doi: 10.3934/naco.2021044
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Modified Dai-Yuan iterative scheme for nonlinear systems and its application

1. 

Department of Mathematical Sciences, Bayero University, Kano, Nigeria

2. 

Department of Mathematics, Sule Lamido University, Kafin Hausa, Nigeria

3. 

Department of Mathematics, Gombe state University, Gombe, Nigeria

4. 

Numerical Optimization Research Group, Bayero University, Kano, Nigeria

* Corresponding author: Mohammed Yusuf Waziri

Received  February 2021 Revised  September 2021 Early access November 2021

By exploiting the idea employed in the spectral Dai-Yuan method by Xue et al. [IEICE Trans. Inf. Syst. 101 (12)2984-2990 (2018)] and the approach applied in the modified Hager-Zhang scheme for nonsmooth optimization [PLos ONE 11(10): e0164289 (2016)], we develop a Dai-Yuan type iterative scheme for convex constrained nonlinear monotone system. The scheme's algorithm is obtained by combining its search direction with the projection method [Kluwer Academic Publishers, pp. 355-369(1998)]. One of the new scheme's attribute is that it is derivative-free, which makes it ideal for solving non-smooth problems. Furthermore, we demonstrate the method's application in image de-blurring problems by comparing its performance with a recent effective method. By employing mild assumptions, global convergence of the scheme is determined and results of some numerical experiments show the method to be favorable compared to some recent iterative methods.

Citation: Mohammed Yusuf Waziri, Kabiru Ahmed, Abubakar Sani Halilu, Aliyu Mohammed Awwal. Modified Dai-Yuan iterative scheme for nonlinear systems and its application. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2021044
References:
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A. B. AbubakarK. MuangchooA. H. IbrahimJ. Abubakar and S. A. Rano, FR-type algorithm for finding approximate solutions to nonlinear monotone operator equations, Arab. J. Math., 10 (2021), 261-270.  doi: 10.1007/s40065-021-00313-5.  Google Scholar

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S. AjiP. KumamA. M. AwwalM. M. Yahaya and K. Sitthithakerngkiet, An efficient DY-type spectral conjugate gradient method for system nonlinear monotone equations with applications in signal recovery, AIMS Math., 6 (2021), 8078-8106.  doi: 10.3934/math.2021469.  Google Scholar

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S. Babaie-Kafaki and R. Ghanbari, A descent family of Dai-Liao conjugate gradient methods, Optim. Methods Softw., 29 (2013), 583-591.  doi: 10.1080/10556788.2013.833199.  Google Scholar

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J. M. Barizilai and M. Borwein, Two point step size gradient methods, IMA J. Numer. Anal., 8 (1988), 141-148.  doi: 10.1093/imanum/8.1.141.  Google Scholar

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W. Cheng, A PRP type method for systems of monotone equations, Math. Comput. Modelling, 50 (2009), 15-20.  doi: 10.1016/j.mcm.2009.04.007.  Google Scholar

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W. La Cruz, A spectral algorithm for large-scale systems of nonlinear monotone equations, Numer. Algor., 76 (2017), 1109-1130.  doi: 10.1007/s11075-017-0299-8.  Google Scholar

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W. La Cruz and M. Raydan, Nonmonotone spectral methods for large-scale nonlinear systems, Optim. Methods Softw., 18 (2003), 583-599.  doi: 10.1080/10556780310001610493.  Google Scholar

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T. Elaine, Y. Wotao and Z. Yin, A fixed-point continuation method for $\ell_1-$regularized minimization with applications to compressed sensing, CAAM TR07-07, Rice University, (2007), 43-44. Google Scholar

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A. S. HaliluA. MajumderM. Y. WaziriA. M. Awwal and K. Ahmed, On solving double direction methods for convex constrained monotone nonlinear equations with image restoration, Comput. Appl. Math., 40 (2021), 239-265.  doi: 10.1007/s40314-021-01624-1.  Google Scholar

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J. K. LiuJ. L. Xu and L. Q. Zhang, Partially symmetrical derivative-free Liu-Storey projection method for convex constrained equations, Inter. J. Comput. Math., 96 (2019), 1787-1798.  doi: 10.1080/00207160.2018.1533122.  Google Scholar

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[31]

J. K. Liu and S. J. Li, Spectral DY-type projection methods for nonlinear monotone system of equations, J. Comput. Math., 33 (2015), 341-355.  doi: 10.4208/jcm.1412-m4494.  Google Scholar

[32]

J. K. Liu and S. J. Li, Multivariate spectral projection method for convex constrained nonlinear monotone equations, Journal of Industrial and Management Optimization, 13 (2017), 283-297.  doi: 10.3934/jimo.2016017.  Google Scholar

[33]

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show all references

References:
[1]

A. B. AbubakarP. KumamH. MohammadA. M. Awwal and K. Sitthithakerngkiet, A modified Fletcher-Reeves conjugate gradient method for monotone nonlinear equations with some applications, Mathematics, 7 (2019), 745.  doi: 10.3390/math7080745.  Google Scholar

[2]

A. B. AbubakarK. MuangchooA. H. IbrahimJ. Abubakar and S. A. Rano, FR-type algorithm for finding approximate solutions to nonlinear monotone operator equations, Arab. J. Math., 10 (2021), 261-270.  doi: 10.1007/s40065-021-00313-5.  Google Scholar

[3]

S. AjiP. KumamA. M. AwwalM. M. Yahaya and K. Sitthithakerngkiet, An efficient DY-type spectral conjugate gradient method for system nonlinear monotone equations with applications in signal recovery, AIMS Math., 6 (2021), 8078-8106.  doi: 10.3934/math.2021469.  Google Scholar

[4]

S. Babaie-Kafaki and R. Ghanbari, A descent family of Dai-Liao conjugate gradient methods, Optim. Methods Softw., 29 (2013), 583-591.  doi: 10.1080/10556788.2013.833199.  Google Scholar

[5]

M. R. Banham and A. K. Katsaggelos, Digital image restoration, IEEE Signal Process Mag., 14 (1997), 24-41.  doi: 10.1109/79.581363.  Google Scholar

[6]

J. M. Barizilai and M. Borwein, Two point step size gradient methods, IMA J. Numer. Anal., 8 (1988), 141-148.  doi: 10.1093/imanum/8.1.141.  Google Scholar

[7]

C. L. ChanA. K. Katsaggelos and A. V. Sahakian, Image sequence filtering in quantum-limited noise with applications to low-dose fluoroscopy, IEEE Trans. Med. Imaging, 12 (1993), 610-621.   Google Scholar

[8]

W. Cheng, A two-term PRP-based descent method, Numer. Funct. Anal. Optim., 28 (2007), 1217-1230.  doi: 10.1080/01630560701749524.  Google Scholar

[9]

W. Cheng, A PRP type method for systems of monotone equations, Math. Comput. Modelling, 50 (2009), 15-20.  doi: 10.1016/j.mcm.2009.04.007.  Google Scholar

[10]

W. La Cruz, A spectral algorithm for large-scale systems of nonlinear monotone equations, Numer. Algor., 76 (2017), 1109-1130.  doi: 10.1007/s11075-017-0299-8.  Google Scholar

[11]

W. La Cruz and M. Raydan, Nonmonotone spectral methods for large-scale nonlinear systems, Optim. Methods Softw., 18 (2003), 583-599.  doi: 10.1080/10556780310001610493.  Google Scholar

[12]

Y. H. Dai and Y. Yuan, A nonlinear conjugate gradient method with a strong global convergence property, SIAM J. Optim., 10 (1999), 177-182.  doi: 10.1137/S1052623497318992.  Google Scholar

[13]

S. P. Dirkse and M. C. Ferris, A collection of nonlinear mixed complementarity problems, Optim. Methods Softw., 5 (1995), 319-345.   Google Scholar

[14]

E. D. Dolan and J. J. Moré, Benchmarking optimization software with performance profiles, Math. Program., 91 (2002), 201-213.  doi: 10.1007/s101070100263.  Google Scholar

[15]

T. Elaine, Y. Wotao and Z. Yin, A fixed-point continuation method for $\ell_1-$regularized minimization with applications to compressed sensing, CAAM TR07-07, Rice University, (2007), 43-44. Google Scholar

[16]

M. Figueiredo, R. Nowak and S. J. Wright, Gradient projection for sparse reconstruction, application to compressed sensing and other inverse problems, IEEE J-STSP, IEEE Press, Piscataway, NJ. (2007), 586-597. Google Scholar

[17]

R. Fletcher and C. Reeves, Function minimization by conjugate gradients, The Computer Journal, 7 (1964), 149-154.  doi: 10.1093/comjnl/7.2.149.  Google Scholar

[18]

R. Fletcher, Practical Method of Optimization, Volume 1: Unconstrained Optimization, 2nd edition, Wiley, New York, 1997.  Google Scholar

[19]

W. W. Hager and H. Zhang, A new conjugate gradient method with guaranteed descent and an efficient line search, SIAM J. Optim., 16 (2005), 170-192.  doi: 10.1137/030601880.  Google Scholar

[20]

A. S. Halilu and M. Y. Waziri, An improved derivative-free method via double direction approach for solving systems of nonlinear equations, J. Ramanujan Math. Soc., 33 (2018), 75-89.  doi: 10.1007/s00180-017-0741-3.  Google Scholar

[21]

A. S. HaliluA. MajumderM. Y. Waziri and K. Ahmed, Signal recovery with convex constrained nonlinear monotone equations through conjugate gradient hybrid approach, Math. Comput. Simulation, 187 (2021), 520-539.  doi: 10.1016/j.matcom.2021.03.020.  Google Scholar

[22]

A. S. HaliluA. MajumderM. Y. WaziriA. M. Awwal and K. Ahmed, On solving double direction methods for convex constrained monotone nonlinear equations with image restoration, Comput. Appl. Math., 40 (2021), 239-265.  doi: 10.1007/s40314-021-01624-1.  Google Scholar

[23]

B. S. HeH. Yang and S. L. Wang, Alternationg direction method with self-adaptive penalty parameters for monotone variational inequalites, J. Optim. Theory Appl., 106 (2000), 337-356.  doi: 10.1023/A:1004603514434.  Google Scholar

[24]

M. R. Hestenes and E. L. Stiefel, Methods of conjugate gradients for solving linear systems, J. Research Nat. Bur. Standards, 49 (1952), 409-436.   Google Scholar

[25]

G. A. Hively, On a class of nonlinear integral equations arising in transport theory, SIAM J. Numer. Anal., 9 (1978), 787-792.  doi: 10.1137/0509060.  Google Scholar

[26]

D. H. Li and X. L. Wang, A modified Fletcher-Reeves-type derivative-free method for symmetric nonlinear equations, Numer. Algebra, Control and Optimization, 1 (2011), 71-82.  doi: 10.3934/naco.2011.1.71.  Google Scholar

[27]

Q. N. Li and D. H. Li, A class of derivative-free methods for large-scale nonlinear monotone equations, IMA J. Numer. Anal., 31 (2011), 1625-1635.  doi: 10.1093/imanum/drq015.  Google Scholar

[28]

J. Liu and Y. Feng, A derivative-free iterative method for nonlinear monotone equations with convex constraints, Numer. Algor., 82 (2019), 245-262.  doi: 10.1007/s11075-018-0603-2.  Google Scholar

[29]

