A Modified PRP-CG Type Derivative-Free Algorithm with Optimal Choices for Solving Large-Scale Nonlinear Symmetric Equations
Abstract
:1. Introduction
2. Modified PRP CG-Type with Optimal Choices
2.1. The First Optimal Choice for
2.2. The Second Optimal Choice for
3. Global Convergence
Algorithm 1: MPRP Algorithm |
step 0 Choose , , , and set . step 1 Check if is satisfied, else go to step 2. step 2 Determine the stepsize by using (11). step 3 Compute step 4 Determine the CG direction as step 5 Set and go back to step 1. |
- (i)
- The set Ψ is bounded.
- (ii)
- The Jacobian is Lipschitz in some neighbourhood of Ψ. The positive constant M exists, such that
4. Numerical Experiments
- , for .
- ,
- for ,
- , for
- for ,
- for ,
- for ,
5. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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PROBLEM 1 | |||||||||||||||||
MPRP() | MPRP() | NDDA | ICGM | ||||||||||||||
DIMENSION | INITIAL POINT | ITER | FVAL | TIME | NORM | ITER | FVAL | TIME | NORM | ITER | FVAL | TIME | NORM | ITER | FVAL | TIME | NORM |
50,000 | 31 | 93 | 1.51757 | 7.72 × 10 | 11 | 44 | 0.517181 | 5.84 × 10 | 19 | 95 | 0.602169 | 8.39 × 10 | 13 | 39 | 0.394323 | 6.33 × 10 | |
25 | 75 | 1.182817 | 7.05 × 10 | 22 | 88 | 0.825475 | 8.39 × 10 | 21 | 105 | 0.641815 | 7.71 × 10 | 18 | 54 | 0.531024 | 4.16 × 10 | ||
33 | 99 | 1.604626 | 8.78 × 10 | 11 | 44 | 0.503097 | 4.82 × 10 | 18 | 90 | 0.558197 | 7.96 × 10 | 19 | 57 | 0.58265 | 4.37 × 10 | ||
39 | 117 | 1.894566 | 5.22 × 10 | 30 | 120 | 1.348329 | 5.4 × 10 | 20 | 100 | 0.608259 | 6.12 × 10 | 27 | 81 | 0.830261 | 4.49 × 10 | ||
39 | 117 | 1.737297 | 5.31 × 10 | 28 | 112 | 1.227482 | 5.55 × 10 | 20 | 100 | 0.614197 | 8.51 × 10 | 27 | 81 | 0.82587 | 6.28 × 10 | ||
43 | 129 | 1.95166 | 8.01 × 10 | 19 | 76 | 0.938165 | 9.28 × 10 | 21 | 105 | 0.643514 | 5.89 × 10 | 21 | 63 | 0.663403 | 5.11 × 10 | ||
39 | 117 | 1.561943 | 5.31 × 10 | 28 | 112 | 1.237306 | 5.55 × 10 | 20 | 100 | 0.611944 | 8.51 × 10 | 27 | 81 | 0.817997 | 6.28 × 10 | ||
24 | 72 | 0.866136 | 5.98 × 10 | 17 | 68 | 0.578612 | 7.32 × 10 | 25 | 125 | 0.744996 | 6.89 × 10 | 26 | 78 | 0.801215 | 2.77 × 10 | ||
100,000 | 33 | 99 | 2.503401 | 4.79 × 10 | 11 | 44 | 1.060649 | 8.26 × 10 | 20 | 100 | 1.261737 | 4.86 × 10 | 13 | 39 | 0.824816 | 8.95 × 10 | |
25 | 75 | 1.828535 | 7.05 × 10 | 22 | 88 | 1.748833 | 8.39 × 10 | 21 | 105 | 1.29851 | 7.71 × 10 | 18 | 54 | 1.103288 | 5.3 × 10 | ||
35 | 105 | 2.51006 | 5.45 × 10 | 11 | 44 | 1.062033 | 6.82 × 10 | 19 | 95 | 1.200428 | 4.61 × 10 | 19 | 57 | 1.206809 | 6.18 × 10 | ||
