, Volume 79, Issue 1, pp 53–60 | Cite as

Comparison of the Hansen-Sengupta and the Frommer-Lang-Schnurr existence tests

  • A. Goldsztejn


The Krawczyk and the Hansen-Sengupta interval operators are closely related to the interval Newton operator. These interval operators can be used as existence tests to prove existence of solutions for systems of equations. It is well known that the Krawczyk operator existence test is less powerful that the Hansen-Sengupta operator existence test, the latter being less powerful than the interval Newton operator existence test. In 2004, Frommer et al. proposed an existence test based on the Poincaré-Miranda theorem and proved that it is more powerful than the Krawczyk existence test. In this paper, we complete the classification of these four existence tests showing that, in practice, the Hansen-Sengupta existence test is actually more powerful than the existence test proposed by Frommer et al.

AMS Subject Classifications

65H10 65G20 65G40 


Nonlinear systems of equations existence test interval analysis 


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  1. Alefeld, G., Frommer, A., Heindel, G., Mayer, J. 2004On the existence theorems of Kantorovich, Miranda and BorsukElectronic Trans. Numer. Anal.17102111zbMATHGoogle Scholar
  2. Alefeld, G., Potra, F. A., Shen, Z.: On the existence theorems of Kantorovich, Moore and Miranda. In: Topics in Numerical Analysis with Special Emphasis on Nonlinear Problems (Alefeld, G., Chen, X., eds.), pp. 21–28. Computing Suppl. 2001Google Scholar
  3. Frommer, A., Lang, B., Schnurr, M. 2004A comparison of the Moore and Miranda existence testsComputing72349354zbMATHCrossRefMathSciNetGoogle Scholar
  4. Hayes, B. 2003A lucid intervalAmerican Scientist91484488CrossRefGoogle Scholar
  5. Kearfott, R. B. 1996Interval computations: Introduction, uses, and resourcesEuromath Bull.295112MathSciNetGoogle Scholar
  6. Miranda, C. 1940Un' osservazione su un teorema di BrouwerBolletino Unione Mathematica Italiana257MathSciNetGoogle Scholar
  7. Moore, R. E., Kioustelidis, J. 1980A simple test for accuracy of approximate solutions to nonlinear (or linear) systemsSIAM J. Numer. Anal.17521529zbMATHCrossRefMathSciNetGoogle Scholar
  8. Neumaier, A. 1990Interval methods for systems of equationsCambridge University PressCambridgezbMATHGoogle Scholar
  9. Neumaier, A. 1999A simple derivation of the Hansen-Bliek-Rohn-Ning-Kearfott enclosure for linear interval equationsReliab. Comp.5131136zbMATHCrossRefMathSciNetGoogle Scholar
  10. Poincaré, H. 1883Sur certaines solutions particulières du problème des trois corpsComptes rendus de l'Académie des sciences97251252Google Scholar
  11. Schnurr, M. 2005On the proofs of some statements concerning the theorems of Kantorovich, Moore and MirandaReliab. Comp.117785zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Wien 2007

Authors and Affiliations

  1. 1.Information and Computer ScienceUniversity of CaliforniaIrvineUSA

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