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2011, 1(1): 71-82. doi: 10.3934/naco.2011.1.71

A modified Fletcher-Reeves-Type derivative-free method for symmetric nonlinear equations

Dong-Hui Li 1, and  Xiao-Lin Wang 2,

1. 

School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China
2. 

College of Mathematics and Econometrics, Hunan University, Changsha, 410082, China

Received  October 2010 Revised  October 2010 Published  February 2011

In this paper, we propose a descent derivative-free method for solving symmetric nonlinear equations. The method is an extension of the modified Fletcher-Reeves (MFR) method proposed by Zhang, Zhou and Li [25] to symmetric nonlinear equations. It can be applied to solve large-scale symmetric nonlinear equations due to lower storage requirement. An attractive property of the method is that the directions generated by the method are descent for the residual function. By the use of some backtracking line search technique, the generated sequence of function values is decreasing. Under appropriate conditions, we show that the proposed method is globally convergent. The preliminary numerical results show that the method is practically effective.
Keywords: Symmetric nonlinear equations, descent direction, derivative-free method, global convergence..
Mathematics Subject Classification: Primary: 65H10; Secondary: 90C3.
Citation: Dong-Hui Li, Xiao-Lin Wang. A modified Fletcher-Reeves-Type derivative-free method for symmetric nonlinear equations. Numerical Algebra, Control & Optimization, 2011, 1 (1) : 71-82. doi: 10.3934/naco.2011.1.71
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[2]

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[3]

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[4]

IMA Journal of Numerical Analysis, 29 (2009), 814-825. doi: 10.1093/imanum/drn019.  Google Scholar

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IMA Journal of Numerical Analysis, 16 (1996), 155-164. doi: 10.1093/imanum/16.2.155.  Google Scholar

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Pacific Journal of Optimization, 2 (2006), 35-58.  Google Scholar

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Optimization Methods and Software, 18 (2003), 583-599. doi: 10.1080/10556780310001610493.  Google Scholar

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Mathematics of Computation, 75 (2006), 1429-1448. doi: 10.1090/S0025-5718-06-01840-0.  Google Scholar

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Applied Mathematics, Journal of Chinese Universities, Ser. B, 10 (1995), 75-82.  Google Scholar

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Hokkaido Journal of Mathematics, 36 (2007), 729-743.  Google Scholar

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SIAM Journal on Numerical Analysis, 37 (1999), 152-172. doi: 10.1137/S0036142998335704.  Google Scholar

[19]

Optimization Methods and Software, 13 (2000), 181-201. doi: 10.1080/10556780008805782.  Google Scholar

[20]

Q. Li and D. H. Li, A class of derivative-free methods for large-scale nonlinear monotone equations,, IMA Journal of Numerical Analysis, ().   Google Scholar

[21]

Mathematical Programming, 11 (1976), 42-49. doi: 10.1007/BF01580369.  Google Scholar

[22]

Mathematical Programming, 2 (1977), 241-254. doi: 10.1007/BF01593790.  Google Scholar

[23]

Journal of Computational and Applied Mathematics, 234 (2010), 649-657. doi: 10.1016/j.cam.2010.01.001.  Google Scholar

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Optimization Methods and Software, 22 (2007), 237-252. doi: 10.1080/10556780500397074.  Google Scholar

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Numerische Mathematik, 104 (2006), 561-572. doi: 10.1007/s00211-006-0028-z.  Google Scholar

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Journal of Computational Mathematics, 25 (2007), 89-96.  Google Scholar

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in "Integer and Nonlinear Programming" (eds. J. Abadie), North-Holland, Amsterdam, (1970), 37-86.  Google Scholar

show all references

References:
[1]

IMA Journal of Numerical Analysis, 5 (1985), 121-124. doi: 10.1093/imanum/5.1.121.  Google Scholar

[2]

SIAM Journal on Scientific Computing, 23 (2001), 940-960. doi: 10.1137/S1064827599363976.  Google Scholar

[3]

Journal of Australia Mathematical Society, Ser. B, 28 (1986), 75-92.  Google Scholar

[4]

IMA Journal of Numerical Analysis, 29 (2009), 814-825. doi: 10.1093/imanum/drn019.  Google Scholar

[5]

Journal of Computational Mathematics, 2 (1996), 142-148.  Google Scholar

[6]

IMA Journal of Numerical Analysis, 16 (1996), 155-164. doi: 10.1093/imanum/16.2.155.  Google Scholar

[7]

Shanghai Science and Technology Publisher, Shanghai, 2000. Google Scholar

[8]

Computer Journal, 7 (1964), 149-154. doi: 10.1093/comjnl/7.2.149.  Google Scholar

[9]

SIAM Journal on Optimization, 2 (1992), 21-42. doi: 10.1137/0802003.  Google Scholar

[10]

SIAM Journal on Numerical Analysis, 40 (2003), 1763-1774. doi: 10.1137/S0036142901397423.  Google Scholar

[11]

Systems Science and Mathematical Science, 11 (1998), 112-116.  Google Scholar

[12]

Pacific Journal of Optimization, 2 (2006), 35-58.  Google Scholar

[13]

Journal of Optimization Theory and Applications, 71 (1991), 399-405. doi: 10.1007/BF00939927.  Google Scholar

[14]

Optimization Methods and Software, 18 (2003), 583-599. doi: 10.1080/10556780310001610493.  Google Scholar

[15]

Mathematics of Computation, 75 (2006), 1429-1448. doi: 10.1090/S0025-5718-06-01840-0.  Google Scholar

[16]

Applied Mathematics, Journal of Chinese Universities, Ser. B, 10 (1995), 75-82.  Google Scholar

[17]

Hokkaido Journal of Mathematics, 36 (2007), 729-743.  Google Scholar

[18]

SIAM Journal on Numerical Analysis, 37 (1999), 152-172. doi: 10.1137/S0036142998335704.  Google Scholar

[19]

Optimization Methods and Software, 13 (2000), 181-201. doi: 10.1080/10556780008805782.  Google Scholar

[20]

Q. Li and D. H. Li, A class of derivative-free methods for large-scale nonlinear monotone equations,, IMA Journal of Numerical Analysis, ().   Google Scholar

[21]

Mathematical Programming, 11 (1976), 42-49. doi: 10.1007/BF01580369.  Google Scholar

[22]

Mathematical Programming, 2 (1977), 241-254. doi: 10.1007/BF01593790.  Google Scholar

[23]

Journal of Computational and Applied Mathematics, 234 (2010), 649-657. doi: 10.1016/j.cam.2010.01.001.  Google Scholar

[24]

Optimization Methods and Software, 22 (2007), 237-252. doi: 10.1080/10556780500397074.  Google Scholar

[25]

Numerische Mathematik, 104 (2006), 561-572. doi: 10.1007/s00211-006-0028-z.  Google Scholar

[26]

Journal of Computational Mathematics, 25 (2007), 89-96.  Google Scholar

[27]

Mathematics of Computation, 77 (2008), 2231-2240. doi: 10.1090/S0025-5718-08-02121-2.  Google Scholar

[28]

in "Integer and Nonlinear Programming" (eds. J. Abadie), North-Holland, Amsterdam, (1970), 37-86.  Google Scholar

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