Abstract
Let f(x) denote a system of n nonlinear functions in m variables, m≥n. Recently, a linearization of f(x) in a box x has been suggested in the form L(x)=Ax+b where A is a real n×m matrix and b is an interval n-dimensional vector. Here, an improved linearization L(x,y)=Ax+By+b, x∈x, y∈y is proposed where y is a p-dimensional vector belonging to the interval vector y while A and B are real matrices of appropriate dimensions and b is a real vector. The new linearization can be employed in solving various nonlinear problems: global solution of nonlinear systems, bounding the solution set of underdetermined systems of equations or systems of equalities and inequalities, global optimization. Numerical examples illustrating the superiority of L(x,y)=Ax+By+b over L(x)=Ax+b have been solved for the case where the problem is the global solution of a system of nonlinear equations (n=m).
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References
I. Braems, M. Kieffer and E. Walter, Set estimation, computation of volumes and data safety, in: Scientific Computing, Validated Numerics, eds. W. Kramer and J. Wolff von Gudenberg (Kluwer Academic/Plenum, New York, 2001) pp. 267–278.
K. Bühler and W. Barth, A new intersection algorithm for parametric surfaces based on linear interval estimations, in: Scientific Computing, Validated Numerics, eds. W. Kramer and J. Wolff von Gudenberg (Kluwer Academic/Plenum, New York, 2001) pp. 179–190.
J.L.D. Comba and J. Stolfi, Affine arithmetic and its applications to computer graphics, in: Proc. of VI SIBGRAPHI (Brasilian Symposium on Computer Graphics and Image Processing), 1990, pp. 9–18.
E.R. Hansen, A Generalised Interval Arithmetic, Lecture Notes in Computer Science (Springer, New York, 1989) pp. 75–82.
E. Hansen, Global Optimization Using Interval Analysis (Marcel Dekker, New York, 1992).
L. Kolev, Use of interval slopes for the irrational part of factorable functions, Reliable Comput. 3 (1997) 83–93.
L. Kolev, An efficient interval method for global analysis of nonlinear resistive circuits, Internat. J. Circuit Theory Appl. 26 (1998) 81–92.
L. Kolev, A new method for global solution of systems of nonlinear equations, Reliable Comput. 4 (1998) 1–21.
L. Kolev, An improved method for global solution of nonlinear systems, Reliable Comput. 5(2) (1999) 103–111.
L. Kolev, A new interval approach to global optimization, Comput. Math. Math. Phys. 39(5) (1999) 727–737.
L. Kolev, Automatic computation of a linear interval enclosure, Reliable Comput. 7 (2001) 17–28.
L. Kolev and V. Mladenov, A linear programming implementation of an interval method for global nonlinear DC analysis, in: IEEE Internat. Conf. Electronics, Circuits and Systems. Surfing the Waves of Science and Technology, Vol. 1, Instituto Superior Tecnico Lisboa, Portugal, 7–10 September 1998, pp. 75–78.
R. Krawczuk and A. Neumaier, Interval slopes for rational functions and associated centred forms, SIAM J. Numer. Anal. 22 (1985) 604–616.
I. Nenov and D. Fylstra, Interval methods for accelerated global search in the microsoft exel solver, in: Extended Abstracts of Validated Computing 2002, ed. R.B. Kearfott, Toronto, Canada, 23–25 May 2002, pp. 145–147.
H. Ratschek and J. Rokne, Experiment using interval analysis for solving a circuit design problem, J. Global Optim. 3 (1993) 546–551.
S. Zuhe and M. A. Wolfe, On interval enclosures using slope arithmetic, Appl. Math. Comput. 39 (1990) 89–105.
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Kolev, L.V. An Improved Interval Linearization for Solving Nonlinear Problems. Numerical Algorithms 37, 213–224 (2004). https://doi.org/10.1023/B:NUMA.0000049468.03595.4c
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DOI: https://doi.org/10.1023/B:NUMA.0000049468.03595.4c