39 | 117 | 2.825586 | 7.37 × 10 | 30 | 120 | 2.821417 | 7.3 × 10 | 20 | 100 | 1.258583 | 8.6 × 10 | 28 | 84 | 1.74041 | 6.85 × 10 | ||
39 | 117 | 2.623883 | 7.38 × 10 | 28 | 112 | 2.570306 | 7.91 × 10 | 20 | 100 | 1.236652 | 9.1 × 10 | 29 | 87 | 1.816512 | 6.83 × 10 | ||
45 | 135 | 3.021735 | 4.97 × 10 | 21 | 84 | 2.200632 | 1.13 × 10 | 21 | 105 | 1.268213 | 8.33 × 10 | 21 | 63 | 1.341347 | 7.22 × 10 | ||
39 | 117 | 2.603868 | 7.38 × 10 | 28 | 112 | 2.621052 | 7.91 × 10 | 20 | 100 | 1.17081 | 9.1 × 10 | 29 | 87 | 1.804864 | 6.83 × 10 | ||
24 | 72 | 1.63477 | 5.98 × 10 | 17 | 68 | 1.197102 | 7.63 × 10 | 25 | 125 | 1.476767 | 6.89 × 10 | 26 | 78 | 1.619349 | 2.77 × 10 | ||
PROBLEM 2 | |||||||||||||||||
MPRP() | MPRP() | NDDA | ICGM | ||||||||||||||
DIMENSION | INITIAL POINT | ITER | FVAL | TIME | NORM | ITER | FVAL | TIME | NORM | ITER | FVAL | TIME | NORM | ITER | FVAL | TIME | NORM |
50,000 | 170 | 510 | 7.931493 | 8.93 × 10 | 150 | 600 | 4.737454 | 8.27 × 10 | 382 | 1910 | 11.20393 | 9.75 × 10 | 237 | 711 | 7.620697 | 9.98 × 10 | |
141 | 423 | 5.058762 | 6.93 × 10 | 269 | 1076 | 9.397728 | 9.33 × 10 | 155 | 775 | 4.362288 | 8.76 × 10 | 244 | 732 | 7.83738 | 9.88 × 10 | ||
102 | 306 | 2.842245 | 8.74 × 10 | 137 | 548 | 4.299041 | 8.93 × 10 | 384 | 1920 | 8.689407 | 9.91 × 10 | 211 | 633 | 4.873246 | 9.22 × 10 | ||
324 | 972 | 9.531927 | 5.72 × 10 | 309 | 1236 | 10.46505 | 9.73 × 10 | 619 | 3095 | 12.28244 | 9.99 × 10 | 1000 | 3000 | 22.33118 | 10.91535 | ||
244 | 732 | 6.331992 | 5.68 × 10 | 268 | 1072 | 7.62583 | 8.86 × 10 | 455 | 2275 | 8.708046 | 9.02 × 10 | 1000 | 3000 | 19.49895 | 10.68423 | ||
67 | 201 | 1.95987 | 7.29 × 10 | 122 | 488 | 3.443245 | 9.85 × 10 | 352 | 1760 | 6.605092 | 9.02 × 10 | 324 | 972 | 6.04279 | 9.73 × 10 | ||
276 | 828 | 9.354755 | 8.37 × 10 | 278 | 1112 | 8.543894 | 8.97 × 10 | 455 | 2275 | 9.163612 | 9.02 × 10 | 1000 | 3000 | 21.8643 | 10.68423 | ||
207 | 621 | 6.630383 | 7.9 × 10 | 157 | 628 | 4.880471 | 6.37 × 10 | 288 | 1440 | 6.438931 | 8.97 × 10 | 166 | 498 | 3.764327 | 9.71 × 10 | ||
100,000 | 152 | 456 | 9.113001 | 7.61 × 10 | 148 | 592 | 8.525471 | 9.9 × 10 | 383 | 1915 | 15.44497 | 9.74 × 10 | 91 | 273 | 4.354032 | 9.74 × 10 | |
163 | 489 | 10.66124 | 9.28 × 10 | 223 | 892 | 15.55174 | 9.09 × 10 | 155 | 775 | 6.307293 | 9.95 × 10 | 399 | 1197 | 17.15205 | 9.74 × 10 | ||
77 | 231 | 3.928872 | 4.92 × 10 | 127 | 508 | 6.798184 | 7.22 × 10 | 388 | 1940 | 14.24754 | 9.96 × 10 | 260 | 780 | 10.00556 | 7.72 × 10 | ||
317 | 951 | 16.27087 | 9.79 × 10 | 292 | 1168 | 14.8396 | 6.16 × 10 | 614 | 3070 | 22.54226 | 9.65 × 10 | 1000 | 3000 | 38.68036 | 10.18736 | ||
