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Article Contents

# Projection method with inertial step for nonlinear equations: Application to signal recovery

• * Corresponding author: Poom Kumam

The first author is supported by the Petchra Pra Jom Klao Ph.D. Research Scholarship from King Mongkut's University of Technology Thonburi (Grant no. 16/2561)

• In this paper, using the concept of inertial extrapolation, we introduce a globally convergent inertial extrapolation method for solving nonlinear equations with convex constraints for which the underlying mapping is monotone and Lipschitz continuous. The method can be viewed as a combination of the efficient three-term derivative-free method of Gao and He [Calcolo. 55(4), 1-17, 2018] with the inertial extrapolation step. Moreover, the algorithm is designed such that at every iteration, the method is free from derivative evaluations. Under standard assumptions, we establish the global convergence results for the proposed method. Numerical implementations illustrate the performance and advantage of this new method. Moreover, we also extend this method to solve the LASSO problems to decode a sparse signal in compressive sensing. Performance comparisons illustrate the effectiveness and competitiveness of our algorithm.

Mathematics Subject Classification: Primary: 65F10, 90C52, 65K05.

 Citation:

• Figure 1.  Performance profile on iteration (Iner.Algo versus Algo)

Figure 2.  Performance profile on function evaluations (Iner.Algo versus Algo)

Figure 3.  Performance profile on CPU time (Iner.Algo versus Algo)

Figure 4.  Numerical results of Problem (36) with $\delta = 0$

Figure 5.  Numerical results of Problem (36) with $\delta = 10^{-4}$

Figure 6.  Numerical results of Problem (36) with $\delta = 10^{-3}$

Figure 7.  Numerical results of Problem (36) with $\delta = 10^{-2}$

Table 1.  Numerical results of Problem (36) with different noise

 Standard deviation Methods RelErr Iter Times (s) $\delta$=0 Iner.Algo 0.0317 279 2.15 Algo 0.0317 468 3.15 $\delta=10^{-4}$ Iner.Algo 0.0281 292 2.21 Algo 0.0281 593 4.22 $\delta=10^{-3}$ Iner.Algo 0.0328 288 2.39 Algo 0.0241 438 2.96 $\delta=10^{-2}$ Iner.Algo 0.0518 337 2.55 Algo 0.0518 939 6.75

Table 2.  Test results for problem 1

 INER. ALGO ALGO. DIM INP NI NF CPU NORM NI NF CPU NORM 1000 Case I 6 24 0.006537 4.32E-08 11 44 0.009092 5.72E-07 Case II 7 28 0.008529 1.46E-07 11 44 0.013711 9.90E-07 Case III 7 28 0.006432 1.93E-07 12 48 0.013874 2.73E-07 Case IV 8 32 0.010414 1.79E-07 13 52 0.022209 3.00E-07 Case V 6 24 0.010836 7.10E-07 13 52 0.013571 6.02E-07 Case VI 8 32 0.00719 1.96E-07 13 52 0.010269 2.99E-07 Case VII 11 44 0.008426 1.12E-07 13 52 0.013525 6.09E-07 5000 Case I 6 24 0.22912 2.95E-08 12 48 0.10538 2.54E-07 Case II 7 28 0.85665 1.13E-07 12 48 0.079825 4.39E-07 Case III 7 28 0.53427 4.32E-07 12 48 0.032032 5.68E-07 Case IV 8 32 0.059319 3.98E-07 13 52 0.044075 4.28E-07 Case V 7 28 0.038857 3.61E-08 13 52 0.044154 4.83E-07 Case VI 8 32 0.065865 4.39E-07 13 52 0.18386 3.91E-07 Case VII 11 44 0.045807 3.86E-07 14 56 0.055375 2.68E-07 10000 Case I 6 24 0.055883 2.62E-08 12 48 0.26245 3.59E-07 Case II 7 28 0.10776 9.11E-08 12 48 0.084684 6.19E-07 Case III 7 28 0.12502 6.11E-07 12 48 0.07673 7.96E-07 Case IV 8 32 0.060193 5.62E-07 13 52 0.082261 5.60E-07 Case V 7 28 0.0602 5.18E-08 13 52 0.068433 4.87E-07 Case VI 8 32 0.05542 6.21E-07 13 52 0.066654 5.01E-07 Case VII 13 52 0.085591 1.80E-07 14 56 0.13534 3.78E-07 50000 Case I 6 24 0.17095 2.91E-08 12 48 0.23645 8.03E-07 Case II 7 28 0.15714 6.26E-08 13 52 0.25119 2.77E-07 Case III 8 32 0.57011 2.73E-08 13 52 0.21362 3.53E-07 Case IV 9 36 0.46934 2.51E-08 14 56 0.26703 2.34E-07 Case V 7 28 0.29169 1.17E-07 13 52 0.2287 7.20E-07 Case VI 9 36 0.2315 2.78E-08 14 56 0.24843 2.05E-07 Case VII 13 52 0.56004 2.37E-08 14 56 0.32785 8.43E-07 100000 Case I 6 24 0.98438 3.58E-08 13 52 0.43828 2.27E-07 Case II 7 28 0.39255 6.17E-08 13 52 0.62503 3.91E-07 Case III 8 32 0.48543 3.86E-08 13 52 0.42684 4.99E-07 Case IV 9 36 0.52322 3.56E-08 14 56 0.50903 3.28E-07 Case V 7 28 0.43716 1.66E-07 13 52 0.62726 9.52E-07 Case VI 9 36 0.54937 3.93E-08 14 56 0.49271 2.86E-07 Case VII 13 52 0.89929 4.99E-08 15 60 0.53354 2.38E-07

