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The Convergence Conditions of Interval Newton’s Method Based on Point Estimates

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Computer Algebra in Scientific Computing (CASC 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10490))

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Abstract

Both Smale’s alpha theory and Rump’s interval theorem provide the conditions which guarantee the existence of a simple solution of a square nonlinear system. In this paper, we generalize the conclusion provided by Rall to reveal the relationship between Smale’s alpha theory and Rump’s interval theorem. By point estimates, we propose the conditions under which the condition of Rump’s interval theorem holds. Furthermore, using only the information of the given system at the initial approximate point, we give the convergence conditions of interval Newton’s algorithm proposed by Rump.

Z. Li—This research was supported by Chinese National Natural Science Foundation under Grant Nos. 11601039, 11671169, 11501051, by the open fund Key Lab. of Symbolic Computation and Knowledge Engineering under Grant No. 93K172015K06, and by the Education Department of Jilin Province, “13th Five-Year” science and technology project under Grant No. JJKH20170618KJ.

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References

  1. Blum, L., Cucker, F., Shub, M., Smale, S.: Complexity and Real Computation. Springer, New York (1998)

    Book  MATH  Google Scholar 

  2. Kantorovich, L.V.: Functional analysis and applied mathematics. Uspehi. Mat. Nauk. 3(6), 89–185 (1948)

    MathSciNet  MATH  Google Scholar 

  3. Giusti, M., Lecerf, G., Salvy, B., Yakoubsohn, J.C.: On location and approximation of clusters of zeros of analytic functions. Found. Comput. Math. 5(3), 257–311 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Giusti, M., Lecerf, G., Salvy, B., Yakoubsohn, J.C.: On location and approximation of clusters of zeros: case of embedding dimension one. Found. Comput. Math. 7(1), 1–58 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Hauenstein, J.D., Levandovskyy, V.: Certifying solutions to square systems of polynomial-exponential equations. J. Symb. Comput. 79(3), 575–593 (2015)

    MathSciNet  MATH  Google Scholar 

  6. Hauenstein, J.D., Sottile, F.: Algorithm 921: alphacertified: certifying solutions to polynomial systems. ACM Trans. Math. Softw. 38(4), 1–20 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Li, N., Zhi, L.: Verified error bounds for isolated singular solutions of polynomial systems: case of breadth one. Theor. Comput. Sci. 479, 163–173 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Li, N., Zhi, L.: Verified error bounds for isolated singular solutions of polynomial systems. SIAM J. Numer. Anal. 52(4), 1623–1640 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Krawczyk, R.: Newton-algorithmen zur bestimmung von nullstellen mit fehlerschranken. Computing 4, 187–201 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  10. Moore, R.E.: A test for existence of solutions to nonlinear system. SIAM J. Numer. Anal. 14(4), 611–615 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  11. Rall, L.B.: A comparison of the existence theorems of Kantorvich and Moore. SIAM J. Numer. Anal. 17(1), 148–161 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  12. Rump, S.M.: Solving algebraic problems with high accuracy. In: Kulisch, W.L., Miranker, W.L. (eds.) A New Approach to Scientific Computation, pp. 51–120. Academic Press, San Diego (1983)

    Chapter  Google Scholar 

  13. Rump, S.M.: INTLAB-Interval Laboratory. Springer, Netherlands, Berlin (1999)

    Book  MATH  Google Scholar 

  14. Rump, S.M., Graillat, S.: Verified error bounds for multiple roots of systems of nonlinear equations. Numer. Algorithm 54, 359–377 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Smale, S.: Newton method estimates from data at one point. The Merging of Disciplines: New Directions in Pure, Applied and Computational Mathematics, pp. 185C–196C. Springer, Berlin (1986)

    Chapter  Google Scholar 

  16. Shub, M., Smale, S.: Computational complexity: on the geometry of polynomials and a theory of cost. I. Ann. Sci. Ècole Norm. Sup. 18(1), 107–142 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  17. Shub, M., Smale, S.: Computational complexity: on the geometry of polynomials and a theory of cost. II. SIAM J. Comput. 15(1), 145–161 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  18. Wang, X.H., Han, D.F.: On dominating sequence method in the point estimate and Smale theorem. Sci. China Ser. A 33(2), 135–144 (1990)

    MathSciNet  MATH  Google Scholar 

  19. Yang, Z., Zhi, L., Zhu, Y.: Verfied error bounds for real solutions of positive-dimensional polynomial systems. In: The 2013 International Symposium on Symbolic and Algebraic Computation, pp. 371–378. ACM press, San Jose (2013)

    Google Scholar 

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Authors and Affiliations

  1. School of Science, Changchun University of Science and Technology, Changchun, 130022, China

    Zhe Li

  2. School of Mathematics, Key Laboratory of Symbolic Computation and Knowledge Engineering, Jilin University, Changchun, 130012, China

    Zhe Li, Baocheng Wan & Shugong Zhang

  3. Jilin Agricultural University, Changchun, 130118, China

    Baocheng Wan

Authors
  1. Zhe Li
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  2. Baocheng Wan
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  3. Shugong Zhang
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Corresponding author

Correspondence to Zhe Li .

Editor information

Editors and Affiliations

  1. Joint Institute of Nuclear Research, Dubna, Russia

    Vladimir P. Gerdt

  2. Universität Kassel, Kassel, Germany

    Wolfram Koepf

  3. Universität Kassel, Kassel, Germany

    Werner M. Seiler

  4. Russian Academy of Sciences, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

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Li, Z., Wan, B., Zhang, S. (2017). The Convergence Conditions of Interval Newton’s Method Based on Point Estimates. In: Gerdt, V., Koepf, W., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2017. Lecture Notes in Computer Science(), vol 10490. Springer, Cham. https://doi.org/10.1007/978-3-319-66320-3_20

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  • DOI: https://doi.org/10.1007/978-3-319-66320-3_20

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-66319-7

  • Online ISBN: 978-3-319-66320-3

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