J. K. LiuJ. L. Xu and L. Q. Zhang, Partially symmetrical derivative-free Liu-Storey projection method for convex constrained equations, Inter. J. Comput. Math., 96 (2019), 1787-1798.  doi: 10.1080/00207160.2018.1533122.  Google Scholar

[30]

J. K. Liu and S. J. Li, A projection method for convex constrained monotone nonlinear equations with applications, Comput. Math. Appl., 70 (2015), 2442-2453.  doi: 10.1016/j.camwa.2015.09.014.  Google Scholar

[31]

J. K. Liu and S. J. Li, Spectral DY-type projection methods for nonlinear monotone system of equations, J. Comput. Math., 33 (2015), 341-355.  doi: 10.4208/jcm.1412-m4494.  Google Scholar

[32]

J. K. Liu and S. J. Li, Multivariate spectral projection method for convex constrained nonlinear monotone equations, Journal of Industrial and Management Optimization, 13 (2017), 283-297.  doi: 10.3934/jimo.2016017.  Google Scholar

[33]

Y. Liu and C. Storey, Efficient generalized conjugate gradient algorithms, Part 1: Theory, J. Optim. Theory Appl., 69 (1991), 129-137.  doi: 10.1007/BF00940464.  Google Scholar

[34]

A. T. MarioR. Figueiredo and D. Nowak, An EM algorithm for wavelet-based image restoration, IEEE Transactions on Image Processing, 12 (2003), 906-916.  doi: 10.1109/TIP.2003.814255.  Google Scholar

[35]

K. Meintjes and A. P. Morgan, A methodology for solving chemical equilibrium systems, Appl. Math. Comput., 22 (1987), 333-361.  doi: 10.1016/0096-3003(87)90076-2.  Google Scholar

[36]

J. S. Pang, Inexact Newton methods for the nonlinear complementarity problem, Math. Program., 36 (1986), 54-71.  doi: 10.1007/BF02591989.  Google Scholar

[37]

G. Peiting and H. Chuanjiang, A derivative-free three-term projection algorithm involving spectral quotient for solving nonlinear monotone equations, Optimization, (2018) 1-18. doi: 10.1080/02331934.2018.1482490.  Google Scholar

[38]

E. Polak and G. Ribi$\acute{e}$re, Note Sur la convergence de directions conjugèes, Rev. Francaise Informat. Recherche Operationelle, 3 (1969), 35-43.   Google Scholar

[39]

B. T. Polyak, The conjugate gradient method in extreme problems, USSR Comp. Math. Math. Phys., 9 (1969), 94-112.   Google Scholar

[40]

J. Sabi'uA. Shah and M. Y. Waziri, Two optimal Hager-Zhang conjugate gradient methods for solving monotone nonlinear equations, Appl. Numer. Math., 153 (2020), 217-233.  doi: 10.1016/j.apnum.2020.02.017.  Google Scholar

[41]

J. Sabi'u, A. Shah, M. Y. Waziri and K. Ahmed, Modified Hager-Zhang conjugate gradient methods via singular value analysis for solving monotone nonlinear equations with convex constraint, Int. J. Comput. Methods, 18 (2021), Paper No. 2050043, 33 pages. doi: 10.1142/S0219876220500437.  Google Scholar

[42]

C. H. Slump, Real-time image restoration in diagnostic X-ray imaging, the effects on quantum noise, in Proceedings 11th IAPR International Conference on Pattern Recognition, Vol.II. Conference B: Pattern Recognition Methodology and Systems, (1992), 693-696. doi: 10.1109/ICPR.1992.201871.  Google Scholar

[43]

V. M. Solodov and A. N. Iusem, Newton-type methods with generalized distances for constrained optimization, Optimization, 41 (1997), 257-277.  doi: 10.1080/02331939708844339.  Google Scholar

[44]

M. V. Solodov and B. F. Svaiter, A globally convergent inexact Newton method for systems of monotone equations, in Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods (eds. M. Fukushima, L. Qi), Kluwer Academic Publishers, (1998), 355-369. doi: 10.1007/978-1-4757-6388-1_18.  Google Scholar

[45]

C. W. WangY. J. Wang and C. L. Xu, A projection method for a system of nonlinear monotone equations with convex constraints, Math. Meth. Oper. Res., 66 (2007), 33-46.  doi: 10.1007/s00186-006-0140-y.  Google Scholar

[46]

