Abstract
The numerical performance of iterative methods applied to discretized operator equations may depend strongly on their theoretical rate of convergence on the underlying problem g(x)=0 in Hilbert space. It is found that the usual invertibility and smoothness assumptions on the Frechet derivative g'(x) are sufficient for local and linear but not necessarily superlinear convergence of secant methods. For both Broyden's Method and Variable Metric Methods it is shown that the asymptotic rate of convergence depends on the essential norm of the discrepancy D0 between the Frechet derivative g' at the solution x* and its initial approximation B0. In particular one obtains local and Q-superlinear convergence if D0 is compact which can be ensured in the case of mildly nonlinear problems where g'(x*) is known up to a compact perturbation.
Keywords
- Secant Methods
- Variational Characterization of Eigenvalues
- Compact Operators
- Running Head
- Secant Methods in Hilbert Space
This work was supported by NSF grant DMS-8401023.
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Griewank, A. (1986). Rates of convergence for secant methods on nonlinear problems in hilbert space. In: Hennart, JP. (eds) Numerical Analysis. Lecture Notes in Mathematics, vol 1230. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0072677
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DOI: https://doi.org/10.1007/BFb0072677
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