JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2023, Vol. 58 ›› Issue (8): 118-126.doi: 10.6040/j.issn.1671-9352.0.2022.597

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A new modified efficient Shamanskii-like Levenberg-Marquardt method for solving systems of nonlinear equations

Minglei FANG(),Defeng DING*(),Ming WANG,Yuting SHENG   

  1. School of Mathmatics and Big Data, Anhui University of Science and Technology, Huainan 232001, Anhui, China
  • Received:2022-11-09 Online:2023-08-20 Published:2023-07-28
  • Contact: Defeng DING E-mail:fmlmath@sina.com;2021201353@edu.aust.cn

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Abstract:

A new Shamanskii-like Levenberg-Marquardt method(SLM) to solve systems of nonlinear equations is presented by introducing parameters in this paper. With m order nonmonotone Armijo line search, the global convergence of the new algorithm is proved and the convergence rate of whom is shown to be m+1. Numerical experiments demonstrate that the new algorithm can solve large scale problems effectively.

Key words: unconstrained optimization, Armijo line search, Shamanskii-like Levenberg-Marquardt method, systems of nonlinear equations

CLC Number: 

  • O221.2

Table 1

Numerical test data and results 1 $(\operatorname{rank}(\hat{\boldsymbol{J}}(\boldsymbol{x}))=n-1, m=4) $"

Problem n x0 算法1 算法2.1
NF NJ Time F NF NJ Time F
Problem1 1 000 1 17 9 76.555 3.19E-06 17 9 81.402 3.21E-06
10 17 9 67.410 3.19E-06 17 9 89.941 3.21E-06
100 17 9 76.029 3.19E-06 17 9 101.496 1.06E-07
Problem2 3 000 1 37 19 13.318 1.06E-07 37 19 16.446 1.06E-07
10 37 19 14.538 1.06E-07 37 19 17.367 1.06E-07
100 37 19 13.652 1.06E-07 37 19 17.522 2.05E-07
10 000 1 37 19 426.502 2.05E-07 37 19 474.798 2.05E-07
10 37 19 429.713 2.05E-07 37 19 521.876 2.05E-07
100 37 19 423.808 2.05E-07 37 19 784.704 3.33E-04
Problem3 3 000 1 23 12 10.889 3.32E-04 23 12 12.365 3.33E-04
10 23 12 11.383 3.32E-04 23 12 13.205 3.33E-04
100 23 12 12.419 3.32E-04 23 12 14.297 1.54E-04
10 000 1 25 13 379.290 1.52E-04 25 13 448.392 1.54E-04
10 25 13 381.506 1.52E-04 25 13 449.274 1.54E-04
100 25 13 402.279 1.52E-04 25 13 457.089 3.59E-05
Problem4 3 000 1 21 11 7.337 3.59E-05 21 11 7.499 4.16E-05
10 29 15 11.570 1.41E-04 31 16 12.383 3.83E-05
100 37 19 21.359 6.29E-05 55 28 32.069 6.55E-05
10 000 1 21 11 235.916 6.55E-05 21 11 297.483 1.08E-04
10 31 16 381.813 6.43E-05 31 16 389.080 2.25E-04
100 37 19 976.738 1.15E-04 89 45 1507.956 3.09E+00

Table 2

Numerical test data and results 2 $ (\operatorname{rank}(\hat{\boldsymbol{J}}(\boldsymbol{x}))=n-1, m=5)$"

Problem n x0 算法1 算法2.1
NF NJ Time F NF NJ Time F
Problem1 1 000 1 17 9 56.153 3.19E-06 17 9 61.007 3.21E-06
10 17 9 69.121 3.19E-06 17 9 97.596 3.21E-06
100 17 9 71.824 3.19E-06 17 9 97.673 3.21E-06
Problem2 1 37 19 15.059 1.06E-07 39 20 28.859 5.70E-08
3 000 10 37 19 14.791 1.06E-07 39 20 31.826 5.70E-08
100 37 19 14.508 1.06E-07 39 20 33.386 5.70E-08
1 37 19 540.234 2.05E-07 39 20 583.387 2.63E-07
10 000 10 37 19 772.284 2.05E-07 39 20 888.355 2.63E-07
100 37 19 749.794 2.05E-07 39 20 860.357 2.63E-07
Problem3 1 23 12 13.445 3.32E-04 23 12 27.244 3.33E-04
3 000 10 23 12 13.146 3.32E-04 23 12 23.083 3.33E-04
100 23 12 14.902 3.32E-04 23 12 26.641 3.33E-04
1 23 12 12.463 3.32E-04 25 13 700.327 1.54E-04
10 000 10 25 13 501.323 1.52E-04 25 13 562.015 1.54E-04
100 25 13 470.546 1.52E-04 25 13 692.879 1.54E-04
Problem4 1 21 11 12.101 3.59E-05 21 11 17.185 3.59E-05
3 000 10 29 15 16.989 1.41E-04 31 16 27.769 4.16E-05
100 37 19 48.289 6.29E-05 55 28 58.192 3.83E-05
1 21 11 261.971 6.55E-05 21 11 384.457 6.55E-05
10 000 10 31 16 431.609 6.43E-05 31 16 666.508 1.08E-04
100 37 19 2 034.931 1.15E-04 89 45 2 414.532 2.25E-04

