# Minimum Energy (ME) Interpolation points on the Sphere S2

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Summary of data for Minimum Energy (ME) points.

 n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

• Minimum energy (ME) points, Degree n = 1, Number of points dn = 4
•  ME points n = 1, dn = 4 Mesh norm function, h = 1.2 Interpolation norm function, ||Ln|| = 2.00 Eigenvalues of G lmin = 0.318, lavg = 0.318, lmax = 0.318
• Minimum energy (ME) points, Degree n = 2, Number of points dn = 9
•  ME points,  n = 2, dn = 9 Mesh norm function, h = 0.806 Interpolation norm function ||Ln|| = 3.19 Eigenvalues of G lmin = 0.439, lavg = 0.716, lmax = 0.895
• Minimum energy (ME) points, Degree n = 3, Number of points dn = 16
•  ME points, n = 3, dn = 16 Mesh norm function, h = 0.580 Interpolation norm function, ||Ln|| = 3.67 Eigenvalues of G lmin = 0.708, lavg = 1.273, lmax = 1.809
• Minimum energy (ME) points, Degree n = 4, Number of points dn = 25
•  ME points n = 4, dn = 25 Mesh norm function h = 0.520 Interpolation norm function ||Ln|| = 5.60 Eigenvalues of G lmin = 0.629, lavg = 1.989, lmax = 3.155

Minimum energy (ME) points, Degree n = 5, Number of points dn = 36
 ME points n = 5, dn = 36 Mesh norm function h = 0.412 Interpolation norm function ||Ln|| = 7.59 Eigenvalues of G lmin = 0.463, lavg = 2.865, lmax = 5.129

• Minimum energy (ME) points, Degree n = 6, Number of points dn = 49
•  ME points n = 6, dn = 49 Mesh norm function h = 0.349 Interpolation norm function ||Ln|| = 7.39 Eigenvalues of G lmin = 0.545, lavg = 3.899, lmax = 6.962
• Minimum energy (ME) points, Degree n = 7, Number of points dn = 64
•  ME points n = 7, dn = 64 Mesh norm function h = 0.304 Interpolation norm function ||Ln|| = 12.54 Eigenvalues of G lmin = 0.251, lavg = 5.093, lmax = 9.854
• Minimum energy (ME) points, Degree n = 8, Number of points dn = 81
•  ME points n = 8, dn = 81 Mesh norm function h = 0.272 Interpolation norm function ||Ln|| = 12.92 Eigenvalues of G lmin = 0.371, lavg = 6.446, lmax = 12.43
• Minimum energy (ME) points, Degree n = 9, Number of points dn = 100
•  ME points n = 9, dn = 100 Mesh norm function h = 0.244 Interpolation norm function ||Ln|| = 9.96 Eigenvalues of G lmin = 1.042, lavg = 7.958, lmax = 14.65
• Minimum energy (ME) points, Degree n = 10, Number of points dn = 121
•  ME points n = 10, dn = 121 Mesh norm function h = 0.340 Interpolation norm function ||Ln|| = 161.8 Eigenvalues of G lmin = 0.0026, lavg = 9.629, lmax = 19.31
• Minimum energy (ME) points, Degree n = 11, Number of points dn = 144
•  ME points n = 11, dn = 144 Mesh norm function h = 0.208 Interpolation norm function ||Ln|| = 373.9 Eigenvalues of G lmin = 0.0006, lavg = 11.46, lmax = 22.86
• Minimum energy (ME) points, Degree n = 12, Number of points dn = 169
•  ME points n = 12, dn = 169 Mesh norm function h = 0.183. Interpolation norm function ||Ln|| = 1504.8 Eigenvalues of G lmin = 0.00005, lavg = 13.45, lmax = 26.86
• Minimum energy (ME) points, Degree n = 13, Number of points dn = 196
•  ME points n = 13, dn = 196 Mesh norm function h = 0.172 Interpolation norm function ||Ln|| = 422.1 Eigenvalues of G lmin = 0.0007, lavg = 15.60, lmax = 31.26
• Minimum energy (ME) points, Degree n = 14, Number of points dn = 225
•  ME points n = 14, dn = 225 Mesh norm function h = 0.164 Interpolation norm function ||Ln|| = 172.6 Eigenvalues of G lmin = 0.0062, lavg = 17.90, lmax = 35.79
• Minimum energy (ME) points, Degree n = 15, Number of points dn = 256
•  ME points n = 15, dn = 256 Mesh norm function h = 0.156 Interpolation norm function ||Ln|| = 267.1 Eigenvalues of G lmin = 0.0035, lavg = 20.37, lmax = 40.73
• Minimum energy (ME) points, Degree n = 16, Number of points dn = 289
•  ME points n = 16, dn = 289 Mesh norm function h = 0.144 Interpolation norm function ||Ln|| = 162.1 Eigenvalues of G lmin = 0.0099, lavg = 23.00, lmax = 45.86
• Minimum energy (ME) points, Degree n = 17, Number of points dn = 324
•  ME points n = 17, dn = 324 Mesh norm function h = 0.142 Interpolation norm function ||Ln|| = 1968.0 Eigenvalues of G lmin = 0.00007, lavg = 25.78, lmax = 51.57
• Minimum energy (ME) points, Degree n = 18, Number of points dn = 361
•  ME points n = 18, dn = 361 Mesh norm function h = 0.138 Interpolation norm function ||Ln|| = 2101.6 Eigenvalues of G lmin = 0.0004, lavg = 28.73, lmax = 57.38
• Minimum energy (ME) points, Degree n = 19, Number of points dn = 400
•  ME points n = 19, dn = 400 Mesh norm function h = 0.126 Interpolation norm function ||Ln|| = 1138.4 Eigenvalues of G lmin = 0.0004, lavg = 31.83, lmax = 63.67
• Minimum energy (ME) points, Degree n = 20, Number of points dn = 441
•  ME points n = 20, dn = 441 Mesh norm function h = 0.119 Interpolation norm function ||Ln|| = 29565.2 Eigenvalues of G lmin = 0.000001, lavg = 35.09, lmax = 70.22
• Minimum energy (ME) points, Degree n = 21, Number of points dn = 484
•  ME points n = 21, dn = 484 Mesh norm function h = 0.115 Interpolation norm function ||Ln|| = 793.1 Eigenvalues of G lmin = 0.00094, lavg = 38.52, lmax = 77.10
• Minimum energy (ME) points, Degree n = 22, Number of points dn = 529
•  ME points n = 22, dn = 529 Mesh norm function h = 0.108 Interpolation norm function ||Ln|| = 1569.3 Eigenvalues of G lmin = 0.00023, lavg = 42.10, lmax = 84.13
• Minimum energy (ME) points, Degree n = 23, Number of points dn = 576
•  ME points n = 23, dn = 576 Mesh norm function h = 0.104 Interpolation norm function ||Ln|| = 1369.5 Eigenvalues of G lmin = 0.00053, lavg = 45.84, lmax = 91.78
• Minimum energy (ME) points, Degree n = 24, Number of points dn = 625
•  ME points n = 24, dn = 625 Mesh norm function h = 0.103 Interpolation norm function ||Ln|| = 5106.1 Eigenvalues of G lmin = 0.000032, lavg = 49.74, lmax = 99.52
• Minimum energy (ME) points, Degree n = 25, Number of points dn = 676
•  ME points n = 25, dn = 676 Mesh norm function h = 0.097 Interpolation norm function ||Ln|| = 2109.8 Eigenvalues of G lmin = 0.00024, lavg = 53.79, lmax = 107.8
• Minimum energy (ME) points, Degree n = 26, Number of points dn = 729
•  ME points n = 26, dn = 729 Mesh norm function h = 0.094 Interpolation norm function ||Ln|| = 9615.7 Eigenvalues of G lmin = 0.000015, lavg = 58.01, lmax = 116.2
• Minimum energy (ME) points, Degree n = 27, Number of points dn = 784
•  ME points n = 27, dn = 784 Mesh norm function h = 0.090 Interpolation norm function ||Ln|| = 20224.5 Eigenvalues of G lmin = 0.000003, lavg = 62.39, lmax = 124.8
• Minimum energy (ME) points, Degree n = 28, Number of points dn = 841
•  ME points n = 28, dn = 841 Mesh norm function h = 0.085 Interpolation norm function ||Ln|| = 878.4 Eigenvalues of G lmin = 0.0020, lavg = 66.93, lmax = 133.9
• Minimum energy (ME) points, Degree n = 29, Number of points dn = 900
•  ME points n = 29, dn = 900 Mesh norm function h = 0.084 Interpolation norm function ||Ln|| = 2127.4 Eigenvalues of G lmin = 0.00026, lavg = 71.62, lmax = 143.4

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Rob Womersley, School of Mathematics, UNSW. Last updated 3-Nov-1999