Extremal (Maximum Determinant)
points on the sphere S2

Last updated 17-Oct-2007

Extremal systems -- Eigenvalues and Lagrangian norms
Deg. No. of points Eigenvalues of G and condition number Lebesgue constant
Lagrangian norms
n
dn = (n+1)2
lmin lavg lmax cond2(G) || Ln|| Reimer
upper
bound
maxx in S2
|| L(x) ||2
maxx in S2
|| L(x) ||inf - 1
1 4 0.3183 0.3183 0.3183 1.00 2.00 2.00 1.00000 0.0e+00
2 9 0.4633 0.7162 0.8950 1.93 3.24 3.73 1.13697 2.2e-16
3 16 0.7967 1.2732 1.9149 2.40 3.60 5.06 1.12051 4.4e-16
4 25 0.8965 1.9894 2.8658 3.20 5.22 7.45 1.27372 6.7e-16
5 36 1.0983 2.8648 4.3354 3.95 6.13 9.69 1.31083 1.3e-15
6 49 1.2027 3.8993 6.1968 5.15 7.46 12.60 1.36945 1.3e-15
7 64 1.3534 5.0930 8.3877 6.20 8.19 15.52 1.35201 1.3e-15
8 81 1.5380 6.4458 10.8123 7.03 8.86 18.42 1.40769 1.3e-15
9 100 1.7882 7.9577 13.5931 7.60 8.60 21.10 1.27795 1.6e-15
10 121 1.8487 9.6289 16.5433 8.95 10.90 25.10 1.43470 1.8e-15
11 144 3.0530 11.4592 19.3570 6.34 11.44 23.25 1.45369 2.7e-15
12 169 2.3608 13.4486 23.5398 9.97 14.38 31.03 1.53730 3.3e-15
13 196 2.5059 15.5972 27.6946 11.05 15.45 34.93 1.55752 3.8e-15
14 225 2.5568 17.9049 31.7788 12.43 20.28 39.69 1.88592 3.6e-15
15 256 2.3949 20.3718 36.4973 15.24 18.23 46.66 1.65131 4.9e-15
16 289 2.9111 22.9979 40.4140 13.88 18.28 47.78 1.61844 4.9e-15
17 324 2.9921 25.7831 46.0495 15.39 21.01 52.84 1.75121 4.7e-15
18 361 3.0030 28.7275 52.0852 17.34 20.51 58.77 1.65585 7.1e-15
19 400 3.3460 31.8310 57.8951 17.30 22.29 61.69 1.69404 5.3e-15
20 441 3.2625 35.0937 63.8296 19.56 23.38 68.87 1.74291 1.0e-14
21 484 3.6911 38.5155 70.6113 19.13 25.73 71.07 1.75259 8.0e-15
22 529 3.5209 42.0965 76.4184 21.70 25.02 79.53 1.72857 1.1e-14
23 576 3.6356 45.8366 84.5816 23.27 27.41 85.22 1.79264 1.2e-14
24 625 4.0385 49.7359 90.9942 22.53 27.71 87.73 1.76238 1.3e-14
25 676 4.3086 53.7944 98.6914 22.91 29.23 91.87 1.93722 1.2e-14
26 729 3.9634 58.0120 106.9091 26.97 29.31 103.30 1.72924 1.6e-14
27 784 4.0445 62.3887 115.0502 28.45 30.78 109.97 1.80435 1.6e-14
28 841 4.0429 66.9247 124.1057 30.70 32.13 117.99 1.86149 1.8e-14
29 900 4.6410 71.6197 133.1534 28.69 32.21 117.85 1.74197 2.0e-14
30 961 4.1986 76.4740 142.7842 34.01 36.28 132.30 1.94961 1.7e-14
31 1024 4.4149 81.4873 151.4716 34.31 36.95 137.48 1.86461 1.6e-14
32 1089 4.5858 86.6599 161.5849 35.24 38.65 143.45 1.85658 2.1e-14
33 1156 3.9609 91.9916 172.5413 43.56 39.48 163.85 1.81736 2.6e-14
34 1225 4.4890 97.4824 181.2408 40.37 40.13 163.10 1.83258 4.5e-14
35 1296 4.6734 103.1324 193.2572 41.35 41.44 169.12 1.88393 2.2e-14
36 1369 4.0042 108.9416 205.1954 51.24 49.03 192.99 2.05478 4.3e-14
37 1444 4.0062 114.9099 216.8890 54.14 43.75 203.51 1.86619 3.3e-14
38 1521 5.4970 121.0373 228.7028 41.60 45.09 183.00 1.89704 3.4e-14
39 1600 4.8968 127.3240 239.7571 48.96 47.81 203.97 1.91683 4.4e-14
