Laszlo Hars’ Homepage
Email: Laszlo at Hars period US
Resume [pdf]
I work in Information Security – Anti Tamper research and development.
Publications: See in my Resume
66+ Issued Patents
80+ Published Patent Applications
Numerical Solutions for the Tammes Problem (N = 2…60) [pdf]
The Tammes Problems ask to place a given number N points on the surface of the unit sphere, such that the minimum distance between these points is maximal. The referred document deals with their numerical (approximate) solutions.
The term “numerical solution” is used for
- finding arrangements of N points on the sphere
- with “large” shortest distances
- by numerical optimization methods
- starting from random initial point sets.
The found shortest distances are close to the possible maxima, and at reasonably high probabilities they are maximal.
The linked document
- summarizes our experiments with random-restart numerical optimizations for the Tammes problems
- compares a few publicly available numerical optimization algorithms
- discusses several possible problem encodings
- presents program code with diagnose/debug/verification options
- investigates the generation of uniform- and homogenous random spherical points
- finds symmetries in the contact graphs
- shows various graphical representations of the results; and the programs plotting them
- lists the best found Tammes point sets (coordinates of the points, and graphic plots)
- presents simple, arbitrary precision methods for computing the exact minimax distances of the Tammes problem for N = 3…16, 24. (The methods for N = 10, 13, 14, 15 and 16 seem to be new.)
Below some interesting facts about the N-point Tammes Graphs, denoted with TG-<N>, are listed. (They are the contact graphs of the found best solutions of the Tammes problem – where points at the minimal distance are connected.)
- TG-19: the smallest Tammes Graph with an isolated point
- TG-20: is the smallest Tammes Graph with 2 isolated points
- TG-21, 22 and 23: the smallest cases with no symmetry
- TG-23 disproves the conjecture that the Tammes set for N = 23 is the same as it is for N = 24, with one point removed
- TG-23: has an isolated point and 2 rhombic faces, which are almost square (their diagonals are of the same length within 0.3%)
- TG-37: has 3 isolated points
- TG-52, 56: have 4 isolated points…
Below are the stereographic projections of a few interesting Tammes Graphs:
Several computer programs are presented in the document for finding numerical solutions and for plotting them. They are placed into the public domain: anyone can use them for any purpose – although without our guaranties for correctness or fitness to any goals. We hope that the presented information is useful.
The table below contains the cosine of the found best minimax angular distances of the Tammes sets.
|
|
0 |
10 |
20 |
30 |
40 |
50 |
|
+1 |
− |
0.4472135955 |
0.6994984311 |
0.7911186133 |
0.8412363430 |
0.8714843649 |
|
+2 |
−1.00000000000 |
0.4472135955 |
0.7103062586 |
0.7936166149 |
0.8433315516 |
0.8729667088 |
|
+3 |
-0.5000000000 |
0.5426364868 |
0.7228469849 |
0.8063976136 |
0.8472088654 |
0.8761898040 |
|
+4 |
-0.3333333333 |
0.5639503003 |
0.7230784683 |
0.8109843372 |
0.8482013783 |
0.8770043064 |
|
+5 |
0.0000000000 |
0.5926059029 |
0.7473986286 |
0.8159377715 |
0.8542494741 |
0.8807850465 |
|
+6 |
0.0000000000 |
0.6122946165 |
0.7542781771 |
0.8172481853 |
0.8575341669 |
0.8817315784 |
|
+7 |
0.2101383127 |
0.6280944150 |
0.7583892108 |
0.8248924802 |
0.8591223614 |
0.8843637376 |
|
+8 |
0.2612038749 |
0.6486958322 |
0.7732302623 |
0.8265832594 |
0.8592922951 |
0.8865557452 |
|
+9 |
0.3333333333 |
0.6731168875 |
0.7802814159 |
0.8339913224 |
0.8666914790 |
0.8878557371 |
|
+10 |
0.4043943252 |
0.6764771381 |
0.7815518751 |
0.8371620725 |
0.8681732091 |
0.8894735676 |
This is a draft document; any feedback is appreciated. (See the latest version here.)
