Modified Dai-Yuan iterative scheme for nonlinear systems and its application

Mohammed Yusuf Waziri; Kabiru Ahmed; Abubakar Sani Halilu; Aliyu Mohammed Awwal

doi:10.3934/naco.2021044

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Modified Dai-Yuan iterative scheme for nonlinear systems and its application

Mohammed Yusuf Waziri ^1,4,, , Kabiru Ahmed ^1,4, , Abubakar Sani Halilu ^2,4, and Aliyu Mohammed Awwal ^3,4,

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.

Keywords: Nonlinear monotone equations, Linesearch, Projection method, Signal processing, Image de-blurring, Convex constraint.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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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show all references

References:

[1]	A. B. Abubakar, P. Kumam, H. Mohammad, A. 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. Abubakar, K. Muangchoo, A. H. Ibrahim, J. 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. Aji, P. Kumam, A. M. Awwal, M. 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. Chan, A. 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. Halilu, A. Majumder, M. 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. Halilu, A. Majumder, M. Y. Waziri, A. 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. He, H. 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. Liu, J. 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. Mario, R. 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'u, A. 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. Wang, Y. 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]	X. Y. Wang, X. J. Li and X. P. Kou, A self-adaptive three-term conjugate gradient method for monotone nonlinear equations with convex constraints, Calcolo, 53 (2016), 133-145. doi: 10.1007/s10092-015-0140-5. Google Scholar
[47]	M. Y. Waziri, K. Ahmed and J. Sabi'u, A family of Hager-Zhang conjugate gradient methods for system of monotone nonlinear equations, Appl. Math. Comput., 361 (2019), 645-660. doi: 10.1016/j.amc.2019.06.012. Google Scholar
[48]	M. Y. Waziri, K. Ahmed and J. Sabi'u, A Dai-Liao conjugate gradient method via modified secant equation for system of nonlinear equations, Arab. J. Math., 9 (2020), 443-457. doi: 10.1007/s40065-019-0264-6. Google Scholar
[49]	M. Y. Waziri, K. Ahmed and J. Sabi'u, Descent Perry conjugate gradient methods for systems of monotone nonlinear equations, Numer. Algor., 85 (2020), 763-785. doi: 10.1007/s11075-019-00836-1. Google Scholar
[50]	M. Y. Waziri, K. Ahmed, J. Sabi'u and A. S. Halilu, Enhanced Dai-Liao conjugate gradient methods for systems of monotone nonlinear equations, SeMA Journal, 78 (2021), 15-51. doi: 10.1007/s40324-020-00228-9. Google Scholar
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Figure 1. Performance profile of Dolan and Moré for number of iterations

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Figure 2. Performance profile of Dolan and Moré for function evaluations

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Figure 3. Performance profile of Dolan and Moré for processing time

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Figure 4. Original and blurred images (First and second columns from the left). Restored images by the two methods (Third and Fourth columns)

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Table 1. Test results of the four methods for problems 4.1

Nvars	IGuess	MDDYM				PDYM				MDY				MFR
Nvars	IGuess	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

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Nvars	IGuess	MDDYM				PDYM				MDY				MFR
Nvars	IGuess	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
Nvars	IGuess	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

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Nvars	IGuess	MDDYM				PDYM				MDY				MFR
Nvars	IGuess	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
Nvars	IGuess	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	**	**	**	**

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Nvars	IGuess	MDDYM				PDYM				MDY				MFR
Nvars	IGuess	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
Nvars	IGuess	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

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Nvars	IGuess	MDDYM				PDYM				MDY				MFR
Nvars	IGuess	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
Nvars	IGuess	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

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Nvars	IGuess	MDDYM				PDYM				MDY				MFR
Nvars	IGuess	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
Nvars	IGuess	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

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Nvars	IGuess	MDDYM				PDYM				MDY				MFR
Nvars	IGuess	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
Nvars	IGuess	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

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Nvars	IGuess	MDDYM				PDYM				MDY				MFR
Nvars	IGuess	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
Nvars	IGuess	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

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Nvars	IGuess	MDDYM				PDYM				MDY				MFR
Nvars	IGuess	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%	-	-	-

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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

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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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Modified Dai-Yuan iterative scheme for nonlinear systems and its application

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American Institute of Mathematical Sciences

Modified Dai-Yuan iterative scheme for nonlinear systems and its application

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