, Volume 50, Issue 4, pp 353–368 | Cite as

On certain computable tests and componentwise error bounds

  • Z. Shen
  • M. A. Wolfe


Improved forms of the existence and uniqueness tests due to Pandian are given and are related to results due to Moore and Kioustelidis and to Shen and Neumaier.

AMS Subject Classifications

65G10 65H10 65R20 

Key words

Interval arithmetic existence uniqueness convergence componentwise error bounds 

Konstruktive Tests und komponentenweise Fehlerschranken


Existenz- und Eindeutigkeitstests von Pandian werden verallgemeinert und in Zusammenhang gebracht mit Ergebnissen von Moore und Kioustelidis sowie von Shen und Neumaire.


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  1. [1]
    Alefeld, G., Herzberger, J.: Introduction to interval computations. London: Academic Press 1983.Google Scholar
  2. [2]
    Gay, D. M.: Perturbation bounds for nonlinear equations. SIAM J. Numer. Anal.18, 654–663 (1981).Google Scholar
  3. [3]
    Griewank, A., Corliss, G. F.: Automatic differentiation of algorithms. Philadelphia: Society for Industrial and Applied Mathematics 1992.Google Scholar
  4. [4]
    Kioustelidis, J. B.: Algorithmic error estimation for approximate solutions of nonlinear systems of equations. Computing19, 313–320 (1978).Google Scholar
  5. [5]
    Miranda, C.: Un osservatione su un teorema di Brower. Bolletino Unione Math. Ital. Ser. II,3, 5–7 (1940).Google Scholar
  6. [6]
    Moore, R. E.: Methods and applications of interval analysis (SIAM Studies, 2). Philadelphia: Society for Industrial and Applied Mathematics 1979.Google Scholar
  7. [7]
    Moore, R. E., Kioustelidis, J. B.: A simple test for accuracy of approximate solutions to nonlinear (or linear) systems. SIAM J. Numer. Anal.17, 521–529 (1980).Google Scholar
  8. [8]
    Neumaier, A.: Interval methods for systems of equations. Cambridge: Cambridge University Press 1990.Google Scholar
  9. [9]
    Ortega, J. M., Rheinboldt, W. C.: Iterative solution of nonlinear equations in several variables. San Diego: Academic Press 1970.Google Scholar
  10. [10]
    Pandian, M. C.: A convergence test and componentwise error estimates for Newton type methods. SIAM J. Numer. Anal.22, 779–791 (1985).Google Scholar
  11. [11]
    Qi, L.: A generalization of the Krawczyk-Moore algorithm. In: Nickel, K. (ed.) Interval Mathematics 1980 pp. 481–488. New York: Academic Press 1980.Google Scholar
  12. [12]
    Qi, L.: A note on the Moore test for nonlinear systems. SIAM J. Numer. Anal.19, 851–857 (1982).Google Scholar
  13. [13]
    Shen Z., Neumaier, A.: A note on Moore's interval test for zeros of nonlinear systems. Computing40, 85–90 (1988).Google Scholar
  14. [14]
    Shen Z., Wolfe, M. A.: Slope tests for Newton-type methods. Applied Mathematics and Computation52, 403–416 (1992).Google Scholar
  15. [15]
    Shen Z., Wolfe, M. A.: A note on the comparison of the Kantorovich and Moore theorems. Nonlinear analysis: theory, methods and applications15, 229–232 (1990).Google Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • Z. Shen
    • 1
  • M. A. Wolfe
    • 2
  1. 1.Mathematics DepartmentNanjing UniversityPeople's Republic of China
  2. 2.Department of Mathematical and Computational SciencesUniversity of St. AndrewsScotland

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