Table 3

Numerical test data and results 3 $(\operatorname{rank}(\hat{\boldsymbol{J}}(\boldsymbol{x}))=n-2, m=4) $"

Problem n x0 算法1 算法2.1
NF NJ Time F NF NJ Time F
Problem1 1 000 1 17 9 56.100 3.19E-06 17 9 81.402 3.21E-06
10 17 9 56.283 3.19E-06 17 9 89.941 3.21E-06
100 17 9 55.768 3.19E-06 17 9 101.496 1.06E-07
Problem2 1 27 14 12.161 1.12E-04 37 19 16.446 1.06E-07
3 000 10 27 14 11.688 1.12E-04 37 19 17.367 1.06E-07
100 27 14 13.202 1.12E-04 37 19 17.522 2.05E-07
1 27 14 423.625 2.04E-04 37 19 474.798 2.05E-07
10 000 10 27 14 650.622 2.04E-04 37 19 521.876 2.05E-07
100 27 14 581.387 2.04E-04 37 19 784.704 3.33E-04
Problem3 1 25 13 12.663 1.17E-04 23 12 12.365 3.33E-04
3 000 10 25 13 14.205 1.17E-04 23 12 13.205 3.33E-04
100 25 13 14.663 1.17E-04 23 12 14.297 1.54E-04
1 25 13 757.739 1.65E-04 25 13 448.392 1.54E-04
10 000 10 25 13 926.005 1.65E-04 25 13 449.274 1.54E-04
100 25 13 488.993 1.65E-04 25 13 457.089 3.59E-05
Problem4 1 21 11 8.214 3.59E-05 21 11 7.498515 4.16E-05
3 000 10 29 15 13.087 1.41E-04 31 16 12.383 3.83E-05
100 37 19 25.485 6.29E-05 55 28 32.069 6.55E-05
1 21 11 268.594 6.55E-05 21 11 297.483 1.08E-04
10 000 10 31 16 441.670 6.43E-05 31 16 389.080 2.25E-04
100 37 19 1 131.421 1.15E-04 89 45 1 507.956 3.09E+00

Table 4

Numerical test data and results 4 $ (\operatorname{rank}(\hat{\boldsymbol{J}}(\boldsymbol{x}))=n-2, m=5)$"

Problem n x0 算法1 算法2.1
NF NJ Time F NF NJ Time F
Problem1 1 000 1 17 9 55.984 3.19E-06 200 100 531.416 5.10E-04
10 17 9 58.558 3.19E-06 200 100 545.202 5.10E-04
100 17 9 58.388 3.19E-06 200 100 546.315 5.10E-04
Problem2 1 27 14 11.443 1.12E-04 27 14 12.28 1.12E-04
3 000 10 27 14 17.987 1.12E-04 27 14 18.058 1.12E-04
100 27 14 16.723 1.12E-04 27 14 17.898 1.12E-04
1 27 14 598.614 2.04E-04 27 14 634.585 2.07E-04
10 000 10 27 14 503.249 2.04E-04 27 14 616.028 2.07E-04
100 27 14 555.029 2.04E-04 27 14 574.702 2.07E-04
Problem3 1 25 13 11.769 1.17E-04 25 13 13.818 1.17E-04
3 000 10 25 13 12.683 1.17E-04 25 13 19.147 1.17E-04
100 25 13 14.274 1.17E-04 25 13 18.621 1.17E-04
1 25 13 458.903 1.65E-04 25 13 754.511 1.64E-04
10 000 10 25 13 455.095 1.65E-04 25 13 739.808 1.64E-04
100 25 13 434.821 1.65E-04 25 13 2794.36 1.64E-04
Problem4 1 21 11 8.238 3.59E-05 21 11 11.469 3.59E-05
3 000 10 29 15 12.516 1.41E-04 31 16 18.836 4.16E-05
100 37 19 23.106 6.29E-05 55 28 38.185 3.83E-05
1 21 11 257.89 6.55E-05 21 11 320.29 6.55E-05
10 000 10 31 16 551.884 6.43E-05 31 16 568.913 1.08E-04
100 37 19 1 987.197 1.15E-04 89 45 7 778.456 2.25E-04
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