278 | 834 | 14.69331 | 9.92 × 10 | 296 | 1184 | 15.86248 | 7.74 × 10 | 547 | 2735 | 19.97045 | 8.39 × 10 | 1000 | 3000 | 37.66586 | 13.37156 | ||
114 | 342 | 5.378932 | 9.74 × 10 | 110 | 440 | 5.0539 | 7.91 × 10 | 354 | 1770 | 12.81065 | 8.79 × 10 | 258 | 774 | 11.08156 | 7.29 × 10 | ||
274 | 822 | 13.97211 | 6.32 × 10 | 275 | 1100 | 15.80533 | 8.28 × 10 | 546 | 2730 | 19.73994 | 8.9 × 10 | 1000 | 3000 | 37.19552 | 13.37156 | ||
208 | 624 | 12.82466 | 8.67 × 10 | 144 | 576 | 9.141726 | 6.49 × 10 | 160 | 800 | 7.659241 | 9.25 × 10 | 516 | 1548 | 21.02884 | 9.86 × 10 |
PROBLEM 3 | |||||||||||||||||
MPRP() | MPRP() | NDDA | ICGM | ||||||||||||||
DIMENSION | INITIAL POINT | ITER | FVAL | TIME | NORM | ITER | FVAL | TIME | NORM | ITER | FVAL | TIME | NORM | ITER | FVAL | TIME | NORM |
50,000 | 11 | 33 | 0.322182 | 5.17 × 10 | 10 | 40 | 0.400361 | 5.96 × 10 | 599 | 2995 | 12.23839 | 8.38× 10 | 13 | 39 | 0.241657 | 3.49 × 10 | |
16 | 48 | 0.607104 | 7.34 × 10 | 20 | 80 | 0.686698 | 8.82 × 10 | 58 | 290 | 1.276429 | 8.28 × 10 | 10 | 30 | 0.211414 | 6.37 × 10 | ||
11 | 33 | 0.320112 | 4.62 × 10 | 15 | 60 | 0.505286 | 7.15× 10 | 495 | 2475 | 10.54336 | 9.96 × 10 | 10 | 30 | 0.240468 | 9.4 × 10 | ||
9 | 27 | 0.292944 | 8.08 × 10 | 6 | 24 | 0.372806 | 764 | 3820 | 13.48614 | 7.08 × 10 | 14 | 42 | 0.410652 | 9.06 × 10 | |||
15 | 45 | 0.588793 | 3.6 × 10 | 15 | 60 | 0.664088 | 3.97 × 10 | 252 | 1260 | 4.319963 | 4.89 × 10 | 12 | 36 | 0.377217 | 9.45 × 10 | ||
11 | 33 | 0.448889 | 5.17 × 10 | 10 | 40 | 0.60674 | 5.96 × 10 | 599 | 2995 | 11.48276 | 8.38× 10 | 13 | 39 | 0.437431 | 3.49 × 10 | ||
15 | 45 | 0.60361 | 3.6 × 10 | 15 | 60 | 0.713857 | 3.97 × 10 | 252 | 1260 | 3.986961 | 4.89 × 10 | 12 | 36 | 0.399796 | 9.45 × 10 | ||
14 | 42 | 0.873261 | 9.37 × 10 | 11 | 44 | 0.432592 | 8.11 × 10 | 61 | 305 | 1.189075 | 5.05 × 10 | 8 | 24 | 0.234699 | 3.02 × 10 | ||
100,000 | 6 | 18 | 0.609567 | 6.69 × 10 | 5 | 20 | 0.322736 | 1.14 × 10 | 15 | 75 | 0.647448 | 5.11 × 10 | 10 | 30 | 0.5706 | 1.66 × 10 | |
12 | 36 | 1.599188 | 7.4 × 10 | 11 | 44 | 0.893855 | 9.9 × 10 | 76 | 380 | 3.81177 | 5.42 × 10 | 8 | 24 | 0.481089 | 4.31 × 10 | ||
10 | 30 | 0.793137 | 8.9 × 10 | 13 | 52 | 1.784466 | 9.06 × 10 | 909 | 4545 | 28.60394 | 8.29 × 10 | 11 | 33 | 0.773752 | 2.2 × 10 | ||
12 | 36 | 0.977956 | 6.37 × 10 | 13 | 52 | 1.161575 | 4.14 × 10 | 404 | 2020 | 14.25168 | 2.02 × 10 | 12 | 36 | 0.857321 | 4.46 × 10 | ||
14 | 42 | 1.261941 | 9.15 × 10 | 11 | 44 | 0.940573 | 7.49 × 10 | 516 | 2580 | 15.59973 | 7.95 × 10 | 12 | 36 | 0.842963 | 6.33 × 10 | ||
6 | 18 | 0.614141 | 6.69 × 10 | 5 | 20 | 0.310198 | 1.14 × 10 | 15 | 75 | 0.482139 | 2.5 × 10 | 10 | 30 | 0.58923 | 1.66 × 10 | ||
