Advertisement

A Neglected Theorem - Numerical Analysis via Variable Interval Computing

  • Tsau Young LinEmail author
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 1000)

Abstract

A neglected theorem in numerical computing is proved. In science labs, the computing of a scientific formula \(y_0 = E(x_0)\) is often approximated by computing an ‘adequately close’ finite decimal f, namely, E(f). In essence, numerical computing is approximate computing: A finite decimal, say \(f = 0.333\), can be rounded from any reals in [0.3325, 0.333), called rounding interval; so a computation of f is an approximate computing for any reals in this rounding interval. A real number, based on Cantor’s infinite decimal representation, can be regarded as a convergent sequence of rounding intervals. The computing of such sequences is variable interval computing – hope to be a useful approach to numerical analysis.

References

  1. 1.
    Birkhof, G., MacLane, S.: A Survey of Modern Algebra. MacMillan Publishers, London (1970)Google Scholar
  2. 2.
    Hopcroft, J., Ullman, J.: Introduction to Automata Theory, Languages and Computation (1979)Google Scholar
  3. 3.
    Kearfott, R.B.: Interval Computations: Introduction, Uses, and Resources. https://interval.louisiana.edu/preprints/survey.pdf
  4. 4.
    Kelley, J.: General Topology, Van Nostrand (Chap. 3, Theorem 1, Item e) (1955)Google Scholar
  5. 5.
    Kuroki, N.: On power semigroups. Proc. Japan Acad. 47, 449 (1971)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Lin, T.-Y.: A mathematical theory of fuzzy numbers - granular computing approach. In: RSFDGrC 2013, pp. 208–215 (2013)Google Scholar
  7. 7.
    Lin, T.-Y.: A paradox in rounding errors approximate computing for big data. In: SMC 2015, pp. 2567–2573 (2015)Google Scholar
  8. 8.
    Scott, L.R.: Numerical Analysis. Princeton University Press, Princeton (2011)CrossRefGoogle Scholar
  9. 9.
    Smale, S.: Some remarks on the foundations of numerical analysis. SIAM Rev. 32(2), 211–220 (1990). Society for Industrial and Applied MathematicsMathSciNetCrossRefGoogle Scholar
  10. 10.
    Wilder, R.: The Foundation of Mathematics, p. 82. Wiley, Hoboken (1952)Google Scholar
  11. 11.
    Zassenhaus, H.: Group Theory. Chelsea Publishing Co. (1958). (Dover 1999)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.San Jose State UniversitySan JoseUSA

Personalised recommendations