Numerical Solutions for Midsize Tammes Problems: N = 61…100 [pdf]
The referred document discusses extensions of the work discussed above, including the results of our experiments and improvements of the computer program for larger cases of the Tammes problem. Somewhat arbitrarily, the range of the problem size 61…100 is referred to as “midsize”, based on their solvability on a cheap home computer. On a computer network or high-performance computers, the presented methods would lead to the numerical solutions of the Tammes problems well above this range, still using reasonable computing time.
The program is placed in the public domain, anyone can use it for any purpose.
Below the results are summarized in a table. There were test-runs for all problem sizes; 5 cases with 150,000 restarts (N = 68, 73, 77, 81, 82), the others with 50,000 restarts, but with different initial population generators. The listed version of the program found better or the same local optima, but the total number of random restarts listed in the second column of the table could be increased by 50,000, and in the aforementioned 5 cases, by 150,000.
|
N |
Restarts |
Opt. Iter# |
Optimum |
|
61 |
60,000 |
1,648 |
0.89200844328203 |
|
62 |
60,000 |
4,392 |
0.89349685056883 |
|
63 |
70,000 |
9,974 |
0.89503618081411 |
|
64 |
70,000 |
5,349 |
0.89698816753708 |
|
65 |
70,000 |
33,199 |
0.89825910858396 |
|
66 |
70,000 |
5,665 |
0.89919577748308 |
|
67 |
80,000 |
23,502 |
0.90119822775634 |
|
68 |
80,000 |
24,435 |
0.90285692152157 |
|
69 |
80,000 |
39,249 |
0.90383460922318 |
|
70 |
80,000 |
30,825 |
0.90504303680279 |
|
71 |
80,000 |
5,768 |
0.90639673677562 |
|
72 |
90,000 |
48,542 |
0.90684928543572 |
|
73 |
90,000 |
67,941 |
0.90957174381548 |
|
74 |
90,000 |
80,847 |
0.91053262350104 |
|
75 |
100,000 |
12,019 |
0.91139090465160 |
|
76 |
100,000 |
74,505 |
0.91268728948391 |
|
77 |
100,000 |
91,891 |
0.91353634468780 |
|
78 |
100,000 |
46,319 |
0.91403444541297 |
|
79 |
110,000 |
85,644 |
0.91619796372558 |
|
80 |
110,000 |
106,955 |
0.91669066788385 |
|
81 |
110,000 |
881 |
0.91812050592972 |
|
82 |
120,000 |
99,829 |
0.91921610673818 |
|
83 |
120,000 |
14,144 |
0.91993788202315 |
|
84 |
130,000 |
25,084 |
0.92015169888905 |
|
85 |
130,000 |
77,276 |
0.92200728603694 |
|
86 |
130,000 |
123,548 |
0.92271617029789 |
|
87 |
140,000 |
97,812 |
0.92356759726451 |
|
88 |
140,000 |
55,055 |
0.92419417978360 |
|
89 |
150,000 |
101,789 |
0.92513747511725 |
|
90 |
150,000 |
148,292 |
0.92623115515484 |
|
91 |
160,000 |
2,417 |
0.92684609213400 |
|
92 |
160,000 |
82,952 |
0.92700341624231 |
|
93 |
160,000 |
21,881 |
0.92853006727172 |
|
94 |
170,000 |
99,547 |
0.92900246577367 |
|
95 |
180,000 |
104,390 |
0.92989681194411 |
|
96 |
180,000 |
135,953 |
0.93046202053222 |
|
97 |
190,000 |
36,320 |
0.93105553713269 |
|
98 |
190,000 |
133,987 |
0.93123999104557 |
|
99 |
200,000 |
4,708 |
0.93276201266923 |
|
100 |
200,000 |
8,416 |
0.93341118057491 |
The best solutions for some N values were found too close to the total number of restarts. Shaded entries in the table indicate where less than 10% fewer restarts would miss the best solution. It shows that more restarts could actually be necessary for finding the global optima, but on a home computer it may not be practical.
This is a draft document; any feedback is appreciated. (See the latest version here.)