14 | 42 | 1.242289 | 9.15 × 10 | 11 | 44 | 0.93746 | 7.49 × 10 | 516 | 2580 | 17.44822 | 7.95 × 10 | 12 | 36 | 0.8428 | 6.33 × 10 | ||
13 | 39 | 1.664243 | 3.68 × 10 | 13 | 52 | 1.209169 | 1.86 × 10 | 75 | 375 | 2.555772 | 8.69 × 10 | 8 | 24 | 0.49346 | 7.81 × 10 | ||
PROBLEM 4 | |||||||||||||||||
MPRP() | MPRP() | NDDA | ICGM | ||||||||||||||
DIMENSION | INITIAL POINT | ITER | FVAL | TIME | NORM | ITER | FVAL | TIME | NORM | ITER | FVAL | TIME | NORM | ITER | FVAL | TIME | NORM |
50,000 | 9 | 27 | 0.212201 | 9.28 × 10 | 11 | 44 | 0.247058 | 4.22 × 10 | 38 | 190 | 0.483188 | 8.05 × 10 | 12 | 36 | 0.22417 | 3.58 × 10 | |
14 | 42 | 0.321678 | 5.07 × 10 | 11 | 44 | 0.263394 | 5.47 × 10 | 27 | 135 | 0.398743 | 8.23 × 10 | 22 | 66 | 0.394745 | 6.26 × 10 | ||
9 | 27 | 0.202813 | 1.19 × 10 | 9 | 36 | 0.215252 | 1.46 × 10 | 26 | 130 | 0.436246 | 8.17 × 10 | 8 | 24 | 0.147041 | 6.88 × 10 | ||
23 | 69 | 0.593432 | 8.53 × 10 | 21 | 84 | 0.50064 | 5.56 × 10 | 39 | 195 | 0.690833 | 8.04 × 10 | 36 | 108 | 0.63236 | 9.86 × 10 | ||
23 | 69 | 0.578756 | 4.6 × 10 | 20 | 80 | 0.491674 | 3.58× 10 | 39 | 195 | 0.714213 | 8.04 × 10 | 39 | 117 | 0.705334 | 9.49 × 10 | ||
17 | 51 | 0.45665 | 9.68× 10 | 19 | 76 | 0.595362 | 3.48× 10 | 37 | 185 | 0.682459 | 8.44 × 10 | 12 | 36 | 0.219326 | 2.67 × 10 | ||
23 | 69 | 0.586717 | 4.6 × 10 | 20 | 80 | 0.488278 | 3.58× 10 | 39 | 195 | 0.732293 | 8.04 × 10 | 39 | 117 | 0.698136 | 9.52 × 10 | ||
18 | 54 | 0.47571 | 4.21 × 10 | 32 | 128 | 0.906585 | 7.41× 10 | 28 | 140 | 0.51624 | 8.54 × 10 | 27 | 81 | 0.479781 | 4.96 × 10 | ||
100,000 | 11 | 33 | 0.476225 | 1.02× 10 | 11 | 44 | 0.484772 | 5.96 × 10 | 39 | 195 | 1.445101 | 7.29 × 10 | 12 | 36 | 0.438499 | 5.06 × 10 | |
14 | 42 | 0.633407 | 5.06 × 10 | 11 | 44 | 0.527568 | 5.47 × 10 | 27 | 135 | 1.000728 | 8.23 × 10 | 19 | 57 | 0.675121 | 9.94 × 10 | ||
9 | 27 | 0.406597 | 1.69 × 10 | 9 | 36 | 0.42648 | 2.06 × 10 | 27 | 135 | 0.974066 | 7.39 × 10 | 8 | 24 | 0.291315 | 9.73 × 10 | ||
24 | 72 | 1.218071 | 8.4 × 10 | 21 | 84 | 1.017136 | 1.52 × 10 | 40 | 200 | 1.431425 | 7.28 × 10 | 30 | 90 | 1.037708 | 2.42 × 10 | ||
23 | 69 | 1.171541 | 7.52 × 10 | 20 | 80 | 0.960639 | 4.96 × 10 | 40 | 200 | 1.427526 | 7.28 × 10 | 43 | 129 | 1.466193 | 7.58 × 10 | ||
17 | 51 | 0.901294 | 1.37× 10 | 19 | 76 | 1.156228 | 4.92× 10 | 38 | 190 | 1.339707 | 7.63 × 10 | 12 | 36 | 0.446339 | 3.78 × 10 | ||
23 | 69 | 1.171954 | 7.52 × 10 | 20 | 80 | 0.942363 | 4.96 × 10 | 40 | 200 | 1.419573 | 7.28 × 10 | 43 | 129 | 1.472821 | 7.58 × 10 | ||
18 | 54 | 0.958594 | 4.21 × 10 | 25 | 100 | 1.289564 | 8.32 × 10 | 28 | 140 | 1.009042 | 8.54 × 10 | 26 | 78 | 0.900203 | 8.57 × 10 |