Table 3.  Test results for problem 2

 INER. ALGO ALGO. DIM INP NI NF CPU NORM NI NF CPU NORM 1000 Case I 4 12 0.087572 7.61E-09 4 12 0.008488 5.17E-07 Case II 5 15 0.016259 6.04E-09 5 15 0.006875 6.04E-09 Case III 5 15 0.011455 4.37E-07 5 15 0.006569 4.37E-07 Case IV 6 18 0.011153 1.52E-07 6 18 0.008092 1.52E-07 Case V 7 21 0.00633 1.10E-09 7 21 0.005713 1.10E-09 Case VI 7 21 0.011702 1.74E-08 7 21 0.004815 1.74E-08 Case VII 11 35 0.012359 2.77E-07 15 49 0.021261 3.66E-07 5000 Case I 4 12 0.064074 8.56E-10 4 12 0.035632 1.75E-07 Case II 5 15 0.076987 6.27E-10 5 15 0.019409 6.27E-10 Case III 5 15 0.027949 1.42E-07 5 15 0.014664 1.42E-07 Case IV 6 18 0.036548 3.94E-08 6 18 0.026202 3.94E-08 Case V 6 18 0.01692 4.05E-07 6 18 0.018243 4.05E-07 Case VI 7 21 0.025943 2.36E-09 7 21 0.027663 2.36E-09 Case VII 13 43 0.094345 3.76E-09 18 62 0.19571 1.75E-07 10000 Case I 4 12 0.022826 3.98E-10 4 12 0.02201 1.21E-07 Case II 5 15 0.029782 2.79E-10 5 15 0.03389 2.79E-10 Case III 5 15 0.036108 9.73E-08 5 15 0.020746 9.73E-08 Case IV 6 18 0.031098 2.56E-08 6 18 0.031954 2.56E-08 Case V 6 18 0.11664 2.93E-07 6 18 0.13272 2.93E-07 Case VI 7 21 0.039807 1.24E-09 7 21 0.056343 1.24E-09 Case VII 13 43 0.37509 1.17E-08 20 69 0.12928 9.06E-08 50000 Case I 4 13 0.36887 1.05E-10 4 12 0.15651 6.32E-08 Case II 5 16 0.42862 6.75E-11 5 16 0.10554 6.75E-11 Case III 5 15 0.24321 4.87E-08 5 15 0.16698 4.87E-08 Case IV 6 18 0.46737 1.11E-08 6 18 0.1099 1.11E-08 Case V 6 18 0.13319 1.84E-07 6 18 0.11779 1.84E-07 Case VI 7 21 0.18161 4.01E-10 7 21 0.1203 4.01E-10 Case VII 16 54 0.46124 1.11E-09 23 81 0.5211 6.89E-07 100000 Case I 4 13 0.26038 6.80E-11 4 12 0.13094 5.40E-08 Case II 5 16 0.30277 4.27E-11 5 16 0.18755 4.27E-11 Case III 5 15 0.4347 4.05E-08 5 15 0.27985 4.05E-08 Case IV 6 18 0.42556 8.15E-09 6 18 0.19207 8.15E-09 Case V 6 18 0.25488 1.80E-07 6 18 0.20156 1.80E-07 Case VI 7 21 0.43576 2.71E-10 7 21 0.21005 2.71E-10 Case VII 15 51 0.82806 1.64E-09 24 86 1.1405 4.14E-09