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Figure 1.  Performance profile of Dolan and Moré for number of iterations
Figure 2.  Performance profile of Dolan and Moré for function evaluations
Figure 3.  Performance profile of Dolan and Moré for processing time
Figure 4.  Original and blurred images (First and second columns from the left). Restored images by the two methods (Third and Fourth columns)
Table 1.  Test results of the four methods for problems 4.1
Nvars IGuess MDDYM PDYM MDY MFR
Niter Nfev Cpt Norm Niter Nfev Cpt Norm Niter Nfev Cpt Norm Niter Nfev Cpt Norm
5000 x1 6 12 0.27234 2.0475E-09 16 35 0.30469 4.1222E-09 11 25 0.13273 2.4989E-09 26 106 0.36405 0.0000E+00
5000 x2 6 12 0.07889 4.0428E-09 16 35 0.05510 8.2456E-09 12 28 0.05936 2.2971E-09 26 106 0.12459 0.0000E+00
5000 x3 3 5 0.01436 8.4985E-09 18 39 0.06173 4.5974E-09 12 28 0.07119 5.7248E-09 30 122 0.17739 0.0000E+00
5000 x4 5 9 0.02079 1.8997E-09 20 43 0.06536 4.6462E-09 14 34 0.05971 2.5758E-09 34 138 0.16181 0.0000E+00
5000 x5 11 23 0.03935 3.9754E-10 20 43 0.07339 8.5913E-09 15 37 0.05510 2.6441E-09 36 146 0.18173 0.0000E+00
5000 x6 20 45 0.08341 8.4879E-09 21 45 0.07166 3.4438E-09 15 36 0.06630 5.1003E-09 35 141 0.17659 0.0000E+00
5000 x7 12 25 0.05233 5.4026E-10 21 46 0.08474 4.3326E-09 15 39 0.09094 5.0139E-09 35 142 0.16383 0.0000E+00
5000 x8 10 21 0.04351 9.5806E-09 21 46 0.07816 5.2005E-09 15 41 0.07043 4.5835E-09 35 142 0.18567 0.0000E+00
10000 x1 6 12 0.04366 2.8956E-09 16 35 0.11302 5.8297E-09 10 23 0.05868 8.5342E-09 26 106 0.22574 0.0000E+00
10000 x2 6 12 0.04652 5.7173E-09 17 37 0.10115 3.8700E-09 12 28 0.13233 3.1978E-09 26 106 0.25446 0.0000E+00
10000 x3 4 6 0.02748 5.9571E-10 18 39 0.10617 6.5017E-09 12 28 0.08322 3.7136E-09 30 122 0.25620 0.0000E+00
10000 x4 5 9 0.03605 2.6866E-09 20 43 0.12137 6.5707E-09 14 33 0.10267 4.4531E-09 34 138 0.26421 0.0000E+00
10000 x5 11 23 0.09120 5.6221E-10 21 45 0.12382 4.0482E-09 15 37 0.11103 5.1673E-09 36 146 0.31887 0.0000E+00
10000 x6 21 48 0.14197 6.1811E-09 20 44 0.15292 5.5936E-09 16 39 0.12905 3.0303E-09 35 141 0.31019 0.0000E+00
10000 x7 12 25 0.08928 7.6404E-10 21 46 0.13180 6.1272E-09 16 43 0.12267 3.0885E-09 35 142 0.29184 0.0000E+00
10000 x8 11 22 0.07167 3.5449E-10 21 47 0.12698 7.3547E-09 15 41 0.14901 2.9293E-09 35 142 0.29874 0.0000E+00
50000 x1 6 12 0.18477 6.4748E-09 17 37 0.43468 4.3261E-09 12 28 0.31329 3.4966E-09 26 106 1.02635 0.0000E+00
50000 x2 7 13 0.18950 4.9344E-10 17 37 0.39080 8.6535E-09 12 28 0.32265 5.2020E-09 26 106 0.99223 0.0000E+00
50000 x3 4 6 0.09771 1.3320E-09 19 41 0.47699 4.8342E-09 13 31 0.37285 4.4055E-09 30 122 1.16776 0.0000E+00
50000 x4 5 9 0.13988 6.0075E-09 21 45 0.48957 4.8925E-09 16 42 0.43939 2.3366E-09 34 138 1.26896 0.0000E+00
50000 x5 11 23 0.29815 1.2571E-09 23 53 0.54401 5.3731E-09 18 52 0.53469 2.9724E-09 36 146 1.33839 0.0000E+00
50000 x6 22 51 0.55331 7.5655E-09 23 54 0.54909 7.2056E-09 19 55 0.59269 3.2582E-09 35 141 1.28348 0.0000E+00
50000 x7 12 25 0.32319 1.7085E-09 23 54 0.58040 4.9765E-09 19 61 0.60155 3.3731E-09 35 142 1.33907 0.0000E+00
50000 x8 11 22 0.26948 7.9265E-10 23 54 0.56286 5.4177E-09 18 56 0.56586 2.7351E-09 35 142 1.31569 0.0000E+00
Nvars IGuess MDDYM PDYM MDY MFR
Niter Nfev Cpt Norm Niter Nfev Cpt Norm Niter Nfev Cpt Norm Niter Nfev Cpt Norm
5000 x1 6 12 0.27234 2.0475E-09 16 35 0.30469 4.1222E-09 11 25 0.13273 2.4989E-09 26 106 0.36405 0.0000E+00
5000 x2 6 12 0.07889 4.0428E-09 16 35 0.05510 8.2456E-09 12 28 0.05936 2.2971E-09 26 106 0.12459 0.0000E+00
5000 x3 3 5 0.01436 8.4985E-09 18 39 0.06173 4.5974E-09 12 28 0.07119 5.7248E-09 30 122 0.17739 0.0000E+00
5000 x4 5 9 0.02079 1.8997E-09 20 43 0.06536 4.6462E-09 14 34 0.05971 2.5758E-09 34 138 0.16181 0.0000E+00
5000 x5 11 23 0.03935 3.9754E-10 20 43 0.07339 8.5913E-09 15 37 0.05510 2.6441E-09 36 146 0.18173 0.0000E+00
5000 x6 20 45 0.08341 8.4879E-09 21 45 0.07166 3.4438E-09 15 36 0.06630 5.1003E-09 35 141 0.17659 0.0000E+00
5000 x7 12 25 0.05233 5.4026E-10 21 46 0.08474 4.3326E-09 15 39 0.09094 5.0139E-09 35 142 0.16383 0.0000E+00
5000 x8 10 21 0.04351 9.5806E-09 21 46 0.07816 5.2005E-09 15 41 0.07043 4.5835E-09 35 142 0.18567 0.0000E+00
10000 x1 6 12 0.04366 2.8956E-09 16 35 0.11302 5.8297E-09 10 23 0.05868 8.5342E-09 26 106 0.22574 0.0000E+00
10000 x2 6 12 0.04652 5.7173E-09 17 37 0.10115 3.8700E-09 12 28 0.13233 3.1978E-09 26 106 0.25446 0.0000E+00
10000 x3 4 6 0.02748 5.9571E-10 18 39 0.10617 6.5017E-09 12 28 0.08322 3.7136E-09 30 122 0.25620 0.0000E+00
10000 x4 5 9 0.03605 2.6866E-09 20 43 0.12137 6.5707E-09 14 33 0.10267 4.4531E-09 34 138 0.26421 0.0000E+00
10000 x5 11 23 0.09120 5.6221E-10 21 45 0.12382 4.0482E-09 15 37 0.11103 5.1673E-09 36 146 0.31887 0.0000E+00
10000 x6 21 48 0.14197 6.1811E-09 20 44 0.15292 5.5936E-09 16 39 0.12905 3.0303E-09 35 141 0.31019 0.0000E+00
10000 x7 12 25 0.08928 7.6404E-10 21 46 0.13180 6.1272E-09 16 43 0.12267 3.0885E-09 35 142 0.29184 0.0000E+00
10000 x8 11 22 0.07167 3.5449E-10 21 47 0.12698 7.3547E-09 15 41 0.14901 2.9293E-09 35 142 0.29874 0.0000E+00
50000 x1 6 12 0.18477 6.4748E-09 17 37 0.43468 4.3261E-09 12 28 0.31329 3.4966E-09 26 106 1.02635 0.0000E+00
50000 x2 7 13 0.18950 4.9344E-10 17 37 0.39080 8.6535E-09 12 28 0.32265 5.2020E-09 26 106 0.99223 0.0000E+00
50000 x3 4 6 0.09771 1.3320E-09 19 41 0.47699 4.8342E-09 13 31 0.37285 4.4055E-09 30 122 1.16776 0.0000E+00
50000 x4 5 9 0.13988 6.0075E-09 21 45 0.48957 4.8925E-09 16 42 0.43939 2.3366E-09 34 138 1.26896 0.0000E+00
50000 x5 11 23 0.29815 1.2571E-09 23 53 0.54401 5.3731E-09 18 52 0.53469 2.9724E-09 36 146 1.33839 0.0000E+00
50000 x6 22 51 0.55331 7.5655E-09 23 54 0.54909 7.2056E-09 19 55 0.59269 3.2582E-09 35 141 1.28348 0.0000E+00
50000 x7 12 25 0.32319 1.7085E-09 23 54 0.58040 4.9765E-09 19 61 0.60155 3.3731E-09 35 142 1.33907 0.0000E+00
50000 x8 11 22 0.26948 7.9265E-10 23 54 0.56286 5.4177E-09 18 56 0.56586 2.7351E-09 35 142 1.31569 0.0000E+00
Table 2.  Test results of the four methods for problems 4.2
Nvars IGuess MDDYM PDYM MDY MFR
Niter Nfev Cpt Norm Niter Nfev Cpt Norm Niter Nfev Cpt Norm Niter Nfev Cpt Norm
5000 x1 1 2 0.01078 0.0000E+00 1 2 0.01097 0.0000E+00 1 2 0.01014 0.0000E+00 1 2 0.01003 0.0000E+00
5000 x2 1 2 0.02040 0.0000E+00 1 2 0.00944 0.0000E+00 1 2 0.01005 0.0000E+00 1 2 0.02019 0.0000E+00
5000 x3 1 2 0.01013 0.0000E+00 1 2 0.00965 0.0000E+00 1 2 0.00947 0.0000E+00 1 2 0.01038 0.0000E+00
5000 x4 1 2 0.00951 0.0000E+00 1 2 0.00966 0.0000E+00 1 2 0.00928 0.0000E+00 ** ** ** **
5000 x5 1 2 0.01045 0.0000E+00 1 2 0.00748 0.0000E+00 1 2 0.00649 0.0000E+00 1 2 0.00520 0.0000E+00
5000 x6 1 2 0.00940 0.0000E+00 1 2 0.00732 0.0000E+00 1 2 0.00662 0.0000E+00 1 2 0.00842 0.0000E+00
5000 x7 1 2 0.00979 0.0000E+00 1 2 0.00660 0.0000E+00 1 2 0.00667 0.0000E+00 1 2 0.00757 0.0000E+00
5000 x8 1 2 0.01111 0.0000E+00 1 2 0.00795 0.0000E+00 1 2 0.00687 0.0000E+00 1 2 0.00779 0.0000E+00
10000 x1 1 2 0.01478 0.0000E+00 1 2 0.01622 0.0000E+00 1 2 0.01650 0.0000E+00 1 2 0.01449 0.0000E+00
10000 x2 1 2 0.01571 0.0000E+00 1 2 0.01633 0.0000E+00 1 2 0.01488 0.0000E+00 1 2 0.01594 0.0000E+00
10000 x3 1 2 0.01490 0.0000E+00 1 2 0.02384 0.0000E+00 1 2 0.03357 0.0000E+00 1 2 0.01579 0.0000E+00
10000 x4 1 2 0.01739 0.0000E+00 1 2 0.01549 0.0000E+00 1 3 0.02037 0.0000E+00 ** ** ** **
10000 x5 1 2 0.01636 0.0000E+00 1 2 0.01000 0.0000E+00 1 2 0.01013 0.0000E+00 1 2 0.00759 0.0000E+00
10000 x6 1 2 0.01637 0.0000E+00 1 2 0.00939 0.0000E+00 1 2 0.01043 0.0000E+00 1 2 0.01118 0.0000E+00
10000 x7 1 2 0.01602 0.0000E+00 1 2 0.00893 0.0000E+00 1 2 0.00994 0.0000E+00 1 2 0.01012 0.0000E+00
