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doi: 10.3934/jimo.2022093
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Solving tensor complementarity problems with $ Z $-tensors via a weighted fixed point method

Liyuan Tian and  Yong Wang ,

School of Mathematics, Tianjin University, Tianjin 300350, China

* Corresponding author: Yong Wang

Received  December 2021 Revised  April 2022 Early access June 2022

Fund Project: The work is partly supported by the National Natural Science Foundation of China (Grants No. 12171357, 11871051)

In this paper, we focus on solving the tensor complementarity problems with $ Z $-tensors. To this end, a weighted fixed point method is proposed for solving such tensor complementarity problems. Then, it is showed that the iterative sequence generated by the algorithm is monotonically decreasing with the help of $ Z $-tensors. Moreover, the limit point of the iterative sequence is a solution of the corresponding tensor complementarity problem. Finally numerical experiments show the effectiveness of the algorithm and testify the theoretical conclusions.

Keywords: Tensor complementarity problem, weighted fixed point method, Z-tensor, L-tensor, monotone convergence.
Mathematics Subject Classification: Primary: 15A69, 90C22.
Citation: Liyuan Tian, Yong Wang. Solving tensor complementarity problems with $ Z $-tensors via a weighted fixed point method. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022093
References:
[1]

B. H. Ahn, Solution of nonsymmetric linear complementarity problems by iterative methods, J. Optim. Theory Appl., 33 (1981), 175-185.  doi: 10.1007/BF00935545.

[2]

X.-L. Bai, Z.-H. Huang and Y. Wang, Global uniqueness and solvability for tensor complementarity problems, J. Optim. Theory Appl., 170 (2016), 72-84.  doi: 10.1007/s10957-016-0903-4.

[3]

P.-F. Dai, A fixed point iterative method for tensor complementarity problems, J. Sci. Comput., 84 (2020), Paper No. 49, 20 pp. doi: 10.1007/s10915-020-01299-6.

[4]

S. Du and L. Zhang, A mixed integer programming approach to the tensor complementarity problem, J. Global Optim., 73 (2019), 789-800.  doi: 10.1007/s10898-018-00731-4.

[5]

M. S. Gowda, Z. Luo, L. Qi and N. Xiu, Z-tensors and complementarity problems, arXiv: 1510.07933v2, 2015.

[6]

H.-B. Guan and D.-H. Li, Linearized methods for tensor complementarity problems, J. Optim. Theory Appl., 184 (2020), 972-987.  doi: 10.1007/s10957-019-01627-3.

[7]

L. Han, A continuation method for tensor complementarity problems, J. Optim. Theory Appl., 180 (2019), 949-963.  doi: 10.1007/s10957-018-1422-2.

[8]

Z.-H. Huang and L. Qi, Formulating an $n$-person noncooperative game as a tensor complementarity problem, Comput. Optim. Appl., 66 (2017), 557-576.  doi: 10.1007/s10589-016-9872-7.

[9]

Z.-H. Huang and L. Qi, Tensor complementarity problems-Part Ⅰ: Basic theory, J. Optim. Theory Appl., 183 (2019), 1-23.  doi: 10.1007/s10957-019-01566-z.

[10]

Z.-H. Huang and L. Qi, Tensor complementarity problems-Part Ⅲ: Applications, J. Optim. Theory Appl., 183 (2019), 771-791.  doi: 10.1007/s10957-019-01573-0.

[11]

D.-H. Li, C.-D. Chen and H.-B. Guan, A lower dimensional linear equation approach to the M-tensor complementarity problem, Calcolo, 58, (2021), Paper No. 5, 21 pp. doi: 10.1007/s10092-021-00397-7.

[12]

D. Liu, W. Li and S.-W. Vong, Tensor complementarity problems: The GUS-property and an algorithm, Linear Multilinear Algebra, 66 (2018), 1726-1749.  doi: 10.1080/03081087.2017.1369929.

[13]

X. Liu and Y. Wang, Weakening convergence conditions of a potential reduction method for tensor complementarity problems, Journal of Industrial and Management Optimization, 2021. doi: 10.3934/jimo.2021080.

[14]

Z. Luo, L. Qi and N. Xiu, The sparsest solutions to $Z$-tensor complementarity problems, Optim. Lett., 11 (2017), 471-482.  doi: 10.1007/s11590-016-1013-9.

[15]

O. L. Mangasarian, Solution of symmetric linear complementarity problems by iterative methods, J. Optim. Theory Appl., 22 (1977), 465-485.  doi: 10.1007/BF01268170.

