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Numerical differential continuation approach for systems of nonlinear equations with singular Jacobian | ||
| AUT Journal of Mathematics and Computing | ||
| مقاله 7، دوره 3، شماره 1، اردیبهشت 2022، صفحه 53-58 اصل مقاله (321.16 K) | ||
| نوع مقاله: Original Article | ||
| شناسه دیجیتال (DOI): 10.22060/ajmc.2021.20487.1068 | ||
| نویسنده | ||
| Mohammad Ali Mehrpouya* | ||
| Department of Mathematics, Tafresh University, 39518-79611, Tafresh, Iran | ||
| چکیده | ||
| It is well known that, one of the useful and rapid methods for a nonlinear system of algebraic equations is Newton’s method. Newton’s method has at least quadratic convergence when the Jacobian is a nonsingular matrix in a neighborhood of the solution. In this paper, a differential continuation method is presented for solving the nonlinear system of algebraic equations whose Jacobian matrix is singular at the solution. For this purpose, at first, an auxiliary equation named the homotopy equation is constructed. Then, by differentiating from the homotopy equation, a system of differential equations is replaced instead of the target problem and solved. In other words, the solution of the nonlinear system of algebraic equations with singular Jacobian is transformed to the solution of a system of differential equations. Some numerical tests are presented at the end and the computational efficiency of the method is described. | ||
| کلیدواژه‌ها | ||
| Nonlinear equations؛ Newton’s method؛ Singular Jacobian؛ Continuation method | ||
| مراجع | ||
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[2] R. L. Burden, J. D. Faires, and A. C. Reynolds, Numerical analysis, Cengage learning, 10 ed., 2016.
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[4] C. T. Kelley, Solving nonlinear equations with Newton’s method, vol. 1 of Fundamentals of Algorithms, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2003.
[5] C.-Y. Ku and Y.-H. Tsai, Solving nonlinear problems with singular initial conditions using a perturbed scalar homotopy method, Int. J. Nonlinear Sci. Numer. Simul., 14 (2013), pp. 367–375.
[6] M. A. Mehrpouya and S. Fallahi, A modified control parametrization method for the numerical solution of bang-bang optimal control problems, J. Vib. Control, 21 (2015), pp. 2407–2415.
[7] M. A. Mehrpouya and M. Khaksar-e Oshagh, An efficient numerical solution for time switching optimal control problems, Comput. Methods Differ. Equ., 9 (2021), pp. 225–243.
[8] M. A. Mehrpouya and M. Shamsi, Gauss pseudospectral and continuation methods for solving two-point boundary value problems in optimal control theory, Appl. Math. Model., 39 (2015), pp. 5047–5057.
[9] M. A. Mehrpouya, M. Shamsi, and V. Azhmyakov, An efficient solution of Hamiltonian boundary value problems by combined Gauss pseudospectral method with differential continuation approach, J. Franklin Inst., 351 (2014), pp. 4765–4785.
[10] W. C. Rheinboldt, Numerical continuation methods: a perspective, vol. 124, 2000, pp. 229–244. Numerical analysis 2000, Vol. IV, Optimization and nonlinear equations. | ||
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