10000 x8 1 2 0.01539 0.0000E+00 1 2 0.01026 0.0000E+00 1 2 0.01045 0.0000E+00 1 2 0.01100 0.0000E+00
50000 x1 1 2 0.05796 0.0000E+00 1 2 0.06032 0.0000E+00 1 2 0.06280 0.0000E+00 1 2 0.05899 0.0000E+00
50000 x2 1 2 0.06203 0.0000E+00 1 2 0.05772 0.0000E+00 1 2 0.06076 0.0000E+00 1 2 0.05922 0.0000E+00
50000 x3 1 2 0.05224 0.0000E+00 1 2 0.05857 0.0000E+00 1 2 0.06821 0.0000E+00 1 2 0.06629 0.0000E+00
50000 x4 1 2 0.06079 0.0000E+00 1 3 0.07697 0.0000E+00 1 5 0.11996 0.0000E+00 ** ** ** **
50000 x5 1 2 0.06053 0.0000E+00 1 2 0.03772 0.0000E+00 1 2 0.03890 0.0000E+00 1 2 0.03026 0.0000E+00
50000 x6 1 2 0.05694 0.0000E+00 1 2 0.03597 0.0000E+00 1 2 0.03897 0.0000E+00 1 2 0.03691 0.0000E+00
50000 x7 1 2 0.06281 0.0000E+00 1 2 0.03922 0.0000E+00 1 2 0.03959 0.0000E+00 1 2 0.03842 0.0000E+00
50000 x8 1 2 0.05664 0.0000E+00 1 2 0.03785 0.0000E+00 1 2 0.03866 0.0000E+00 1 2 0.04029 0.0000E+00
Nvars IGuess MDDYM PDYM MDY MFR
Niter Nfev Cpt Norm Niter Nfev Cpt Norm Niter Nfev Cpt Norm Niter Nfev Cpt Norm
5000 x1 1 2 0.01078 0.0000E+00 1 2 0.01097 0.0000E+00 1 2 0.01014 0.0000E+00 1 2 0.01003 0.0000E+00
5000 x2 1 2 0.02040 0.0000E+00 1 2 0.00944 0.0000E+00 1 2 0.01005 0.0000E+00 1 2 0.02019 0.0000E+00
5000 x3 1 2 0.01013 0.0000E+00 1 2 0.00965 0.0000E+00 1 2 0.00947 0.0000E+00 1 2 0.01038 0.0000E+00
5000 x4 1 2 0.00951 0.0000E+00 1 2 0.00966 0.0000E+00 1 2 0.00928 0.0000E+00 ** ** ** **
5000 x5 1 2 0.01045 0.0000E+00 1 2 0.00748 0.0000E+00 1 2 0.00649 0.0000E+00 1 2 0.00520 0.0000E+00
5000 x6 1 2 0.00940 0.0000E+00 1 2 0.00732 0.0000E+00 1 2 0.00662 0.0000E+00 1 2 0.00842 0.0000E+00
5000 x7 1 2 0.00979 0.0000E+00 1 2 0.00660 0.0000E+00 1 2 0.00667 0.0000E+00 1 2 0.00757 0.0000E+00
5000 x8 1 2 0.01111 0.0000E+00 1 2 0.00795 0.0000E+00 1 2 0.00687 0.0000E+00 1 2 0.00779 0.0000E+00
10000 x1 1 2 0.01478 0.0000E+00 1 2 0.01622 0.0000E+00 1 2 0.01650 0.0000E+00 1 2 0.01449 0.0000E+00
10000 x2 1 2 0.01571 0.0000E+00 1 2 0.01633 0.0000E+00 1 2 0.01488 0.0000E+00 1 2 0.01594 0.0000E+00
10000 x3 1 2 0.01490 0.0000E+00 1 2 0.02384 0.0000E+00 1 2 0.03357 0.0000E+00 1 2 0.01579 0.0000E+00
10000 x4 1 2 0.01739 0.0000E+00 1 2 0.01549 0.0000E+00 1 3 0.02037 0.0000E+00 ** ** ** **
10000 x5 1 2 0.01636 0.0000E+00 1 2 0.01000 0.0000E+00 1 2 0.01013 0.0000E+00 1 2 0.00759 0.0000E+00
10000 x6 1 2 0.01637 0.0000E+00 1 2 0.00939 0.0000E+00 1 2 0.01043 0.0000E+00 1 2 0.01118 0.0000E+00
10000 x7 1 2 0.01602 0.0000E+00 1 2 0.00893 0.0000E+00 1 2 0.00994 0.0000E+00 1 2 0.01012 0.0000E+00
10000 x8 1 2 0.01539 0.0000E+00 1 2 0.01026 0.0000E+00 1 2 0.01045 0.0000E+00 1 2 0.01100 0.0000E+00
50000 x1 1 2 0.05796 0.0000E+00 1 2 0.06032 0.0000E+00 1 2 0.06280 0.0000E+00 1 2 0.05899 0.0000E+00
50000 x2 1 2 0.06203 0.0000E+00 1 2 0.05772 0.0000E+00 1 2 0.06076 0.0000E+00 1 2 0.05922 0.0000E+00
50000 x3 1 2 0.05224 0.0000E+00 1 2 0.05857 0.0000E+00 1 2 0.06821 0.0000E+00 1 2 0.06629 0.0000E+00
50000 x4 1 2 0.06079 0.0000E+00 1 3 0.07697 0.0000E+00 1 5 0.11996 0.0000E+00 ** ** ** **
50000 x5 1 2 0.06053 0.0000E+00 1 2 0.03772 0.0000E+00 1 2 0.03890 0.0000E+00 1 2 0.03026 0.0000E+00
50000 x6 1 2 0.05694 0.0000E+00 1 2 0.03597 0.0000E+00 1 2 0.03897 0.0000E+00 1 2 0.03691 0.0000E+00
50000 x7 1 2 0.06281 0.0000E+00 1 2 0.03922 0.0000E+00 1 2 0.03959 0.0000E+00 1 2 0.03842 0.0000E+00
50000 x8 1 2 0.05664 0.0000E+00 1 2 0.03785 0.0000E+00 1 2 0.03866 0.0000E+00 1 2 0.04029 0.0000E+00
Table 3.  Test results of the four methods for problems 4.3
Nvars IGuess MDDYM PDYM MDY MFR
Niter Nfev Cpt Norm Niter Nfev Cpt Norm Niter Nfev Cpt Norm Niter Nfev Cpt Norm
5000 x1 44 271 0.62935 3.1455E-09 42 258 0.60676 6.9529E-09 33 161 0.45597 7.0484E-09 ** ** ** **
5000 x2 39 243 0.63451 5.8143E-09 42 258 0.63349 6.8440E-09 49 253 0.65486 6.8393E-09 ** ** ** **
5000 x3 40 252 0.60480 7.5147E-09 41 252 0.58345 9.1694E-09 31 147 0.42323 8.9989E-09 ** ** ** **
5000 x4 37 233 0.57729 3.7953E-09 35 216 0.53482 9.0123E-09 40 194 0.51740 7.4457E-09 ** ** ** **
5000 x5 33 202 0.46893 9.9461E-09 35 217 0.53483 8.7841E-09 42 204 0.52178 7.4199E-09 ** ** ** **
5000 x6 34 212 0.52856 8.7562E-09 40 247 0.57437 6.6311E-09 64 358 0.89216 7.2966E-09 ** ** ** **
5000 x7 43 267 0.70223 9.1513E-09 37 228 0.55987 7.0629E-09 33 160 0.43868 9.8882E-09 ** ** ** **
5000 x8 39 244 0.62983 5.5424E-09 38 236 0.56374 9.7994E-09 39 212 0.51684 4.9792E-09 ** ** ** **
10000 x1 46 283 1.24261 8.9039E-09 41 252 1.06589 8.7049E-09 43 201 0.99828 6.2921E-09 ** ** ** **
10000 x2 43 270 1.20866 6.3134E-09 41 252 1.09452 8.5691E-09 33 125 0.65028 8.5659E-09 ** ** ** **
10000 x3 35 213 0.98029 7.5116E-09 41 252 1.11311 7.5636E-09 38 200 0.90475 9.7816E-09 ** ** ** **
10000 x4 41 256 1.13918 6.2659E-09 35 216 0.95719 7.4878E-09 ** ** ** ** ** ** ** **
10000 x5 38 232 1.09266 4.3520E-09 35 217 0.95539 7.2962E-09 32 165 0.78884 6.7343E-09 ** ** ** **
10000 x6 38 232 1.05984 8.2915E-09 39 241 1.02612 8.3641E-09 40 189 0.88094 8.8953E-09 ** ** ** **
10000 x7 41 262 1.15504 6.3728E-09 32 200 0.91508 9.2095E-09 42 207 1.01358 9.6843E-09 ** ** ** **
10000 x8 44 263 1.22024 4.4229E-09 45 278 1.23987 7.9323E-09 ** ** ** ** ** ** ** **
50000 x1 51 316 6.68143 6.4806E-09 42 260 5.45556 7.9101E-09 38 187 4.12196 7.2817E-09 ** ** ** **
50000 x2 48 301 6.33527 8.4813E-09 42 260 5.42795 7.8102E-09 40 181 4.31980 5.7514E-09 ** ** ** **
50000 x3 36 218 4.66628 7.4951E-09 42 260 5.36750 7.0264E-09 44 190 4.36631 3.5019E-09 ** ** ** **
50000 x4 33 206 4.39647 3.1500E-09 34 210 4.33667 7.4626E-09 34 143 3.36253 3.8370E-09 ** ** ** **
50000 x5 36 226 4.71243 9.0560E-09 34 211 4.36821 7.2827E-09 32 132 3.13728 9.7006E-09 ** ** ** **
50000 x6 38 235 4.97988 9.5093E-09 40 249 5.25738 8.0717E-09 36 164 3.76700 5.7396E-09 ** ** ** **
50000 x7 44 282 5.86905 6.3230E-09 37 234 4.87686 6.5499E-09 40 200 4.41362 5.0952E-09 ** ** ** **
50000 x8 37 231 4.86337 9.4485E-09 33 210 4.50616 9.9184E-09 36 138 3.29200 6.6412E-09 ** ** ** **
Nvars IGuess MDDYM PDYM MDY MFR
Niter Nfev Cpt Norm Niter Nfev Cpt Norm Niter Nfev Cpt Norm Niter Nfev Cpt Norm
5000 x1 44 271 0.62935 3.1455E-09 42 258 0.60676 6.9529E-09 33 161 0.45597 7.0484E-09 ** ** ** **
5000 x2 39 243 0.63451 5.8143E-09 42 258 0.63349 6.8440E-09 49 253 0.65486 6.8393E-09 ** ** ** **
5000 x3 40 252 0.60480 7.5147E-09 41 252 0.58345 9.1694E-09 31 147 0.42323 8.9989E-09 ** ** ** **
5000 x4 37 233 0.57729 3.7953E-09 35 216 0.53482 9.0123E-09 40 194 0.51740 7.4457E-09 ** ** ** **
5000 x5 33 202 0.46893 9.9461E-09 35 217 0.53483 8.7841E-09 42 204 0.52178 7.4199E-09 ** ** ** **
5000 x6 34 212 0.52856 8.7562E-09 40 247 0.57437 6.6311E-09 64 358 0.89216 7.2966E-09 ** ** ** **
5000 x7 43 267 0.70223 9.1513E-09 37 228 0.55987 7.0629E-09 33 160 0.43868 9.8882E-09 ** ** ** **
5000 x8 39 244 0.62983 5.5424E-09 38 236 0.56374 9.7994E-09 39 212 0.51684 4.9792E-09 ** ** ** **
10000 x1 46 283 1.24261 8.9039E-09 41 252 1.06589 8.7049E-09 43 201 0.99828 6.2921E-09 ** ** ** **
10000 x2 43 270 1.20866 6.3134E-09 41 252 1.09452 8.5691E-09 33 125 0.65028 8.5659E-09 ** ** ** **
10000 x3 35 213 0.98029 7.5116E-09 41 252 1.11311 7.5636E-09 38 200 0.90475 9.7816E-09 ** ** ** **
10000 x4 41 256 1.13918 6.2659E-09 35 216 0.95719 7.4878E-09 ** ** ** ** ** ** ** **
10000 x5 38 232 1.09266 4.3520E-09 35 217 0.95539 7.2962E-09 32 165 0.78884 6.7343E-09 ** ** ** **
10000 x6 38 232 1.05984 8.2915E-09 39 241 1.02612 8.3641E-09 40 189 0.88094 8.8953E-09 ** ** ** **
10000 x7 41 262 1.15504 6.3728E-09 32 200 0.91508 9.2095E-09 42 207 1.01358 9.6843E-09 ** ** ** **
10000 x8 44 263 1.22024 4.4229E-09 45 278 1.23987 7.9323E-09 ** ** ** ** ** ** ** **
50000 x1 51 316 6.68143 6.4806E-09 42 260 5.45556 7.9101E-09 38 187 4.12196 7.2817E-09 ** ** ** **
50000 x2 48 301 6.33527 8.4813E-09 42 260 5.42795 7.8102E-09 40 181 4.31980 5.7514E-09 ** ** ** **
50000 x3 36 218 4.66628 7.4951E-09 42 260 5.36750 7.0264E-09 44 190 4.36631 3.5019E-09 ** ** ** **
50000 x4 33 206 4.39647 3.1500E-09 34 210 4.33667 7.4626E-09 34 143 3.36253 3.8370E-09 ** ** ** **
50000 x5 36 226 4.71243 9.0560E-09 34 211 4.36821 7.2827E-09 32 132 3.13728 9.7006E-09 ** ** ** **
50000 x6 38 235 4.97988 9.5093E-09 40 249 5.25738 8.0717E-09 36 164 3.76700 5.7396E-09 ** ** ** **
50000 x7 44 282 5.86905 6.3230E-09 37 234 4.87686 6.5499E-09 40 200 4.41362 5.0952E-09 ** ** ** **
50000 x8 37 231 4.86337 9.4485E-09 33 210 4.50616 9.9184E-09 36 138 3.29200 6.6412E-09 ** ** ** **
Table 4.  Test results of the four methods for problems 4.4
Nvars IGuess MDDYM PDYM MDY MFR
Niter Nfev Cpt Norm Niter Nfev Cpt Norm Niter Nfev Cpt Norm Niter Nfev Cpt Norm
5000 x1 4 6 0.01613 1.1694E-09 16 35 0.06067 4.1367E-09 11 25 0.04204 2.7254E-09 13 22 0.05005 2.5649E-09
5000 x2 4 6 0.01681 8.3703E-09 16 35 0.06441 8.3035E-09 6 14 0.02475 3.7553E-09 15 25 0.05655 1.3159E-09
5000 x3 3 5 0.01304 3.3286E-09 18 39 0.05937 4.7509E-09 12 28 0.04195 4.8432E-09 17 28 0.05855 5.2831E-09
5000 x4 15 33 0.05422 1.7766E-09 20 43 0.06423 3.6850E-09 13 30 0.05059 2.4951E-09 21 35 0.09843 3.9220E-09
5000 x5 10 23 0.05208 1.0831E-09 16 35 0.05671 5.9886E-09 15 36 0.06009 5.0128E-09 21 37 0.07153 3.9833E-09
5000 x6 8 17 0.04271 6.9887E-09 20 44 0.06183 7.2667E-09 16 41 0.06403 4.8801E-09 22 42 0.09015 3.9830E-09
5000 x7 20 44 0.06693 1.8089E-09 20 46 0.07138 9.6847E-09 15 43 0.05725 3.8385E-09 22 43 0.08100 3.9830E-09
5000 x8 8 16 0.03136 2.2485E-09 20 46 0.06917 6.3318E-09 15 44 0.52479 4.4557E-09 22 44 0.08352 3.9830E-09
10000 x1 4 6 0.02494 1.6538E-09 16 35 0.08973 5.8502E-09 11 25 0.06575 2.1388E-09 13 22 0.07581 3.6273E-09
10000 x2 5 7 0.02770 1.0276E-10 17 37 0.08415 3.8974E-09 12 28 0.06498 3.1362E-09 15 25 0.12385 1.8610E-09
10000 x3 3 5 0.02101 4.7073E-09 18 39 0.08491 6.7188E-09 11 26 0.07826 6.4560E-09 17 28 0.12292 7.4714E-09
10000 x4 15 33 0.10815 2.5125E-09 20 43 0.09236 5.2114E-09 15 37 0.09462 2.7562E-09 21 35 0.13054 5.5465E-09
10000 x5 10 23 0.06283 1.5317E-09 17 38 0.09884 9.9575E-09 16 40 0.21861 3.1439E-09 21 37 0.13807 5.6333E-09
10000 x6 8 17 0.05651 9.8835E-09 21 48 0.11028 7.9275E-09 16 42 0.10622 4.2741E-09 22 42 0.13112 5.6328E-09
10000 x7 20 44 0.10776 2.5582E-09 21 49 0.12883 4.5754E-09 16 46 0.08559 5.0752E-09 22 43 0.12979 5.6328E-09
10000 x8 8 16 0.05015 3.1798E-09 20 47 0.11975 8.9545E-09 16 47 0.11307 5.1334E-09 22 44 0.12479 5.6328E-09
50000 x1 4 6 0.09720 3.6981E-09 17 37 0.36221 4.3415E-09 12 28 0.27787 3.4805E-09 13 22 0.29020 8.1110E-09
50000 x2 5 7 0.11296 2.2973E-10 17 37 0.32769 8.7148E-09 12 28 0.27745 5.2196E-09 15 25 0.48431 4.1613E-09
50000 x3 4 7 0.09722 5.7871E-09 19 41 0.37649 4.9966E-09 13 31 0.30385 4.6998E-09 19 31 0.40698 2.0700E-09
50000 x4 15 33 0.34967 5.6182E-09 22 49 0.42467 3.5662E-09 16 41 0.39497 5.2693E-09 23 38 0.48994 1.5367E-09
50000 x5 10 23 0.25464 3.4249E-09 22 52 0.44383 8.0423E-09 18 50 0.44577 4.6654E-09 23 40 0.49968 1.5607E-09
50000 x6 9 18 0.19377 6.5941E-10 23 55 0.46324 4.6940E-09 19 54 0.46182 5.3105E-09 24 45 0.53573 1.5606E-09
50000 x7 20 44 0.43657 5.7203E-09 23 58 0.53040 9.1580E-09 18 57 0.47142 3.8366E-09 24 46 0.54006 1.5606E-09
50000 x8 8 16 0.17111 7.1103E-09 23 58 0.49150 8.6064E-09 18 57 0.48898 3.4965E-09 24 47 0.56776 1.5606E-09
Nvars IGuess MDDYM PDYM MDY MFR
Niter Nfev Cpt Norm Niter Nfev Cpt Norm Niter Nfev Cpt Norm Niter Nfev Cpt Norm
5000 x1 4 6 0.01613 1.1694E-09 16 35 0.06067 4.1367E-09 11 25 0.04204 2.7254E-09 13 22 0.05005 2.5649E-09
5000 x2 4 6 0.01681 8.3703E-09 16 35 0.06441 8.3035E-09 6 14 0.02475 3.7553E-09 15 25 0.05655 1.3159E-09
5000 x3 3 5 0.01304 3.3286E-09 18 39 0.05937 4.7509E-09 12 28 0.04195 4.8432E-09 17 28 0.05855 5.2831E-09
5000 x4 15 33 0.05422 1.7766E-09 20 43 0.06423 3.6850E-09 13 30 0.05059 2.4951E-09 21 35 0.09843 3.9220E-09
5000 x5 10 23 0.05208 1.0831E-09 16 35 0.05671 5.9886E-09 15 36 0.06009 5.0128E-09 21 37 0.07153 3.9833E-09
5000 x6 8 17 0.04271 6.9887E-09 20 44 0.06183 7.2667E-09 16 41 0.06403 4.8801E-09 22 42 0.09015 3.9830E-09
5000 x7 20 44 0.06693 1.8089E-09 20 46 0.07138 9.6847E-09 15 43 0.05725 3.8385E-09 22 43 0.08100 3.9830E-09
5000 x8 8 16 0.03136 2.2485E-09 20 46 0.06917 6.3318E-09 15 44 0.52479 4.4557E-09 22 44 0.08352 3.9830E-09
10000 x1 4 6 0.02494 1.6538E-09 16 35 0.08973 5.8502E-09 11 25 0.06575 2.1388E-09 13 22 0.07581 3.6273E-09
10000 x2 5 7 0.02770 1.0276E-10 17 37 0.08415 3.8974E-09 12 28 0.06498 3.1362E-09 15 25 0.12385 1.8610E-09
10000 x3 3 5 0.02101 4.7073E-09 18 39 0.08491 6.7188E-09 11 26 0.07826 6.4560E-09 17 28 0.12292 7.4714E-09
10000 x4 15 33 0.10815 2.5125E-09 20 43 0.09236 5.2114E-09 15 37 0.09462 2.7562E-09 21 35 0.13054 5.5465E-09
10000 x5 10 23 0.06283 1.5317E-09 17 38 0.09884 9.9575E-09 16 40 0.21861 3.1439E-09 21 37 0.13807 5.6333E-09
10000 x6 8 17 0.05651 9.8835E-09 21 48 0.11028 7.9275E-09 16 42 0.10622 4.2741E-09 22 42 0.13112 5.6328E-09
10000 x7 20 44 0.10776 2.5582E-09 21 49 0.12883 4.5754E-09 16 46 0.08559 5.0752E-09 22 43 0.12979 5.6328E-09
10000 x8 8 16 0.05015 3.1798E-09 20 47 0.11975 8.9545E-09 16 47 0.11307 5.1334E-09 22 44 0.12479 5.6328E-09
50000 x1 4 6 0.09720 3.6981E-09 17 37 0.36221 4.3415E-09 12 28 0.27787 3.4805E-09 13 22 0.29020 8.1110E-09
50000 x2 5 7 0.11296 2.2973E-10 17 37 0.32769 8.7148E-09 12 28 0.27745 5.2196E-09 15 25 0.48431 4.1613E-09
50000 x3 4 7 0.09722 5.7871E-09 19 41 0.37649 4.9966E-09 13 31 0.30385 4.6998E-09 19 31 0.40698 2.0700E-09
50000 x4 15 33 0.34967 5.6182E-09 22 49 0.42467 3.5662E-09 16 41 0.39497 5.2693E-09 23 38 0.48994 1.5367E-09
50000 x5 10 23 0.25464 3.4249E-09 22 52 0.44383 8.0423E-09 18 50 0.44577 4.6654E-09 23 40 0.49968 1.5607E-09
50000 x6 9 18 0.19377 6.5941E-10 23 55 0.46324 4.6940E-09 19 54 0.46182 5.3105E-09 24 45 0.53573 1.5606E-09
50000 x7 20 44 0.43657 5.7203E-09 23 58 0.53040 9.1580E-09 18 57 0.47142 3.8366E-09 24 46 0.54006 1.5606E-09
50000 x8 8 16 0.17111 7.1103E-09 23 58 0.49150 8.6064E-09 18 57 0.48898 3.4965E-09 24 47 0.56776 1.5606E-09
Table 5.  Test results of the four methods for problems 4.5
Nvars IGuess MDDYM PDYM MDY MFR
Niter Nfev Cpt Norm Niter Nfev Cpt Norm Niter Nfev Cpt Norm Niter Nfev Cpt Norm
5000 x1 13 29 0.08409 6.1206E-09 21 45 0.11835 4.5793E-09 13 34 0.07926 8.7285E-09 27 57 0.14177 9.2581E-09
5000 x2 13 29 0.08607 6.0981E-09 21 45 0.11653 4.5624E-09 13 34 0.07905 8.6940E-09 27 57 0.14553 9.2256E-09
5000 x3 13 29 0.08117 5.9183E-09 21 45 0.09188 4.4271E-09 13 34 0.09186 8.4192E-09 27 57 0.13343 8.9623E-09
5000 x4 13 29 0.07941 4.4547E-09 20 43 0.14263 9.9982E-09 12 29 0.09527 8.3184E-09 27 57 0.14216 6.6834E-09
5000 x5 13 29 0.07700 3.3263E-09 20 43 0.11080 7.4583E-09 15 38 0.09559 3.4919E-09 27 57 0.12148 4.9324E-09
5000 x6 12 27 0.07471 8.0821E-09 20 43 0.12521 4.9185E-09 14 34 0.08074 3.5022E-09 26 55 0.12577 7.3575E-09
5000 x7 12 27 0.11041 3.8935E-09 19 41 0.11220 7.1229E-09 13 31 0.07748 4.0165E-09 25 53 0.13185 8.0804E-09
5000 x8 11 25 0.06216 4.7068E-09 18 39 0.10240 9.9853E-09 13 31 0.08459 3.3404E-09 24 51 0.51949 8.5601E-09
10000 x1 13 30 0.15350 2.3785E-09 23 53 0.19978 5.1107E-09 15 46 0.16562 8.1229E-09 28 58 0.31646 0.0000E+00
10000 x2 15 37 0.15897 5.1927E-09 23 53 0.19735 5.0918E-09 15 46 0.17351 8.0936E-09 28 58 0.24591 0.0000E+00