[16]

M. Ng, L. Qi and G. Zhou, Finding the largest eigenvalue of a nonnegative tensor, SIAM J. Matrix Anal. Appl., 31 (2009), 1090-1099.  doi: 10.1137/09074838X.

[17]

Y. Song and L. Qi, Properties of some classes of structured tensors, J. Optim. Theory Appl., 165 (2015), 854-873.  doi: 10.1007/s10957-014-0616-5.

[18]

Y. Wang, Z.-H. Huang and X.-L. Bai, Exceptionally regular tensors and tensor complementarity problems, Optim. Methods Softw., 31 (2016), 815-828.  doi: 10.1080/10556788.2016.1180386.

[19]

S.-L. Xie, D.-H. Li and H.-R. Xu, An iterative method for finding the least solution to the tensor complementarity problem, J. Optim. Theory Appl., 175 (2017), 119-136.  doi: 10.1007/s10957-017-1157-5.

[20]

S.-L. Xie and H.-R. Xu, A two-level additive Schwarz method for a kind of tensor complementarity problem, Linear Algebra Appl., 584 (2020), 394-408.  doi: 10.1016/j.laa.2019.09.025.

[21]

H.-R. Xu, D.-H. Li and S.-L. Xie, An equivalent tensor equation to the tensor complementarity problem with positive semi-definite $Z$-tensor, Optim. Lett., 13 (2019), 685-694.  doi: 10.1007/s11590-018-1268-4.

[22]

K. Zhang, H. Chen and P. Zhao, A potential reduction method for tensor complementarity problems, J. Ind. Manag. Optim., 15 (2019), 429-443.  doi: 10.3934/jimo.2018049.

[23]

X. Zhao and J. Fan, A semidefinite method for tensor complementarity problems, Optim. Methods Softw., 34 (2019), 758-769.  doi: 10.1080/10556788.2018.1439489.

show all references

References:
[1]

B. H. Ahn, Solution of nonsymmetric linear complementarity problems by iterative methods, J. Optim. Theory Appl., 33 (1981), 175-185.  doi: 10.1007/BF00935545.

[2]

X.-L. Bai, Z.-H. Huang and Y. Wang, Global uniqueness and solvability for tensor complementarity problems, J. Optim. Theory Appl., 170 (2016), 72-84.  doi: 10.1007/s10957-016-0903-4.

[3]

P.-F. Dai, A fixed point iterative method for tensor complementarity problems, J. Sci. Comput., 84 (2020), Paper No. 49, 20 pp. doi: 10.1007/s10915-020-01299-6.

[4]

S. Du and L. Zhang, A mixed integer programming approach to the tensor complementarity problem, J. Global Optim., 73 (2019), 789-800.  doi: 10.1007/s10898-018-00731-4.

[5]

M. S. Gowda, Z. Luo, L. Qi and N. Xiu, Z-tensors and complementarity problems, arXiv: 1510.07933v2, 2015.

[6]

H.-B. Guan and D.-H. Li, Linearized methods for tensor complementarity problems, J. Optim. Theory Appl., 184 (2020), 972-987.  doi: 10.1007/s10957-019-01627-3.

[7]

L. Han, A continuation method for tensor complementarity problems, J. Optim. Theory Appl., 180 (2019), 949-963.  doi: 10.1007/s10957-018-1422-2.

[8]

Z.-H. Huang and L. Qi, Formulating an $n$-person noncooperative game as a tensor complementarity problem, Comput. Optim. Appl., 66 (2017), 557-576.  doi: 10.1007/s10589-016-9872-7.

[9]

Z.-H. Huang and L. Qi, Tensor complementarity problems-Part Ⅰ: Basic theory, J. Optim. Theory Appl., 183 (2019), 1-23.  doi: 10.1007/s10957-019-01566-z.

[10]

Z.-H. Huang and L. Qi, Tensor complementarity problems-Part Ⅲ: Applications, J. Optim. Theory Appl., 183 (2019), 771-791.  doi: 10.1007/s10957-019-01573-0.

[11]

D.-H. Li, C.-D. Chen and H.-B. Guan, A lower dimensional linear equation approach to the M-tensor complementarity problem, Calcolo, 58, (2021), Paper No. 5, 21 pp. doi: 10.1007/s10092-021-00397-7.

[12]

D. Liu, W. Li and S.-W. Vong, Tensor complementarity problems: The GUS-property and an algorithm, Linear Multilinear Algebra, 66 (2018), 1726-1749.  doi: 10.1080/03081087.2017.1369929.

[13]

X. Liu and Y. Wang, Weakening convergence conditions of a potential reduction method for tensor complementarity problems, Journal of Industrial and Management Optimization, 2021. doi: 10.3934/jimo.2021080.

[14]

Z. Luo, L. Qi and N. Xiu, The sparsest solutions to $Z$-tensor complementarity problems, Optim. Lett., 11 (2017), 471-482.  doi: 10.1007/s11590-016-1013-9.

[15]

O. L. Mangasarian, Solution of symmetric linear complementarity problems by iterative methods, J. Optim. Theory Appl., 22 (1977), 465-485.  doi: 10.1007/BF01268170.

[16]

M. Ng, L. Qi and G. Zhou, Finding the largest eigenvalue of a nonnegative tensor, SIAM J. Matrix Anal. Appl., 31 (2009), 1090-1099.  doi: 10.1137/09074838X.

[17]

Y. Song and L. Qi, Properties of some classes of structured tensors, J. Optim. Theory Appl., 165 (2015), 854-873.  doi: 10.1007/s10957-014-0616-5.

[18]

Y. Wang, Z.-H. Huang and X.-L. Bai, Exceptionally regular tensors and tensor complementarity problems, Optim. Methods Softw., 31 (2016), 815-828.  doi: 10.1080/10556788.2016.1180386.

[19]

S.-L. Xie, D.-H. Li and H.-R. Xu, An iterative method for finding the least solution to the tensor complementarity problem, J. Optim. Theory Appl., 175 (2017), 119-136.  doi: 10.1007/s10957-017-1157-5.

[20]

S.-L. Xie and H.-R. Xu, A two-level additive Schwarz method for a kind of tensor complementarity problem, Linear Algebra Appl., 584 (2020), 394-408.  doi: 10.1016/j.laa.2019.09.025.

[21]

H.-R. Xu, D.-H. Li and S.-L. Xie, An equivalent tensor equation to the tensor complementarity problem with positive semi-definite $Z$-tensor, Optim. Lett., 13 (2019), 685-694.  doi: 10.1007/s11590-018-1268-4.

[22]

K. Zhang, H. Chen and P. Zhao, A potential reduction method for tensor complementarity problems, J. Ind. Manag. Optim., 15 (2019), 429-443.  doi: 10.3934/jimo.2018049.

[23]

X. Zhao and J. Fan, A semidefinite method for tensor complementarity problems, Optim. Methods Softw., 34 (2019), 758-769.  doi: 10.1080/10556788.2018.1439489.