Numerical Solutions of the Thomson-P Problems [pdf]
This document reports related work to the numerical solutions of the Tammes problem, the Thomson-P family of problems.
The latest version of this document can be downloaded from here.
The objective of the Thomson problem (Thomson-1) is to determine the minimum electrostatic potential energy configuration of N electrons constrained to the surface of a unit sphere. The electrons repel each other with a force given by Coulomb's law. The physicist J. J. Thomson posed the problem in 1904. Mathematically, the objective of the Thomson problem is to minimize the sum of the reciprocals of the distances between pairs of points.
A generalization of the Thomson problem, the Thomson-P problem, is minimizing the sum of the P-th powers of the pairwise distances. For P, negative even numbers are interesting. At large -P values, the sum is dominated by the distances of the closest point-pairs, approximating the Tammes problems.
For the Thomson-1 problem of N ≤ 257 points the first presented program (written in Julia) found the best published point sets in reasonable time. The second, multithreaded program is written in C for Windows 10. It is over 40 times faster, therefore it could handle more points: N ≤ 322 in reasonable running times.
Listing the coordinates of the points would be too long. Instead, the iteration number of the found best solutions are given in the table below, together with the corresponding values of the objective function. From the iteration numbers as input the described programs generate the coordinates of the known best Thomson sets, in mere seconds on a home computer. The computation slows down with larger number of points, but it still takes only a couple of minutes for up to 322 points, where we stopped the experiments, due to the lack of reliable data to compare the results to.
The program in the document is placed in the public domain, anyone can use it for any purpose.
This is a draft document; any feedback is appreciated.
The tables below contain our numerical optimization results for the Thomson-1 problem, illustrating the viability of the chosen approach.
· The first column (N) shows the number of points in the Thomson-1 set investigated.
· The second column (Ei) contains the best solutions found so far according to https://en.wikipedia.org/wiki/Thomson_problem (taken 12/30/2020) − for comparison.
· The third column shows the best Thomson solutions found with the programs listed in the document, using a cheap home computer.
· The 4th column of the table (Iter#) has the iteration numbers of our optimization program, at which the optimum was found. The coordinates of the corresponding Thomson sets can be printed with the presented programs in just seconds on a home PC.
