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Experiments using interval analysis for solving a circuit design problem. (English) Zbl 0793.90077
Summary: An already classical attempt at solving a circuit design problem leads to a system of 9 nonlinear equations in 9 variables. The sensitivity of the problem to small perturbations is extraordinarily high. Since 1974 several investigations have been made into this problem and they hint at one solution in the restricted domain of the nonnegative reals. The investigations did not give error estimates nor did they present conclusive evidence that the solution found is the only one in the domain of the nonnegative reals. Our paper reports on experimental computations which used various kinds of interval analytic methods while also sometimes reflecting on Wright-Cutteridge’s philosophy and theses. The computations resulted in a guarantee that in the domain of consideration, that is, the interval \([0,10]\) for each of the 9 variables, exactly one solution did exist, which was near the solution known up to now. Finally, our solution could be localized within a parallelepiped with edge lengths between \(10^{-6}\) and \(3.2\cdot 10^{-4}\).

MSC:
90C30 Nonlinear programming
65G30 Interval and finite arithmetic
Software:
INTBIS
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References:
[1] Alefeld, G. and Herzberger, J. (1983),Introduction to Interval Computations, Academic Press, New York. · Zbl 0552.65041
[2] Cutteridge, O. P. D. (1974), Powerful 2-part program for solution of nonlinear simultaneous equations,Electronics Letters 10, 182-184. · doi:10.1049/el:19740140
[3] Dimmer, P. R. and Cutteridge, O. P. D. (1980), Second derivative Gauss-Newton-based method for solving nonlinear simultaneous equations,IEE Proc. 127, 278-283
[4] Ebers, J. J. and Moll, J. L. (1954), Large-scale behavior of junction transistors,IEE Proc. 42, 1761-1772.
[5] Hansen, E. R. (1992),Global Optimization Using Interval Analysis, Marcel Dekker, New York. · Zbl 0762.90069
[6] Hansen, E. R. and Greenberg, R. I. (1983), An interval Newton method,Applied Math, and Comp. 12, 89-98. · Zbl 0526.65040 · doi:10.1016/0096-3003(83)90001-2
[7] Horst, R. and Tuy, H. (1990),Global Optimization, Springer-Verlag, Berlin. · Zbl 0704.90057
[8] IEEE (1985), IEEE standard for binary floating-point arithmetic. IEEE Standard 754-1985, IEEE, New York.
[9] IEEE (1987), IEEE standard for radix-independent floating-point arithmetic. IEEE Standard 854-1987, IEEE, New York.
[10] Kearfott, B. (1979), An efficient degree-computation method for a generalized method of bisection,Numerische Mathematik 32, 109-127. · Zbl 0393.65021 · doi:10.1007/BF01404868
[11] Kearfott, B. (1987), Some tests of generalized bisection,ACM Trans. Math. Software 13, 197-220. · Zbl 0632.65056 · doi:10.1145/29380.29862
[12] Kearfott, B. (1987), Abstract generalized bisection and a cost bound,Math, of Comput. 49, 187-202. · Zbl 0632.65055 · doi:10.1090/S0025-5718-1987-0890261-9
[13] Kearfott, B. (1990), INTBIS, a portable interval Newton/bisection package,ACM. Trans. Math. Software 16, 152-157. · Zbl 0900.65152 · doi:10.1145/78928.78931
[14] Kulisch, U. (ed.) (1986),PASCAL-SC Manual and System Disks, Wiley-Teubner Series in Computer Science, Stuttgart.
[15] Moore, R. E. (1979),Methods and Applications of Interval Analysis, SIAM, Philadelphia. · Zbl 0417.65022
[16] Moore, R. E. (1990), Interval tools for computer aided proofs in analysis, inComputer Aided Proofs in Analysis, ed. K. R. Meyer and D. S. Schmidt, IMA Series, vol. 28, Springer-Verlag, Berlin.
[17] Neumaier, A. (1990),Interval Methods for Systems of Equations, Cambridge University Press, Cambridge. · Zbl 0715.65030
[18] Price, W. L. (1978), A controlled random search procedure for global optimization, inTowards Global Optimization 2, ed. L. C. W. Dixon and G. P. Szegö, North-Holland, Amsterdam, pp. 71-84.
[19] Rall, L. B. (1981),Automatic Differentiation, Springer-Verlag, Berlin. · Zbl 0473.68025
[20] Ratschek, H. (1975), Nichtnumerische Aspekte der Intervallarithmetik, inInterval Mathematics, ed. by K. Nickel, Springer-Verlag, Berlin, pp. 48-74.
[21] Ratschek, H. and Rokne, J. (1984),Computer Methods for the Range of Functions, Ellis Horwood, Chichester. · Zbl 0584.65019
[22] Ratschek, H. and Rokne, J. (1988),New Computer Methods for Global Optimization, Ellis Horwood, Chichester. · Zbl 0648.65049
[23] Ratschek, H. and Rokne, J. (1991), Interval tools for global optimization,Computers and Mathematics with Applications 21, 41-50. · Zbl 0728.65060 · doi:10.1016/0898-1221(91)90159-2
[24] Ratschek, H. and Voller, R. (1991), What can interval analysis do for global optimization?Journal of Global Optimization 9, 111-130. · Zbl 0752.65054 · doi:10.1007/BF00119986
[25] Wright, D. J. and Cutteridge, O. P. D. (1976), Applied optimization and circuit design,Computer Aided Design 8, 70-76. · doi:10.1016/0010-4485(76)90087-7
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