml.org/doc/219251},
volume = {26},
year = {1999},
}

TY - JOUR
AU - Argyros, Ioannis
TI - Local convergence of inexact Newton methods under affine invariant conditions and hypotheses on the second Fréchet derivative
JO - Applicationes Mathematicae
PY - 1999
VL - 26
IS - 4
SP - 457
EP - 465
AB - We use inexact Newton iterates to approximate a solution of a nonlinear equation in a Banach space. Solving a nonlinear equation using Newton iterates at each stage is very expensive in general. That is why we consider inexact Newton methods, where the Newton equations are solved only approximately, and in some unspecified manner. In earlier works [2], [3], natural assumptions under which the forcing sequences are uniformly less than one were given based on the second Fréchet derivative of the operator involved. This approach showed that the upper error bounds on the distances involved are smaller compared with the corresponding ones using hypotheses on the first Fréchet derivative. However, the conditions on the forcing sequences were not given in affine invariant form. The advantages of using conditions given in affine invariant form were explained in [3], [10]. Here we reproduce all the results obtained in [3] but using affine invariant conditions.
LA - eng
KW - superlinear; Fréchet derivative; weak convergence; inexact Newton method; strong; forcing sequence; Banach space; nonlinear operator equations; inexact Newton methods; second Fréchet derivative; error bounds; affine invariant conditions
UR - http://eudml.org/doc/219251
ER -

## References

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1. [1] I. K. Argyros, On the convergence of some projection methods with perturbation, J. Comput. Appl. Math. 36 (1991), 255-258. Zbl0755.65056
2. [2] I. K. Argyros, Comparing the radii of some balls appearing in connection to three local convergence theorems for Newton's method, Southwest J. Pure Appl. Math. 1 (1998), 32-43.
3. [3] I. K. Argyros, Relations between forcing sequences and inexact Newton iterates in Banach space, Computing 62 (1999), 71-82. Zbl0937.65062
4. [4] I. K. Argyros and F. Szidarovszky, The Theory and Application of Iteration Methods, CRC Press, Boca Raton, FL, 1993. Zbl0844.65052
5. [5] P. N. Brown, A local convergence theory for combined inexact-Newton/finite-difference projection methods, SIAM J. Numer. Anal. 24 (1987), 407-434. Zbl0618.65037
6. [6] R. S. Dembo, S. C. Eisenstat and T. Steihaug, Inexact Newton methods, SIAM J. Numer. Anal. 19, (1982), 400-408. Zbl0478.65030
7. [7] J. M. Gutierrez, A new semilocal convergence theorem for Newton's method, J. Comput. Appl. Math. 79 (1997), 131-145. Zbl0872.65045
8. [8] L. V. Kantorovich and G. P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982. Zbl0484.46003
9. [9] F. A. Potra, On Q-order and R-order of convergence, SIAM J. Optim. Theory Appl. 63 (1989), 415-431. Zbl0663.65049
10. [10] T. J. Ypma, Local convergence of inexact Newton methods, SIAM J. Numer. Anal. 21 (1984), 583-590. Zbl0566.65037

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