Table 4.  Test results for problem 3

 INER. ALGO ALGO. DIM INP NI NF CPU NORM NI NF CPU NORM 1000 Case I 13 52 0.075605 3.36E-07 31 124 0.023401 6.98E-07 Case II 14 56 0.033438 9.91E-07 32 126 0.016722 2.09E-22 Case III 15 59 0.014509 6.78E-07 34 135 0.029896 7.24E-07 Case IV 15 59 0.022327 8.58E-07 35 140 0.033452 8.69E-07 Case V 15 59 0.012348 4.93E-07 35 140 0.024649 9.73E-07 Case VI 16 64 0.011655 6.63E-07 36 142 0.041312 0 Case VII 15 60 0.019587 7.37E-07 34 136 0.018809 8.10E-07 5000 Case I 13 51 0.084308 7.51E-07 30 118 0.095447 0 Case II 15 60 0.052927 6.65E-07 31 122 0.50476 5.62E-21 Case III 16 62 0.08685 0 35 140 0.084126 9.71E-07 Case IV 16 62 0.054943 1.40E-21 36 143 0.095079 7.25E-21 Case V 16 64 0.10298 3.31E-07 35 138 0.26686 1.40E-21 Case VI 17 66 0.121 2.34E-22 36 142 0.16331 0 Case VII 16 64 0.82826 5.03E-07 36 144 0.16091 6.28E-07 10000 Case I 14 55 0.094948 3.19E-07 31 122 0.12401 6.62E-22 Case II 15 60 0.17919 9.40E-07 32 126 0.14805 0 Case III 16 62 0.085448 3.31E-22 36 143 0.13867 8.24E-07 Case IV 16 64 0.12333 8.14E-07 35 138 0.15826 2.65E-21 Case V 16 62 0.1041 3.31E-22 35 138 0.27449 6.62E-22 Case VI 17 66 0.11764 0 36 143 0.16026 2.65E-21 Case VII 16 64 0.081495 7.03E-07 36 144 0.19776 8.97E-07 50000 Case I 12 46 0.4807 1.15E-19 31 122 0.53862 2.96E-21 Case II 16 62 0.85283 0 31 122 0.5534 2.96E-21 Case III 16 62 0.43053 3.70E-21 33 130 0.66305 0 Case IV 15 58 0.34538 1.78E-20 34 134 0.59923 2.37E-20 Case V 17 66 0.45061 0 35 138 0.82508 1.48E-21 Case VI 18 72 0.46072 4.22E-07 36 143 0.62746 2.96E-21 Case VII 17 68 0.40219 4.73E-07 38 152 0.6653 7.24E-07 100000 Case I 15 60 0.73965 3.02E-07 31 122 1.139 2.09E-21 Case II 16 62 1.1267 5.23E-22 37 148 1.361 6.48E-07 Case III 17 68 0.58078 6.10E-07 38 151 1.5385 9.38E-07 Case IV 17 68 0.64576 7.73E-07 37 146 1.3166 0 Case V 17 68 0.77823 4.44E-07 40 158 1.4507 0 Case VI 17 66 0.67135 3.14E-21 40 159 1.4451 8.14E-07 Case VII 17 68 0.67963 6.70E-07 39 156 1.5115 6.14E-07

