Skip to main content
SpringerLink
Log in

Real Zeros of Polynomials

  • Chapter
Computer Algebra

Part of the book series: Computing Supplementa ((COMPUTING,volume 4))

Abstract

Let A be a polynomial over Z, Q or Q(α) where α is a real algebraic number. The problem is to compute a sequence of disjoint intervals with rational endpoints, each containing exactly one real zero of A and together containing all real zeros of A. We describe an algorithm due to Kronecker based on the minimum root Separation, Sturm’s algorithm, an algorithm based on Rolle’s theorem due to Collins and Loos and the modified Uspensky algorithm due to Collins and Aritas. For the last algorithm a recursive version with correctness proof is given which appears in print for the first time.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Collins, G. E.: Infallible Calculation of Polynomial Zeros to Specifled Precision, Mathematical Software III (Rice, J. R., ed.), pp. 35–68. New York: Academic Press 1977.

    Google Scholar 

  2. Collins, G. E.: Quantifier Elimination for Real Closed Fields by Cylindrical Algebraic Decomposition. Lecture Notes in Computer Science, Vol. 33, pp. 134–183. Berlin-Heidelberg- New York: Springer 1975.

    Google Scholar 

  3. Collins, G. E.: Real Root Subsystem, in SAC-2 System of Algebraic Algorithms. Available at request.

    Google Scholar 

  4. Collins, G. E., Akritas, A. G.: Polynomial Real Root Isolation Using Descartes’ Rule of Signs. SYMSAC 1976, 272–275.

    Google Scholar 

  5. Collins, G. E., Horowitz, E.: The Minimum Root Separation of a Polynomial. Math. Comp. 28, 589–597 (1974).

    Article  MATH  MathSciNet  Google Scholar 

  6. Collins, G. E., Loos, R.: Polynomial Real Root Isolation by Differentiation. SYMSAC 1976, 15–25.

    Google Scholar 

  7. Heindel, L. E.: Integer Arithmetic Algorithms for Polynomial Real Zero Determination. JACM 18,533–548 (1971).

    Article  MATH  MathSciNet  Google Scholar 

  8. Henrici, P.: Methods of Search for Solving Polynomial Equations. JACM 17, 273–283 (1970).

    Article  MATH  MathSciNet  Google Scholar 

  9. Henrici, P., Gargantini, I.: Uniformly Convergent Algorithms for the Simultaneous Approximation of All Zeros of a Polynomial. In: Constructive Aspects of the Fundamental Theorem of Algebra (Dejon, B., Henrici, P., eds.), pp. 77–114. London: Wiley-Interscience 1969.

    Google Scholar 

  10. Knuth, D. E.: The Art of Computer Programming, Vol. 2, Ist ed. (Seminumerical Algorithms). Reading, Mass.: Addison-Wesley 1969.

    MATH  Google Scholar 

  11. Kronecker, L.: Über den Zahlbegriff. Crelle J. reine und angew. Mathematik 101, 337–395 (1887).

    Google Scholar 

  12. Pinkert, J. R.: An Exact Method for Finding the Roots of a Complex Polynomial. TOMS 2, 351–363 (1976).

    Google Scholar 

  13. Uspensky, J. V.: Theory of Equations. New York: McGraw-Hill 1948.

    Google Scholar 

  14. Vincent, M.: Sur la Resolution des Equations Numeriques. J. de Mathematiques Pures et Appliquees 1, 341–372 (1836).

    Google Scholar 

  15. Weyl, H.: Randbemerkungen zu Hauptproblemen der Mathematik, II. Fundamentalsatz der Algebra und Grundlagen der Mathematik. Math. Z. 20, 131–150 (1924)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Computer Science Department, University of Wisconsin-Madison, 1210 West Dayton Street, Madison, WI, USA

    Prof. Dr. G. E. Collins

  2. Institut für Informatik I, Universität Karlsruhe Zirkel 2, D-7500, Karlsruhe, Federal Republic of Germany

    Prof. Dr. R. Loos

Authors
  1. Prof. Dr. G. E. Collins
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Prof. Dr. R. Loos
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institut für Mathematik, Johannes Kepler Universität, Linz, Austria

    Bruno Buchberger

  2. Computer Science Department, University of Wisconsin-Madison, Madison, Wis., USA

    George Edwin Collins

  3. Institut für Informatik I, Universität Karlsruhe, Federal Republic of Germany

    Rüdiger Loos

  4. Institut für Informatik und Numerische Mathematik, Universität Innsbruck, Austria

    Rudolf Albrecht

Rights and permissions

Reprints and permissions

Copyright information

© 1983 Springer-Verlag/Wien

About this chapter

Cite this chapter

Collins, G.E., Loos, R. (1983). Real Zeros of Polynomials. In: Buchberger, B., Collins, G.E., Loos, R., Albrecht, R. (eds) Computer Algebra. Computing Supplementa, vol 4. Springer, Vienna. https://doi.org/10.1007/978-3-7091-7551-4_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-7091-7551-4_7

  • Publisher Name: Springer, Vienna

  • Print ISBN: 978-3-211-81776-6

  • Online ISBN: 978-3-7091-7551-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation