Skip to main content

Some Results Relevant to Smale’s Reports

  • Chapter

Abstract

The background of this paper is Smale’s report to the 20th International Congress of Mathematicians in 1986 and his material [1] written for this congress. We mainly list results about iterative convergence, estimates from data at one point, and complexity of numerical integrals from [7–13].

The subject was supported by Zhejiang Province Natural Funds.

This is a preview of subscription content, log in via an institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • 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.

References

  1. Smale, S., Algorithms for solving equations, Proceedings of the International Congress of Mathematicians Berkeley, CA. American Mathematical Society, Providence, RI, 1986, 172–195.

    Google Scholar 

  2. Smale, S., Newton’s method estimates from data at one point, in The Merging of Disciplines; New Directions in Pure and Applied Mathematics, Ewing, R., Cross, K., and Martin, C. (Eds), Springer-Verlag, New York, 1986.

    Google Scholar 

  3. Smale, S., On the efficiency of algorithms of analysis, Bull. Amer. Math. Soc. (New Series) 13 (1985), 87–121.

    Article  MathSciNet  MATH  Google Scholar 

  4. Smale, S., The fundamental theorem of algebra and complexity theory, Bull. Amer. Math. Soc., 4 (1981), 1–36.

    Article  MathSciNet  MATH  Google Scholar 

  5. Shub, M. and Smale, S., On the existence of generally convergent algorithms, J Complexity 2 (1986), 2–11.

    Article  MathSciNet  MATH  Google Scholar 

  6. Wang Xinghua and Xuan Xiaohua, Random polynomial space and computational complexity theory, Scientia Sinica (Series A) 7 (1987), 637–684.

    Google Scholar 

  7. Wang Xinghua and Han Danfu, On dominating sequence method in the point estimate and Smale theorem, Sci. China (Ser. A) 33 (2) (1990), 135–144.

    MathSciNet  MATH  Google Scholar 

  8. Wang Xinghua and Han Danfu, Computational complexity in numerical integrals, preprint.

    Google Scholar 

  9. Wang Xinghua and Han Danfu, Domain and point estimates on Newton’s iteration, Chinese J. Numer. Math. Appl. 12 (3) (1990), 1–8.

    MathSciNet  Google Scholar 

  10. Wang Xinghua and Han Danfu, Precise judgement of approximate zero of the second kind for Newton’s iteration, A Friendly Collection of Mathematics Papers I150 Jilin University Press, Changchun, China, 19XX, pp. 22–24.

    Google Scholar 

  11. Wang Xinghua, Han Danfu, and Sun Fangyu, Point estimates on some eformed Newton’s methods, Chinese J. Numer. Math. Appl. 12 (4) (1990).

    Google Scholar 

  12. Wang Xinghua, Zheng Shiming, and Han Danfu, The convergence of Euler’s series, Euler’s and Halley’s families (Chinese), Acta Math. Sinica, 6 (1990)

    Google Scholar 

  13. Wang Xinghua, Shen Guangxing, and Han Danfu, Some remarks on the algorithms of solving equations Acta Math Sinica (New Series) (Chinese J. Math.) 8 (92), 337–348.

    Google Scholar 

  14. Wang Xinghua, Convergence on a iterative procedure (in Chinese), Ke Xue Tong Bao 20 (1975), 558–559.

    Google Scholar 

  15. Wang Xinghua, Error estimates on some numerical methods of equation finding (in Chinese), Acta Math. Sinica 5 (1979), 638–642.

    Google Scholar 

  16. Wang Xinghua, Convergent neighbourhood on Newton’s method (in Chinese), Ke Xue Tong Bao, Special Issue of Mathematics, Physics and Chemistry (1980), 36–37.

    Google Scholar 

  17. Wang Xinghua and Zheng Shiming, A family of parallel and interval iterations for finding simultaneously all roots of a polynomial with rapid convergences, J. Comput. Math. 1 (1984), 70–76.

    Google Scholar 

  18. Xuan Xiaohua, The computational complexity of the resultant method for solving polynomial equations, J. Comput. Math. 3 (1985), 162–166.

    Google Scholar 

  19. Xuan Xiaohua, Computational complexity theory, Master’s paper, Hangzhou University.

    Google Scholar 

  20. Gragg, G.W. and Tapia, R.A., Optimal error bounds for the Newton-Kantorovich theorem, SI AM J. Numer. Anal. 11 (1974), 10–13.

    Article  MathSciNet  MATH  Google Scholar 

  21. Ostrowski, A.M., Solutions of Equations in Eulidean and Banach Spaces, Academic Press, New York, 1973.

    Google Scholar 

  22. Traub, J. and Wozniakowski, H., Convergence and complexity of Newton iteration for operator equation, J. Assoc. Comput. Mach. 29 (1979), 250–258.

    Article  MathSciNet  Google Scholar 

  23. Kim, M.H., On approximate zeros and root finding algorithms for a complex polynomial, to appear

    Google Scholar 

Download references

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1993 Springer-Verlag New York, Inc.

About this chapter

Cite this chapter

Wang, X. (1993). Some Results Relevant to Smale’s Reports. In: Hirsch, M.W., Marsden, J.E., Shub, M. (eds) From Topology to Computation: Proceedings of the Smalefest. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-2740-3_41

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-2740-3_41

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-7648-7

  • Online ISBN: 978-1-4612-2740-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics