Skip to main content
Log in

Interval iteration for zeros of systems of equations

  • Part II Numerical Mathematics
  • Invited Paper
  • Published:
BIT Numerical Mathematics Aims and scope Submit manuscript

Abstract

Easily verifiable existence and convergence conditions are given for a class of interval iteration algorithms for the enclosure of a zero of a system of nonlinear equations. In particular, a quadratically convergent method is obtained which throughout the iteration uses the same interval enclosure of the derivative.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. G. Alefeld, Über die Durchführbarkeit des Gaußschen Algorithmus bei Gleichungen mit Intervallen als Koeffizienten, Computing Suppl. 1 (1977), 15–19.

    Google Scholar 

  2. G. Alefeld,Intervallanalytische Methoden bei nichtlinearen Gleichungen, in:Jahrbuch Überblicke Mathematik, Bibl. Inst., Mannheim 1979, pp. 63–78.

    Google Scholar 

  3. G. Alefeld,On the convergence of some intervalarithmetic modifications of Newton's method, in:Scientific Computing (R. S. Stepleman, ed.), North-Holland, Amsterdam 1983, pp. 223–230.

    Google Scholar 

  4. G. Alefeld and J. Herzberger,Introduction to Interval Computations. Acad. Press, New York-London 1983.

    Google Scholar 

  5. H. Cornelius,Untersuchungen zu einem intervallarithmetischen Iterationsverfahren mit Anwendungen auf eine Klasse nichtlinearer Gleichungssysteme. Dissertation, Technische Universität Berlin, 1981.

  6. H. Cornelius and G. Alefeld,A device for the acceleration of convergence of a monotonously enclosing iteration method, in:Iterative solution of nonlinear systems of equations, Springer Lecture Notes in Mathematics 953, 1982, pp. 68–79.

  7. E. Hansen and R. Smith,Interval arithmetic in matrix computations, Part II, SIAM J. Numer. Anal. 4 (1967), 1–9.

    Article  Google Scholar 

  8. W. M. Kahan,A more complete interval arithmetic. Lecture Notes, University of Michigan, 1968.

  9. R. Krawczyk,Interval iterations for including a set of solutions, Computing, 32 (1984), 13–31.

    Google Scholar 

  10. R. Krawczyk and A. Neumaier,An improved interval Newton operator, to appear.

  11. R. E. Moore,Interval Analysis, Prentice-Hall, Englewood Cliffs, N.Y., 1966.

    Google Scholar 

  12. A. Neumaier,New techniques for the analysis of linear interval equations, Linear Algebra Appl., 58 (1984), 273–325.

    Article  Google Scholar 

  13. A. Neumaier,An interval version of the secant method, BIT 24 (1984), 366–372.

    Google Scholar 

  14. K. Nickel,On the Newton method in interval analysis, MRC Techn. Sum. Rep. 1136, University of Wisconsin, Madison 1971.

  15. J. M. Ortega and W. C. Rheinboldt,Iterative Solution of Nonlinear Equations in Several Variables. Acad. Press, New York-London 1970.

    Google Scholar 

  16. K. Reichmann, Abbruch beim Intervall-Gauß-Algorithmus, Computing 22 (1979), 355–361.

    Google Scholar 

  17. K. Reichmann, Ein hinreichendes Kriterium für die Durchführbarkeit des Intervall-Gauss-Algorithmus bei Intervall-Hessenbergmatrizen ohne Pivotsuche, Z. Angew. Math. Mech. 59 (1979), 373–379.

    Google Scholar 

  18. H. Schwetlick,Numerische Lösung nichtlinearer Gleichungen. R. Oldenburg, München-Wien 1979.

    Google Scholar 

  19. J. M. Yohe,Software for interval arithmetic: a reasonably portable package, ACM Trans. Math. Software 5 (1979), 50–63.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Neumaier, A. Interval iteration for zeros of systems of equations. BIT 25, 256–273 (1985). https://doi.org/10.1007/BF01935003

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01935003

AMS Subject Classifications

Navigation