10000 x3 19 54 0.23207 2.7569E-09 23 53 0.21008 4.9409E-09 15 46 0.15890 7.8592E-09 28 58 0.23659 0.0000E+00
10000 x4 13 32 0.13566 2.2501E-09 21 45 0.18849 4.7069E-09 14 39 0.16351 6.5113E-09 27 57 0.37153 9.3333E-09
10000 x5 14 35 0.13436 7.6581E-09 21 45 0.16255 3.5112E-09 13 34 0.12313 6.3921E-09 27 57 0.23985 6.9447E-09
10000 x6 12 28 0.12876 7.1509E-09 20 43 0.18314 6.9563E-09 11 25 0.12049 8.3813E-09 27 56 0.25423 0.0000E+00
10000 x7 9 18 0.08808 5.6775E-09 20 43 0.21754 3.3642E-09 12 29 0.13141 4.5477E-09 26 54 0.21101 0.0000E+00
10000 x8 7 13 0.06663 5.1294E-09 19 41 0.16822 4.6958E-09 13 31 0.14696 4.3632E-09 25 52 0.23674 0.0000E+00
50000 x1 8 14 0.31637 9.7835E-10 26 67 1.02269 9.0422E-09 23 107 1.49375 8.1128E-09 24 50 0.92306 0.0000E+00
50000 x2 8 14 0.29982 9.7482E-10 26 67 1.06621 9.0088E-09 23 107 1.41950 8.0837E-09 24 50 0.92547 0.0000E+00
50000 x3 8 14 0.29072 9.4607E-10 26 67 1.07177 8.7417E-09 23 104 1.38072 6.5311E-09 24 50 0.89133 0.0000E+00
50000 x4 8 14 0.29903 7.1236E-10 25 63 1.00707 7.9619E-09 20 80 1.19303 6.3556E-09 24 50 0.94953 0.0000E+00
50000 x5 8 14 0.29453 5.3210E-10 23 53 0.87334 6.1957E-09 17 58 0.90448 6.6277E-09 24 50 0.88540 0.0000E+00
50000 x6 8 14 0.31612 3.5134E-10 22 49 0.86319 7.0213E-09 14 40 0.63377 7.7993E-09 22 46 0.86539 0.0000E+00
50000 x7 8 14 0.28685 1.7015E-10 20 43 0.74746 7.5227E-09 15 38 0.65625 3.5221E-09 22 46 0.85031 0.0000E+00
50000 x8 7 13 0.27290 5.4616E-09 20 43 0.74983 3.5066E-09 11 27 0.48111 6.0062E-09 22 46 0.85129 0.0000E+00
Nvars IGuess MDDYM PDYM MDY MFR
Niter Nfev Cpt Norm Niter Nfev Cpt Norm Niter Nfev Cpt Norm Niter Nfev Cpt Norm
5000 x1 13 29 0.08409 6.1206E-09 21 45 0.11835 4.5793E-09 13 34 0.07926 8.7285E-09 27 57 0.14177 9.2581E-09
5000 x2 13 29 0.08607 6.0981E-09 21 45 0.11653 4.5624E-09 13 34 0.07905 8.6940E-09 27 57 0.14553 9.2256E-09
5000 x3 13 29 0.08117 5.9183E-09 21 45 0.09188 4.4271E-09 13 34 0.09186 8.4192E-09 27 57 0.13343 8.9623E-09
5000 x4 13 29 0.07941 4.4547E-09 20 43 0.14263 9.9982E-09 12 29 0.09527 8.3184E-09 27 57 0.14216 6.6834E-09
5000 x5 13 29 0.07700 3.3263E-09 20 43 0.11080 7.4583E-09 15 38 0.09559 3.4919E-09 27 57 0.12148 4.9324E-09
5000 x6 12 27 0.07471 8.0821E-09 20 43 0.12521 4.9185E-09 14 34 0.08074 3.5022E-09 26 55 0.12577 7.3575E-09
5000 x7 12 27 0.11041 3.8935E-09 19 41 0.11220 7.1229E-09 13 31 0.07748 4.0165E-09 25 53 0.13185 8.0804E-09
5000 x8 11 25 0.06216 4.7068E-09 18 39 0.10240 9.9853E-09 13 31 0.08459 3.3404E-09 24 51 0.51949 8.5601E-09
10000 x1 13 30 0.15350 2.3785E-09 23 53 0.19978 5.1107E-09 15 46 0.16562 8.1229E-09 28 58 0.31646 0.0000E+00
10000 x2 15 37 0.15897 5.1927E-09 23 53 0.19735 5.0918E-09 15 46 0.17351 8.0936E-09 28 58 0.24591 0.0000E+00
10000 x3 19 54 0.23207 2.7569E-09 23 53 0.21008 4.9409E-09 15 46 0.15890 7.8592E-09 28 58 0.23659 0.0000E+00
10000 x4 13 32 0.13566 2.2501E-09 21 45 0.18849 4.7069E-09 14 39 0.16351 6.5113E-09 27 57 0.37153 9.3333E-09
10000 x5 14 35 0.13436 7.6581E-09 21 45 0.16255 3.5112E-09 13 34 0.12313 6.3921E-09 27 57 0.23985 6.9447E-09
10000 x6 12 28 0.12876 7.1509E-09 20 43 0.18314 6.9563E-09 11 25 0.12049 8.3813E-09 27 56 0.25423 0.0000E+00
10000 x7 9 18 0.08808 5.6775E-09 20 43 0.21754 3.3642E-09 12 29 0.13141 4.5477E-09 26 54 0.21101 0.0000E+00
10000 x8 7 13 0.06663 5.1294E-09 19 41 0.16822 4.6958E-09 13 31 0.14696 4.3632E-09 25 52 0.23674 0.0000E+00
50000 x1 8 14 0.31637 9.7835E-10 26 67 1.02269 9.0422E-09 23 107 1.49375 8.1128E-09 24 50 0.92306 0.0000E+00
50000 x2 8 14 0.29982 9.7482E-10 26 67 1.06621 9.0088E-09 23 107 1.41950 8.0837E-09 24 50 0.92547 0.0000E+00
50000 x3 8 14 0.29072 9.4607E-10 26 67 1.07177 8.7417E-09 23 104 1.38072 6.5311E-09 24 50 0.89133 0.0000E+00
50000 x4 8 14 0.29903 7.1236E-10 25 63 1.00707 7.9619E-09 20 80 1.19303 6.3556E-09 24 50 0.94953 0.0000E+00
50000 x5 8 14 0.29453 5.3210E-10 23 53 0.87334 6.1957E-09 17 58 0.90448 6.6277E-09 24 50 0.88540 0.0000E+00
50000 x6 8 14 0.31612 3.5134E-10 22 49 0.86319 7.0213E-09 14 40 0.63377 7.7993E-09 22 46 0.86539 0.0000E+00
50000 x7 8 14 0.28685 1.7015E-10 20 43 0.74746 7.5227E-09 15 38 0.65625 3.5221E-09 22 46 0.85031 0.0000E+00
50000 x8 7 13 0.27290 5.4616E-09 20 43 0.74983 3.5066E-09 11 27 0.48111 6.0062E-09 22 46 0.85129 0.0000E+00
Table 6.  Test results of the four methods for problems 4.6
Nvars IGuess MDDYM PDYM MDY MFR
Niter Nfev Cpt Norm Niter Nfev Cpt Norm Niter Nfev Cpt Norm Niter Nfev Cpt Norm
5000 x1 11 29 0.05830 5.9634E-09 21 66 0.09121 4.9608E-09 12 33 0.05388 5.3440E-09 60 245 0.30692 9.2918E-09
5000 x2 11 29 0.05493 4.3213E-09 21 66 0.07995 4.7837E-09 12 33 0.05358 5.8173E-09 60 245 0.29193 9.1558E-09
5000 x3 10 27 0.04793 1.1728E-09 20 63 0.08563 9.6409E-09 12 33 0.05520 4.2467E-09 60 245 0.30544 7.9678E-09
5000 x4 8 21 0.03816 8.8309E-09 6 15 0.02606 1.5409E-09 10 25 0.04814 5.5576E-09 59 241 0.27576 8.6175E-09
5000 x5 17 52 0.08657 6.7905E-09 22 69 0.10643 3.8403E-09 12 32 0.06375 3.6002E-09 61 248 0.29183 8.0471E-09
5000 x6 12 31 0.04854 6.6177E-09 22 68 0.11060 4.6406E-09 13 42 0.10346 5.9588E-09 62 250 0.29224 8.1876E-09
5000 x7 13 36 0.05865 5.1067E-09 23 72 0.11025 5.6953E-09 11 31 0.05111 2.8649E-09 62 250 0.27411 7.6296E-09
5000 x8 14 38 0.06893 9.5514E-10 7 18 0.03452 1.0412E-09 11 29 0.05033 7.4314E-09 62 250 0.31480 8.3682E-09
10000 x1 11 29 0.12935 8.4335E-09 21 66 0.14482 7.0156E-09 13 37 0.09583 1.4459E-09 61 249 0.49287 9.5333E-09
10000 x2 11 29 0.08446 6.1113E-09 21 66 0.18801 6.7652E-09 13 37 0.10481 1.7840E-09 61 249 0.50182 9.3937E-09
10000 x3 10 27 0.08571 1.6586E-09 21 66 0.15276 4.9435E-09 12 33 0.10223 4.7426E-09 61 249 0.48216 8.1748E-09
10000 x4 9 23 0.07085 1.8142E-09 6 15 0.05041 2.1791E-09 10 25 0.08289 9.5969E-09 60 245 0.46931 8.8414E-09
10000 x5 17 52 0.14664 9.6032E-09 22 69 0.18159 5.4309E-09 11 28 0.08064 8.7915E-09 62 252 0.50543 8.2562E-09
10000 x6 12 31 0.08264 9.3589E-09 23 73 0.15916 6.3474E-09 14 46 0.10846 1.7662E-09 63 254 0.51418 8.4004E-09
10000 x7 13 36 0.12906 7.2220E-09 23 73 0.25665 8.5323E-09 14 43 0.10606 2.1715E-09 63 254 0.51744 7.8279E-09
10000 x8 14 38 0.10402 1.3508E-09 7 18 0.07598 1.4725E-09 14 42 0.09957 2.1096E-09 63 254 0.49114 8.5857E-09
50000 x1 12 31 0.33042 9.3840E-10 22 69 0.62427 5.6816E-09 14 41 0.42098 2.3754E-09 64 261 2.32850 8.1397E-09
50000 x2 12 31 0.36379 5.6107E-10 22 69 0.62287 5.4788E-09 14 41 0.42743 2.1971E-09 64 261 2.27368 8.0206E-09
50000 x3 10 27 0.29782 3.7088E-09 22 69 0.64410 4.0033E-09 13 39 0.41428 8.9237E-09 63 257 2.26461 9.6210E-09
50000 x4 9 23 0.26084 4.0568E-09 6 15 0.17896 4.8726E-09 13 38 0.39586 3.2734E-09 63 257 2.21871 7.5490E-09
50000 x5 18 54 0.54905 3.5244E-09 24 77 0.69938 4.1819E-09 17 62 0.58483 2.4241E-09 64 260 2.27889 9.7167E-09
50000 x6 13 33 0.38900 1.9096E-09 24 77 0.74722 5.1440E-09 15 50 0.45349 5.4303E-09 65 262 2.28170 9.8864E-09
50000 x7 14 38 0.38756 1.1869E-09 24 77 0.67827 6.9131E-09 14 39 0.40820 1.4432E-09 65 262 2.33294 9.2126E-09
50000 x8 14 38 0.40306 3.0204E-09 24 77 0.69055 7.2256E-09 14 38 0.38383 1.4634E-09 66 266 2.30200 7.3306E-09
Nvars IGuess MDDYM PDYM MDY MFR
Niter Nfev Cpt Norm Niter Nfev Cpt Norm Niter Nfev Cpt Norm Niter Nfev Cpt Norm
5000 x1 11 29 0.05830 5.9634E-09 21 66 0.09121 4.9608E-09 12 33 0.05388 5.3440E-09 60 245 0.30692 9.2918E-09
5000 x2 11 29 0.05493 4.3213E-09 21 66 0.07995 4.7837E-09 12 33 0.05358 5.8173E-09 60 245 0.29193 9.1558E-09
5000 x3 10 27 0.04793 1.1728E-09 20 63 0.08563 9.6409E-09 12 33 0.05520 4.2467E-09 60 245 0.30544 7.9678E-09
