Complexity theory and numerical analysis

@article{Smale1997ComplexityTA,
  title={Complexity theory and numerical analysis},
  author={Stephen Smale},
  journal={Acta Numerica},
  year={1997},
  volume={6},
  pages={523 - 551}
}
  • S. Smale
  • Published 1 January 1997
  • Mathematics
  • Acta Numerica
Complexity theory of numerical analysis is the study of the number of arithmetic operations required to pass from the input to the output of a numerical problem. 
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References

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The problem of increasing the understanding of algorithms by considering the foundations of numerical analysis and computer science is considered and the legitimacy and importance of models of machines that accept real numbers is considered.
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An optimal convergence condition for Newton iteration in a Banach space is established and which stronger condition must be imposed to also assure good complexity.
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A new invariant, the sparse condition number, is introduced and it is shown that a sparse polynomial system analysis in terms of this invariant is easier to solve than a non sparse one.
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