Figure 1.  The relationship between $ \gamma $ and the number of iterative with $ \lambda = 0.5 $. From left to right and from top to bottom, the 4 subgraphs are corresponding to $ m = 3 $, $ m = 4 $, $ m = 5 $ and $ m = 6 $, respectively
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Figure 2.  The relationship between $ \eta $ and the number of iteration with $ \lambda = 0.5 $. From left to right and from top to bottom, the 4 subgraphs are corresponding to $ m = 3 $, $ m = 4 $, $ m = 5 $ and $ m = 6 $, respectively
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Table 1.  Numerical results for Example 4.1
$ q $ $ (177.8912, 0)^\top $ $ (-8.2037,166.0649)^\top $ $ (-30.8545, 0)^\top $
$ x^0 $ $ (4.3398, 7.8482)^\top $ $ (5.2790, 2.5602)^\top $ $ (58.1740, 9.9800)^\top $
$ x^* $ $ (2.6254, 7.8482)^\top $ $ (2.8642, 0.0000)^\top $ $ (17.9916, 9.8800)^\top $
$ \lambda=0.2 $
IT 122 237 138
Err $ 8.801848\mathrm{e}-13 $ $ 9.918555\mathrm{e}-13 $ $ 9.308110\mathrm{e}-13 $
Val $ 4.857622\mathrm{e}-11 $ $ 1.395496\mathrm{e}-09 $ $ 2.412048\mathrm{e}-09 $
$ \lambda=0.5 $
IT 42 80 47
Err $ 5.382361\mathrm{e}-13 $ $ 9.645002\mathrm{e}-13 $ $ 6.323830\mathrm{e}-13 $
Val $ 7.461785\mathrm{e}-12 $ $ 3.866887\mathrm{e}-10 $ $ 4.106157\mathrm{e}-10 $
$ \lambda=0.8 $
IT 19 36 21
Err $ 5.102585\mathrm{e}-13 $ $ 8.295835\mathrm{e}-13 $ $ 7.709389\mathrm{e}-13 $
Val $ 1.790828\mathrm{e}-12 $ $ 1.114540\mathrm{e}-10 $ $ 1.252813\mathrm{e}-10 $
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$ q $ $ (177.8912, 0)^\top $ $ (-8.2037,166.0649)^\top $ $ (-30.8545, 0)^\top $
$ x^0 $ $ (4.3398, 7.8482)^\top $ $ (5.2790, 2.5602)^\top $ $ (58.1740, 9.9800)^\top $
$ x^* $ $ (2.6254, 7.8482)^\top $ $ (2.8642, 0.0000)^\top $ $ (17.9916, 9.8800)^\top $
$ \lambda=0.2 $
IT 122 237 138
Err $ 8.801848\mathrm{e}-13 $ $ 9.918555\mathrm{e}-13 $ $ 9.308110\mathrm{e}-13 $
Val $ 4.857622\mathrm{e}-11 $ $ 1.395496\mathrm{e}-09 $ $ 2.412048\mathrm{e}-09 $
$ \lambda=0.5 $
IT 42 80 47
Err $ 5.382361\mathrm{e}-13 $ $ 9.645002\mathrm{e}-13 $ $ 6.323830\mathrm{e}-13 $
Val $ 7.461785\mathrm{e}-12 $ $ 3.866887\mathrm{e}-10 $ $ 4.106157\mathrm{e}-10 $
$ \lambda=0.8 $
IT 19 36 21
Err $ 5.102585\mathrm{e}-13 $ $ 8.295835\mathrm{e}-13 $ $ 7.709389\mathrm{e}-13 $
Val $ 1.790828\mathrm{e}-12 $ $ 1.114540\mathrm{e}-10 $ $ 1.252813\mathrm{e}-10 $
Table 2.  Numerical results for Example 4.3 (ⅰ)
$ k $ $ x^k $ Err Val
1 $ (6.9596\mathrm{e}+00, 3.7592\mathrm{e}+00) $ $ 4.6777\mathrm{e}-01 $ $ 1.8146\mathrm{e}+02 $
2 $ (6.4918\mathrm{e}+00, 3.7592\mathrm{e}+00) $ $ 6.2133\mathrm{e}-01 $ $ 2.0821\mathrm{e}+02 $
3 $ (5.8717\mathrm{e}+00, 3.7199\mathrm{e}+00) $ $ 8.7813\mathrm{e}-01 $ $ 2.2988\mathrm{e}+02 $
4 $ (4.9991\mathrm{e}+00, 3.6220\mathrm{e}+00) $ $ 1.3984\mathrm{e}+00 $ $ 2.2460\mathrm{e}+02 $
5 $ (3.6133\mathrm{e}+00, 3.4348\mathrm{e}+00) $ $ 1.1093\mathrm{e}+00 $ $ 1.9417\mathrm{e}+02 $
$ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $
83 $ (6.5725\mathrm{e}-12, 9.3919\mathrm{e}-12) $ $ 3.3575\mathrm{e}-12 $ $ 4.8035\mathrm{e}-10 $
84 $ (4.6474\mathrm{e}-12, 6.6410\mathrm{e}-12) $ $ 2.3741\mathrm{e}-12 $ $ 3.3966\mathrm{e}-10 $
85 $ (3.2862\mathrm{e}-12, 4.6959\mathrm{e}-12) $ $ 1.6787\mathrm{e}-12 $ $ 2.4018\mathrm{e}-10 $
86 $ (2.3237\mathrm{e}-12, 3.3205\mathrm{e}-12) $ $ 1.1871\mathrm{e}-12 $ $ 1.6983\mathrm{e}-10 $
87 $ (1.6431\mathrm{e}-12, 2.3480\mathrm{e}-12) $ $ 8.3937\mathrm{e}-13 $ $ 1.2009\mathrm{e}-10 $
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$ k $ $ x^k $ Err Val
1 $ (6.9596\mathrm{e}+00, 3.7592\mathrm{e}+00) $ $ 4.6777\mathrm{e}-01 $ $ 1.8146\mathrm{e}+02 $
2 $ (6.4918\mathrm{e}+00, 3.7592\mathrm{e}+00) $ $ 6.2133\mathrm{e}-01 $ $ 2.0821\mathrm{e}+02 $
3 $ (5.8717\mathrm{e}+00, 3.7199\mathrm{e}+00) $ $ 8.7813\mathrm{e}-01 $ $ 2.2988\mathrm{e}+02 $
4 $ (4.9991\mathrm{e}+00, 3.6220\mathrm{e}+00) $ $ 1.3984\mathrm{e}+00 $ $ 2.2460\mathrm{e}+02 $
5 $ (3.6133\mathrm{e}+00, 3.4348\mathrm{e}+00) $ $ 1.1093\mathrm{e}+00 $ $ 1.9417\mathrm{e}+02 $
$ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $
83 $ (6.5725\mathrm{e}-12, 9.3919\mathrm{e}-12) $ $ 3.3575\mathrm{e}-12 $ $ 4.8035\mathrm{e}-10 $
84 $ (4.6474\mathrm{e}-12, 6.6410\mathrm{e}-12) $ $ 2.3741\mathrm{e}-12 $ $ 3.3966\mathrm{e}-10 $
85 $ (3.2862\mathrm{e}-12, 4.6959\mathrm{e}-12) $ $ 1.6787\mathrm{e}-12 $ $ 2.4018\mathrm{e}-10 $
86 $ (2.3237\mathrm{e}-12, 3.3205\mathrm{e}-12) $ $ 1.1871\mathrm{e}-12 $ $ 1.6983\mathrm{e}-10 $
87 $ (1.6431\mathrm{e}-12, 2.3480\mathrm{e}-12) $ $ 8.3937\mathrm{e}-13 $ $ 1.2009\mathrm{e}-10 $
Table 3.  Numerical results for Example 4.3 (ⅱ)
$ k $ $ x^k $ Err Val
1 $ (4.1635\mathrm{e}+00, 3.4391\mathrm{e}+00) $ $ 1.2195\mathrm{e}+00 $ $ 2.9115\mathrm{e}+02 $
2 $ (2.9440\mathrm{e}+00, 3.4391\mathrm{e}+00) $ $ 8.6832\mathrm{e}-01 $ $ 2.2710\mathrm{e}+02 $
3 $ (2.0818\mathrm{e}+00, 3.3369\mathrm{e}+00) $ $ 6.3071\mathrm{e}-01 $ $ 1.7722\mathrm{e}+02 $
4 $ (1.4720\mathrm{e}+00, 3.1756\mathrm{e}+00) $ $ 4.7134\mathrm{e}-01 $ $ 1.3617\mathrm{e}+02 $
5 $ (1.0409\mathrm{e}+00, 2.9851\mathrm{e}+00) $ $ 3.6426\mathrm{e}-01 $ $ 1.0285\mathrm{e}+02 $
$ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $
103 $ (1.8490\mathrm{e}-15, 1.4078\mathrm{e}+00) $ $ 2.3244\mathrm{e}-12 $ $ 7.0445\mathrm{e}-11 $
104 $ (1.3074\mathrm{e}-15, 1.4078\mathrm{e}+00) $ $ 1.7666\mathrm{e}-12 $ $ 5.3529\mathrm{e}-11 $
105 $ (9.2449\mathrm{e}-16, 1.4078\mathrm{e}+00) $ $ 1.3425\mathrm{e}-12 $ $ 4.0675\mathrm{e}-11 $
