Reliable Computing

, Volume 9, Issue 3, pp 241–250 | Cite as

Finding All Solution Sets of Piecewise-Trapezoidal Equations Described by Set-Valued Functions

  • Kiyotaka Yamamura


This letter deals with the problem of bounding all solution sets to systems of nonlinear equations where nonlinear terms are described by set-valued functions termed piecewise-trapezoidal functions. Such a problem is important in the numerical computation with guaranteed accuracy and in the analysis of fluctuated systems (such as the tolerance analysis of electronic circuits). It is shown that the proposed algorithm could find all solution sets to a system of 300 piecewise-trapezoidal equations approximately in about 30 hours using a 360 MHz computer.


Mathematical Modeling Numerical Computation Computational Mathematic Industrial Mathematic Nonlinear Equation 
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  1. 1.
    Alefeld, G. and Herzberger, J.: Introduction to Interval Computations, Academic Press, New York, 1983.Google Scholar
  2. 2.
    Berz, M. and Hoffstätter, G.: Computation and Application of Taylor Polynomials with Interval Remainder Bounds, Reliable Computing 4(1) (1998), pp. 83-97.Google Scholar
  3. 3.
    Kashiwagi, M.: Interval Arithmetic with Linear Programming-Extension of Yamamura's Idea, in: Proc. of the 1996 Int. Symp. on Nonlinear Theory and Its Applications, Kochi, Japan, October 7-9, 1996, pp. 61-64.Google Scholar
  4. 4.
    Kashiwagi, M.: Simplex Method for Calculating Optimal Value with Guaranteed Accuracy, in: Proc. of the 1997 Int. Symp. on Nonlinear Theory and Its Applications, Hawaii, November 29-December 3, 1997, pp. 317-320.Google Scholar
  5. 5.
    Kearfott, R. B. and Kreinovich, V. (eds): Applications of Interval Computations, Kluwer Academic Publishers, Dordrecht, 1996.Google Scholar
  6. 6.
    Moore, R. E.: Methods and Applications of Interval Analysis, SIAM Studies in Applied Mathematics, Philadelphia, 1979.Google Scholar
  7. 7.
    Neumaier, A.: Interval Methods for Systems of Equations, Cambridge University Press, Cambridge, 1990.Google Scholar
  8. 8.
    Ushida, A. and Chua, L. O.: Tracing Solution Curves of Non-Linear Equations with Sharp Turning Points, Int. J. Circuit Theory and Applications 12(1) (1984), pp. 1-21.Google Scholar
  9. 9.
    Yamamura, K.: An Algorithm for Representing Functions of Many Variables by Superpositions of Functions of One Variable and Addition, IEEE Trans. Circuits and Systems-I 43(4) (1996), pp. 338-340.Google Scholar
  10. 10.
    Yamamura, K., Kawata, H., and Tokue, A.: Interval Solution of Nonlinear Equations Using Linear Programming, BIT 38(1) (1998), pp. 186-199.Google Scholar
  11. 11.
    Yamamura, K. and Tanaka, S.: Finding All Solutions of Systems of Nonlinear Equations Using the Dual Simplex Method, BIT 42(1) (2002), pp. 214-230.Google Scholar
  12. 12.
    Yamamura, K. and Yuasa, T.: Finding All Solutions of Piecewise-Trapezoidal Circuits Described by Set-Valued Mappings, in: Proc. of the 1998 Engineering Sciences Society Conference of IEICE, Kofu, Japan, September 29-October 2, 1998, p. 29 (in Japanese).Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Kiyotaka Yamamura
    • 1
  1. 1.Department of Electrical, Electronic, and Communication Engineering, Faculty of Science and EngineeringChuo University, Bunkyo-kuTokyoJapan

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