5000 x4 8 21 0.03816 8.8309E-09 6 15 0.02606 1.5409E-09 10 25 0.04814 5.5576E-09 59 241 0.27576 8.6175E-09
5000 x5 17 52 0.08657 6.7905E-09 22 69 0.10643 3.8403E-09 12 32 0.06375 3.6002E-09 61 248 0.29183 8.0471E-09
5000 x6 12 31 0.04854 6.6177E-09 22 68 0.11060 4.6406E-09 13 42 0.10346 5.9588E-09 62 250 0.29224 8.1876E-09
5000 x7 13 36 0.05865 5.1067E-09 23 72 0.11025 5.6953E-09 11 31 0.05111 2.8649E-09 62 250 0.27411 7.6296E-09
5000 x8 14 38 0.06893 9.5514E-10 7 18 0.03452 1.0412E-09 11 29 0.05033 7.4314E-09 62 250 0.31480 8.3682E-09
10000 x1 11 29 0.12935 8.4335E-09 21 66 0.14482 7.0156E-09 13 37 0.09583 1.4459E-09 61 249 0.49287 9.5333E-09
10000 x2 11 29 0.08446 6.1113E-09 21 66 0.18801 6.7652E-09 13 37 0.10481 1.7840E-09 61 249 0.50182 9.3937E-09
10000 x3 10 27 0.08571 1.6586E-09 21 66 0.15276 4.9435E-09 12 33 0.10223 4.7426E-09 61 249 0.48216 8.1748E-09
10000 x4 9 23 0.07085 1.8142E-09 6 15 0.05041 2.1791E-09 10 25 0.08289 9.5969E-09 60 245 0.46931 8.8414E-09
10000 x5 17 52 0.14664 9.6032E-09 22 69 0.18159 5.4309E-09 11 28 0.08064 8.7915E-09 62 252 0.50543 8.2562E-09
10000 x6 12 31 0.08264 9.3589E-09 23 73 0.15916 6.3474E-09 14 46 0.10846 1.7662E-09 63 254 0.51418 8.4004E-09
10000 x7 13 36 0.12906 7.2220E-09 23 73 0.25665 8.5323E-09 14 43 0.10606 2.1715E-09 63 254 0.51744 7.8279E-09
10000 x8 14 38 0.10402 1.3508E-09 7 18 0.07598 1.4725E-09 14 42 0.09957 2.1096E-09 63 254 0.49114 8.5857E-09
50000 x1 12 31 0.33042 9.3840E-10 22 69 0.62427 5.6816E-09 14 41 0.42098 2.3754E-09 64 261 2.32850 8.1397E-09
50000 x2 12 31 0.36379 5.6107E-10 22 69 0.62287 5.4788E-09 14 41 0.42743 2.1971E-09 64 261 2.27368 8.0206E-09
50000 x3 10 27 0.29782 3.7088E-09 22 69 0.64410 4.0033E-09 13 39 0.41428 8.9237E-09 63 257 2.26461 9.6210E-09
50000 x4 9 23 0.26084 4.0568E-09 6 15 0.17896 4.8726E-09 13 38 0.39586 3.2734E-09 63 257 2.21871 7.5490E-09
50000 x5 18 54 0.54905 3.5244E-09 24 77 0.69938 4.1819E-09 17 62 0.58483 2.4241E-09 64 260 2.27889 9.7167E-09
50000 x6 13 33 0.38900 1.9096E-09 24 77 0.74722 5.1440E-09 15 50 0.45349 5.4303E-09 65 262 2.28170 9.8864E-09
50000 x7 14 38 0.38756 1.1869E-09 24 77 0.67827 6.9131E-09 14 39 0.40820 1.4432E-09 65 262 2.33294 9.2126E-09
50000 x8 14 38 0.40306 3.0204E-09 24 77 0.69055 7.2256E-09 14 38 0.38383 1.4634E-09 66 266 2.30200 7.3306E-09
Table 7.  Test results of the four methods for problems 4.7
Nvars IGuess MDDYM PDYM MDY MFR
Niter Nfev Cpt Norm Niter Nfev Cpt Norm Niter Nfev Cpt Norm Niter Nfev Cpt Norm
5000 x1 16 59 0.09432 1.3050E-09 30 124 0.14592 7.0705E-09 10 32 0.04996 7.5384E-09 57 349 0.32182 9.4550E-09
5000 x2 14 53 0.07568 4.0067E-09 30 124 0.13148 6.9353E-09 10 30 0.04617 9.1316E-09 57 349 0.35728 9.3912E-09
5000 x3 16 60 0.08765 6.4133E-09 30 124 0.13896 5.8827E-09 9 30 0.03916 7.4550E-09 57 349 0.30529 8.7821E-09
5000 x4 16 62 0.07775 5.5391E-09 11 37 0.06037 1.8485E-09 9 29 0.04321 4.7455E-09 53 325 0.31769 7.4711E-09
5000 x5 15 57 0.07999 9.8573E-09 12 40 0.05478 9.3263E-09 11 37 0.05279 5.2468E-09 57 346 0.31532 7.5276E-09
5000 x6 15 58 0.07809 7.8166E-09 13 43 0.06090 2.5833E-09 ** ** ** ** 58 349 0.29898 8.0592E-09
5000 x7 16 59 0.08286 1.3795E-09 14 44 0.07650 3.3787E-09 ** ** ** ** 56 333 0.32301 7.5313E-09
5000 x8 12 42 0.06033 2.4066E-10 14 43 0.06044 6.0314E-09 ** ** ** ** 54 321 0.29418 8.5637E-09
10000 x1 16 59 0.14692 1.8456E-09 30 124 0.26347 9.9993E-09 11 33 0.06356 7.5740E-09 58 355 0.54602 9.3986E-09
10000 x2 14 53 0.10996 5.6664E-09 30 124 0.23092 9.8079E-09 11 34 0.07902 4.5397E-11 58 355 0.55024 9.3352E-09
10000 x3 16 60 0.13400 9.0698E-09 30 124 0.21597 8.3194E-09 13 43 0.10110 9.3470E-09 58 355 0.53853 8.7297E-09
10000 x4 16 62 0.13363 7.8334E-09 11 37 0.30166 2.6142E-09 6 19 0.04657 4.5079E-10 54 331 0.52338 7.4265E-09
10000 x5 16 61 0.12671 4.8938E-09 13 43 0.07998 1.9679E-09 11 39 0.08054 2.4455E-09 58 352 0.52326 7.4827E-09
10000 x6 16 62 0.12887 5.9649E-09 13 43 0.09948 3.6534E-09 ** ** ** ** 59 355 0.61184 8.0112E-09
10000 x7 16 59 0.14343 1.9510E-09 14 44 0.07941 4.7783E-09 ** ** ** ** 57 339 0.51324 7.4864E-09
10000 x8 12 42 0.09692 3.4034E-10 14 43 0.09317 8.5297E-09 167 1820 2.60568 4.7429E-11 55 327 0.52918 8.5127E-09
50000 x1 16 59 0.52241 4.1268E-09 32 132 0.95785 4.9313E-09 14 54 0.46701 1.1179E-09 61 373 2.61151 7.2981E-09
50000 x2 15 56 0.46085 5.5395E-09 32 132 0.98458 4.8369E-09 16 60 0.51123 4.0239E-10 61 373 2.58425 7.2489E-09
50000 x3 17 64 0.56293 7.8990E-09 31 128 0.97264 8.7364E-09 16 58 0.52341 3.8080E-09 60 367 2.53142 9.6440E-09
50000 x4 17 65 0.52033 1.7365E-09 11 37 0.33492 5.8455E-09 10 32 0.30108 3.4982E-09 56 343 2.30189 8.2044E-09
50000 x5 17 64 0.56208 1.1172E-09 13 43 0.40677 4.4003E-09 87 897 5.90688 5.5572E-09 60 364 2.46086 8.2665E-09
50000 x6 17 66 0.57559 4.9142E-09 13 43 0.38142 8.1692E-09 61 577 4.09402 3.9378E-09 61 367 2.51340 8.8502E-09
50000 x7 16 59 0.51799 4.3625E-09 17 54 0.49638 1.9299E-09 66 643 4.51278 7.7185E-09 59 351 2.42242 8.2705E-09
50000 x8 12 42 0.40991 7.6102E-10 35 137 1.04612 6.9307E-09 185 1959 12.83235 7.1829E-09 57 339 2.30068 9.4043E-09
Nvars IGuess MDDYM PDYM MDY MFR
Niter Nfev Cpt Norm Niter Nfev Cpt Norm Niter Nfev Cpt Norm Niter Nfev Cpt Norm
5000 x1 16 59 0.09432 1.3050E-09 30 124 0.14592 7.0705E-09 10 32 0.04996 7.5384E-09 57 349 0.32182 9.4550E-09
5000 x2 14 53 0.07568 4.0067E-09 30 124 0.13148 6.9353E-09 10 30 0.04617 9.1316E-09 57 349 0.35728 9.3912E-09
5000 x3 16 60 0.08765 6.4133E-09 30 124 0.13896 5.8827E-09 9 30 0.03916 7.4550E-09 57 349 0.30529 8.7821E-09
5000 x4 16 62 0.07775 5.5391E-09 11 37 0.06037 1.8485E-09 9 29 0.04321 4.7455E-09 53 325 0.31769 7.4711E-09
5000 x5 15 57 0.07999 9.8573E-09 12 40 0.05478 9.3263E-09 11 37 0.05279 5.2468E-09 57 346 0.31532 7.5276E-09
5000 x6 15 58 0.07809 7.8166E-09 13 43 0.06090 2.5833E-09 ** ** ** ** 58 349 0.29898 8.0592E-09
5000 x7 16 59 0.08286 1.3795E-09 14 44 0.07650 3.3787E-09 ** ** ** ** 56 333 0.32301 7.5313E-09
5000 x8 12 42 0.06033 2.4066E-10 14 43 0.06044 6.0314E-09 ** ** ** ** 54 321 0.29418 8.5637E-09
10000 x1 16 59 0.14692 1.8456E-09 30 124 0.26347 9.9993E-09 11 33 0.06356 7.5740E-09 58 355 0.54602 9.3986E-09
10000 x2 14 53 0.10996 5.6664E-09 30 124 0.23092 9.8079E-09 11 34 0.07902 4.5397E-11 58 355 0.55024 9.3352E-09
10000 x3 16 60 0.13400 9.0698E-09 30 124 0.21597 8.3194E-09 13 43 0.10110 9.3470E-09 58 355 0.53853 8.7297E-09
10000 x4 16 62 0.13363 7.8334E-09 11 37 0.30166 2.6142E-09 6 19 0.04657 4.5079E-10 54 331 0.52338 7.4265E-09
10000 x5 16 61 0.12671 4.8938E-09 13 43 0.07998 1.9679E-09 11 39 0.08054 2.4455E-09 58 352 0.52326 7.4827E-09
10000 x6 16 62 0.12887 5.9649E-09 13 43 0.09948 3.6534E-09 ** ** ** ** 59 355 0.61184 8.0112E-09
10000 x7 16 59 0.14343 1.9510E-09 14 44 0.07941 4.7783E-09 ** ** ** ** 57 339 0.51324 7.4864E-09
10000 x8 12 42 0.09692 3.4034E-10 14 43 0.09317 8.5297E-09 167 1820 2.60568 4.7429E-11 55 327 0.52918 8.5127E-09
50000 x1 16 59 0.52241 4.1268E-09 32 132 0.95785 4.9313E-09 14 54 0.46701 1.1179E-09 61 373 2.61151 7.2981E-09
50000 x2 15 56 0.46085 5.5395E-09 32 132 0.98458 4.8369E-09 16 60 0.51123 4.0239E-10 61 373 2.58425 7.2489E-09
50000 x3 17 64 0.56293 7.8990E-09 31 128 0.97264 8.7364E-09 16 58 0.52341 3.8080E-09 60 367 2.53142 9.6440E-09
50000 x4 17 65 0.52033 1.7365E-09 11 37 0.33492 5.8455E-09 10 32 0.30108 3.4982E-09 56 343 2.30189 8.2044E-09
50000 x5 17 64 0.56208 1.1172E-09 13 43 0.40677 4.4003E-09 87 897 5.90688 5.5572E-09 60 364 2.46086 8.2665E-09
50000 x6 17 66 0.57559 4.9142E-09 13 43 0.38142 8.1692E-09 61 577 4.09402 3.9378E-09 61 367 2.51340 8.8502E-09
50000 x7 16 59 0.51799 4.3625E-09 17 54 0.49638 1.9299E-09 66 643 4.51278 7.7185E-09 59 351 2.42242 8.2705E-09