106 $ (6.5371\mathrm{e}-16, 1.4078\mathrm{e}+00) $ $ 1.0203\mathrm{e}-12 $ $ 3.0910\mathrm{e}-11 $
107 $ (4.6224\mathrm{e}-16, 1.4078\mathrm{e}+00) $ $ 7.7538\mathrm{e}-13 $ $ 2.3488\mathrm{e}-11 $
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$ k $ $ x^k $ Err Val
1 $ (4.1635\mathrm{e}+00, 3.4391\mathrm{e}+00) $ $ 1.2195\mathrm{e}+00 $ $ 2.9115\mathrm{e}+02 $
2 $ (2.9440\mathrm{e}+00, 3.4391\mathrm{e}+00) $ $ 8.6832\mathrm{e}-01 $ $ 2.2710\mathrm{e}+02 $
3 $ (2.0818\mathrm{e}+00, 3.3369\mathrm{e}+00) $ $ 6.3071\mathrm{e}-01 $ $ 1.7722\mathrm{e}+02 $
4 $ (1.4720\mathrm{e}+00, 3.1756\mathrm{e}+00) $ $ 4.7134\mathrm{e}-01 $ $ 1.3617\mathrm{e}+02 $
5 $ (1.0409\mathrm{e}+00, 2.9851\mathrm{e}+00) $ $ 3.6426\mathrm{e}-01 $ $ 1.0285\mathrm{e}+02 $
$ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $
103 $ (1.8490\mathrm{e}-15, 1.4078\mathrm{e}+00) $ $ 2.3244\mathrm{e}-12 $ $ 7.0445\mathrm{e}-11 $
104 $ (1.3074\mathrm{e}-15, 1.4078\mathrm{e}+00) $ $ 1.7666\mathrm{e}-12 $ $ 5.3529\mathrm{e}-11 $
105 $ (9.2449\mathrm{e}-16, 1.4078\mathrm{e}+00) $ $ 1.3425\mathrm{e}-12 $ $ 4.0675\mathrm{e}-11 $
106 $ (6.5371\mathrm{e}-16, 1.4078\mathrm{e}+00) $ $ 1.0203\mathrm{e}-12 $ $ 3.0910\mathrm{e}-11 $
107 $ (4.6224\mathrm{e}-16, 1.4078\mathrm{e}+00) $ $ 7.7538\mathrm{e}-13 $ $ 2.3488\mathrm{e}-11 $
Table 4.  Numerical results for Example 4.3 (ⅲ)
$ k $ $ x^k $ Err Val
1 $ (2.5510\mathrm{e}+00, 5.0596\mathrm{e}+00) $ $ 7.4716\mathrm{e}-01 $ $ 1.5451\mathrm{e}+02 $
2 $ (1.8038\mathrm{e}+00, 5.0596\mathrm{e}+00) $ $ 6.2157\mathrm{e}-01 $ $ 1.3847\mathrm{e}+02 $
3 $ (1.2755\mathrm{e}+00, 4.7321\mathrm{e}+00) $ $ 7.5567\mathrm{e}-01 $ $ 1.2059\mathrm{e}+02 $
4 $ (9.0190\mathrm{e}-01, 4.0753\mathrm{e}+00) $ $ 1.1283\mathrm{e}+00 $ $ 9.4788\mathrm{e}+01 $
5 $ (6.3774\mathrm{e}-01, 2.9783\mathrm{e}+00) $ $ 8.9211\mathrm{e}-01 $ $ 7.0090\mathrm{e}+01 $
$ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $
81 $ (2.3201\mathrm{e}-12, 1.0835\mathrm{e}-11) $ $ 3.2455\mathrm{e}-12 $ $ 2.6613\mathrm{e}-10 $
82 $ (1.6405\mathrm{e}-12, 7.6616\mathrm{e}-12) $ $ 2.2949\mathrm{e}-12 $ $ 1.8819\mathrm{e}-10 $
83 $ (1.1600\mathrm{e}-12, 5.4176\mathrm{e}-12) $ $ 1.6227\mathrm{e}-12 $ $ 1.3307\mathrm{e}-10 $
84 $ (8.2027\mathrm{e}-13, 3.8308\mathrm{e}-12) $ $ 1.1474\mathrm{e}-12 $ $ 9.4093\mathrm{e}-11 $
85 $ (5.8002\mathrm{e}-13, 2.7088\mathrm{e}-12) $ $ 8.1137\mathrm{e}-13 $ $ 6.6534\mathrm{e}-11 $
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$ k $ $ x^k $ Err Val
1 $ (2.5510\mathrm{e}+00, 5.0596\mathrm{e}+00) $ $ 7.4716\mathrm{e}-01 $ $ 1.5451\mathrm{e}+02 $
2 $ (1.8038\mathrm{e}+00, 5.0596\mathrm{e}+00) $ $ 6.2157\mathrm{e}-01 $ $ 1.3847\mathrm{e}+02 $