40 1681 4.3601 133.7697 251.8577 57.76 48.91 227.10 1.87516 3.2e-14
41 1764 5.0396 140.3747 263.9665 52.38 49.46 221.66 1.92758 5.3e-14
42 1849 5.1035 147.1387 278.4061 54.55 51.76 230.89 1.99834 5.0e-14
43 1936 4.9733 154.0620 291.8666 58.69 51.19 244.89 1.87166 3.3e-14
44 2025 5.4097 161.1444 304.9891 56.38 53.68 245.60 2.05367 5.5e-14
45 2116 5.3542 168.3859 318.3880 59.47 57.88 257.97 1.98281 4.5e-14
46 2209 5.5893 175.7866 332.7618 59.54 63.25 263.58 2.12681 5.5e-14
47 2304 6.0740 183.3465 347.3021 57.18 62.18 263.72 2.07050 5.2e-14
48 2401 6.4091 191.0655 361.3585 56.38 57.19 267.54 1.95802 6.9e-14
49 2500 5.4655 198.9437 376.9594 68.97 61.02 301.66 1.99455 6.5e-14
50 2601 5.7598 206.9810 392.2796 68.11 69.47 305.72 2.18637 5.3e-14
51 2704 5.0107 215.1775 408.2328 81.47 64.59 340.76 2.00418 NaN
52 2809 5.5264 223.5331 425.9326 77.07 71.86 337.07 2.24074 NaN
53 2916 5.7470 232.0479 442.1840 76.94 61.58 343.13 1.96329 NaN
54 3025 5.5924 240.7219 458.3554 81.96 68.49 360.85 2.14552 NaN
55 3136 5.6602 249.5550 474.0778 83.76 65.56 371.84 1.96489 NaN
56 3249 5.7970 258.5472 491.0405 84.71 78.30 380.67 2.10573 NaN
57 3364 6.0176 267.6986 510.5602 84.84 70.83 386.85 2.02033 NaN
58 3481 6.2211 277.0092 528.7921 85.00 67.09 393.70 2.06086 NaN
59 3600 5.3259 286.4789 545.5791 102.44 79.00 440.05 2.15567 NaN
60 3721 5.8826 296.1078 564.0016 95.88 72.48 432.78 1.99447 NaN
61 3844 5.9371 305.8958 583.8493 98.34 77.20 445.03 2.14970 NaN
62 3969 5.1869 315.8430 603.4712 116.34 80.25 491.61 2.17089 NaN
63 4096 5.8962 325.9493 622.7176 105.61 89.08 475.85 2.31825 NaN
64 4225 5.8691 336.2148 645.3358 109.95 82.10 491.97 2.09441 NaN
65 4356 4.7658 346.6395 660.8337 138.66 88.90 562.88 2.18188 NaN
66 4489 5.9766 357.2233 682.7040 114.23 87.46 517.99 2.19517 NaN
67 4624 5.3136 367.9662 702.9972 132.30 86.66 565.87 2.20978 NaN
68 4761 5.6062 378.8683 725.6956 129.44 84.34 567.23 2.08889 NaN
69 4900 6.2579 389.9296 746.1165 119.23 93.42 552.56 2.18018 NaN
70 5041 4.9456 401.1500 770.7363 155.84 97.40 639.44 2.27247 NaN
71 5184 5.9782 412.5296 790.6031 132.25 97.20 598.10 2.17940 NaN
72 5329 5.8956 424.0683 810.8480 137.53 103.78 619.12 2.30687 NaN
73 5476 5.8408 435.7662 834.4194 142.86 97.31 639.18 2.21494 NaN
74 5625 5.2493 447.6233 857.0787 163.28 112.79 692.58 2.37001 NaN
75 5776 5.0266 459.6395 880.8615 175.24 106.28 726.75 2.28395 NaN
76 5929 5.2780 471.8148 905.4028 171.54 112.75 728.01 2.32748 NaN
77 6084 5.5389 484.1493 929.3957 167.80 110.69 729.25 2.35370 NaN
78 6241 4.9176 496.6430 952.0096 193.59 112.88 793.92 2.31499 NaN
79 6400 5.0803 509.2958 978.4245 192.59 120.11 801.00 2.43080 NaN
80 6561 6.4534 522.1078 1000.4037 155.02 109.75 728.57 2.27143 NaN
81 6724 5.3994 535.0789 1024.6148 189.76 113.46 816.30 2.28688 NaN
82 6889 4.3690 548.2092 1052.5396 240.91 130.79 929.74 2.49930 NaN
83 7056 5.5540 561.4986 1077.9846 194.09 121.97 844.60 2.30834 NaN
84 7225 4.5505 574.9472 1103.1769 242.43 122.30 955.44 2.34502 NaN
85 7396 5.2101 588.5550 1128.2724 216.56 133.35 914.05 2.38641 NaN
86 7569 5.6928 602.3219 1158.6285 203.52 126.25 894.89 2.30357 NaN
87 7744 6.4212 616.2479 1183.7750 184.35 149.56 862.09 2.59059 NaN
88 7921 5.8139 630.3332 1210.3505 208.18 118.06 926.70 2.26672 NaN
89 8100 5.6521 644.5775 1237.9005 219.02 128.93 961.12 2.35685 NaN
90 8281 4.8047 658.9810 1269.7183 264.27 134.40 1065.72 2.38080 NaN
91 8464 5.4190 673.5437 1292.6883 238.55 136.75 1025.68 2.30578 NaN
92 8649 5.3637 688.2656 1321.1745 246.32 137.99 1053.49 2.45431 NaN
93 8836 6.7344 703.1465 1349.8431 200.44 126.20 960.51 2.27708 NaN
94 9025 5.3260 718.1867 1381.1389 259.32 156.07 1103.17 2.51637 NaN
95 9216 4.5723 733.3860 1412.4979 308.93 143.81 1215.83 2.35272 NaN
96 9409 5.3136 748.7444 1435.3258 270.12 145.30 1151.45 2.45307 NaN
97 9604 5.3017 764.2620 1469.4535 277.16 158.51 1176.62 2.40737 NaN
98 9801 5.6733 779.9388 1497.3942 263.94 147.29 1160.78 2.40431 NaN
99 10000 5.8646 795.7747 1532.8388 261.37 143.03 1164.86 2.38640 NaN
100 10201 5.2953 811.7698 1561.3621 294.86 160.69 1250.52 2.60540 NaN
101 10404 5.6569 827.9240 1593.3205 281.66 165.54 1233.97 2.51537 NaN
102 10609 4.0368 844.2374 1623.9962 402.29 161.12 1489.53 2.59200 NaN
103 10816 5.6885 860.7099 1653.6621 290.70 168.08 1279.27 2.54357 NaN
104 11025 5.7265 877.3416 1685.7891 294.39 176.69 1299.66 2.64938 NaN
105 11236 4.6385 894.1325 1724.0981 371.69 171.40 1471.69 2.53021 NaN
106 11449 6.0645 911.0825 1753.7034 289.18 163.24 1311.50 2.45828 NaN
107 11664 4.5297 928.1916 1791.9294 395.60 170.57 1546.00 2.48749 NaN
108 11881 5.3149 945.4599 1819.9626 342.43 174.45 1453.79 2.59018 NaN
109 12100 4.7199 962.8874 1854.8774 392.99 189.07 1571.14 2.57713 NaN
110 12321 4.1465 980.4740 1888.0850 455.34 199.27 1706.87 2.63039 NaN
111 12544 4.5565 998.2198 1920.5568 421.50 178.08 1657.74 2.55843 NaN
112 12769 4.5981 1016.1247 1957.3535 425.69 212.52 1679.82 2.76167 NaN
113 12996 4.0981 1034.1888 1990.2019 485.64 245.14 1810.97 3.13592 NaN
114 13225 4.5747 1052.4121 2028.1184 443.33 188.55 1744.25 2.67367 NaN
115 13456 4.4669 1070.7945 2066.3110 462.58 189.54 1796.00 2.53222 NaN
116 13689 4.1074 1089.3360 2100.5248 511.40 201.23 1905.38 2.68323 NaN
117 13924 4.8839 1108.0367 2131.6950 436.47 199.91 1777.36 2.70599 NaN
118 14161 3.2183 1126.8966 2171.1221 674.62 208.43 2226.77 2.63573 NaN
119 14400 4.0325 1145.9156 2207.1617 547.34 200.46 2022.88 2.59177 NaN
120 14641 4.2942 1165.0938 2246.1202 523.06 220.45 1993.09 2.83948 NaN


Last updated 17-Oct-2007   |   Page contact: R.Womersley (AT) unsw.edu.au   |   Top   |   Contents