The solutions found by the discussed program agreed with the best published results for all N ≤ 322. For problem sizes, for which there are no reliable published solutions, some of our tabulated results could be suboptimal, to be improved with more restarts on faster computational platforms.
|
N |
Ei |
Num-Opt |
Iter# |
|
2 |
0.500000000 |
|
|
|
3 |
1.732050808 |
1.7320508075689 |
1 |
|
4 |
3.674234614 |
3.6742346141748 |
1 |
|
5 |
6.474691495 |
6.4746914946882 |
1 |
|
6 |
9.985281374 |
9.9852813742386 |
1 |
|
7 |
14.452977414 |
14.452977414221 |
1 |
|
8 |
19.675287861 |
19.675287861233 |
1 |
|
9 |
25.759986531 |
25.759986531270 |
1 |
|
10 |
32.716949460 |
32.716949460148 |
1 |
|
11 |
40.596450510 |
40.596450508191 |
1 |
|
12 |
49.165253058 |
49.165253057629 |
1 |
|
58.853230612 |
58.853230611702 |
1 |
|
|
14 |
69.306363297 |
69.306363296626 |
1 |
|
15 |
80.670244114 |
80.670244114294 |
1 |
|
16 |
92.911655302 |
92.911655302545 |
1 |
|
17 |
106.050404829 |
106.05040482862 |
1 |
|
18 |
120.084467447 |
120.08446744749 |
1 |
|
19 |
135.089467557 |
135.08946755668 |
1 |
|
20 |
150.881568334 |
150.88156833376 |
1 |
|
21 |
167.641622399 |
167.64162239927 |
1 |
|
22 |
185.287536149 |
185.28753614931 |
1 |
|
23 |
203.930190663 |
203.93019066288 |
1 |
|
24 |
223.347074052 |
223.34707405181 |
1 |
|
25 |
243.812760299 |
243.81276029877 |
1 |
|
26 |
265.133326317 |
265.13332631736 |
1 |
|
27 |
287.302615033 |
287.30261503304 |
1 |
|
28 |
310.491542358 |
310.49154235820 |
1 |
|
29 |
334.634439920 |
334.63443992042 |
1 |
|
30 |
359.603945904 |
359.60394590376 |
1 |
|
31 |
385.530838063 |
385.53083806330 |
1 |
|
32 |
412.261274651 |
412.26127465053 |
1 |
|
33 |
440.204057448 |
440.20405744765 |
1 |
|
N |
Ei |
Num-Opt |
Iter# |
|
34 |
468.904853281 |
468.90485328134 |
1 |
|
35 |
498.569872491 |
498.56987249065 |
1 |
|
36 |
529.122408375 |
529.12240837541 |
1 |
|
37 |
560.618887731 |
560.61888773104 |
9 |
|
38 |
593.038503566 |
593.03850356645 |
10 |
|
39 |
626.389009017 |
626.38900901682 |
3 |
|
40 |
660.675278835 |
660.67527883462 |
1 |
|
41 |
695.916744342 |
695.91674434189 |
1 |
|
42 |
732.078107544 |
732.07810754367 |
1 |
|
43 |
769.190846459 |
769.19084645916 |
1 |
|
44 |
807.174263085 |
807.17426308463 |
1 |
|
45 |
846.188401061 |
846.18840106108 |
1 |
|
46 |
886.167113639 |
886.16711363919 |
1 |
|
47 |
927.059270680 |
927.05927067971 |
1 |
|
48 |
968.713455344 |
968.71345534379 |
1 |
|
49 |
1011.557182654 |
1011.5571826536 |
1 |
|
50 |
1055.182314726 |
1055.1823147263 |
1 |
|
51 |
1099.819290319 |
1099.8192903189 |
1 |
|
52 |
1145.418964319 |
1145.4189643193 |
2 |
|
53 |
1191.922290416 |
1191.9222904162 |
1 |
|
54 |
1239.361474729 |
1239.3614747292 |
1 |
|
55 |
1287.772720783 |
1287.7727207827 |
1 |
|
56 |
1337.094945276 |
1337.0949452757 |
3 |
|
57 |
1387.383229253 |
1387.3832292528 |
1 |
|
58 |
1438.618250640 |
1438.6182506404 |
1 |
|
59 |
1490.773335279 |
1490.7733352787 |
4 |
|
60 |
1543.830400976 |
1543.8304009764 |
1 |
|
61 |
1597.941830199 |
