Skip to main content

Nonlinear problems

Part of the Lecture Notes in Mathematics book series (LNM,volume 1733)

Keywords

  • Adaptive Method
  • Wiener Measure
  • Average Case Analysis
  • Symmetric Classis
  • Residual Sense

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

1.1. Notes and References

  • Kung, H. T. (1976), The complexity of obtaining starting points for solving operator equations by Newton's method, in: Analytic computational complexity, J. F. Traub, ed., pp. 35–57, Academic Press, New York.

    Google Scholar 

  • Sikorski, K. (1982), Bisection is optimal, Numer. Math. 40, 111–117.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Sikorski, K. (1989), Study of linear information for classes of polynomial equations, Aequationes Math. 37, 1–14.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Graf, S., Novak, E., and Papageorgiou, A. (1989), Bisection is not optimal on the average, Numer. Math. 55, 481–491.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Novak, E. (1989), Average-case results for zero finding, J. Complexity 5, 489–501.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Novak, E., and Ritter, K. (1992), Average errors for zero finding: lower bounds, Math. Z. 211, 671–686.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Ritter, K. (1994), Average errors for zero finding: lower bounds for smooth or monotone functions, Aequationes Math. 48, 194–219.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Novak, E., Ritter, K., and Woźniakowski, H. (1995), Average case optimality of a hybrid secant-bisection method, Math. Comp. 64, 1517–1539.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Novak, E. (1996b), The Bayesian approach to numerical problems: results for zero finding, in: Proc. IMACS-GAMM Int. Symp. Numerical Methods and Error Bounds, G. Alefeld, J. Herzberger, eds., pp. 164–171, Akademie Verlag, Berlin.

    Google Scholar 

  • Novak, E., and Ritter, K. (1993), Some complexity results for zero finding for univariate functions, J. Complexity 9, 15–40.

    CrossRef  MathSciNet  MATH  Google Scholar 

2.1. Notes and References

  • Wasilkowski, G. W. (1984), Some nonlinear problems are as easy as the approximation problem, Comput. Math. Appl. 10, 351–363.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Novak, E. (1988), Deterministic and stochastic error bounds in numerical analysis, Lect. Notes in Math. 1349, Springer-Verlag, Berlin.

    MATH  Google Scholar 

  • Kushner, H. J. (1962), A versatile stochastic model of a function of unknown and time varying form, J. Math. Anal. Appl. 5, 150–167.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Žilinskas, A. (1985), Axiomatic characterization of a global optimization algorithm and investigation of its search strategy, OR Letters 4, 35–39.

    MathSciNet  MATH  Google Scholar 

  • Mockus, J. (1989), Bayesian approach to global optimization, Kluwer, Dordrecht.

    CrossRef  MATH  Google Scholar 

  • Törn, A., and Žilinskas, A. (1989), Global optimization, Lect. Notes in Comp. Sci. 350, Springer-Verlag, Berlin.

    MATH  Google Scholar 

  • Boender, C. G. E., and Romeijn, H. E. (1995), Stochastic methods, in: Handbook of global optimization, R. Horst and P. M. Pardalos, eds., pp. 829–869, Kluwer, Dordrecht.

    CrossRef  Google Scholar 

  • Ritter, K. (1990), Approximation and optimization on the Wiener space, J. Complexity 6, 337–364.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Calvin, J. M. (1996), An asymptotically optimal non-adaptive algorithm for minimization of Brownian motion, in: Mathematics of Numerical Analysis, J. Renegar, M. Shub, S. Smale, eds., Lectures in Appl. Math. 32, pp. 157–163, AMS, Providence.

    Google Scholar 

  • Calvin, J. M. (1997), Average performance of a class of adaptive algorithms for global optimization, Ann. Appl. Prob. 7, 711–730.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Calvin, J. M. (1999), A one-dimensional optimization algorithm and its convergence rate under the Wiener measure, submitted for publication.

    Google Scholar 

Download references

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2000 Springer-Verlag

About this chapter

Cite this chapter

Ritter, K. (2000). Nonlinear problems. In: Ritter, K. (eds) Average-Case Analysis of Numerical Problems. Lecture Notes in Mathematics, vol 1733. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0103942

Download citation

  • DOI: https://doi.org/10.1007/BFb0103942

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-67449-8

  • Online ISBN: 978-3-540-45592-9

  • eBook Packages: Springer Book Archive