3 $ (1.2755\mathrm{e}+00, 4.7321\mathrm{e}+00) $ $ 7.5567\mathrm{e}-01 $ $ 1.2059\mathrm{e}+02 $
4 $ (9.0190\mathrm{e}-01, 4.0753\mathrm{e}+00) $ $ 1.1283\mathrm{e}+00 $ $ 9.4788\mathrm{e}+01 $
5 $ (6.3774\mathrm{e}-01, 2.9783\mathrm{e}+00) $ $ 8.9211\mathrm{e}-01 $ $ 7.0090\mathrm{e}+01 $
$ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $
81 $ (2.3201\mathrm{e}-12, 1.0835\mathrm{e}-11) $ $ 3.2455\mathrm{e}-12 $ $ 2.6613\mathrm{e}-10 $
82 $ (1.6405\mathrm{e}-12, 7.6616\mathrm{e}-12) $ $ 2.2949\mathrm{e}-12 $ $ 1.8819\mathrm{e}-10 $
83 $ (1.1600\mathrm{e}-12, 5.4176\mathrm{e}-12) $ $ 1.6227\mathrm{e}-12 $ $ 1.3307\mathrm{e}-10 $
84 $ (8.2027\mathrm{e}-13, 3.8308\mathrm{e}-12) $ $ 1.1474\mathrm{e}-12 $ $ 9.4093\mathrm{e}-11 $
85 $ (5.8002\mathrm{e}-13, 2.7088\mathrm{e}-12) $ $ 8.1137\mathrm{e}-13 $ $ 6.6534\mathrm{e}-11 $
Table 5.  Numerical results for Example 4.3 (ⅳ)
$ k $ $ (x^k_1, x^k_2) $ Err Val
$ 1 $ $ (3.6587\mathrm{e}-01, 1.0791\mathrm{e}+00) $ $ 5.5271\mathrm{e}-03 $ $ 6.0490\mathrm{e}+03 $
$ 2 $ $ (3.6587\mathrm{e}-01, 1.0791\mathrm{e}+00) $ $ 1.7842\mathrm{e}-02 $ $ 3.4454\mathrm{e}+04 $
$ 3 $ $ (3.6280\mathrm{e}-01, 1.0791\mathrm{e}+00) $ $ 3.8265\mathrm{e}-02 $ $ 1.0020\mathrm{e}+05 $
$ 4 $ $ (3.4071\mathrm{e}-01, 1.0791\mathrm{e}+00) $ $ 5.1704\mathrm{e}-02 $ $ 1.7255\mathrm{e}+05 $
$ 5 $ $ (3.0859\mathrm{e}-01, 1.0790\mathrm{e}+00) $ $ 4.6910\mathrm{e}-02 $ $ 2.2193\mathrm{e}+05 $
$ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $
$ 485 $ $ (7.0361\mathrm{e}-22, 1.5492\mathrm{e}-11) $ $ 1.4605\mathrm{e}-12 $ $ 2.2067\mathrm{e}-06 $
$ 486 $ $ (6.3727\mathrm{e}-22, 1.4031\mathrm{e}-11) $ $ 1.3228\mathrm{e}-12 $ $ 2.0014\mathrm{e}-06 $
$ 487 $ $ (5.7719\mathrm{e}-22, 1.2709\mathrm{e}-11) $ $ 1.1981\mathrm{e}-12 $ $ 1.8158\mathrm{e}-06 $
$ 488 $ $ (5.2278\mathrm{e}-22, 1.1510\mathrm{e}-11) $ $ 1.0852\mathrm{e}-12 $ $ 1.6461\mathrm{e}-06 $
$ 489 $ $ (4.7349\mathrm{e}-22, 1.0425\mathrm{e}-11) $ $ 9.8286\mathrm{e}-13 $ $ 1.4943\mathrm{e}-06 $
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$ k $ $ (x^k_1, x^k_2) $ Err Val
$ 1 $ $ (3.6587\mathrm{e}-01, 1.0791\mathrm{e}+00) $ $ 5.5271\mathrm{e}-03 $ $ 6.0490\mathrm{e}+03 $
$ 2 $ $ (3.6587\mathrm{e}-01, 1.0791\mathrm{e}+00) $ $ 1.7842\mathrm{e}-02 $ $ 3.4454\mathrm{e}+04 $
$ 3 $ $ (3.6280\mathrm{e}-01, 1.0791\mathrm{e}+00) $ $ 3.8265\mathrm{e}-02 $ $ 1.0020\mathrm{e}+05 $
$ 4 $ $ (3.4071\mathrm{e}-01, 1.0791\mathrm{e}+00) $ $ 5.1704\mathrm{e}-02 $ $ 1.7255\mathrm{e}+05 $
$ 5 $ $ (3.0859\mathrm{e}-01, 1.0790\mathrm{e}+00) $ $ 4.6910\mathrm{e}-02 $ $ 2.2193\mathrm{e}+05 $
$ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $
$ 485 $ $ (7.0361\mathrm{e}-22, 1.5492\mathrm{e}-11) $ $ 1.4605\mathrm{e}-12 $ $ 2.2067\mathrm{e}-06 $
