Minimum Energy
points on the sphere S2

Last updated 24-Jan-2003
2601 minimum energy points and cubature weights

Minimum Energy Systems

  • Minimum energy systems are sets of m points xj, j = 1,...,m on

    S2 = {x in R3 : |x| = 1}

    that minimize the potential energy

    sumi=1:m   sumj=i+1:m 1 / |  xi - xj  |

    where |x| denotes the Euclidean norm in R3.

  • There are many local miminma, often with objective values close to the global minimum.
  • A lower bound on the potential energy is

    (1/2)m2 + c m(3/2)
    where

    c = 3(sqrt(3)/(8pi))1/2 zeta(1/2)*L-3(1/2),

    zeta is the Riemann zeta function and L-3is the Dirichlet L-function with characteristic (1, -1, 0).

    c = -0.553051...

  • The coefficient of the m3/2 term in the potential energy expansion can be estimated by

    (PE - 0.5 m2) / m3/2.

  • The norm of the gradient with respect to a spherical parametrization of the points gives an idea how close it is to a stationary point (gradient = 0). Keep in mind the size of the objective when thinking about how small it is numerically possible to get this.
  • The Hessian is positive definite (smallest eigenvalue > 0) so these are strict local minima.

Local minima of potential energy

potential energy

potential energy - lower bound

infinity norm of gradient of potential energy

Interpolatory Cubature

  • The space Pn of polynomials of degree at most n on S2 has dimension

    dn = (n+1)2.

  • Cubature rule

    sumj = 1 to dn   wj f(xj)

    approximates the integral of f(x) over the surface of the unit sphere.
  • Cubature weights w such that the cubature rule is exact for all polynomials of degree <= n satsify G w = e, e = vector of 1s. G is the reproducing kernel Gram matrix.
  • Solving Gw = e is theoretically possible for any fundamental system (G nonsingular) but in practice depends on the condition of G, which is very poor for many (local) minima of the potential energy.
  • Cubature rule exact for p(x) = 1 implies that the sum of cubature weights = |S2| = 4 p.
  • Average, or equal, cubature weight is

    wavg = |S2| / dn

  • Scaled cubature weights are wi / wavg
  • ,
    smallest   wmin / wavg <= 1,
    largest   wmax / wavg >= 1.
Scaled minimum and maximum cubature weights

Eigenvalues and Condition numbers

  • Many systems have very large condition numbers, making solving a linear system for interpolation or cubature weights unreliable.
  • 1-norm condition number is

    cond1(G) = || G ||1   || G-1 ||1.

  • As Gii = dn / |S2|, trace(G) = sum of eigenvalues of G = dn2 / |S2|.
  • lmin(G) = smallest eigenvalue.
    lavg(G) = average eigenvalue = dn / |S2|,
    lmax(G) = largest eigenvalue.
  • 2-norm condition number is

    cond2(G) = lmax(G) / lmin(G)

  • The 2-norm condition number is generally an order of magnitude less than the 1-norm condition number
Eigenvalues of G

Lagrangian norms

  • Lagranians

    Lj(x) = ejT G-1 g(x)

    for x in S2.
  • Lebesgue constant = ||Ln|| = interpolation operator norm (as a map from C(S2) to C(S2)) is

    maxx in S2 ||L(x)||1.
  • Reimer upper bound on the Lebesgue constant is

    R = (n+1) sqrt(lavg / lmin).

  • Lagrangian 2-norm,

    maxx in S2 || Lj(x) ||2

    and infinity norm,

    maxx in S2 maxj | Lj(x) |,

    are calculated by finding the global maximum over S2.
  • The Lagrangians satisfy

    Lj(xj) = 1 for j = 1,...,dn

    and

    Li(xj) = 0 for i not equal to j.

    Thus the norms of the Lagrangians are >= 1.
Lebesgue constant

Geometrical properties

  • The geodesic dsitance between two points x and y on the unit sphere S2 is dist(x,y) = cos-1(xT y).
  • The spherical cap with center y and radius r is

    C(y, r) = {x in S2 : dist(x, y) <= r }

  • The sphere packing problem is to place m points on a sphere to as to maximize the the minimal distance

    mini neq j   dist(xi, xj)

    between them. The minimum distance is twice the packing radius for packing spherical caps of of the same radius at each point xj, j = 1,...,m.
  • The mesh norm is

    h = maxx in S2   minj=1,...,m   dist(x - xj).

    This is also the covering radius for covering the sphere by spherical caps of radius h centered at each of the points xj, j = 1,..., m.
  • The Voronoi cell Vj about the point xj in S2 is

    Vj = {x in S2 : dist(x, xj <= dist(x, xi)
    for all i not equal to j

  • The sum of the areas of all Voronoi cells is equal to the surface area |S2| Thus the average area of m Voronoi cells is Vavg = |S2| / m and the scaled areas of the Voronoi cells is |Vj| / Vavg
  • Cui and Freeden discrepancy is D =
    (1 - 2 log(1 + sqrt((1 - xjT xk)/2)) ]1/2.
  • Volume of the convex hull of a set of points on S2 is always less than the volume of the unit sphere = 4 p / 3 = 4.18879...
Minimum angle between points

Mesh norm (covering radius)

Caveat: All points are only approximate local minimizers of the potential energy. Attempts to find global minima have been made, with decreasing reliability as n increases.

Notes on point sets

Minimum energy systems -- Potential energy
Deg. No. of points Files Potential energy
n dn = (n+1)2 Points and weights Lower bound
0.5 m2 + c m3/2
Value Value - Lower bound (Value - 0.5 m2)
m3/2
||gradient||
1 4 me01.0004 3.57559e+00 3.674234614175e+00 0.10 -0.540721 3.33e-16
2 9 me02.0009 2.55676e+01 2.575998653127e+01 0.19 -0.545926 1.11e-15
3 16 me03.0016 9.26047e+01 9.291165530254e+01 0.31 -0.548255 4.44e-15
4 25 me04.0025 2.43369e+02 2.438127602988e+02 0.44 -0.549498 7.99e-15
5 36 me05.0036 5.28541e+02 5.291224083754e+02 0.58 -0.550359 7.99e-15
6 49 me06.0049 1.01080e+03 1.011557182654e+03 0.75 -0.550854 1.78e-14
7 64 me07.0064 1.76484e+03 1.765802577927e+03 0.96 -0.551167 3.55e-14
8 81 me08.0081 2.87733e+03 2.878522829664e+03 1.20 -0.551409 4.44e-14
9 100 me09.0100 4.44695e+03 4.448350634331e+03 1.40 -0.551649 5.68e-14
10 121 me10.0121 6.58439e+03 6.586121949584e+03 1.73 -0.551749 5.52e-06
11 144 me11.0144 9.41233e+03 9.414371794460e+03 2.04 -0.551868 8.58e-06
12 169 me12.0169 1.30654e+04 1.306800645113e+04 2.56 -0.551886 5.31e-06
13 196 me13.0196 1.76904e+04 1.769346054808e+04 3.03 -0.551946 6.82e-06
14 225 me14.0225 2.34460e+04 2.344943646067e+04 3.48 -0.552019 1.04e-05
15 256 me15.0256 3.05027e+04 3.050668751585e+04 3.99 -0.552078 9.01e-05
16 289 me16.0289 3.90434e+04 3.904804918674e+04 4.69 -0.552097 2.42e-05
17 324 me17.0324 4.92626e+04 4.926792503237e+04 5.32 -0.552139 8.60e-05
18 361 me18.0361 6.13671e+04 6.137330603860e+04 6.18 -0.552150 5.83e-05
19 400 me19.0400 7.55756e+04 7.558244851221e+04 6.86 -0.552194 3.80e-05
20 441 me20.0441 9.21187e+04 9.212648427313e+04 7.79 -0.552210 5.23e-05
21 484 me21.0484 1.11239e+05 1.112476088853e+05 8.50 -0.552253 9.40e-05
22 529 me22.0529 1.33192e+05 1.332010101693e+05 9.49 -0.552272 1.35e-12
23 576 me23.0576 1.58243e+05 1.582530228917e+05 10.40 -0.552299 1.07e-04
24 625 me24.0625 1.86671e+05 1.866822831916e+05 11.21 -0.552334 3.18e-04
25 676 me25.0676 2.18768e+05 2.187798554905e+05 12.29 -0.552352 1.38e-04
26 729 me26.0729 2.54835e+05 2.548483694560e+05 13.58 -0.552361 2.35e-12
27 784 me27.0784 2.95187e+05 2.952026868575e+05 15.27 -0.552356 4.86e-04
28 841 me28.0841 3.40152e+05 3.401686202055e+05 16.49 -0.552375 4.81e-04
29 900 me29.0900 3.90068e+05 3.900854469030e+05 17.83 -0.552391 6.38e-04
30 961 me30.0961 4.45285e+05 4.453034320125e+05 18.88 -0.552417 1.99e-03
31 1024 me31.1024 5.06166e+05 5.061856600997e+05 20.04 -0.552440 1.01e-04
32 1089 me32.1089 5.73085e+05 5.731075707256e+05 22.08 -0.552437 3.87e-12
33 1156 me33.1156 6.46431e+05 6.464542972937e+05 23.43 -0.552455 4.65e-04
34 1225 me34.1225 7.26600e+05 7.266258102664e+05 25.38 -0.552459 1.94e-04
35 1296 me35.1296 8.14005e+05 8.140313598249e+05 26.52 -0.552483 1.32e-03
36 1369 me36.1369 9.09067e+05 9.090955445014e+05 28.75 -0.552484 2.66e-04
37 1444 me37.1444 1.01222e+06 1.012251500074e+06 30.53 -0.552495 2.34e-03
38 1521 me38.1521 1.12391e+06 1.123945552468e+06 31.50 -0.552520 8.08e-04
39 1600 me39.1600 1.24460e+06 1.244638811686e+06 34.09 -0.552519 1.38e-03
40 1681 me40.1681 1.37476e+06 1.374799891193e+06 36.24 -0.552525 1.22e-03
41 1764 me41.1764 1.51487e+06 1.514911635966e+06 38.10 -0.552537 1.12e-03
42 1849 me42.1849 1.66543e+06 1.665469200966e+06 40.15 -0.552546 1.52e-03
43 1936 me43.1936 1.82694e+06 1.826979297385e+06 42.42 -0.552553 2.11e-03
44 2025 me44.2025 1.99992e+06 1.999959570264e+06 43.87 -0.552570 2.29e-03
45 2116 me45.2116 2.18490e+06 2.184943361901e+06 47.16 -0.552567 3.89e-03
46 2209 me46.2209 2.38242e+06 2.382471416401e+06 50.36 -0.552566 8.58e-03
47 2304 me47.2304 2.59304e+06 2.593096636755e+06 51.69 -0.552584 4.35e-03
48 2401 me48.2401 2.81733e+06 2.817390001317e+06 55.43 -0.552580 1.39e-02
49 2500 me49.2500 3.05587e+06 3.055925552085e+06 56.96 -0.552596 6.35e-03
50 2601 me50.2601 3.30924e+06 3.309297199487e+06 59.51 -0.552603 4.21e-03
51 2704 me51.2704 3.57804e+06 3.578106675232e+06 62.11 -0.552610 4.37e-03
52 2809 me52.2809 3.86290e+06 3.862969514151e+06 65.63 -0.552610 2.41e-03
53 2916 me53.2916 4.16444e+06 4.164510527908e+06 68.20 -0.552618 5.77e-03
54 3025 me54.3025 4.48330e+06 4.483370440031e+06 71.85 -0.552619 5.43e-03
55 3136 me55.3136 4.82012e+06 4.820197407030e+06 74.06 -0.552630 7.72e-03
56 3249 me56.3249 5.17558e+06 5.175656774941e+06 77.50 -0.552633 1.17e-02
57 3364 me57.3364 5.55034e+06 5.550422560486e+06 81.50 -0.552634 8.10e-03
58 3481 me58.3481 5.94510e+06 5.945180149944e+06 84.77 -0.552639 1.26e-02
59 3600 me59.3600 6.36054e+06 6.360629252575e+06 88.33 -0.552642 1.40e-02
60 3721 me60.3721 6.79739e+06 6.797479205028e+06 90.84 -0.552651 1.10e-02
61 3844 me61.3844 7.25636e+06 7.256455726056e+06 95.33 -0.552651 1.07e-02
62 3969 me62.3969 7.73819e+06 7.738291245960e+06 99.56 -0.552653 1.13e-02
63 4096 me63.4096 8.24363e+06 8.243732062366e+06 103.14 -0.552658 1.80e-02
64 4225 me64.4225 8.77343e+06 8.773537252974e+06 106.46 -0.552664 1.78e-02
65 4356 me65.4356 9.32837e+06 9.328475746202e+06 107.78 -0.552676 8.50e-02
66 4489 me66.4489 9.90922e+06 9.909335860759e+06 112.73 -0.552676 2.13e-02
67 4624 me67.4624 1.05168e+07 1.051690791302e+07 116.94 -0.552679 1.77e-02
68 4761 me68.4761 1.11519e+07 1.115199843207e+07 120.26 -0.552685 4.14e-02
69 4900 me69.4900 1.18153e+07 1.181542881745e+07 125.41 -0.552686 3.74e-02
70 5041 me70.5041 1.25079e+07 1.250802673398e+07 129.38 -0.552690 2.50e-02
71 5184 me71.5184 1.32305e+07 1.323064054123e+07 137.83 -0.552682 1.91e-02
72 5329 me72.5329 1.39840e+07 1.398411500017e+07 140.86 -0.552689 3.28e-02
73 5476 me73.5476 1.47692e+07 1.476932354637e+07 145.20 -0.552693 2.11e-02
74 5625 me74.5625 1.55870e+07 1.558714259927e+07 148.61 -0.552699 2.65e-02
75 5776 me75.5776 1.64383e+07 1.643846360312e+07 151.85 -0.552705 1.41e-01
76 5929 me76.5929 1.73240e+07 1.732419458039e+07 160.25 -0.552700 3.74e-02
77 6084 me77.6084 1.82451e+07 1.824524108179e+07 164.68 -0.552704 3.31e-02
78 6241 me78.6241 1.92024e+07 1.920253234277e+07 167.70 -0.552711 8.81e-02
79 6400 me79.6400 2.01968e+07 2.019701511522e+07 177.38 -0.552705 2.10e-02
80 6561 me80.6561 2.12294e+07 2.122962105036e+07 174.68 -0.552723 4.28e-02

Minimum energy systems -- Geometric properties
Deg. No. of points Minimum distance Mesh norm Volume of convex hull Voronoi cells Cui & Freeden Disc. Worst case error
n dn = (n+1)2 Degrees Degrees   Tri. Rect. Pent. Hex. Hept. Vmin
Vavg
Vmax
Vavg
   
1 4 109.4712 70.5288 0.51320 4 0 0 0 0 1.0000 1.0000 0.0913 1.1467
2 9 69.1898 46.3353 2.03963 0 3 6 0 0 0.9448 1.0276 0.0493 0.6192
3 16 48.9362 33.2718 2.87733 0 0 12 4 0 0.9736 1.0792 0.0319 0.4010
4 25 39.6105 30.1185 3.31012 0 0 12 13 0 0.9349 1.0516 0.0227 0.2859
5 36 33.2292 23.6350 3.56603 0 0 12 24 0 0.9540 1.0384 0.0173 0.2170
6 49 28.3866 20.1054 3.72882 0 0 12 37 0 0.9457 1.0294 0.0137 0.1720
7 64 24.9200 17.5451 3.83385 0 0 12 52 0 0.9401 1.0255 0.0112 0.1407
8 81 21.8918 15.7474 3.90654 0 0 12 69 0 0.9507 1.0359 0.0094 0.1179
9 100 20.2968 14.0246 3.95967 0 0 12 88 0 0.9498 1.0257 0.0080 0.1005
10 121 18.1987 12.0502 4.00028 0 0 12 109 0 0.9259 1.0241 0.0069 0.0910
11 144 16.9534 11.0220 4.02994 0 0 12 132 0 0.9283 1.0212 0.0061 0.0764
12 169 15.5375 10.9155 4.05294 0 0 13 155 1 0.9243 1.0224 0.0054 0.0678
13 196 14.2509 9.8540 4.07175 0 0 12 184 0 0.9229 1.0186 0.0048 0.0606
14 225 13.2136 9.1750 4.08676 0 0 12 213 0 0.9239 1.0207 0.0044 0.0547
15 256 12.5722 8.1523 4.09919 0 0 12 244 0 0.9240 1.0137 0.0039 0.0496
16 289 11.8033 7.8412 4.10935 0 0 12 277 0 0.9208 1.0158 0.0036 0.0464
17 324 11.2582 7.2437 4.11789 0 0 12 312 0 0.9220 1.0117 0.0033 0.0416
18 361 10.5358 6.9450 4.12511 0 0 12 349 0 0.9213 1.0134 0.0031 0.0528
19 400 10.0681 6.5092 4.13136 0 0 12 388 0 0.9206 1.0100 0.0028 0.0355
20 441 9.3582 7.0390 4.13670 0 0 23 407 11 0.9194 1.0689 0.0026 0.0466
21 484 8.9376 6.7293 4.14133 0 0 20 456 8 0.9188 1.0618 0.0024 0.0310
22 529 8.5559 6.5163 4.14538 0 0 24 493 12 0.9182 1.0691 0.0023 0.0678
23 576 8.1775 6.1352 4.14892 0 0 24 540 12 0.9183 1.0658 0.0021 0.0326
24 625 7.8578 5.8926 4.15205 0 0 23 591 11 0.9173 1.0631 0.0020 0.0260
25 676 7.5602 5.6440 4.15482 0 0 24 640 12 0.9205 1.0630 0.0019 0.0273
26 729 7.2650 5.4618 4.15729 0 0 24 693 12 0.9166 1.0837 0.0018 0.0888
27 784 7.0099 5.3452 4.15948 0 0 29 738 17 0.9146 1.0800 0.0017 0.0223
28 841 6.7659 5.1608 4.16148 0 0 33 787 21 0.9152 1.0809 0.0016 0.0207
29 900 6.5470 5.0092 4.16327 0 0 36 840 24 0.9158 1.0719 0.0015 0.0198
30 961 6.3091 4.7635 4.16490 0 0 36 901 24 0.9160 1.0706 0.0015 0.0344
31 1024 6.1245 4.6265 4.16637 0 0 36 964 24 0.9166 1.0695 0.0014 0.0235
32 1089 5.9470 4.4354 4.16771 0 0 36 1029 24 0.9157 1.0841 0.0013 0.0179
33 1156 5.7751 4.4324 4.16893 0 0 36 1096 24 0.9181 1.0794 0.0013 0.0233
34 1225 5.6028 4.1922 4.17005 0 0 43 1151 31 0.9163 1.0842 0.0012 0.0157
35 1296 5.4289 4.1583 4.17108 0 0 39 1230 27 0.9192 1.0816 0.0012 0.0150
36 1369 5.3027 3.9395 4.17203 0 0 43 1295 31 0.9188 1.0837 0.0011 0.0145
37 1444 5.1563 3.9751 4.17290 0 0 50 1356 38 0.9140 1.0802 0.0011 0.0140
38 1521 5.0376 3.7878 4.17371 0 0 58 1417 46 0.9158 1.0816 0.0010 0.0133
39 1600 4.9024 3.7585 4.17445 0 0 52 1508 40 0.9146 1.0792 0.0010 0.0146
40 1681 4.7887 3.6278 4.17514 0 0 53 1587 41 0.9154 1.0826 0.0010 0.0758
41 1764 4.6607 3.5269 4.17579 0 0 50 1676 38 0.9156 1.0826 0.0009 0.0178
42 1849 4.5478 3.4742 4.17638 0 0 49 1763 37 0.9172 1.0820 0.0009 0.0173
43 1936 4.4560 3.3709 4.17694 0 0 50 1848 38 0.9164 1.0819 0.0009 0.0306
44 2025 4.3563 3.3222 4.17746 0 0 52 1933 40 0.9137 1.0810 0.0008 0.0111
45 2116 4.2569 3.1760 4.17795 0 0 59 2010 47 0.9140 1.0813 0.0008 0.0700
46 2209 4.1682 3.1660 4.17841 0 0 63 2095 51 0.9168 1.0811 0.0008 0.0116
47 2304 4.0691 3.1093 4.17884 0 0 58 2200 46 0.9161 1.0801 0.0008 0.0251
48 2401 3.9964 2.9854 4.17924 0 0 64 2285 52 0.9150 1.0814 0.0007 0.0128
49 2500 3.9128 2.9891 4.17962 0 0 66 2380 54 0.9158 1.0809 0.0007 0.0185
50 2601 3.8387 2.9181 4.17998 0 0 65 2483 53 0.9163 1.0799 0.0007 0.0108
51 2704 3.7572 2.8398 4.18031 0 0 63 2590 51 0.9161 1.0809 0.0007 0.4838
52 2809 3.6663 2.8018 4.18063 0 0 68 2685 56 0.9147 1.0837 0.0007 0.0920
53 2916 3.6252 2.7206 4.18093 0 0 68 2792 56 0.9148 1.0807 0.0006 0.0112
54 3025 3.5631 2.6868 4.18121 0 0 72 2893 60 0.9150 1.0808 0.0006 0.0129
55 3136 3.4922 2.6566 4.18148 0 0 70 3008 58 0.9168 1.0808 0.0006 0.0087
56 3249 3.4329 2.6201 4.18173 0 0 74 3113 62 0.9143 1.0851 0.0006 0.0692
57 3364 3.3556 2.5669 4.18198 0 0 76 3224 64 0.9153 1.0845 0.0006 0.0607
58 3481 3.3075 2.5197 4.18221 0 0 78 3337 66 0.9140 1.0802 0.0006 0.0225
59 3600 3.2418 2.4749 4.18242 0 0 79 3454 67 0.9150 1.0839 0.0005 0.0526
60 3721 3.2010 2.4625 4.18263 0 0 78 3577 66 0.9145 1.0827 0.0005 0.0225
61 3844 3.1301 2.4129 4.18283 0 0 83 3690 71 0.9143 1.0838 0.0005 0.0170
62 3969 3.0916 2.3915 4.18302 0 0 85 3811 73 0.9148 1.0796 0.0005 0.0102
63 4096 3.0617 2.3263 4.18320 0 0 89 3930 77 0.9148 1.0810 0.0005 0.0180
64 4225 2.9912 2.3006 4.18337 0 0 88 4061 76 0.9144 1.0828 0.0005 0.0104
65 4356 2.9568 2.2559 4.18353 0 0 89 4190 77 0.9157 1.0819 0.0005 0.2814
66 4489 2.8627 2.2394 4.18369 0 0 87 4327 75 0.9155 1.0819 0.0005 0.0137
67 4624 2.8731 2.1919 4.18383 0 0 92 4452 80 0.9132 1.0814 0.0005 0.0124
68 4761 2.8321 2.1496 4.18398 0 0 87 4599 75 0.9140 1.0822 0.0004 0.0162
69 4900 2.7906 2.1008 4.18411 0 0 98 4716 86 0.9144 1.0819 0.0004 0.0277
70 5041 2.7359 2.0935 4.18425 0 0 97 4859 85 0.9147 1.0835 0.0004 0.0284
71 5184 2.6729 2.0568 4.18437 0 0 106 4984 94 0.9146 1.0901 0.0004 0.0408
72 5329 2.6624 2.0414 4.18449 0 0 103 5135 91 0.9136 1.0838 0.0004 0.0412
73 5476 2.6306 2.0104 4.18461 0 0 107 5274 95 0.9128 1.0827 0.0004 0.0206
74 5625 2.5830 1.9984 4.18472 0 0 108 5421 96 0.9138 1.0847 0.0004 0.0102
75 5776 2.5679 1.9795 4.18482 0 0 105 5578 93 0.9142 1.0818 0.0004 0.0407
76 5929 2.5362 1.9062 4.18493 0 0 111 5719 99 0.9155 1.0829 0.0004 0.0166
77 6084 2.4794 1.9026 4.18503 0 0 113 5870 101 0.9152 1.0853 0.0004 0.0075
78 6241 2.4654 1.8770 4.18512 0 0 114 6025 102 0.9138 1.0828 0.0004 0.1069
79 6400 2.4250 1.8673 4.18521 0 0 119 6174 107 0.9133 1.0854 0.0004 0.0067
80 6561 2.4042 1.8516 4.18530 0 0 112 6349 100 0.9138 1.0835 0.0003 0.0223

Minimum energy systems -- Determinants and cubature weights
Deg. No. of points log det G Cubature weights
1-norms and condition number
n dn = (n+1)2 Lower bound Value Upper bound Infinity norm
of gradient
wmin
wavg
wmax
wavg
||G||1 ||G-1||1 cond1(G)
1 4 -6.9 -4.58 -4.6 8.9e-16 1.0000 1.0000 0.3 3.1e+00 1.0e+00
2 9 -10.0 -3.22 -3.0 8.8e-02 0.9803 1.0099 1.1 2.6e+00 2.8e+00
3 16 -9.8 3.23 3.9 4.1e-01 0.9622 1.1135 2.4 1.9e+00 4.5e+00
4 25 -5.3 15.81 17.2 9.7e-01 0.9113 1.0511 4.9 2.4e+00 1.2e+01
5 36 4.6 35.43 37.9 1.5e+00 0.9607 1.0440 7.7 2.7e+00 2.1e+01
6 49 20.5 63.31 66.7 1.5e+00 0.9289 1.0439 12.1 3.0e+00 3.7e+01
7 64 43.2 98.90 104.2 3.2e+00 0.9464 1.0428 16.9 5.5e+00 9.3e+01
8 81 73.1 143.21 150.9 2.7e+00 0.9253 1.0594 24.3 4.8e+00 1.2e+02
9 100 110.6 199.89 207.4 1.8e+00 0.9581 1.0435 29.3 1.8e+00 5.3e+01
10 121 156.4 241.22 274.0 1.3e+02 0.6555 1.4985 41.9 1.5e+04 6.1e+05
11 144 210.6 333.32 351.2 4.9e+00 0.9328 1.0347 51.4 1.8e+01 9.2e+02
12 169 273.7 395.96 439.2 2.1e+05 0.8945 1.0694 64.4 2.9e+10 1.9e+12
13 196 346.0 512.13 538.4 1.1e+01 0.9224 1.0318 79.1 4.1e+01 3.2e+03
14 225 427.8 619.22 649.1 8.0e+00 0.9122 1.0366 95.0 2.8e+01 2.7e+03
15 256 519.3 726.06 771.6 1.5e+01 0.9610 1.0295 110.3 5.8e+01 6.4e+03
16 289 620.9 832.12 906.1 1.5e+02 0.7420 1.3403 130.6 9.2e+03 1.2e+06
17 324 732.7 1006.60 1052.9 2.4e+01 0.9294 1.0170 149.6 1.7e+02 2.6e+04
18 361 855.0 1119.11 1212.2 6.3e+02 -0.5837 2.8619 172.7 4.6e+05 8.0e+07
19 400 988.1 1318.30 1384.2 1.7e+01 0.9476 1.0317 197.6 7.6e+01 1.5e+04
20 441 1132.1 1480.16 1569.1 7.7e+02 -2.1444 3.2736 221.6 1.4e+05 3.1e+07
21 484 1287.1 1692.91 1767.1 3.1e+01 0.7322 1.2175 250.3 1.2e+02 3.1e+04
22 529 1453.5 1848.12 1978.4 6.5e+03 -3.2046 5.9363 279.1 5.3e+06 1.5e+09
23 576 1631.3 2120.57 2203.2 1.3e+02 -0.2980 2.0730 306.5 4.0e+03 1.2e+06
24 625 1820.8 2283.57 2441.7 6.9e+02 0.6519 1.3935 346.9 5.1e+04 1.8e+07
25 676 2022.2 2509.25 2694.0 1.9e+03 0.0614 2.2492 383.3 1.8e+05 6.7e+07
26 729 2235.4 2750.71 2960.2 2.6e+04 -7.4925 11.2233 436.6 1.1e+08 4.6e+10
27 784 2460.8 3101.46 3240.6 4.1e+02 0.0936 1.7373 471.3 3.1e+04 1.5e+07
28 841 2698.5 3378.91 3535.2 1.5e+03 0.3652 1.4996 518.7 2.4e+05 1.2e+08
29 900 2948.6 3700.18 3844.2 2.8e+02 0.6120 1.5734 569.1 2.1e+04 1.2e+07
30 961 3211.2 4002.04 4167.8 2.6e+03 -2.6604 4.8742 609.0 1.9e+06 1.2e+09
31 1024 3486.4 4268.37 4506.1 9.1e+02 -0.9636 2.6498 661.5 3.5e+05 2.3e+08
32 1089 3774.5 4660.57 4859.1 4.9e+02 0.1750 1.7953 721.6 3.9e+04 2.8e+07
33 1156 4075.5 4991.27 5227.1 8.3e+02 -1.4129 2.9801 777.4 1.5e+05 1.2e+08
34 1225 4389.6 5359.12 5610.1 5.9e+02 0.5032 1.4410 884.7 8.0e+04 7.0e+07
35 1296 4716.8 5795.76 6008.3 8.6e+02 0.5913 1.3877 938.9 4.5e+04 4.2e+07
36 1369 5057.3 6179.77 6421.7 6.6e+02 0.4924 1.6289 1028.9 3.7e+04 3.8e+07
37 1444 5411.1 6570.77 6850.5 1.4e+03 0.2564 1.8012 1106.5 5.5e+04 6.1e+07
38 1521 5778.4 7012.34 7294.9 4.4e+03 0.5502 1.3587 1191.0 9.3e+05 1.1e+09
39 1600 6159.4 7476.98 7754.8 1.4e+03 -0.4349 2.6099 1235.2 1.8e+05 2.2e+08
40 1681 6554.0 7932.02 8230.4 1.5e+04 -12.2352 13.2884 1334.9 2.9e+07 3.9e+10
41 1764 6962.4 8346.47 8721.8 1.6e+04 -1.2302 3.8081 1425.3 3.2e+07 4.6e+10
42 1849 7384.7 8855.03 9229.1 3.2e+03 -1.1056 4.3469 1451.3 2.0e+06 2.8e+09
43 1936 7821.0 9387.50 9752.3 5.0e+03 -6.2001 6.9283 1534.1 4.9e+06 7.5e+09
44 2025 8271.4 9877.37 10291.7 6.0e+03 0.2852 1.7702 1707.6 1.3e+06 2.3e+09
45 2116 8735.9 10409.00 10847.2 1.0e+04 -15.1523 16.9427 1783.4 1.4e+07 2.5e+10
46 2209 9214.7 10917.92 11418.9 6.0e+03 -0.7020 2.4957 1867.1 6.7e+06 1.2e+10
47 2304 9707.8 11513.99 12007.0 2.3e+04 -5.3338 8.2631 1984.2 1.5e+07 2.9e+10
48 2401 10215.3 12153.12 12611.5 3.8e+03 -1.5003 3.0237 2017.1 6.7e+05 1.4e+09
49 2500 10737.4 12712.21 13232.6 1.0e+04 -3.0461 6.1436 2184.5 1.8e+07 3.9e+10
50 2601 11274.0 13282.56 13870.2 1.0e+04 -0.6082 3.0961 2285.2 8.7e+06 2.0e+10
51 2704 11825.3 13888.99 14524.4 1.9e+05 -151.2597 163.8320 2401.9 2.4e+09 5.7e+12
52 2809 12391.3 14540.17 15195.5 4.4e+04 -32.9395 33.1241 2460.2 2.7e+08 6.8e+11
53 2916 12972.2 15299.31 15883.3 6.6e+03 -1.8647 3.5592 2671.7 5.2e+06 1.4e+10
54 3025 13567.9 15959.43 16588.0 3.0e+04 -2.2288 4.4262 2757.1 1.7e+08 4.7e+11
55 3136 14178.7 16647.34 17309.7 5.3e+03 -0.7356 3.2890 2823.6 1.9e+06 5.4e+09
56 3249 14804.4 17380.97 18048.4 4.8e+05 -36.2336 33.2610 2934.5 4.1e+10 1.2e+14
57 3364 15445.3 18135.37 18804.3 5.6e+04 -22.2915 22.0547 3107.6 1.8e+08 5.5e+11
58 3481 16101.3 18842.45 19577.3 8.1e+04 -5.8154 7.2894 3246.0 1.2e+09 3.9e+12
59 3600 16772.6 19564.99 20367.6 3.4e+04 -25.8169 21.5153 3344.5 3.2e+07 1.1e+11
60 3721 17459.2 20329.79 21175.2 8.2e+04 -10.5095 11.9865 3512.4 2.5e+08 8.7e+11
61 3844 18161.2 21104.81 22000.2 2.3e+04 -6.7622 7.6425 3649.3 3.3e+07 1.2e+11
62 3969 18878.6 21919.77 22842.6 9.3e+04 -4.1852 4.5986 3833.1 7.4e+07 2.8e+11
63 4096 19611.6 22869.21 23702.5 1.5e+04 -7.4266 9.4118 3951.3 1.6e+07 6.4e+10
64 4225 20360.1 23693.86 24580.0 1.4e+04 -2.8640 5.0777 4160.2 1.1e+07 4.5e+10
65 4356 21124.2 24536.11 25475.1 4.9e+05 -160.5521 142.6096 4349.2 1.2e+10 5.2e+13
66 4489 21904.1 25421.53 26388.0 7.6e+04 -4.7071 8.3746 4549.7 1.1e+08 5.0e+11
67 4624 22699.7 26317.06 27318.6 4.3e+04 -7.0115 8.1176 4628.7 3.1e+07 1.4e+11
68 4761 23511.1 27142.06 28267.0 9.6e+04 -7.0709 12.5308 4802.6 3.1e+08 1.5e+12
69 4900 24338.4 28197.99 29233.2 7.3e+04 -12.4103 15.1928 4954.0 7.7e+07 3.8e+11
70 5041 25181.6 29130.63 30217.4 1.3e+05 -13.6762 16.2956 5116.5 2.5e+08 1.3e+12
71 5184 26040.8 30059.58 31219.6 4.1e+05 -54.5313 50.9244 5330.2 3.7e+11 2.0e+15
72 5329 26916.1 31000.89 32239.9 6.3e+04 -20.4756 29.9878 5491.0 3.7e+08 2.0e+12
73 5476 27807.5 32088.36 33278.2 4.8e+04 -15.2585 20.9113 5684.6 9.6e+07 5.4e+11
74 5625 28715.0 33066.09 34334.7 7.9e+04 -5.8653 6.6677 5937.5 4.2e+08 2.5e+12
75 5776 29638.7 34198.21 35409.4 9.5e+04 -26.5346 23.3370 6155.0 8.4e+07 5.2e+11
76 5929 30578.7 35264.77 36502.4 1.3e+04 -8.6041 10.4092 6309.2 2.6e+07 1.6e+11
77 6084 31535.0 36316.86 37613.7 1.8e+04 -2.4123 4.4925 6497.3 1.1e+07 7.0e+10
78 6241 32507.6 37353.13 38743.3 5.6e+05 -89.3507 94.7180 6703.0 4.4e+09 3.0e+13
79 6400 33496.7 38533.45 39891.4 2.7e+04 -2.4068 3.5775 6936.0 2.6e+07 1.8e+11
80 6561 34502.2 39605.72 41057.9 3.2e+04 -16.8310 17.3538 7135.9 4.8e+07 3.4e+11

Minimum energy 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 3.18e-01 0.3183 0.3183 1.00e+00 2.00e+00 2.00e+00 1.00e+00 0.00e+00
2 9 4.39e-01 0.7162 0.8952 2.04e+00 3.19e+00 3.83e+00 1.17e+00 5.76e-04
3 16 7.08e-01 1.2732 1.8088 2.56e+00 3.67e+00 5.36e+00 1.16e+00 6.10e-03
4 25 6.29e-01 1.9894 3.1551 5.02e+00 5.60e+00 8.90e+00 1.35e+00 2.64e-02
5 36 4.63e-01 2.8648 5.1295 1.11e+01 7.59e+00 1.49e+01 1.47e+00 3.24e-02
6 49 5.45e-01 3.8993 6.9616 1.28e+01 7.39e+00 1.87e+01 1.40e+00 2.35e-02
7 64 2.51e-01 5.0930 9.8536 3.92e+01 1.25e+01 3.60e+01 1.81e+00 7.56e-02
8 81 3.71e-01 6.4458 12.4273 3.35e+01 1.29e+01 3.75e+01 1.76e+00 6.34e-02
9 100 1.04e+00 7.9577 14.6452 1.41e+01 9.93e+00 2.76e+01 1.36e+00 1.78e-02
10 121 1.40e-04 9.6289 19.5887 1.40e+05 6.14e+02 2.88e+03 7.34e+01 1.72e+01
11 144 8.48e-02 11.4592 22.8128 2.69e+02 3.48e+01 1.39e+02 3.38e+00 9.59e-02
12 169 5.96e-11 13.4486 26.9825 4.52e+11 1.30e+06 6.17e+06 1.12e+05 1.69e+04
13 196 5.29e-02 15.5972 31.1141 5.88e+02 5.14e+01 2.40e+02 4.36e+00 2.42e-01
14 225 7.82e-02 17.9049 35.7755 4.57e+02 4.61e+01 2.27e+02 3.77e+00 1.58e-01
15 256 4.14e-02 20.3718 40.7973 9.86e+02 6.25e+01 3.55e+02 4.79e+00 3.17e-01
16 289 2.35e-04 22.9979 46.0211 1.96e+05 1.13e+03 5.31e+03 8.07e+01 1.21e+01
17 324 1.16e-02 25.7831 51.6087 4.45e+03 1.55e+02 8.49e+02 1.05e+01 5.57e-01
18 361 4.89e-06 28.7275 57.4754 1.18e+07 7.46e+03 4.61e+04 4.84e+02 7.16e+01
19 400 3.27e-02 31.8310 63.6782 1.95e+03 9.50e+01 6.24e+02 6.03e+00 2.94e-01
20 441 2.41e-05 35.0937 70.1997 2.91e+06 4.83e+03 2.53e+04 3.01e+02 6.26e+01
21 484 2.17e-02 38.5155 76.9654 3.54e+03 1.65e+02 9.26e+02 9.67e+00 7.85e-01
22 529 4.86e-07 42.0965 84.3312 1.73e+08 3.68e+04 2.14e+05 2.08e+03 3.02e+02
23 576 5.48e-04 45.8366 91.6337 1.67e+05 1.01e+03 6.94e+03 5.10e+01 4.81e+00
24 625 4.24e-05 49.7359 99.5539 2.35e+06 4.92e+03 2.71e+04 2.40e+02 2.58e+01
25 676 1.54e-05 53.7944 107.5482 6.96e+06 6.34e+03 4.85e+04 3.12e+02 4.34e+01
26 729 2.90e-08 58.0120 116.1891 4.01e+09 1.58e+05 1.21e+06 7.64e+03 1.13e+03
27 784 9.74e-05 62.3887 124.8114 1.28e+06 4.08e+03 2.24e+04 1.91e+02 2.54e+01
28 841 1.14e-05 66.9247 133.9237 1.17e+07 8.80e+03 7.02e+04 4.26e+02 5.49e+01
29 900 1.25e-04 71.6197 143.2660 1.15e+06 2.82e+03 2.27e+04 1.22e+02 1.22e+01
30 961 1.54e-06 76.4740 152.9658 9.94e+07 2.79e+04 2.19e+05 1.11e+03 1.29e+02
31 1024 7.69e-06 81.4873 162.9908 2.12e+07 1.54e+04 1.04e+05 6.11e+02 6.18e+01
32 1089 7.14e-05 86.6599 173.3315 2.43e+06 4.76e+03 3.63e+04 1.85e+02 1.82e+01
33 1156 1.75e-05 91.9916 183.9666 1.05e+07 9.37e+03 7.80e+04 3.52e+02 3.31e+01
34 1225 3.69e-05 97.4824 195.1407 5.29e+06 6.42e+03 5.69e+04 2.46e+02 2.45e+01
35 1296 5.18e-05 103.1324 206.2731 3.98e+06 7.18e+03 5.08e+04 2.62e+02 2.44e+01
36 1369 6.59e-05 108.9416 217.9261 3.30e+06 5.55e+03 4.76e+04 1.90e+02 1.58e+01
37 1444 4.43e-05 114.9099 230.0219 5.19e+06 7.33e+03 6.12e+04 2.77e+02 2.78e+01
38 1521 2.76e-06 121.0373 242.0488 8.76e+07 3.47e+04 2.58e+05 1.16e+03 9.92e+01
39 1600 1.51e-05 127.3240 254.7923 1.69e+07 1.50e+04 1.16e+05 4.80e+02 3.97e+01
40 1681 8.52e-08 133.7697 267.7668 3.14e+09 1.88e+05 1.62e+06 5.64e+03 4.19e+02