Table 5.  Test results for problem 4

 INER. ALGO ALGO. DIM INP NI NF CPU NORM NI NF CPU NORM 1000 Case I 13 51 0.06059 3.25E-07 31 124 0.02262 6.43E-07 Case II 14 56 0.008665 7.01E-07 32 128 0.020618 7.10E-07 Case III 14 56 0.023062 7.14E-07 33 132 0.030138 8.25E-07 Case IV 14 56 0.009094 9.53E-07 33 132 0.022234 9.09E-07 Case V 15 60 0.019868 8.47E-07 32 128 0.020698 6.86E-07 Case VI 15 60 0.032655 7.79E-07 35 140 0.033957 6.08E-07 Case VII 16 64 0.016672 4.18E-07 33 132 0.030958 7.44E-07 5000 Case I 13 51 0.057064 7.27E-07 32 127 0.083999 8.63E-07 Case II 15 59 0.047515 4.70E-07 33 132 0.15239 9.52E-07 Case III 15 59 0.032093 4.79E-07 35 140 0.061537 6.64E-07 Case IV 15 59 0.062007 6.39E-07 35 140 0.13051 7.32E-07 Case V 16 63 0.045188 5.68E-07 33 132 0.071242 9.20E-07 Case VI 16 64 0.037473 5.22E-07 36 144 0.060172 8.16E-07 Case VII 16 64 0.030254 8.70E-07 35 140 0.072475 6.05E-07 10000 Case I - - - - 33 131 0.12064 7.32E-07 Case II 15 59 0.048278 6.65E-07 34 135 0.63105 8.08E-07 Case III 15 59 0.057155 6.77E-07 35 140 0.11101 9.39E-07 Case IV 15 60 0.060725 9.04E-07 36 143 0.1951 6.21E-07 Case V 16 63 0.053938 8.04E-07 34 135 0.12425 7.81E-07 Case VI 16 63 0.066593 7.39E-07 37 147 0.20795 6.92E-07 Case VII 17 68 0.057214 3.60E-07 35 140 0.12255 8.50E-07 50000 Case I 14 55 1.0077 6.89E-07 34 134 0.42136 0 Case II 16 63 0.61724 4.46E-07 35 138 0.43246 0 Case III 16 63 0.32509 4.54E-07 37 147 1.0982 7.56E-07 Case IV 16 63 0.19589 6.06E-07 37 148 0.59668 8.33E-07 Case V 17 66 0.24309 0 - - - - Case VI - - - - - - - - Case VII 17 68 0.2842 8.32E-07 37 148 0.47886 6.85E-07 100000 Case I 14 54 0.32344 0 - - - - Case II 16 63 0.51789 6.31E-07 35 138 0.81304 0 Case III 16 62 0.5148 0 38 150 1.2176 0 Case IV - - - - 38 151 1.0824 7.07E-07 Case V 17 68 0.56565 7.63E-07 35 138 1.0007 0 Case VI 17 68 0.57103 7.01E-07 - - - - Case VII 18 72 0.53447 3.50E-07 37 148 0.88698 9.70E-07

