Abstract.
We extend Smale’s concept of approximate zeros of an analytic function on a Banach space to two computational models that account for errors in the computation: first, the weak model where the computations are done with a fixed precision; and second, the strong model where the computations are done with varying precision. For both models, we develop a notion of robust approximate zero and derive a corresponding robust point estimate.
A useful specialization of an analytic function on a Banach space is a system of integer polynomials. Given such a zero-dimensional system, we bound the complexity of computing an absolute approximation to a root of the system using the strong model variant of Newton’s method initiated from a robust approximate zero. The bound is expressed in terms of the condition number of the system and is a generalization of a well-known bound of Brent to higher dimensions.
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Sharma, V. Robust Approximate Zeros in Banach Space. Math.comput.sci. 1, 71–109 (2007). https://doi.org/10.1007/s11786-007-0005-7
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DOI: https://doi.org/10.1007/s11786-007-0005-7