41 1764 8.59e-08 140.3747 281.0060 3.27e+09 1.69e+05 1.70e+06 5.12e+03 4.28e+02
42 1849 1.52e-06 147.1387 294.4799 1.94e+08 4.18e+04 4.23e+05 1.45e+03 1.51e+02
43 1936 6.32e-07 154.0620 308.3720 4.88e+08 8.06e+04 6.87e+05 2.37e+03 2.10e+02
44 2025 2.15e-06 161.1444 322.3437 1.50e+08 4.16e+04 3.90e+05 1.23e+03 1.02e+02
45 2116 2.01e-07 168.3859 337.0583 1.68e+09 1.24e+05 1.33e+06 3.41e+03 2.58e+02
46 2209 4.11e-07 175.7866 351.8381 8.57e+08 1.05e+05 9.72e+05 2.78e+03 1.84e+02
47 2304 2.10e-07 183.3465 366.8872 1.75e+09 1.94e+05 1.42e+06 5.51e+03 4.71e+02
48 2401 4.58e-06 191.0655 382.4351 8.35e+07 3.36e+04 3.16e+05 8.83e+02 7.43e+01
49 2500 1.61e-07 198.9437 398.2200 2.47e+09 2.24e+05 1.76e+06 5.72e+03 4.00e+02
50 2601 3.60e-07 206.9810 414.2664 1.15e+09 1.13e+05 1.22e+06 2.81e+03 2.38e+02
51 2704 1.32e-09 215.1775 430.6426 3.27e+11 1.67e+06 2.10e+07 4.23e+04 NaN
52 2809 1.12e-08 223.5331 447.5840 4.00e+10 6.85e+05 7.49e+06 1.88e+04 NaN
53 2916 7.07e-07 232.0479 464.5131 6.57e+08 1.17e+05 9.78e+05 3.09e+03 NaN
54 3025 1.72e-08 240.7219 481.8113 2.80e+10 5.26e+05 6.51e+06 1.29e+04 NaN
55 3136 1.44e-06 249.5550 499.5599 3.48e+08 6.13e+04 7.38e+05 1.48e+03 NaN
56 3249 9.31e-11 258.5472 517.7701 5.56e+12 8.02e+06 9.50e+07 2.01e+05 NaN
57 3364 2.07e-08 267.6986 535.9924 2.59e+10 6.28e+05 6.59e+06 1.47e+04 NaN
58 3481 2.74e-09 277.0092 554.7248 2.02e+11 1.84e+06 1.88e+07 4.02e+04 NaN
59 3600 1.24e-07 286.4789 573.9159 4.62e+09 2.93e+05 2.88e+06 6.65e+03 NaN
60 3721 1.48e-08 296.1078 592.9162 4.02e+10 8.93e+05 8.64e+06 2.02e+04 NaN
61 3844 1.08e-07 305.8958 612.3406 5.69e+09 3.23e+05 3.31e+06 7.06e+03 NaN
62 3969 4.38e-08 315.8430 632.4677 1.44e+10 5.60e+05 5.35e+06 1.25e+04 NaN
63 4096 2.04e-07 325.9493 652.4762 3.20e+09 2.68e+05 2.56e+06 5.85e+03 NaN
64 4225 2.46e-07 336.2148 673.2525 2.73e+09 2.89e+05 2.40e+06 6.03e+03 NaN
65 4356 2.89e-10 346.6395 693.9861 2.40e+12 5.33e+06 7.22e+07 1.13e+05 NaN
66 4489 3.57e-08 357.2233 715.2489 2.01e+10 6.14e+05 6.71e+06 1.27e+04 NaN
67 4624 1.28e-07 367.9662 736.8191 5.75e+09 3.67e+05 3.64e+06 7.89e+03 NaN
68 4761 1.05e-08 378.8683 758.6132 7.21e+10 1.08e+06 1.31e+07 2.16e+04 NaN
69 4900 4.15e-08 389.9296 780.7191 1.88e+10 7.12e+05 6.78e+06 1.35e+04 NaN
70 5041 1.10e-08 401.1500 803.2813 7.33e+10 1.42e+06 1.36e+07 2.77e+04 NaN
71 5184 7.23e-12 412.5296 826.1952 1.14e+14 5.08e+06 5.44e+08 1.03e+05 NaN
72 5329 1.07e-08 424.0683 849.4022 7.93e+10 1.35e+06 1.45e+07 2.52e+04 NaN
73 5476 5.00e-08 435.7662 872.7025 1.75e+10 4.26e+05 6.91e+06 8.52e+03 NaN
74 5625 9.32e-09 447.6233 896.7092 9.63e+10 1.49e+06 1.64e+07 2.98e+04 NaN
75 5776 4.05e-08 459.6395 920.3514 2.27e+10 8.92e+05 8.09e+06 1.62e+04 NaN
76 5929 1.16e-07 471.8148 945.3499 8.13e+09 3.51e+05 4.91e+06 6.13e+03 NaN
77 6084 3.30e-07 484.1493 969.4340 2.94e+09 2.51e+05 2.99e+06 4.59e+03 NaN
78 6241 8.96e-10 496.6430 994.6321 1.11e+12 3.83e+06 5.88e+07 7.09e+04 NaN
79 6400 1.33e-07 509.2958 1019.9393 7.65e+09 4.04e+05 4.94e+06 7.16e+03 NaN
80 6561 8.50e-08 522.1078 1045.6178 1.23e+10 5.07e+05 6.35e+06 9.24e+03 NaN

References

Acknowlegements

The use of the high performance computing facilities of is gratefully acknowledged. Last updated: