The Continuous Analogy of Newton’s Method for Solving a System of Linear Algebraic Equations

Affiliation(s)

School of Mathematics and Computer Science, National University of Mongolia, Ulan-Bator, Mongolia.

School of Mathematics and Computer Science, National University of Mongolia, Ulan-Bator, Mongolia&Joint Institute for Nuclear Research, Dubna, Moscow region, Russia.

School of Mathematics and Computer Science, National University of Mongolia, Ulan-Bator, Mongolia.

School of Mathematics and Computer Science, National University of Mongolia, Ulan-Bator, Mongolia&Joint Institute for Nuclear Research, Dubna, Moscow region, Russia.

ABSTRACT

We propose a continuous analogy of Newton’s method with inner iteration for solving a system of linear algebraic equations. Implementation of inner iterations is carried out in two ways. The former is to fix the number of inner iterations in advance. The latter is to use the inexact Newton method for solution of the linear system of equations that arises at each stage of outer iterations. We give some new choices of iteration parameter and of forcing term, that ensure the convergence of iterations. The performance and efficiency of the proposed iteration is illustrated by numerical examples that represent a wide range of typical systems.

Cite this paper

T. Zhanlav, O. Chuluunbaatar and G. Ankhbayar, "The Continuous Analogy of Newton’s Method for Solving a System of Linear Algebraic Equations,"*Applied Mathematics*, Vol. 4 No. 1, 2013, pp. 210-216. doi: 10.4236/am.2013.41A032.

T. Zhanlav, O. Chuluunbaatar and G. Ankhbayar, "The Continuous Analogy of Newton’s Method for Solving a System of Linear Algebraic Equations,"

References

[1] R. Kress, “Numerical Analysis,” Springer-Verlag, Berlin, 1998. doi:10.1007/978-1-4612-0599-9

[2] T. Zhanlav, “On the Iteration Method with Minimal Defect for Solving a System of Linear Algebraic Equations,” Scientific Transaction, No. 8, 2001, pp. 59-64.

[3] T. Zhanlav and I. V. Puzynin, “The Convergence of Iterations Based on a Continuous Analogy Newton’s Method,” Journal of Computational Mathematics and Mathematical Physics, Vol. 32, No. 6, 1992, pp. 729-737.

[4] R. S. Dembo, S. C. Eisenstat and T. Steihaug, “Inexact Newton Methods,” SIAM Journal on Numerical Analysis, Vol. 19, No. 2, 1982, pp. 400-408. doi:10.1137/0719025

[5] H. B. An, Z. Y. Mo and X. P. Liu, “A Choice of Forcing Terms in Inexact Newton Method,” Journal of Computational and Applied Mathematics, Vol. 200, No. 1, 2007, pp. 47-60. doi:10.1016/j.cam.2005.12.030

[6] T. Zhanlav, O. Chuluunbaatar and G. Ankhbayar, “Relationship between the Inexact Newton Method and the Continuous Analogy of Newton Method,” Revue D’Analyse Numerique et de Theorie de L’Approximation, Vol. 40, No. 2, 2011, pp. 182-189.

[7] T. Zhanlav and O. Chuluunbaatar, “The Local and Global Convergence of the Continuous Analogy of Newton’s Method,” Bulletin of PFUR, Series Mathematics, Information Sciences, Physics, No. 1, 2012, pp. 34-43.

[8] J. W. Demmel, “Applied Numerical Linear Algebra,” SIAM, Philadelphia, 1997, pp. 265-360. doi:10.1137/1.9781611971446

[9] W. Hackbusch, “Elliptic Differential Equations: Theory and Numerical Treatment,” Springer Series in Computational Mathematics, Vol. 18, Springer, Berlin, 1992.

[1] R. Kress, “Numerical Analysis,” Springer-Verlag, Berlin, 1998. doi:10.1007/978-1-4612-0599-9

[2] T. Zhanlav, “On the Iteration Method with Minimal Defect for Solving a System of Linear Algebraic Equations,” Scientific Transaction, No. 8, 2001, pp. 59-64.

[3] T. Zhanlav and I. V. Puzynin, “The Convergence of Iterations Based on a Continuous Analogy Newton’s Method,” Journal of Computational Mathematics and Mathematical Physics, Vol. 32, No. 6, 1992, pp. 729-737.

[4] R. S. Dembo, S. C. Eisenstat and T. Steihaug, “Inexact Newton Methods,” SIAM Journal on Numerical Analysis, Vol. 19, No. 2, 1982, pp. 400-408. doi:10.1137/0719025

[5] H. B. An, Z. Y. Mo and X. P. Liu, “A Choice of Forcing Terms in Inexact Newton Method,” Journal of Computational and Applied Mathematics, Vol. 200, No. 1, 2007, pp. 47-60. doi:10.1016/j.cam.2005.12.030

[6] T. Zhanlav, O. Chuluunbaatar and G. Ankhbayar, “Relationship between the Inexact Newton Method and the Continuous Analogy of Newton Method,” Revue D’Analyse Numerique et de Theorie de L’Approximation, Vol. 40, No. 2, 2011, pp. 182-189.

[7] T. Zhanlav and O. Chuluunbaatar, “The Local and Global Convergence of the Continuous Analogy of Newton’s Method,” Bulletin of PFUR, Series Mathematics, Information Sciences, Physics, No. 1, 2012, pp. 34-43.

[8] J. W. Demmel, “Applied Numerical Linear Algebra,” SIAM, Philadelphia, 1997, pp. 265-360. doi:10.1137/1.9781611971446

[9] W. Hackbusch, “Elliptic Differential Equations: Theory and Numerical Treatment,” Springer Series in Computational Mathematics, Vol. 18, Springer, Berlin, 1992.