Table 6.  Test results for problem 5

 INER. ALGO ALGO. DIM INP NI NF CPU NORM NI NF CPU NORM 1000 Case I 25 95 0.1329 6.45E-07 36 139 0.019135 6.42E-07 Case II 19 72 0.014534 4.04E-07 36 140 0.018994 6.62E-07 Case III 22 86 0.022553 9.29E-07 47 175 0.044522 6.46E-07 Case IV 22 88 0.036427 4.00E-07 40 156 0.03131 6.54E-07 Case V 25 100 0.037173 4.90E-07 37 146 0.023349 7.12E-07 Case VI 41 164 0.071153 6.01E-07 36 143 0.037977 9.25E-07 Case VII 42 167 0.12029 4.55E-07 49 184 0.060357 9.87E-07 5000 Case I 22 83 0.13573 6.53E-07 36 139 0.11871 6.40E-07 Case II 19 72 0.11048 8.54E-07 37 144 0.088934 9.52E-07 Case III 25 98 0.090732 3.03E-07 51 189 0.10334 9.00E-07 Case IV 24 96 0.061785 8.38E-07 44 169 0.44477 9.18E-07 Case V 30 120 0.1862 4.11E-07 40 157 0.11434 6.51E-07 Case VI 54 216 0.33113 9.45E-07 41 160 0.10301 8.14E-07 Case VII 69 275 0.5637 3.29E-07 47 180 0.107 9.84E-07 10000 Case I 27 103 0.14553 4.01E-07 36 139 0.1347 7.33E-07 Case II 20 76 0.086196 3.64E-07 38 147 0.15397 8.23E-07 Case III 25 98 0.20935 8.75E-07 47 177 0.17983 7.72E-07 Case IV 24 96 0.19624 5.28E-07 44 170 0.19929 7.87E-07 Case V 35 140 0.1977 6.41E-07 42 163 0.18619 9.45E-07 Case VI 60 240 0.65802 4.12E-07 49 185 0.47874 7.04E-07 Case VII 84 335 0.96443 8.09E-07 43 169 0.1934 7.22E-07 50000 Case I 55 215 2.6998 9.96E-07 65 255 1.3087 8.37E-07 Case II 20 76 0.2954 8.35E-07 48 179 0.92835 6.84E-07 Case III 27 106 0.47985 8.73E-07 67 240 0.92921 6.31E-07 Case IV 41 164 1.7421 9.71E-07 47 181 0.72302 6.44E-07 Case V 36 144 1.5835 8.46E-07 45 174 0.86921 7.83E-07 Case VI 79 316 4.0512 6.61E-07 54 201 0.94527 9.68E-07 Case VII 103 411 6.5137 7.57E-07 52 204 0.79431 7.54E-07 100000 Case I 74 291 7.1551 8.21E-07 83 327 3.7002 7.04E-07 Case II 21 80 0.671 3.61E-07 46 173 1.4825 9.75E-07 Case III 26 102 0.89984 6.93E-07 55 204 1.6728 8.97E-07 Case IV 44 176 2.6239 8.75E-07 52 196 1.5811 9.15E-07 Case V 43 172 2.2944 8.30E-07 48 184 1.5047 6.71E-07 Case VI 99 396 12.5198 3.58E-07 47 181 1.4726 8.26E-07 Case VII 111 443 12.3882 5.20E-07 69 272 2.5982 7.84E-07

Table 7.  Test results for problem 6

 INER. ALGO ALGO. DIM INP NI NF CPU NORM NI NF CPU NORM 1000 Case I 17 68 0.070165 3.66E-07 37 148 0.086194 8.53E-07 Case II 17 68 0.019243 3.42E-07 37 148 0.049573 8.20E-07 Case III 17 68 0.022663 3.01E-07 37 148 0.051928 7.22E-07 Case IV 16 64 0.016058 6.87E-07 36 144 0.074625 8.24E-07 Case V 16 64 0.028711 5.52E-07 36 144 0.080704 6.61E-07 Case VI 16 64 0.036264 3.25E-07 35 140 0.098138 6.50E-07 Case VII 17 68 0.029906 3.02E-07 37 148 0.057573 7.28E-07 5000 Case I 17 68 0.11976 8.21E-07 39 156 0.16301 6.87E-07 Case II 17 68 0.13738 7.66E-07 39 156 0.1438 6.61E-07 Case III 17 68 0.070333 6.75E-07 38 152 0.14908 9.71E-07 Case IV 17 68 0.097123 4.62E-07 38 152 0.16691 6.64E-07 Case V 17 68 0.070428 3.71E-07 37 148 0.15817 8.88E-07 Case VI 16 64 0.065808 7.29E-07 36 144 0.36454 8.73E-07 Case VII 17 68 0.089469 6.80E-07 38 152 0.19723 9.77E-07 10000 Case I 18 72 0.1274 3.48E-07 39 156 0.35497 9.72E-07 Case II 18 72 0.12939 3.25E-07 39 156 0.50582 9.35E-07 Case III 17 68 0.13873 9.55E-07 39 156 0.46728 8.24E-07 Case IV 17 68 0.13388 6.54E-07 38 152 0.2839 9.40E-07 Case V 17 68 0.20628 5.24E-07 38 152 0.27388 7.54E-07 Case VI 17 68 0.16002 3.09E-07 37 148 0.56098 7.41E-07 Case VII 17 68 0.16743 9.63E-07 39 156 0.46017 8.28E-07 50000 Case I 18 70 0.74334 0 40 158 1.2103 0 Case II 18 70 0.54759 0 40 158 1.4011 0 Case III 18 70 0.46758 0 39 154 1.3023 0 Case IV 18 70 0.47821 0 39 154 1.2953 0 Case V 18 70 0.51731 0 38 150 1.2347 0 Case VI 17 66 0.52485 0 37 146 1.2817 0 Case VII 18 70 0.55512 0 40 158 1.3476 0 100000 Case I 18 70 1.0765 0 39 154 2.4974 0 Case II 18 70 1.3017 0 39 154 2.5376 0 Case III 18 70 1.2128 0 38 150 2.45 0 Case IV 17 66 1.2676 0 38 150 2.3559 0 Case V 17 66 1.1563 0 37 146 2.4861 0 Case VI 17 66 1.0464 0 36 142 2.2564 0 Case VII 18 70 1.0587 0 39 154 2.6041 0

