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Finding all solutions of a class of nonlinear equations using an improved LP test

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Abstract

Recently, a new computational test has been proposed for nonexistence of a solution to a system of nonlinear equations using linear programming. This test is termed the LP test. It has been shown that the LP test is much more powerful than the conventional nonexistence test if the system of nonlinear equations consists of many linear terms and a relatively small number of nonlinear terms. By introducing the LP test to interval analysis, all solutions of nonlinear equations can be found very efficiently. In this paper, we propose some techniques for improving the computational efficiency of the LP test in some special cases. Using the proposed techniques, all solutions of a special class of nonlinear equations (including circuit equations) can be found very efficiently.

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References

  1. G. Alefeld and J. Herzberger, Introduction to Interval Computations. Academic Press, New York, 1983.

    MATH  Google Scholar 

  2. G. Alefeld and J. Herzberger, A quadratically convergent Krawczyk-like algorithm. SIAM J. Numer. Anal.,20 (1983), 210–219.

    Article  MATH  MathSciNet  Google Scholar 

  3. G. Alefeld and F. Potra, A new class of interval methods with higher order of convergence. Computing,42 (1989), 69–80.

    Article  MATH  MathSciNet  Google Scholar 

  4. E. Allgower and K. Georg, Simplicial and continuation methods for approximating fixed points and solutions to systems of equations. SIAM Rev.,22 (1980), 28–85.

    Article  MATH  MathSciNet  Google Scholar 

  5. L.O. Chua and P.M. Lin, Computer-Aided Analysis of Electronic Circuits: Algorithms and Computational Techniques. Prentice-Hall, Englewood Cliffs, New Jersey, 1975.

    MATH  Google Scholar 

  6. M. Hanada and M. Kashiwagi, An interval analysis using linear programming (simplex method). IEICE Technical Report, NLP96-96, Nov. 1996, 9–16 (in Japanese).

  7. E.R. Hansen, A globally convergent interval method for computing and bounding real roots. BIT,18 (1978), 415–424.

    Article  MATH  MathSciNet  Google Scholar 

  8. E.R. Hansen and S. Sengupta, Bounding solutions of systems of equations using interval analysis. BIT,21 (1981), 203–211.

    Article  MATH  MathSciNet  Google Scholar 

  9. M. Kashiwagi and S. Oishi, Numerical validation method for nonlinear equations using interval analysis and rational arithmetic. IEICE Trans.,J77-A (1994), 1372–1382 (in Japanese).

    Google Scholar 

  10. M. Kashiwagi, Interval arithmetic with linear programming — Extension of Yamamura’s idea —. Proceedings of 1996 International Symposium on Nonlinear Theory and its Applications, Kochi, Japan, October 1996, 61–64.

  11. R.B. Kearfott, Some tests of generalized bisection. ACM Trans. Math. Software,13 (1987), 197–220.

    Article  MATH  MathSciNet  Google Scholar 

  12. R.B. Kearfott, Preconditioners for the interval Gauss-Seidel method. SIAM J. Numer. Anal.,27 (1990), 804–822.

    Article  MATH  MathSciNet  Google Scholar 

  13. R.B. Kearfott, Interval arithmetic techniques in the computational solution of nonlinear systems of equations: Introduction, examples, and comparisons. Computational Solution of Nonlinear Systems of Equations (eds. E.L. Allgower and K. Georg), Lectures in Applied Mathematics,26, American Mathematical Society, Providence, RI, 1990, 337–357.

    Google Scholar 

  14. R.B. Kearfott, Decomposition of arithmetic expressions to improve the behavior of interval iteration for nonlinear systems. Computing,47 (1991), 169–191.

    Article  MATH  MathSciNet  Google Scholar 

  15. R.B. Kearfott and V. Kreinovich, eds., Applications of Interval Computations. Kluwer Academic Publishers, Dordrecht, 1996.

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  18. R.E. Moore, Methods and Applications of Interval Analysis. SIAM Studies in Applied Mathematics, Philadelphia, 1979.

  19. R.E. Moore and S.T. Jones, Safe starting regions for iterative methods. SIAM J. Numer. Anal.,14 (1977), 1051–1065.

    Article  MATH  MathSciNet  Google Scholar 

  20. R.E. Moore and L. Qi, A successive interval test for nonlinear systems. SIAM J. Numer. Anal.,19 (1982), 845–850.

    Article  MATH  MathSciNet  Google Scholar 

  21. J.J. More, A collection of nonlinear model problems. Computational Solution of Nonlinear Systems of Equations (eds. E.L. Allgower and K. Georg), Lectures in Applied Mathematics,26, American Mathematical Society, Providence, RI, 1990, 723–762.