PROBLEM 5 | |||||||||||||||||
MPRP() | MPRP() | NDDA | ICGM | ||||||||||||||
DIMENSION | INITIAL POINT | ITER | FVAL | TIME | NORM | ITER | FVAL | TIME | NORM | ITER | FVAL | TIME | NORM | ITER | FVAL | TIME | NORM |
50,000 | 69 | 207 | 2.267497 | 9.33 × 10 | 105 | 420 | 3.891633 | 7.84 × 10 | 191 | 955 | 4.947418 | 9.93 × 10 | 135 | 405 | 3.611205 | 9.12 × 10 | |
87 | 261 | 2.940586 | 8.23 × 10 | 109 | 436 | 3.97732 | 8.31 × 10 | 207 | 1035 | 5.307014 | 9.87 × 10 | 220 | 660 | 5.912201 | 9.97 × 10 | ||
86 | 258 | 2.994068 | 9.81 × 10 | 79 | 316 | 2.592411 | 9.4 × 10 | 214 | 1070 | 5.430976 | 9.9 × 10 | 998 | 2994 | 21.5748 | 9.96 × 10 | ||
79 | 237 | 2.599857 | 7.72 × 10 | 101 | 404 | 3.660022 | 7.37 × 10 | 205 | 1025 | 4.680842 | 9.23 × 10 | 152 | 456 | 2.266204 | 9.89 × 10 | ||
80 | 240 | 2.659913 | 9.52 × 10 | 111 | 444 | 3.90133 | 9.93 × 10 | 205 | 1025 | 4.295369 | 9.23 × 10 | 704 | 2112 | 12.14497 | 9.93 × 10 | ||
86 | 258 | 2.632166 | 9.8 × 10 | 114 | 456 | 4.071767 | 9.07 × 10 | 224 | 1120 | 4.554054 | 9.19 × 10 | 1000 | 3000 | 15.05272 | 2.1 × 10 | ||
79 | 237 | 2.337165 | 8.46 × 10 | 103 | 412 | 3.302599 | 9.43 × 10 | 205 | 1025 | 3.89815 | 9.23 × 10 | 719 | 2157 | 12.18638 | 9.86 × 10 | ||
86 | 258 | 2.501353 | 6.97 × 10 | 122 | 488 | 3.74481 | 4.86 × 10 | 216 | 1080 | 4.080285 | 9.5 × 10 | 172 | 516 | 2.941251 | 7.99 × 10 | ||
100,000 | 80 | 240 | 4.991337 | 6.4 × 10 | 93 | 372 | 5.303749 | 8.04 × 10 | 192 | 960 | 7.479403 | 9.02 × 10 | 656 | 1968 | 21.91715 | 9.86 × 10 | |
86 | 258 | 5.350999 | 9.84 × 10 | 104 | 416 | 5.60915 | 9.53 × 10 | 207 | 1035 | 7.35847 | 9.01 × 10 | 713 | 2139 | 22.61809 | 9.95 × 10 | ||
80 | 240 | 4.52125 | 7.67 × 10 | 94 | 376 | 4.548221 | 7.97 × 10 | 212 | 1060 | 7.260601 | 9.45 × 10 | 758 | 2274 | 25.39184 | 9.98 × 10 | ||
82 | 246 | 4.482321 | 8.53 × 10 | 83 | 332 | 3.911853 | 9.59 × 10 | 205 | 1025 | 8.032846 | 9.62 × 10 | 197 | 591 | 6.705366 | 9.9 × 10 | ||
81 | 243 | 4.063491 | 8.48 × 10 | 79 | 316 | 4.167463 | 8.7 × 10 | 205 | 1025 | 7.737062 | 9.62 × 10 | 173 | 519 | 6.832254 | 9.52 × 10 | ||
83 | 249 | 4.060558 | 6.42 × 10 | 104 | 416 | 6.16264 | 8.65 × 10 | 214 | 1070 | 7.483374 | 9.19 × 10 | 790 | 2370 | 27.09478 | 9.86 × 10 | ||
82 | 246 | 4.173369 | 7.61 × 10 | 85 | 340 | 4.378839 | 8.54 × 10 | 205 | 1025 | 7.096066 | 9.62 × 10 | 431 | 1293 | 14.45969 | 9.87 × 10 | ||
87 | 261 | 4.104345 | 9.47 × 10 | 114 | 456 | 6.210763 | 9.51 × 10 | 215 | 1075 | 7.667577 | 9.88 × 10 | 150 | 450 | 4.905177 | 5.81 × 10 | ||
PROBLEM 6 | |||||||||||||||||