$ 486 $ $ (6.3727\mathrm{e}-22, 1.4031\mathrm{e}-11) $ $ 1.3228\mathrm{e}-12 $ $ 2.0014\mathrm{e}-06 $
$ 487 $ $ (5.7719\mathrm{e}-22, 1.2709\mathrm{e}-11) $ $ 1.1981\mathrm{e}-12 $ $ 1.8158\mathrm{e}-06 $
$ 488 $ $ (5.2278\mathrm{e}-22, 1.1510\mathrm{e}-11) $ $ 1.0852\mathrm{e}-12 $ $ 1.6461\mathrm{e}-06 $
$ 489 $ $ (4.7349\mathrm{e}-22, 1.0425\mathrm{e}-11) $ $ 9.8286\mathrm{e}-13 $ $ 1.4943\mathrm{e}-06 $
Table 6.  Numerical results for Example 4.4
$ (m, n) $ Algorithm 3.1 Linearized method
AveIt AveCpu AveErr AveIt AveCpu AveErr
$ (3, 10) $ $ 46.2 $ $ 0.0406 $ $ 8.1513\mathrm{e}-07 $ $ 33.5 $ $ 0.0500 $ $ 6.4154\mathrm{e}-07 $
$ (3, 30) $ $ 70.6 $ $ 0.3969 $ $ 8.5449\mathrm{e}-07 $ $ 60.3 $ $ 0.4109 $ $ 7.3853\mathrm{e}-07 $
$ (3, 40) $ $ 86.4 $ $ 1.1688 $ $ 8.0033\mathrm{e}-07 $ $ 76.7 $ $ 1.5922 $ $ 7.4773\mathrm{e}-07 $
$ (3,100) $ $ 93.8 $ $ 9.9734 $ $ 8.6858\mathrm{e}-07 $ $ 88.3 $ $ 1.6283 $ $ 8.7046\mathrm{e}-07 $
$ (3,200) $ $ 87.5 $ $ 100.60 $ $ 8.5287\mathrm{e}-07 $ $ 84.3 $ $ 105.96 $ $ 7.9336\mathrm{e}-07 $
$ (3,300) $ $ 106.8 $ $ 425.52 $ $ 8.5950\mathrm{e}-07 $ $ 103.6 $ $ 426.80 $ $ 8.4008\mathrm{e}-07 $
$ (4, 10) $ $ 35.2 $ $ 0.1375 $ $ 6.3347\mathrm{e}-07 $ $ 29.8 $ $ 0.1438 $ $ 5.1179\mathrm{e}-07 $
$ (4, 20) $ $ 35.1 $ $ 0.9688 $ $ 7.0821\mathrm{e}-07 $ $ 32.2 $ $ 0.9234 $ $ 5.8668\mathrm{e}-07 $
$ (4, 50) $ $ 44.4 $ $ 47.389 $ $ 5.6155\mathrm{e}-07 $ $ 42.0 $ $ 48.802 $ $ 6.0793\mathrm{e}-07 $
$ (4,100) $ 40.9 $ 817.98 $ $ 7.7532\mathrm{e}-07 $ $ 38.8 $ $ 823.74 $ $ 7.4341\mathrm{e}-07 $
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$ (m, n) $ Algorithm 3.1 Linearized method
AveIt AveCpu AveErr AveIt AveCpu AveErr
$ (3, 10) $ $ 46.2 $ $ 0.0406 $ $ 8.1513\mathrm{e}-07 $ $ 33.5 $ $ 0.0500 $ $ 6.4154\mathrm{e}-07 $
$ (3, 30) $ $ 70.6 $ $ 0.3969 $ $ 8.5449\mathrm{e}-07 $ $ 60.3 $ $ 0.4109 $ $ 7.3853\mathrm{e}-07 $
$ (3, 40) $ $ 86.4 $ $ 1.1688 $ $ 8.0033\mathrm{e}-07 $ $ 76.7 $ $ 1.5922 $ $ 7.4773\mathrm{e}-07 $
$ (3,100) $ $ 93.8 $ $ 9.9734 $ $ 8.6858\mathrm{e}-07 $ $ 88.3 $ $ 1.6283 $ $ 8.7046\mathrm{e}-07 $
$ (3,200) $ $ 87.5 $ $ 100.60 $ $ 8.5287\mathrm{e}-07 $ $ 84.3 $ $ 105.96 $ $ 7.9336\mathrm{e}-07 $
$ (3,300) $ $ 106.8 $ $ 425.52 $ $ 8.5950\mathrm{e}-07 $ $ 103.6 $ $ 426.80 $ $ 8.4008\mathrm{e}-07 $
$ (4, 10) $ $ 35.2 $ $ 0.1375 $ $ 6.3347\mathrm{e}-07 $ $ 29.8 $ $ 0.1438 $ $ 5.1179\mathrm{e}-07 $
$ (4, 20) $ $ 35.1 $ $ 0.9688 $ $ 7.0821\mathrm{e}-07 $ $ 32.2 $ $ 0.9234 $ $ 5.8668\mathrm{e}-07 $
$ (4, 50) $ $ 44.4 $ $ 47.389 $ $ 5.6155\mathrm{e}-07 $ $ 42.0 $ $ 48.802 $ $ 6.0793\mathrm{e}-07 $
$ (4,100) $ 40.9 $ 817.98 $ $ 7.7532\mathrm{e}-07 $ $ 38.8 $ $ 823.74 $ $ 7.4341\mathrm{e}-07 $
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