    Google Scholar 

  22. G.L. Nemhauser, A.H.G. Rinnooy Kan, M.J. Todd, eds., Handbooks in Operations Research and Management Science (Vol. 1), Optimization. Elsevier Science Publishers B. V., Netherlands, 1989.

    Google Scholar 

  23. A. Neumaier, Interval iteration for zeros of systems of equations. BIT,25 (1985), 256–273.

    Article  MATH  MathSciNet  Google Scholar 

  24. A. Neumaier, Interval Methods for Systems of Equations. Cambridge University Press, Cambridge, England, 1990.

    MATH  Google Scholar 

  25. S. Oishi, A little bit difficult problem in numerical analysis with guaranteed accuracy is just solved. J. IEICE,79 (1996), 693–695 (in Japanese).

    Google Scholar 

  26. S. Pastore and A. Premoli, Polyhedral elements: A new algorithm for capturing all the equilibrium points of piecewise-linear circuits. IEEE Trans. Circuits and Systems-I,40 (1993), 124–132.

    Article  MATH  Google Scholar 

  27. W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical Recipes in C, The Art of Scientific Computing, Second Edition. Cambridge University Press, New York, 1992.

    MATH  Google Scholar 

  28. L. Qi, A note on the Moore test for nonlinear systems. SIAM J. Numer. Anal.,19 (1982), 851–857.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  30. H. Schwandt, Accelerating Krawczyk-like interval algorithms for the solution of nonlinear systems of equations by using second derivatives. Computing,35 (1985), 355–367.

    Article  MATH  MathSciNet  Google Scholar 

  31. H. Schwandt, Krawczyk-like algorithms for the solution of systems of nonlinear equations. SIAM J. Numer. Anal.,22 (1985), 792–810.

    Article  MATH  MathSciNet  Google Scholar 

  32. J.M. Shearer and M.A. Wolfe, An improved form of the Krawczyk-Moore algorithm. Appl. Math. Comp.,17 (1985), 229–239.

    MATH  MathSciNet  Google Scholar 

  33. J.M. Shearer and M.A. Wolfe, Some computable existence, uniqueness, and convergence tests for nonlinear systems. SIAM J. Numer. Anal.,22 (1985), 1200–1207.

    Article  MATH  MathSciNet  Google Scholar 

  34. M.A. Wolfe, A modification of Krawczyk’s algorithm. SIAM J. Numer. Anal.,17 (1980), 376–379.

    Article  MATH  MathSciNet  Google Scholar 

  35. K. Yamamura, Simple algorithms for tracing solution curves. IEEE Trans. Circuits and Systems-I,40 (1993), 537–541.

    Article  MATH  Google Scholar 

  36. K. Yamamura, Finding all solutions of piecewise-linear resistive circuits using simple sign tests. IEEE Trans. Circuits and Systems-I,40 (1993), 546–551.

    Article  MATH  Google Scholar 

  37. K. Yamamura and K. Horiuchi, A globally and quadratically convergent algorithm for solving nonlinear resistive networks. IEEE Trans. Computer-Aided Design of Integrated Circuits and Systems,9 (1990), 487–499.

    Article  Google Scholar 

  38. K. Yamamura, H. Kawata, and A. Tokue, Interval solution of nonlinear equations using linear programming. BIT,38 (1998), 186–199.

    Article  MATH  MathSciNet  Google Scholar 

  39. K. Yamamura and M. Mishina, An algorithm for finding all solutions of piecewise-linear resistive circuits. Int. J. Circuit Theory and Applications,24 (1996), 223–231.

    Article  MATH  Google Scholar 

  40. K. Yamamura and M. Ochiai, An efficient algorithm for finding all solutions of piecewise-linear resistive circuits. IEEE Trans. Circuits and Systems-I,39 (1992), 213–221.

    Article  MATH  Google Scholar 

  41. K. Yamamura and T. Ohshima, Finding all solutions of piecewise-linear resistive circuits using linear programming. IEEE Trans. Circuits and Systems-I,45 (1998), 434–445.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Electrical and Electronic Engineering, Faculty of Science and Engineering, Chuo University, 1-13-27, Kasuga, Bunkyo-ku, 112-8551, Tokyo, Japan

    Kiyotaka Yamamura

  2. NEC Corporation Ltd., Tokyo, Japan

    Masaki Nishizawa

Authors
  1. Kiyotaka Yamamura
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  2. Masaki Nishizawa
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Yamamura, K., Nishizawa, M. Finding all solutions of a class of nonlinear equations using an improved LP test. Japan J. Indust. Appl. Math. 16, 349–368 (1999). https://doi.org/10.1007/BF03167362

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  • DOI: https://doi.org/10.1007/BF03167362

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