zbMATH — the first resource for mathematics

On locating all roots of systems of nonlinear equations inside bounded domain using global optimization methods. (English) Zbl 1193.65078
Summary: A novel method of locating all real roots of systems of nonlinear equations is presented. The root finding problem is transformed to an optimization problem, enabling the application of global optimization methods. Among many methods that exist in the global optimization literature, multistart and minfinder are applied because of their ability to locate not only the global minimum but also all local minima of the objective function. This procedure enables to locate all the possible roots of the system.
Various test cases are examined in order to validate the proposed procedure. This methodology does not make use of a priori knowledge of the number of the existing roots in the same manner as the corresponding global optimization methodology which does not make use of a priori knowledge of the existed number of local minima. The application of the new methodology results in finding all the roots in all test cases. The proposed methodology is general enough to be applied in any root finding problem.

65H10 Numerical computation of solutions to systems of equations
65K05 Numerical mathematical programming methods
90C30 Nonlinear programming
PDF BibTeX Cite
Full Text: DOI
[1] Kowalski, K.; Jankowski, K., Towards complete solutions to systems of nonlinear equations of many-electron theories, Physical review letters, 81, 1195-1198, (1998)
[2] Barbashov, B.M.; Nesterenko, V.V.; Chervyakov, A.M., General solutions of nonlinear equations in the geometric theory of the relativistic string, Communications in mathematical physics, 84, 471-481, (1982) · Zbl 0506.53008
[3] Bickham, S.R.; Kiselev, S.A.; Sievers, A.J., Stationary and moving intrinsic localized modes in one-dimensional monatomic lattices with cubic and quartic anharmonicity, Physical review B, 47, 14206-14211, (1993)
[4] Holstad, A., Numerical solution of nonlinear equations in chemical speciation calculations, Computational geosciences, 3, 229-257, (1999) · Zbl 0964.76074
[5] Argyros, K., On the solution of undetermined systems of nonlinear equations in Euclidean spaces, Pure mathematics and applications, 4, 199-209, (1993) · Zbl 0809.47053
[6] Semenov, V.S., The method of determining all real nonmultiple roots of systems of nonlinear equations, Computational mathematics and mathematical physics, 47, 1428-1434, (2007)
[7] Dennis, J.E.; Schnable, R., Numerical methods for unconstrained optimization and nonlinear equations, (1983), Prentice-Hall Englewood Cliffs, NJ, Mir, Moscow, 1988
[8] Bulatov, V.P., Numerical methods for finding all real roots of systems of nonlinear equations, Zhurnal vychislitel’noi mathematiki i matematicheskoi fiziki, Comput. math. phys., 40, 331-338, (2000) · Zbl 0990.65055
[9] Alolyan, I., An algorithm for finding all zeros of vector functions, Bulletin of the Australian mathematical society, 77, 353-363, (2008) · Zbl 1175.65057
[10] Denis, J.E., On newtons methods and nonlinear simulataneus replacements, SIAM journal of numerical analysis, 4, 103-108, (1967)
[11] Martinez, J.M., Algorithms for solving nonlinear systems of equations, (), 81-108 · Zbl 0828.90125
[12] Deuflhard, P., A modified Newton method for the solution of ill-conditioned systems of nonlinear equations with application to multiple shooting, Numerische Mathematik, 22, 289-315, (1974) · Zbl 0313.65070
[13] Sherman, A.H., On Newton-iterative methods for the solution of systems of nonlinear equations, SIAM journal on numerical analysis, 15, 755-771, (1978) · Zbl 0396.65019
[14] Bellavia, S.; Macconi, M.; Morini, B., An affine scaling trust-region approach to bound-constrained nonlinear systems, Applied numerical mathematics, 44, 257-280, (2003) · Zbl 1018.65067
[15] Lukšan, L., Inexact trust region method for large sparse systems of nonlinear equations, Journal of optimization theory and applicationsand applications, 81, 569-590, (1994) · Zbl 0803.65071
[16] Schnabel, R.B.; Frank, P.D., Tensor methods for nonlinear equations, SIAM journal on numerical analysis, 21, 815-843, (1994) · Zbl 0562.65029
[17] Bouaricha, A.; Schnabel, R.B., Tensor methods for large sparse systems of nonlinear equations, Mathematical programming, 82, 377-400, (1998) · Zbl 0951.65046
[18] Karr, C.L.; Weck, B.; Freeman, L.M., Solutions to systems of nonlinear equations via a genetic algorithm, Engineering applications of artificial intelligence, 11, 369-375, (1998)
[19] Mousa, A.A.; El-Desoky, I.M., GENLS: co-evolutionary algorithm for nonlinear systems of equations, Applied mathematics and computation, 197, 633-642, (2008) · Zbl 1135.65325
[20] Tsoulos, I.G.; Lagaris, I.E., Minfinder: locating all the local minima of a function, Computer physics communications, 174, 166-179, (2006) · Zbl 1196.90087
[21] Hirsch, M.J.; Pardalos, P.M.; Resende, M.G.C., Solving systems of nonlinear equations with continuous GRASP, Nonlinear analysis: real world applications, 10, 2000-2006, (2009) · Zbl 1163.90750
[22] Lagaris, I.E.; Tsoulos, I.G., Stopping rules for box-constrained stochastic global optimization, Applied mathematics and computation, 197, 622-632, (2008) · Zbl 1135.65330
[23] Effati, S.; Nazemi, A.R., A new method for solving a system of the nonlinear equations, Applied mathematics and computation, 168, 877-894, (2005) · Zbl 1081.65044
[24] Crosan, C.; Abraham, A.; Chang, T.G.; Kim, D.H., Solving nonlinear equation systems using an evolutionary multiobjective multiobjective optimization approach, ()
[25] Kubicek, M.; Hofmann, H.; Hlavacek, V.; Sinkule, J., Multiplicity and stability in a sequence of two nonadiabatic nonisothermal CSTR, Chemical engineering sciences, 35, 987-996, (1980)
[26] Floudas, C.A.; Pardalos, P.M.; Adjiman, C.; Esposito, W.; Gumus, Z.; Harding, S.; Klepeis, J.; Meyer, C.; Schweiger, C., Handbook of test problems in local and global optimization, (1999), Kluwer Acedemic Publishers Dordrecht · Zbl 0943.90001
[27] Floudas, C.A., Recent advances in global optimization for process synthesis, design and control: enclosure of all solutions, Computers and chemical engineering, 23, 963-973, (1999)
[28] Pramanik, S., Kinematic synthesis of a six-member mechanism for automotive steering, ASME journal of mechanical design, 124, 642-645, (2002)
[29] Wikipedia, Ackerman Steering Geometry, http://en.wikipedia.org/wiki/Ackerman_steering_geometry, 2006
[30] J.P. Merlet, The CORPIN examples page, http://www-sop.inria.gr/corpin/logiciels/ALIAS/Benches/benches.html, 2006
[31] Powell, M.J.D, A tolerant algorithm for linearly constrained optimization calculations, Mathematical programming, 45, 547-566, (1989) · Zbl 0695.90084
[32] Allgower, E.; Georg, K., Simplicial and continuation methods for approximating fixed points and solutions to systems of equations, SIAM review, 22, 28-85, (1980) · Zbl 0432.65027
[33] Yamamura, K.; Kawata, H.; Tokue, A., Interval solution of nonlinear equations using linear programming, BIT numerical mathematics, 38, 186-199, (1998) · Zbl 0908.65038
[34] Moré, J.J., A collection of nonlinear model problems, (), 723-762
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.