MPRP() | MPRP() | NDDA | ICGM | ||||||||||||||
DIMENSION | INITIAL POINT | ITER | FVAL | TIME | NORM | ITER | FVAL | TIME | NORM | ITER | FVAL | TIME | NORM | ITER | FVAL | TIME | NORM |
50,000 | 3 | 9 | 0.081907 | 1.46 × 10 | 3 | 12 | 0.074954 | 1.46 × 10 | 32 | 160 | 0.510629 | 6.43 × 10 | 2 | 6 | 0.04543 | 2.36 × 10 | |
1 | 3 | 0.033824 | 8 × 10 | 1 | 4 | 0.032205 | 8.08 × 10 | 1 | 5 | 0.032867 | 8 × 10 | 1 | 3 | 0.031592 | 8 × 10 | ||
1 | 3 | 0.033946 | 8.57 × 10 | 1 | 4 | 0.032505 | 8.64 × 10 | 1 | 5 | 0.032653 | 8.57 × 10 | 1 | 3 | 0.03293 | 8.57 × 10 | ||
1 | 3 | 0.034092 | 9.94 × 10 | 1 | 4 | 0.035 | 9.99 × 10 | 1 | 5 | 0.033783 | 9.94 × 10 | 1 | 3 | 0.036741 | 9.94 × 10 | ||
1 | 3 | 0.037075 | 9.94 × 10 | 1 | 4 | 0.037857 | 9.99 × 10 | 1 | 5 | 0.037706 | 9.94 × 10 | 1 | 3 | 0.037229 | 9.94 × 10 | ||
1 | 3 | 0.037861 | 1.3 × 10 | 1 | 4 | 0.036916 | 1.16 × 10 | 1 | 5 | 0.038647 | 1.3 × 10 | 1 | 3 | 0.03646 | 1.3 × 10 | ||
1 | 3 | 0.037923 | 9.94 × 10 | 1 | 4 | 0.039097 | 9.99 × 10 | 1 | 5 | 0.038561 | 9.94 × 10 | 1 | 3 | 0.038467 | 9.94 × 10 | ||
1 | 3 | 0.037427 | 8.01 × 10 | 1 | 4 | 0.03926 | 8.08 × 10 | 1 | 5 | 0.040607 | 8.01 × 10 | 1 | 3 | 0.037036 | 8.01 × 10 | ||
100,000 | 1 | 3 | 0.077721 | 3.79 × 10 | 1 | 4 | 0.075644 | 3.8 × 10 | 1 | 5 | 0.075443 | 3.79 × 10 | 1 | 3 | 0.073561 | 3.79 × 10 | |
1 | 3 | 0.079765 | 2.83 × 10 | 1 | 4 | 0.076144 | 2.86 × 10 | 1 | 5 | 0.077985 | 2.83 × 10 | 1 | 3 | 0.076044 | 2.83 × 10 | ||
1 | 3 | 0.077061 | 3.03 × 10 | 1 | 4 | 0.076729 | 3.05 × 10 | 1 | 5 | 0.076138 | 3.03 × 10 | 1 | 3 | 0.074692 | 3.03 × 10 | ||
1 | 3 | 0.081273 | 3.52 × 10 | 1 | 4 | 0.075837 | 3.53 × 10 | 1 | 5 | 0.075578 | 3.52 × 10 | 1 | 3 | 0.078154 | 3.52 × 10 | ||
1 | 3 | 0.081851 | 3.52 × 10 | 1 | 4 | 0.076603 | 3.53 × 10 | 1 | 5 | 0.075728 | 3.52 × 10 | 1 | 3 | 0.076166 | 3.52 × 10 | ||
1 | 3 | 0.077072 | 4.59× 10 | 1 | 4 | 0.076258 | 4.1× 10 | 1 | 5 | 0.075658 | 4.59× 10 | 1 | 3 | 0.075048 | 4.59× 10 | ||
1 | 3 | 0.07455 | 3.52 × 10 | 1 | 4 | 0.075514 | 3.53 × 10 | 1 | 5 | 0.076933 | 3.52 × 10 | 1 | 3 | 0.074581 | 3.52 × 10 | ||
1 | 3 | 0.087563 | 2.83 × 10 | 1 | 4 | 0.077531 | 2.86 × 10 | 1 | 5 | 0.074854 | 2.83 × 10 | 1 | 3 | 0.074646 | 2.83 × 10 |
PROBLEM 7 | |||||||||||||||||
MPRP() | MPRP() | NDDA | ICGM | ||||||||||||||
DIMENSION | INITIAL POINT | ITER | FVAL | TIME | NORM | ITER | FVAL | TIME | NORM | ITER | FVAL | TIME | NORM | ITER | FVAL | TIME | NORM |
50,000 | 0 | 0 | 0.011313 | 0 | 0 | 0 | 0.01209 | 0 | 0 | 0 | 0.016223 | 0 | 0 | 0 | 0.012003 | 0 | |
87 | 261 | 2.348944 | 9.16 × 10 | 79 | 316 | 2.273501 | 8.88 × 10 | 154 | 770 | 4.291607 | 9.45 × 10 | 1000 | 3000 | 26.32372 | 0.016563 | ||
