Simple "Residual-Norm" Based Algorithms, for the solution of a large system of non-linear algebraic equations, which converge faster than the Newton's method

CMES - Computer Modeling in Engineering and Sciences

Volumn 71, Issue 3, 2011, Pages 279-304

  Liu, Chein Shan ^a   Atluri, Satya N ^b  

^a NATIONAL TAIWAN UNIVERSITY   (Taiwan)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Fictitious time integration method (FTIM); Iterative algorithm; Non linear ordinary differential equations; Non linear partial differential equations; Nonlinear algebraic equations; Residual Norm Based Algorithms (RNBAs)

Indexed keywords

FICTITIOUS TIME INTEGRATION METHODS; ITERATIVE ALGORITHM; NON-LINEAR ORDINARY DIFFERENTIAL EQUATIONS; NON-LINEAR PARTIAL DIFFERENTIAL EQUATIONS; NONLINEAR ALGEBRAIC EQUATIONS; RESIDUAL-NORM BASED ALGORITHMS (RNBAS);

ALGORITHMS; CONVERGENCE OF NUMERICAL METHODS; INTEGRAL EQUATIONS; JACOBIAN MATRICES; LINEAR EQUATIONS; NEWTON-RAPHSON METHOD; ORDINARY DIFFERENTIAL EQUATIONS; PARTIAL DIFFERENTIAL EQUATIONS; VECTOR SPACES;

NONLINEAR EQUATIONS;

EID: 79955519618     PISSN: 15261492     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (24)

1. Allgower, E. L.; Georg, K. (1990): Numerical Continuation Methods: an Introduction. Springer, New York.
2. Atluri, S. N. (2002): Methods of Computer Modeling in Engineering and Sciences. Tech. Science Press, 1400 pages.
3. Atluri, S. N.; Liu, H. T.; Han, Z. D. (2006): Meshless local Petrov-Galerkin (MLPG) mixed collocation method for elasticity problems. CMES: Computer Modeling in Engineering & Sciences, vol. 14, pp. 141-152. (Pubitemid 44562740)
4. Atluri, S. N.; Shen, S. (2002): The meshless local Petrov-Galerkin (MLPG) method: a simple & less-costly alternative to the finite and boundary element methods. CMES: Computer Modeling in Engineering & Sciences, vol. 3, pp. 11-51. (Pubitemid 135695145)
5. Atluri, S. N.; Zhu, T. L. (1998a): A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics. Comp. Mech., vol. 22, pp. 117-127. (Pubitemid 128620889)
6. Atluri, S. N.; Zhu, T. L. (1998b): A new meshless local Petrov-Galerkin (MLPG) approach to nonlinear problems in computer modeling and simulation. Comp. Model. Simul. Eng., vol. 3, pp. 187-196.
7. Boggs, P. T. (1971): The solution of nonlinear systems of equations by A-stable integration technique. SIAM J. Numer. Anal., vol. 8, pp. 767-785.
8. Brown, K. M. (1973): Computer oriented algorithms for solving systems of simultaneous nonlinear algebraic equations. In Numerical Solution of Systems of Nonlinear Algebraic Equations, Byrne, G. D. and Hall C. A. Eds., pp. 281-348, Academic Press, New York.
9. Davidenko, D. (1953): On a new method of numerically integrating a system of nonlinear equations. Doklady Akad. Nauk SSSR, vol. 88, pp. 601-604.
10. Deuflhard, P. (2004): Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms. Springer, New York.
11. Han, T.; Han Y. (2010): Solving large scale nonlinear equations by a new ODE numerical integration method. Appl. Math., vol. 1, pp. 222-229.
12. Hirsch, M.; Smale, S. (1979): On algorithms for solving f (x)=0. Commun. Pure Appl. Math., vol. 32, pp. 281-312.
13. Ku, C. Y.; Yeih, W.; Liu, C.-S. (2010): Solving non-linear algebraic equations by a scalar Newton-homotopy continuation method. Int. J. Non-Linear Sci. Num. Simul., vol. 11, pp. 435-450.
14. Liu, C.-S. (2001): Cone of nonlinear dynamical system and group preserving schemes. Int. J. Non-Linear Mech., vol. 36, pp. 1047-1068. (Pubitemid 32457387)
15. Liu, C.-S. (2008): A time-marching algorithm for solving non-linear obstacle problems with the aid of an NCP-function. CMC: Computers, Materials & Continua, vol. 8, pp. 53-65.
16. Liu, C.-S. (2009a): A fictitious time integration method for a quasilinear elliptic boundary value problem, defined in an arbitrary plane domain. CMC: Computers, Materials & Continua, vol. 11, pp. 15-32.
17. Liu, C.-S. (2009b): A fictitious time integration method for the Burgers equation. CMC: Computers, Materials & Continua, vol. 9, pp. 229-252.
18. Liu, C.-S. (2009c): A fictitious time integration method for solving delay ordinary differential equations. CMC: Computers, Materials & Continua, vol. 10, pp. 97-116.
19. Liu, C.-S.; Atluri, S. N. (2008): A novel time integration method for solving a large system of non-linear algebraic equations. CMES: Computer Modeling in Engineering & Sciences, vol. 31, pp. 71-83.
20. Liu, C.-S.; Chang, C. W. (2009): Novel methods for solving severely ill-posed linear equations system. J. Marine Sciences & Tech., vol. 17, pp. 216-227.
21. Liu, C.-S.; Yeih,W.; Kuo, C. L.; Atluri, S. N. (2009): A scalar homotopy method for solving an over/under-determined system of non-linear algebraic equations. CMES: Computer Modeling in Engineering & Sciences, vol. 53, pp. 47-71.
22. Ramm, A. G. (2007): Dynamical System Methods for Solving Operator Equations. Mathematics in Science and Engineering; Vol. 208 (series editor: Chu, C.K.), Elsevier, Amsterdam, Netherlands.
23. Zhu, T.; Zhang, J.; Atluri, S. N. (1998): A meshless local boundary integral equation (LBIE) method for solving nonlinear problems. Comp. Mech., vol. 22, pp. 174-186. (Pubitemid 128620894)
24. Zhu, T.; Zhang, J.; Atluri, S. N. (1999): A meshless numerical method based on the local boundary integral equation (LBIE) to solve linear and non-linear boundary value problems. Eng. Anal. Bound. Elem., vol. 23, pp. 375-389.