Table 8.  Test results for problem 7

 INER. ALGO ALGO. DIM INP NI NF CPU NORM NI NF CPU NORM 1000 Case I 9 36 0.029847 1.25E-07 14 56 0.010049 4.58E-07 Case II 8 32 0.01614 7.06E-07 14 56 0.009322 3.19E-07 Case III 7 28 0.036217 2.03E-07 11 44 0.014851 6.44E-07 Case IV 8 32 0.00896 1.86E-07 15 60 0.009756 2.52E-07 Case V 9 36 0.012299 2.91E-07 15 60 0.014388 3.91E-07 Case VI 9 35 0.012932 3.19E-07 15 59 0.01453 2.82E-07 Case VII 9 36 0.015981 3.43E-07 14 56 0.017579 3.00E-07 5000 Case I 9 36 0.040232 2.79E-07 15 60 0.06212 2.57E-07 Case II 9 36 0.038549 1.30E-07 14 56 0.035553 7.13E-07 Case III 7 28 0.02036 4.53E-07 12 48 0.037788 3.61E-07 Case IV 8 32 0.025654 4.15E-07 15 60 0.035425 5.64E-07 Case V 9 36 0.026546 6.51E-07 15 60 0.12771 8.73E-07 Case VI 9 35 0.096229 7.14E-07 15 59 0.044662 6.31E-07 Case VII 9 36 0.093227 7.84E-07 14 56 0.053641 6.71E-07 10000 Case I 9 36 0.054105 3.95E-07 15 60 0.055328 3.64E-07 Case II 9 36 0.10633 1.84E-07 15 60 0.10212 2.53E-07 Case III 7 28 0.037857 6.41E-07 12 48 0.066777 5.11E-07 Case IV 8 32 0.082416 5.87E-07 15 60 0.071481 7.98E-07 Case V 9 36 0.18582 9.20E-07 16 64 0.059125 3.10E-07 Case VI 10 39 0.070373 8.34E-08 15 59 0.081851 8.93E-07 Case VII 10 40 0.071963 9.12E-08 14 56 0.31755 9.38E-07 50000 Case I 9 36 0.36461 8.84E-07 15 60 0.32342 8.14E-07 Case II 9 36 0.27196 4.12E-07 15 60 0.3605 5.66E-07 Case III 8 32 0.25501 1.18E-07 13 52 0.22338 2.87E-07 Case IV 9 36 0.22228 1.08E-07 16 64 0.26331 4.48E-07 Case V 10 40 0.25618 1.70E-07 16 64 0.27305 6.94E-07 Case VI 10 39 0.27278 1.87E-07 16 63 0.28175 5.01E-07 Case VII 10 40 0.22836 2.04E-07 15 60 0.75345 5.27E-07 100000 Case I 10 40 0.57002 1.03E-07 16 64 0.56535 2.89E-07 Case II 9 36 0.34715 5.83E-07 15 60 0.66238 8.01E-07 Case III 8 32 0.41892 1.67E-07 13 52 0.45765 4.06E-07 Case IV 9 36 0.34291 1.53E-07 16 64 0.68567 6.34E-07 Case V 10 40 0.53349 2.40E-07 16 64 0.51067 9.81E-07 Case VI 10 39 0.35763 2.64E-07 16 63 0.5497 7.09E-07 Case VII 10 40 0.55468 2.88E-07 15 60 0.6995 7.46E-07

Table 9.  Test results for problem 8

 INER. ALGO ALGO. DIM INP NI NF CPU NORM NI NF CPU NORM 1000 Case I 11 38 0.032713 3.88E-07 16 58 0.00921 9.66E-07 Case II 11 38 0.010107 3.88E-07 16 58 0.00733 9.66E-07 Case III 11 39 0.0097 3.88E-07 16 59 0.010495 9.66E-07 Case IV 11 39 0.008542 3.88E-07 16 59 0.013631 9.66E-07 Case V 11 39 0.010247 3.88E-07 16 59 0.009248 9.66E-07 Case VI 11 39 0.014019 3.88E-07 16 59 0.01325 9.66E-07 Case VII 11 39 0.010456 3.88E-07 16 59 0.015507 9.66E-07 5000 Case I 8 29 0.029519 8.46E-07 12 46 0.03298 6.18E-07 Case II 8 30 0.028146 8.46E-07 12 46 0.028464 6.18E-07 Case III 8 30 0.031133 8.46E-07 12 46 0.026699 6.18E-07 Case IV 8 30 0.032266 8.46E-07 12 46 0.065305 6.18E-07 Case V 8 30 0.043717 8.46E-07 12 46 0.033667 6.18E-07 Case VI 8 30 0.075426 8.46E-07 12 46 0.032206 6.18E-07 Case VII 8 30 0.043783 8.46E-07 12 46 0.052994 6.18E-07 10000 Case I 7 26 0.070454 5.16E-07 9 35 0.045216 7.70E-07 Case II 7 27 0.053641 5.16E-07 9 35 0.059089 7.70E-07 Case III 7 27 0.094187 5.16E-07 9 35 0.045507 7.70E-07 Case IV 7 27 0.20473 5.16E-07 9 35 0.074929 7.70E-07 Case V 7 27 0.071282 5.16E-07 9 35 0.075532 7.70E-07 Case VI 7 27 0.064401 5.16E-07 9 35 0.065486 7.70E-07 Case VII 7 27 0.098105 5.16E-07 9 35 0.055328 7.70E-07 50000 Case I 6 24 0.22768 1.49E-07 12 48 0.35733 5.55E-07 Case II 6 24 0.28215 1.49E-07 12 48 0.34089 5.55E-07 Case III 6 24 0.72157 1.49E-07 12 48 0.30864 5.55E-07 Case IV 6 24 0.53086 1.49E-07 12 48 0.37903 5.55E-07 Case V 6 24 0.39711 1.49E-07 12 48 0.34614 5.55E-07 Case VI 6 24 0.48753 1.49E-07 12 48 0.41103 5.55E-07 Case VII 6 24 0.29061 1.49E-07 12 48 0.33605 5.55E-07 100000 Case I 6 24 0.66013 6.95E-07 6 24 0.43192 5.77E-07 Case II 6 24 0.60689 6.95E-07 6 24 0.30724 5.77E-07 Case III 6 24 0.78423 6.95E-07 6 24 0.46109 5.77E-07 Case IV 6 24 0.7826 6.95E-07 6 24 0.37262 5.77E-07 Case V 6 24 0.70311 6.95E-07 6 24 0.40775 5.77E-07 Case VI 6 24 0.9559 6.95E-07 6 24 0.38522 5.77E-07 Case VII 6 24 0.68726 6.95E-07 6 24 0.48134 5.77E-07
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Tables(9)