14 | 42 | 0.516938 | 7.44 × 10 | 6 | 24 | 0.220502 | 4.02 × 10 | 3 | 15 | 0.080141 | 4.41 × 10 | 5 | 15 | 0.084522 | 6.44 × 10 | ||
34 | 102 | 0.926885 | 8.95 × 10 | 33 | 132 | 0.980913 | 8.11 × 10 | 41 | 205 | 1.188357 | 9.99 × 10 | 1000 | 3000 | 27.79381 | 3.13 × 10 | ||
36 | 108 | 1.009782 | 8.7 × 10 | 36 | 144 | 1.084175 | 7.95 × 10 | 49 | 245 | 1.316967 | 9.56 × 10 | 1000 | 3000 | 45.34329 | 3.2 × 10 | ||
18 | 54 | 0.660205 | 8.81 × 10 | 8 | 32 | 0.29928 | 9.78 × 10 | 4 | 20 | 0.114832 | 1.58 × 10 | 7 | 21 | 0.221179 | 8.72 × 10 | ||
36 | 108 | 1.03182 | 8.7 × 10 | 36 | 144 | 1.087501 | 7.95 × 10 | 49 | 245 | 1.318538 | 9.56 × 10 | 936 | 2808 | 42.23109 | NaN | ||
90 | 270 | 2.548954 | 9.86 × 10 | 88 | 352 | 2.584854 | 7.43 × 10 | 163 | 815 | 4.589919 | 9.82 × 10 | 1000 | 3000 | 29.78786 | 0.037358 | ||
100,000 | 0 | 0 | 0.016449 | 0 | 0 | 0 | 0.017713 | 0 | 0 | 0 | 0.019086 | 0 | 0 | 0 | 0.014114 | 0 | |
87 | 261 | 4.427758 | 8.58 × 10 | 80 | 320 | 4.971012 | 9.72 × 10 | 154 | 770 | 7.208364 | 9.4 × 10 | 1000 | 3000 | 49.27 | 0.017351 | ||
12 | 36 | 0.706042 | 6.35 × 10 | 6 | 24 | 0.4356 | 1.65 × 10 | 3 | 15 | 0.121382 | 2.2 × 10 | 4 | 12 | 0.164811 | 7.8 × 10 | ||
26 | 78 | 1.188328 | 9.34 × 10 | 30 | 120 | 1.787787 | 9.37 × 10 | 34 | 170 | 1.501828 | 9.85 × 10 | 925 | 2775 | 48.35269 | NaN | ||
29 | 87 | 1.375369 | 9.64 × 10 | 31 | 124 | 1.774428 | 9.55 × 10 | 45 | 225 | 1.998629 | 9.4 × 10 | 1000 | 3000 | 53.17642 | 2.14 × 10 | ||
16 | 48 | 1.08696 | 7.41 × 10 | 6 | 24 | 0.420024 | 5.49 × 10 | 3 | 15 | 0.144028 | 7.36 × 10 | 6 | 18 | 0.221337 | 4.8 × 10 | ||
29 | 87 | 1.496431 | 9.64 × 10 | 31 | 124 | 1.761076 | 9.55 × 10 | 45 | 225 | 2.231545 | 9.4 × 10 | 1000 | 3000 | 54.97467 | 1.99 × 10 | ||
92 | 276 | 4.563438 | 9.67 × 10 | 85 | 340 | 4.913522 | 8.77 × 10 | 163 | 815 | 7.319916 | 9.37 × 10 | 1000 | 3000 | 48.95704 | 0.032224 | ||
PROBLEM 8 | |||||||||||||||||
MPRP() | MPRP() | NDDA | ICGM | ||||||||||||||
DIMENSION | INITIAL POINT | ITER | FVAL | TIME | NORM | ITER | FVAL | TIME | NORM | ITER | FVAL | TIME | NORM | ITER | FVAL | TIME | NORM |
1000 | 82 | 246 | 0.455454 | 9.96 × 10 | 105 | 420 | 0.597806 | 8.96 × 10 | 224 | 1120 | 0.94912 | 9.83 × 10 | 135 | 405 | 0.58535 | 9.26 × 10 | |
88 | 264 | 0.59479 | 9.39 × 10 | 98 | 392 | 0.576119 | 8.76 × 10 | 228 | 1140 | 1.088754 | 9.84 × 10 | 148 | 444 | 0.678534 | 9.94 × 10 | ||
76 | 228 | 0.447432 | 8.79 × 10 | 107 | 428 | 0.625566 | 8.22 × 10 | 215 | 1075 | 1.036161 | 9.92 × 10 | 953 | 2859 | 4.435764 | 9.97 × 10 | ||
83 | 249 | 0.571694 | 8.7 × 10 | 84 | 336 | 0.544969 | 7.1 × 10 | 218 | 1090 | 1.264795 | 9.9 × 10 | 952 | 2856 | 5.6264 | 9.97 × 10 | ||
74 | 222 | 0.714045 | 7.74 × 10 | 100 | 400 | 0.991748 | 9.97 × 10 | 218 | 1090 | 1.652656 | 9.9 × 10 | 722 | 2166 | 5.116361 | 1 × 10 | ||
78 | 234 | 0.864799 | 9.45 × 10 | 104 | 416 | 1.442508 | 6.02 × 10 | 188 | 940 | 1.717357 | 1 × 10 | 114 | 342 | 0.906444 | 8.28 × 10 | ||
74 | 222 | 0.872029 | 7.75 × 10 | 93 | 372 | 1.332749 | 8.21 × 10 | 218 | 1090 | 2.532429 | 9.9 × 10 | 151 | 453 | 1.475109 | 7.62 × 10 | ||
90 | 270 | 1.006991 | 7.67 × 10 | 117 | 468 | 1.869405 | 8.9 × 10 | 236 | 1180 | 2.831446 | 9.84 × 10 | 162 | 486 | 1.559799 | 8.39 × 10 | ||
2000 | 82 | 246 | 1.001497 | 9.96 × 10 | 91 | 364 | 1.851274 | 8.63 × 10 | 223 | 1115 | 3.376192 | 9.27 × 10 | 787 | 2361 | 10.66835 | 9.86 × 10 | |
88 | 264 | 1.205998 | 9.39 × 10 | 90 | 360 | 1.751596 | 9.71 × 10 | 228 | 1140 | 3.306594 | 9.9 × 10 | 750 | 2250 | 6.84631 | 1 × 10 | ||
76 | 228 | 1.125966 | 8.79 × 10 | 92 | 368 | 1.842185 | 9.36 × 10 | 214 | 1070 | 3.016198 | 9.96 × 10 | 159 | 477 | 1.140194 | 9.44 × 10 | ||
83 | 249 | 1.167799 | 8.7 × 10 | 144 | 576 | 3.194961 | 8.2 × 10 | 219 | 1095 | 2.994653 | 9.73 × 10 | 885 | 2655 | 8.149794 | 9.97 × 10 | ||
74 | 222 | 1.045681 | 7.74 × 10 | 109 | 436 | 2.276981 | 9.94 × 10 | 219 | 1095 | 2.892295 | 9.75 × 10 | 396 | 1188 | 3.821894 | 9.98 × 10 | ||
78 | 234 | 1.174389 | 9.45 × 10 | 82 | 328 | 1.611686 | 9.42 × 10 | 190 | 950 | 2.546115 | 9.85 × 10 | 108 | 324 | 1.188268 | 8.95 × 10 | ||
74 | 222 | 1.069987 | 7.75 × 10 | 114 | 456 | 2.431602 | 9.96 × 10 | 219 | 1095 | 2.956538 | 9.75 × 10 | 682 | 2046 | 6.50311 | 9.86 × 10 | ||
90 | 270 | 1.224429 | 7.67 × 10 | 117 | 468 | 2.359865 | 7.39 × 10 | 235 | 1175 | 3.148771 | 9.56 × 10 | 272 | 816 | 3.325016 | 1 × 10 |
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Sabi’u, J.; Muangchoo, K.; Shah, A.; Abubakar, A.B.; Jolaoso, L.O. A Modified PRP-CG Type Derivative-Free Algorithm with Optimal Choices for Solving Large-Scale Nonlinear Symmetric Equations. Symmetry 2021, 13, 234. https://doi.org/10.3390/sym13020234
Sabi’u J, Muangchoo K, Shah A, Abubakar AB, Jolaoso LO. A Modified PRP-CG Type Derivative-Free Algorithm with Optimal Choices for Solving Large-Scale Nonlinear Symmetric Equations. Symmetry. 2021; 13(2):234. https://doi.org/10.3390/sym13020234
Chicago/Turabian StyleSabi’u, Jamilu, Kanikar Muangchoo, Abdullah Shah, Auwal Bala Abubakar, and Lateef Olakunle Jolaoso. 2021. "A Modified PRP-CG Type Derivative-Free Algorithm with Optimal Choices for Solving Large-Scale Nonlinear Symmetric Equations" Symmetry 13, no. 2: 234. https://doi.org/10.3390/sym13020234