50000 x8 12 42 0.40991 7.6102E-10 35 137 1.04612 6.9307E-09 185 1959 12.83235 7.1829E-09 57 339 2.30068 9.4043E-09
Table 8.  Test results of the four methods for problems 4.8
Nvars IGuess MDDYM PDYM MDY MFR
Niter Nfev Cpt Norm Niter Nfev Cpt Norm Niter Nfev Cpt Norm Niter Nfev Cpt Norm
5000 x1 14 105 0.16757 0.0000E+00 ** ** ** ** 8 76 0.08422 0.0000E+00 3 19 0.03360 0.0000E+00
5000 x2 14 102 0.14631 0.0000E+00 ** ** ** ** 8 76 0.10084 0.0000E+00 5 32 0.05756 0.0000E+00
5000 x3 14 96 0.13992 0.0000E+00 4 34 0.03438 0.0000E+00 15 141 0.16698 0.0000E+00 4 20 0.06854 0.0000E+00
5000 x4 17 126 0.17229 0.0000E+00 5 44 0.06671 0.0000E+00 16 90 0.14892 0.0000E+00 2 16 0.03029 0.0000E+00
5000 x5 7 55 0.08252 0.0000E+00 6 44 0.07206 0.0000E+00 22 125 0.19471 0.0000E+00 3 30 0.06496 0.0000E+00
5000 x6 4 30 0.04918 0.0000E+00 3 19 0.04112 0.0000E+00 13 95 0.14016 0.0000E+00 2 19 0.03237 0.0000E+00
5000 x7 4 30 0.05837 0.0000E+00 5 36 0.06107 0.0000E+00 19 111 0.18132 0.0000E+00 2 20 0.04827 0.0000E+00
5000 x8 6 55 0.08952 0.0000E+00 4 24 0.06101 0.0000E+00 68 499 0.65635 0.0000E+00 3 22 0.03871 0.0000E+00
10000 x1 10 73 1.18135 0.0000E+00 5 35 0.11482 0.0000E+00 8 76 0.16320 0.0000E+00 3 19 0.05802 0.0000E+00
10000 x2 11 75 0.20685 0.0000E+00 212 1703 3.57595 0.0000E+00 8 76 0.20896 0.0000E+00 3 19 0.06237 0.0000E+00
10000 x3 14 94 0.24852 0.0000E+00 5 37 0.11556 0.0000E+00 5 51 0.13497 0.0000E+00 4 20 0.06374 0.0000E+00
10000 x4 10 74 0.20559 0.0000E+00 3 20 0.05981 0.0000E+00 26 243 0.57624 0.0000E+00 2 16 0.05057 0.0000E+00
10000 x5 4 35 0.09897 0.0000E+00 5 31 0.09720 0.0000E+00 31 267 0.65314 0.0000E+00 3 30 0.08397 0.0000E+00
10000 x6 3 18 0.05165 0.0000E+00 5 34 0.10333 0.0000E+00 12 108 0.25737 0.0000E+00 2 19 0.05356 0.0000E+00
10000 x7 5 42 0.13123 0.0000E+00 4 23 0.07361 0.0000E+00 33 288 0.65692 0.0000E+00 2 20 0.05280 0.0000E+00
10000 x8 4 25 0.07246 0.0000E+00 4 23 0.08236 0.0000E+00 7 64 0.17944 0.0000E+00 3 22 0.06539 0.0000E+00
50000 x1 10 72 0.86301 0.0000E+00 7 42 0.49996 0.0000E+00 8 76 0.82304 0.0000E+00 3 19 0.24206 0.0000E+00
50000 x2 9 71 0.85537 0.0000E+00 7 41 0.49430 0.0000E+00 8 76 0.85630 0.0000E+00 3 19 0.24635 0.0000E+00
50000 x3 10 71 0.82141 0.0000E+00 7 39 0.46821 0.0000E+00 4 28 0.35046 0.0000E+00 3 19 0.23377 0.0000E+00
50000 x4 8 51 0.64090 0.0000E+00 ** ** ** ** 7 60 0.69240 0.0000E+00 2 16 0.16730 0.0000E+00
50000 x5 3 24 0.29724 0.0000E+00 7 38 0.40553 0.0000E+00 9 69 0.73666 0.0000E+00 4 31 0.32924 0.0000E+00
50000 x6 3 18 0.25253 0.0000E+00 7 40 0.47899 0.0000E+00 10 73 0.88118 0.0000E+00 2 19 0.20408 0.0000E+00
50000 x7 3 18 0.22632 0.0000E+00 ** ** ** ** 8 64 0.74632 0.0000E+00 2 20 0.23963 0.0000E+00
50000 x8 4 24 0.30785 0.0000E+00 6 32 0.33168 0.0000E+00 36 246 2.80129 0.0000E+00 3 22 0.27244 0.0000E+00
Nvars IGuess MDDYM PDYM MDY MFR
Niter Nfev Cpt Norm Niter Nfev Cpt Norm Niter Nfev Cpt Norm Niter Nfev Cpt Norm
5000 x1 14 105 0.16757 0.0000E+00 ** ** ** ** 8 76 0.08422 0.0000E+00 3 19 0.03360 0.0000E+00
5000 x2 14 102 0.14631 0.0000E+00 ** ** ** ** 8 76 0.10084 0.0000E+00 5 32 0.05756 0.0000E+00
5000 x3 14 96 0.13992 0.0000E+00 4 34 0.03438 0.0000E+00 15 141 0.16698 0.0000E+00 4 20 0.06854 0.0000E+00
5000 x4 17 126 0.17229 0.0000E+00 5 44 0.06671 0.0000E+00 16 90 0.14892 0.0000E+00 2 16 0.03029 0.0000E+00
5000 x5 7 55 0.08252 0.0000E+00 6 44 0.07206 0.0000E+00 22 125 0.19471 0.0000E+00 3 30 0.06496 0.0000E+00
5000 x6 4 30 0.04918 0.0000E+00 3 19 0.04112 0.0000E+00 13 95 0.14016 0.0000E+00 2 19 0.03237 0.0000E+00
5000 x7 4 30 0.05837 0.0000E+00 5 36 0.06107 0.0000E+00 19 111 0.18132 0.0000E+00 2 20 0.04827 0.0000E+00
5000 x8 6 55 0.08952 0.0000E+00 4 24 0.06101 0.0000E+00 68 499 0.65635 0.0000E+00 3 22 0.03871 0.0000E+00
10000 x1 10 73 1.18135 0.0000E+00 5 35 0.11482 0.0000E+00 8 76 0.16320 0.0000E+00 3 19 0.05802 0.0000E+00
10000 x2 11 75 0.20685 0.0000E+00 212 1703 3.57595 0.0000E+00 8 76 0.20896 0.0000E+00 3 19 0.06237 0.0000E+00
10000 x3 14 94 0.24852 0.0000E+00 5 37 0.11556 0.0000E+00 5 51 0.13497 0.0000E+00 4 20 0.06374 0.0000E+00
10000 x4 10 74 0.20559 0.0000E+00 3 20 0.05981 0.0000E+00 26 243 0.57624 0.0000E+00 2 16 0.05057 0.0000E+00
10000 x5 4 35 0.09897 0.0000E+00 5 31 0.09720 0.0000E+00 31 267 0.65314 0.0000E+00 3 30 0.08397 0.0000E+00
10000 x6 3 18 0.05165 0.0000E+00 5 34 0.10333 0.0000E+00 12 108 0.25737 0.0000E+00 2 19 0.05356 0.0000E+00
10000 x7 5 42 0.13123 0.0000E+00 4 23 0.07361 0.0000E+00 33 288 0.65692 0.0000E+00 2 20 0.05280 0.0000E+00
10000 x8 4 25 0.07246 0.0000E+00 4 23 0.08236 0.0000E+00 7 64 0.17944 0.0000E+00 3 22 0.06539 0.0000E+00
50000 x1 10 72 0.86301 0.0000E+00 7 42 0.49996 0.0000E+00 8 76 0.82304 0.0000E+00 3 19 0.24206 0.0000E+00
50000 x2 9 71 0.85537 0.0000E+00 7 41 0.49430 0.0000E+00 8 76 0.85630 0.0000E+00 3 19 0.24635 0.0000E+00
50000 x3 10 71 0.82141 0.0000E+00 7 39 0.46821 0.0000E+00 4 28 0.35046 0.0000E+00 3 19 0.23377 0.0000E+00
50000 x4 8 51 0.64090 0.0000E+00 ** ** ** ** 7 60 0.69240 0.0000E+00 2 16 0.16730 0.0000E+00
50000 x5 3 24 0.29724 0.0000E+00 7 38 0.40553 0.0000E+00 9 69 0.73666 0.0000E+00 4 31 0.32924 0.0000E+00
50000 x6 3 18 0.25253 0.0000E+00 7 40 0.47899 0.0000E+00 10 73 0.88118 0.0000E+00 2 19 0.20408 0.0000E+00
50000 x7 3 18 0.22632 0.0000E+00 ** ** ** ** 8 64 0.74632 0.0000E+00 2 20 0.23963 0.0000E+00
50000 x8 4 24 0.30785 0.0000E+00 6 32 0.33168 0.0000E+00 36 246 2.80129 0.0000E+00 3 22 0.27244 0.0000E+00
Table 9.  Summary of result from tables 1-4 showing the number of problems/percentage solved with least number of iteration, function values and processing time by each of the four methods
Method Niter Percentage fev Percentage Ptime Percentage Fails
MDDYM 83 43.22% 91 47.40% 81 40.19% -
PDYM 19 9.90% 14 7.29% 30 15.63% 4
MDY 36 18.75% 40 20.83% 55 28.65% 7
MFR 23 11.98% 20 10.42% 26 13.53% 27
Undecided 31 16.15% 27 14.06% - - -
Method Niter Percentage fev Percentage Ptime Percentage Fails
MDDYM 83 43.22% 91 47.40% 81 40.19% -
PDYM 19 9.90% 14 7.29% 30 15.63% 4
MDY 36 18.75% 40 20.83% 55 28.65% 7
MFR 23 11.98% 20 10.42% 26 13.53% 27
Undecided 31 16.15% 27 14.06% - - -
Table 10.  Performance results for MDDYM and MFRM methods based on objective function (ObjFun) value, mean square error (MSE), SNR and SSIM index
Image & size ObjFun MSE SNR SSIM
MDDYM MFRM MDDYM MFRM MDDYM MFRM MDDYM MFRM
Barbara $ 256\times256 $ $ 1.530\times 10^6 $ $ 1.585\times 10^6 $ $ 1.5967\times 10^2 $ $ 2.0627\times 10^2 $ 20.66 19.55 0.80 0.75
Girl $ 256\times256 $ $ 6.224\times 10^6 $ $ 6.321\times 10^6 $ $ 1.4978\times 10^2 $ $ 1.6071\times 10^2 $ 21.73 21.42 0.77 0.75
Lena $ 256\times256 $ $ 1.459\times 10^6 $ $ 1.513\times 10^6 $ $ 6.5691\times 10^1 $ $ 9.0026\times 10^1 $ 24.29 22.93 0.90 0.87
Cameraman $ 256\times256 $ $ 1.415\times 10^6 $ $ 1.473\times 10^6 $ $ 1.2579\times 10^2 $ $ 1.7757\times 10^2 $ 21.55 20.05 0.87 0.83
Image & size ObjFun MSE SNR SSIM
MDDYM MFRM MDDYM MFRM MDDYM MFRM MDDYM MFRM
Barbara $ 256\times256 $ $ 1.530\times 10^6 $ $ 1.585\times 10^6 $ $ 1.5967\times 10^2 $ $ 2.0627\times 10^2 $ 20.66 19.55 0.80 0.75
Girl $ 256\times256 $ $ 6.224\times 10^6 $ $ 6.321\times 10^6 $ $ 1.4978\times 10^2 $ $ 1.6071\times 10^2 $ 21.73 21.42 0.77 0.75
Lena $ 256\times256 $ $ 1.459\times 10^6 $ $ 1.513\times 10^6 $ $ 6.5691\times 10^1 $ $ 9.0026\times 10^1 $ 24.29 22.93 0.90 0.87
Cameraman $ 256\times256 $ $ 1.415\times 10^6 $ $ 1.473\times 10^6 $ $ 1.2579\times 10^2 $ $ 1.7757\times 10^2 $ 21.55 20.05 0.87 0.83
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