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A comparison of the Newton-Krylov method with high order Newton-like methods to solve nonlinear systems. (English) Zbl 1204.65055

Summary: We compare the CPU time and error estimates of some variants of Newton method of the third and fourth-order convergence with those of the Newton-Krylov method used to solve systems of nonlinear equations. By expanding some numerical experiments we show that the use of Newton-Krylov method is better in the cost and accuracy points of view than the use of other high order Newton-like methods when the system is sparse and its size is large.

MSC:

65H10 Numerical computation of solutions to systems of equations

Software:

CGS
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References:

[1] Dembo, R. S.; Eisenstat, S. C.; Steihaug, T., Inexact Newton methods, SIAM J. Numer. Anal., 19, 400-408 (1982) · Zbl 0478.65030
[2] Frontini, M.; Sormani, E., Third-order methods from quadrature formulae for solving systems of nonlinear equations, Appl. Math. Comput., 149, 771-782 (2004) · Zbl 1050.65055
[3] Cordero, A.; Torregrosa, J. R., Variants of Newton’s method for functions of several variables, Appl. Math. Comput., 183, 199-208 (2006) · Zbl 1123.65042
[4] Darvishi, M. T.; Barati, A., A third-order Newton-type method to solve systems of nonlinear equations, Appl. Math. Comput., 187, 630-635 (2007) · Zbl 1116.65060
[5] Darvishi, M. T.; Barati, A., A fourth-order method from quadrature formulae to solve systems of nonlinear equations, Appl. Math. Comput., 188, 257-261 (2007) · Zbl 1118.65045
[6] Darvishi, M. T.; Barati, A., Super cubic iterative methods to solve systems of nonlinear equations, Appl. Math. Comput., 188, 1678-1685 (2007) · Zbl 1119.65045
[7] Babajee, D. K.R.; Dauhoo, M. Z.; Darvishi, M. T.; Barati, A., A note on the local convergence of iterative methods based on Adomian decomposition method and 3-node quadrature rule, Appl. Math. Comput., 200, 452-458 (2008) · Zbl 1160.65018
[8] Hestenes, M. R.; Stiefel, E., Methods of conjugate gradients for solving linear systems, J. Res. Nat. Bur. Stand., 49, 409-435 (1952) · Zbl 0048.09901
[9] Saad, Y.; Schultz, M. H., GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 7, 856-869 (1986) · Zbl 0599.65018
[10] van der Vorst, H. A., Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 13, 631-644 (1992) · Zbl 0761.65023
[11] Freund, R. W., A transpose-free quasi-minimal residual algorithm for non-Hermitian linear systems, SIAM J. Sci. Comput., 14, 470-482 (1993) · Zbl 0781.65022
[12] Eisenstat, S. C.; Walker, H. F., Globally convergent inexact Newton methods, SIAM J. Optim., 4, 393-422 (1994) · Zbl 0814.65049
[13] Bai, Z.-Z.; Golub, G. H.; Lu, L.-Z.; Yin, J.-F., Block triangular and skew-Hermitian splitting methods for positive-definite linear systems, SIAM J. Sci. Comput., 26, 3, 844-863 (2005) · Zbl 1079.65028
[14] Bai, Z.-Z.; Golub, G. H.; Ng, M. K., Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl., 24, 603-626 (2003) · Zbl 1036.65032
[15] Bai, Z.-Z.; Sun, J.-C.; Wang, D.-R., A unified framework for the construction of various matrix multisplitting iterative methods for large sparse system of linear equations, Comput. Math. Appl., 32, 12, 51-76 (1996) · Zbl 0870.65025
[16] Saad, Y., Iterative Methods for Sparse Linear Systems (1996), PWS Publishing Company: PWS Publishing Company Boston · Zbl 1002.65042
[17] An, H.-B.; Bai, Z.-Z., A globally convergent Newton-GMRES method for large sparse systems of nonlinear equations, Appl. Numer. Math., 57, 235-252 (2007) · Zbl 1123.65040
[18] Dennis, J. E.; Schnabel, R. B., Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Series in Automatic Computation (1983), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0579.65058
[19] Eisenstat, S. C.; Walker, H. F., Choosing the forcing terms in an inexact Newton method, SIAM J. Sci. Comput., 17, 16-32 (1996) · Zbl 0845.65021
[20] Shadid, J. N.; Tuminaro, R. S.; Walker, H. F., An inexact Newton method for fully coupled solution of the Navier-Stokes equations with heat and mass transport, J. Comput. Phys., 137, 155-185 (1997) · Zbl 0898.76066
[21] Tuminaro, R. S.; Walker, H. F.; Shadid, J. N., On backtracking failure in Newton-GMRES methods with a demonstration for the Navier-Stokes equations, J. Comput. Phys., 180, 549-558 (2002) · Zbl 1143.76489
[22] Pawlowski, R. P.; Shadid, J. N.; Simonis, J. P.; Walker, H. F., Globalization techniques for Newton-Krylov methods and applications to the fully coupled solution of the Navier-Stokes equations, SIAM Rev., 48, 4, 700-721 (2006) · Zbl 1110.65039
[23] Sonneveld, P., CGS, a fast Lanczos-type solver for nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 10, 36-52 (1989) · Zbl 0666.65029
[24] Walker, H. F.; Zhou, L., A simpler GMRES, Numer. Linear Algebra Appl., 1, 6, 571-581 (1994) · Zbl 0838.65030
[25] Babajee, D. K.R.; Dauhoo, M. Z.; Darvishi, M. T.; Karami, A.; Barati, A., Analysis of two Chebyshev-like third order methods free from second derivatives for solving systems of nonlinear equations, J. Comput. Appl. Math., 233, 2002-2012 (2010) · Zbl 1204.65050
[26] Nedzhibov, G. H., A family of multi-point iterative methods for solving systems of nonlinear equations, J. Comput. Appl. Math., 222, 2, 244-250 (2008) · Zbl 1154.65037
[27] Kelley, C. T., Solution of the Chandrasekhar H-equation by Newton’s method, J. Math. Phys., 21, 1625-1628 (1980) · Zbl 0439.45020
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