1597.9418301990 |
1 |
|
62 |
1652.909409898 |
1652.9094098983 |
2 |
|
63 |
1708.879681503 |
1708.8796815033 |
1 |
|
64 |
1765.802577927 |
1765.8025779273 |
1 |
|
65 |
1823.667960264 |
1823.6679602639 |
1 |
|
66 |
1882.441525304 |
1882.4415253042 |
1 |
|
N |
Ei |
Num-Opt |
Iter# |
|
67 |
1942.122700406 |
1942.1227004055 |
1 |
|
68 |
2002.874701749 |
2002.8747017487 |
3 |
|
69 |
2064.533483235 |
2064.5334832348 |
6 |
|
70 |
2127.100901551 |
2127.1009015506 |
1 |
|
71 |
2190.649906425 |
2190.6499064258 |
1 |
|
72 |
2255.001190975 |
2255.0011909750 |
1 |
|
73 |
2320.633883745 |
2320.6338837454 |
3 |
|
74 |
2387.072981838 |
2387.0729818383 |
1 |
|
75 |
2454.369689040 |
2454.3696890396 |
1 |
|
76 |
2522.674871841 |
2522.6748718414 |
1 |
|
77 |
2591.850152354 |
2591.8501523539 |
1 |
|
78 |
2662.046474566 |
2662.0464745663 |
52 |
|
79 |
2733.248357479 |
2733.2483574788 |
5 |
|
80 |
2805.355875981 |
2805.3558759812 |
1 |
|
81 |
2878.522829664 |
2878.5228296641 |
1 |
|
82 |
2952.569675286 |
2952.5696752865 |
1 |
|
83 |
3027.528488921 |
3027.5284889211 |
2 |
|
84 |
3103.465124431 |
3103.4651244308 |
2 |
|
85 |
3180.361442939 |
3180.3614429386 |
1 |
|
86 |
3258.211605713 |
3258.2116057128 |
4 |
|
87 |
3337.000750014 |
3337.0007500145 |
17 |
|
88 |
3416.720196758 |
3416.7201967584 |
4 |
|
89 |
3497.439018625 |
3497.4390186247 |
2 |
|
90 |
3579.091222723 |
3579.0912227228 |
2 |
|
91 |
3661.713699320 |
3661.7136993200 |
1 |
|
92 |
3745.291636241 |
3745.2916362407 |
1 |
|
93 |
3829.844338421 |
3829.8443384214 |
2 |
|
94 |
3915.309269620 |
3915.3092696204 |
9 |
|
95 |
4001.771675565 |
4001.7716755650 |
1 |
|
96 |
4089.154010060 |
4089.1540100556 |
1 |
|
97 |
4177.533599622 |
4177.5335996222 |
1 |
|
98 |
4266.822464156 |
4266.8224641562 |
1 |
|
99 |
4357.139163132 |
4357.1391631318 |
1 |
|
N |
Ei |
Num-Opt |
Iter# |
|
100 |
4448.350634331 |
4448.3506343313 |
1 |
|
101 |
4540.590051694 |
4540.5900516944 |
1 |
|
102 |
4633.736565899 |
4633.7365658987 |
1 |
|
103 |
4727.836616833 |
4727.8366168330 |
5 |
|
104 |
4822.876522746 |
4822.8765227487 |
9 |
|
105 |
4919.000637616 |
4919.0006376157 |
18 |
|
106 |
5015.984595705 |
5015.9845957047 |
7 |
|
107 |
5113.953547724 |
5113.9535477138 |
8 |
|
108 |
5212.813507831 |
5212.8135078306 |
1 |
|
109 |
5312.735079920 |
5312.7350799202 |
1 |
|
110 |
5413.549294192 |
5413.5492941924 |
6 |
|
111 |
5515.293214587 |
5515.2932145866 |
1 |
|
112 |
5618.044882327 |
5618.0448823266 |
3 |
|
113 |
5721.824978027 |
5721.8249780271 |
12 |
|
114 |
5826.521572163 |
5826.5215721627 |
4 |
|
115 |
5932.181285777 |
5932.1812857773 |
8 |
|
116 |
6038.815593579 |
6038.8155935785 |
76 |
|
117 |
6146.342446579 |
6146.3424465786 |
7 |
|
118 |
6254.877027790 |
6254.8770277896 |
10 |
|
119 |
6364.347317479 |
6364.3473174792 |
22 |
|
120 |
6474.756324980 |
6474.7563249797 |
8 |
|
121 |
6586.121949584 |
6586.1219495841 |
1 |
|
122 |
6698.374499261 |
6698.3744992606 |
105 |
|
123 |
6811.827228174 |
6811.8272281741 |
7 |
|
124 |
6926.169974193 |
6926.1699741935 |
1 |
|
125 |
7041.473264023 |
7041.4732640232 |
24 |
|
126 |
7157.669224867 |
7157.6692248670 |
9 |
|
127 |
7274.819504675 |
7274.8195046756 |
12 |
|
128 |
7393.007443068 |
7393.0074430680 |
6 |
|
129 |
7512.107319268 |
7512.1073192683 |
19 |
|
130 |
7632.167378912 |
7632.1673789125 |
1 |
|
131 |
7753.205166941 |
7753.2051669406 |
10 |
|
132 |
7875.045342797 |
7875.0453427971 |
8 |
|
133 |
7998.179212898 |
7998.1792128984 |
9 |
|
N |
Ei |
Num-Opt |
Iter# |
|
134 |
8122.089721194 |
8122.0897211942 |
7 |
|
135 |
8246.909486992 |
8246.9094869920 |
1 |
|
136 |
8372.743302539 |
8372.7433025386 |
1 |
|
137 |
8499.534494782 |
8499.5344947815 |
4 |
|
138 |
8627.406389880 |
8627.4063898801 |
3 |
|
139 |
8756.227056057 |
8756.2270569530 |
9 |
|
140 |
8885.980609041 |
8885.9806090406 |
1 |
|
141 |
9016.615349190 |
9016.6153491899 |
8 |
|
142 |
9148.271579993 |
9148.2715799935 |
9 |
|
143 |
9280.839851192 |
9280.8398511922 |
52 |
|
144 |
9414.371794460 |
9414.3717944599 |
7 |
|
145 |
9548.928837232 |
9548.9288372318 |
10 |
|
146 |
9684.381825575 |
9684.3818255749 |
3 |
|
147 |
9820.932378373 |
9820.9323783732 |
1 |
|
148 |
9958.406004270 |
9958.4060042699 |
6 |
|
149 |
10096.859907397 |
10096.859907397 |
1 |
|
150 |
10236.196436701 |
10236.196436701 |
7 |
|
151 |
10376.571469275 |
10376.571469275 |
6 |
|
152 |
10517.867592878 |
10517.867592878 |
83 |
|
153 |
10660.082748237 |
10660.082748236 |
54 |
|
154 |
10803.372421141 |
10803.372421141 |
8 |
|
155 |
10947.574692279 |
10947.574692279 |
8 |
|
156 |
11092.798311456 |
11092.798311456 |
143 |
|
157 |
11238.903041156 |
11238.903041156 |
237 |
|
158 |
11385.990186197 |
11385.990186197 |
1 |
|
159 |
11534.023960956 |
11534.023960956 |
27 |
|
160 |
11683.054805549 |
11683.054805549 |
11 |
|
161 |
11833.084739465 |
11833.084739465 |
28 |
|
162 |
11984.050335814 |
11984.050335814 |
37 |
|
163 |
12136.013053220 |
12136.013053220 |
12 |
|
164 |
12288.930105320 |
12288.930105320 |
14 |
|
165 |
12442.804451373 |
12442.804451373 |
16 |
|
166 |
12597.649071323 |
12597.649071323 |
123 |
|
N |
Ei |
Num-Opt |
Iter# |
|
167 |
12753.469429750 |
12753.469429750 |
5 |
|
168 |
12910.212672268 |
12910.212672268 |
10 |
|
169 |
13068.006451127 |
13068.006451127 |
11 |
|
170 |
13226.681078541 |
13226.681078540 |
165 |
|
171 |
13386.355930717 |
13386.355930717 |
21 |
|
172 |
13547.018108787 |
13547.018108787 |
62 |
|
173 |
13708.635243034 |
13708.635243034 |
22 |
|
174 |
13871.187092292 |
13871.187092292 |
1 |
|
175 |
14034.781306929 |
14034.781306929 |
32 |
|
176 |
14199.354775632 |
14199.354775632 |
17 |
|
177 |
14364.837545298 |
14364.837545298 |
569 |
|
178 |
14531.309552587 |
14531.309552588 |
70 |
|
179 |
14698.754594220 |
14698.754594220 |
8 |
|
180 |
14867.099927525 |
14867.099927525 |
82 |
|
181 |
15036.467239769 |
15036.467239769 |
6 |
|
182 |
15206.730610906 |
15206.730610906 |
4 |
|
183 |
15378.166571028 |
15378.166571028 |
8 |
|
184 |
15550.421450311 |
15550.421450311 |
1036 |
|
185 |
15723.720074072 |
15723.720074072 |
280 |
|
186 |
15897.897437048 |
15897.897437048 |
1 |
|
187 |
16072.975186320 |
16072.975186320 |
9 |
|
188 |
16249.222678879 |
16249.222678879 |
514 |
|
189 |
16426.371938862 |
16426.371938864 |
22 |
|
190 |
16604.428338501 |
16604.428338501 |
19 |
|
191 |
16783.452219362 |
16783.452219363 |
29 |
|
192 |
16963.338386460 |
16963.338386461 |
7 |
|
193 |
17144.564740880 |
17144.564740880 |
81 |
|
194 |
17326.616136471 |
17326.616136471 |
184 |
|
195 |
17509.489303930 |
17509.489303930 |
18 |
|
196 |
17693.460548082 |
17693.460548082 |
70 |
|
197 |
17878.340162571 |
17878.340162571 |
|
|
198 |
18064.262177195 |
18064.262177195 |
113 |
|
199 |
18251.082495640 |
18251.082495640 |
196 |
|
200 |
18438.842717530 |
18438.842717530 |
245 |
|
N |
Ei |
Num-Opt |
Iter# |
|
18627.591226244 |
18627.591226244 |
3386 |
|
|
202 |
18817.204718262 |
18817.204718262 |
60 |
|
203 |
19007.981204580 |
19007.981204580 |
274 |
|
204 |
19199.540775603 |
19199.540775603 |
651 |
|
205 |
|
19392.369152388 |
12 |
|
206 |
|
19585.955856549 |
272 |
|
207 |
|
19780.656909314 |
62 |
|
208 |
|
19976.203261203 |
237 |
|
|
20172.754680661 |
578 |
|
|
|
20370.251615129 |
693 |
|
|
211 |
|
20568.740602776 |
280 |
|
212 |
20768.053085964 |
20768.053085964 |
488 |
|
213 |
|
20968.612025491 |
74 |
|
214 |
21169.910410375 |
21169.910410375 |
363 |
|
215 |
|
21372.348789341 |
366 |
|
216 |
21575.596377869 |
21575.596377869 |
53 |
|
217 |
21779.856080418 |
21779.856080418 |
27 |
|
218 |
|
21985.263948921 |
182 |
|
219 |
|
22191.485474814 |
155 |
|
220 |
|
22398.655602531 |
174 |
|
221 |
|
22606.881547259 |
23 |
|
222 |
|
22816.025570249 |
57 |
|
223 |
|
23026.165881160 |
8 |
|
224 |
|
23237.244873487 |
3 |
|
225 |
|
23449.436460667 |
12 |
|
226 |
|
23662.511122654 |
221 |
|
227 |
|
23876.576893718 |
136 |
|
228 |
|
24091.578985867 |
36 |
|
229 |
|
24307.599313290 |
288 |
|
230 |
|
24524.485350945 |
139 |
|
231 |
|
24742.382494768 |
126 |
|
232 |
24961.252318934 |
24961.252318934 |
608 |
|
233 |
|
25181.057399530 |
1 |
|
N |
Ei |
Num-Opt |
Iter# |
|
234 |
|
25401.931786640 |
92 |
|
235 |
|
25623.763144201 |
37 |
|
236 |
|
25846.500563152 |
136 |
|
|
26070.367015459 |
898 |
|
|
|
26295.126047904 |
1221 |
|
|
239 |
|
26520.874117442 |
444 |
|
240 |
|
26747.508205092 |
9 |
|
241 |
|
26975.190283978 |
756 |
|
242 |
|
27203.799285553 |
1886 |
|
243 |
|
27433.367352376 |
418 |
|
244 |
|
27663.905112181 |
1355 |
|
245 |
|
27895.540745391 |
355 |
|
246 |
|
28128.051464319 |
5421 |
|
247 |
|
28361.532777263 |
17 |
|
248 |
|
28596.052102948 |
1213 |
|
249 |
|
28831.473909851 |
575 |
|
250 |
|
29067.889163423 |
940 |
|
251 |
|
29305.233861625 |
5335 |
|
252 |
|
29543.522868125 |
197 |
|
253 |
|
29782.917364395 |
84 |
|
254 |
|
30023.222861843 |
4728 |
|
255 |
30264.424251281 |
30264.424251281 |
307 |
|
30506.687515847 |
30506.687515847 |
6649 |
|
|
257 |
30749.941417346 |
30749.941417346 |
9864 |
Best Results of the Julia Program
|
N |
Ei |
Num-Opt |
Iter# |
|
258 |
|
30994.2135774868 |
611 |
|
259 |
|
31239.4423068662 |
1123 |
|
260 |
|
31485.5781917802 |
1760 |
|
261 |
|
31732.7427877295 |
3223 |
|
262 |
|
31980.8518238660 |
1199 |
|
263 |
|
32229.9148956472 |
2913 |
|
264 |
|
32479.9081411766 |
249 |
|
265 |
|
32731.0123974668 |
6659 |
|
266 |
|
32982.9836290832 |
2588 |
|
267 |
|
33235.9620660390 |
633 |
|
268 |
|
33489.8744849436 |
3243 |
|
269 |
|
33744.8007327701 |
2598 |
|
270 |
|
34000.6484970195 |
40677 |
|
271 |
|
34257.4417381447 |
8915 |
|
272 |
34515.193292681 |
34515.1932926817 |
10303 |
|
273 |
|
34774.2578509450 |
658 |
|
274 |
|
35034.0334667000 |
2803 |
|
275 |
|
35294.8940512130 |
14388 |
|
276 |
|
35556.6333035795 |
17841 |
|
277 |
|
35819.3408109533 |
1006 |
|
278 |
|
36083.0368937559 |
2103 |
|
279 |
|
36347.6584257292 |
1587 |
|
N |
Ei |
Num-Opt |
Iter# |
|
280 |
|
36613.2695900773 |
1005 |
|
281 |
|
36879.8056153435 |
3166 |
|
282 |
37147.294418462 |
37147.2944184620 |
14968 |
|
283 |
|
37416.0757427823 |
1101 |
|
284 |
|
37685.6897932422 |
1326 |
|
285 |
|
37956.2394559865 |
8052 |
|
286 |
|
38227.6733205442 |
4850 |
|
287 |
|
38500.1343322855 |
1856 |
|
288 |
|
38773.5812836889 |
17850 |
|
289 |
|
39048.0491867372 |
15313 |
|
290 |
|
39323.4100620378 |
22241 |
|
291 |
|
39599.7118168330 |
10848 |
|
292 |
39877.008012909 |
39877.0080129083 |
1624 |
|
293 |
|
40155.3490760206 |
1974 |
|
294 |
|
40434.7622974846 |
1280 |
|
295 |
|
40715.1417167072 |
2166 |
|
296 |
|
40996.4139085292 |
654 |
|
297 |
|
41278.7032930107 |
1894 |
|
298 |
|
41561.9207687963 |
25409 |
|
299 |
|
41846.0811834311 |
2832 |
|
300 |
|
42131.2641372452 |
44534 |
|
N |
Ei |
Num-Opt |
Iter# |
|
301 |
|
42417.3566298396 |
45153 |
|
302 |
|
42704.4937584797 |
10025 |
|
303 |
|
42992.5289668779 |
6886 |
|
304 |
|
43281.5133649633 |
22859 |
|
305 |
|
43571.5504536883 |
14785 |
|
306 |
43862.569780797 |
43862.5697807962 |
78567 |
|
307 |
|
44154.5477059508 |
8530 |
|
308 |
|
44447.4530563589 |
1376 |
|
309 |
|
44741.5642838065 |
2714 |
|
310 |
|
45036.4985722224 |
25326 |
|
311 |
|
45332.4021055010 |
1139 |
|
312 |
45629.313804002 |
45629.3138040044 |
48767 |
|
313 |
|
45927.1925470308 |
38645 |
|
314 |
|
46225.9797575717 |
35275 |
|
315 |
46525.825643432 |
46525.8256434321 |
173295 |
|
316 |
|
46826.5884028806 |
48055 |
|
317 |
47128.310344520 |
47128.3103445197 |
69909 |
|
318 |
47431.056020043 |
47431.0560200432 |
31743 |
|
319 |
|
47734.8137639722 |
9808 |
|
320 |
|
48039.4689301648 |
4665 |
|
321 |
|
48345.1609527838 |
65053 |
|
322 |
|
48651.7621823975 |
7427 |
Best Results of the Multithreaded C Program