# Maximum Determinant (MD)  Interpolation points on the Sphere S2

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Caveat: All point sets are only approximate local optimizers of their respective criteria.

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• Maximum determinant (MD) points, Degree n = 1, Number of points dn = 4
•  MD points n = 1, dn = 4 Mesh norm function, h = 1.231 Interpolation norm function, ||Ln|| = 2 Eigenvalues of G lmin = 0.318, lavg = 0.318, lmax = 0.318
• Maximum determinant (MD) points, Degree n = 2, Number of points dn = 9
•  MD points,  n = 2, dn = 9 Mesh norm function, h = 0.818 Interpolation norm function ||Ln|| = 3.24 Eigenvalues of G lmin = 0.463, lavg = 0.716, lmax = 0.895
• Maximum determinant (MD) points, Degree n = 3, Number of points dn = 16
•  MD points, n = 3, dn = 16 Mesh norm function, h = 0.575 Interpolation norm function, ||Ln|| = 3.60 Eigenvalues of G lmin = 0.797, lavg = 1.273, lmax = 1.915
• Maximum determinant (MD) points, Degree n = 4, Number of points dn = 25
•  MD points n = 4, dn = 25 Mesh norm function h = 0.518 Interpolation norm function ||Ln|| = 5.22 Eigenvalues of G lmin = 0.897, lavg = 1.989, lmax = 2.866

Maximum determinant (MD) points, Degree n = 5, Number of points dn = 36
 MD points n = 5, dn = 36 Mesh norm function h = 0.439 Interpolation norm function ||Ln|| = 6.20 Eigenvalues of G lmin = 1.062, lavg = 2.865, lmax = 4.346

• Maximum determinant (MD) points, Degree n = 6, Number of points dn = 49
•  MD points n = 6, dn = 49 Mesh norm function h = 0.345 Interpolation norm function ||Ln|| = 7.16 Eigenvalues of G lmin = 1.034, lavg = 3.899, lmax = 6.162
• Maximum determinant (MD) points, Degree n = 7, Number of points dn = 64
•  MD points n = 7, dn = 64 Mesh norm function h = 0.308 Interpolation norm function ||Ln|| = 8.19 Eigenvalues of G lmin = 1.353, lavg = 5.093, lmax = 8.388
• Maximum determinant (MD) points, Degree n = 8, Number of points dn = 81
•  MD points n = 8, dn = 81 Mesh norm function h = 0.280 Interpolation norm function ||Ln|| = 8.86 Eigenvalues of G lmin = 1.538, lavg = 6.446, lmax = 10.812
• Maximum determinant (MD) points, Degree n = 9, Number of points dn = 100
•  MD points n = 9, dn = 100 Mesh norm function h = 0.236 Interpolation norm function ||Ln|| = 8.61 Eigenvalues of G lmin = 1.788, lavg = 7.958, lmax = 13.593
• Maximum determinant (MD) points, Degree n = 10, Number of points dn = 121
•  MD points n = 10, dn = 121 Mesh norm function h = 0.234 Interpolation norm function ||Ln|| = 11.84 Eigenvalues of G lmin = 1.958, lavg = 9.629, lmax = 16.425
• Maximum determinant (MD) points, Degree n = 11, Number of points dn = 144
•  MD points n = 11, dn = 144 Mesh norm function h = 0.224 Interpolation norm function ||Ln|| = 14.05 Eigenvalues of G lmin = 1.733, lavg = 11.459, lmax = 20.087
• Maximum determinant (MD) points, Degree n = 12, Number of points dn = 169
•  MD points n = 12, dn = 169 Mesh norm function h = 0.210 Interpolation norm function ||Ln|| = 14.70 Eigenvalues of G lmin = 2.327, lavg = 13.449, lmax = 23.255
• Maximum determinant (MD) points, Degree n = 13, Number of points dn = 196
•  MD points n = 13, dn = 196 Mesh norm function h = 0.181 Interpolation norm function ||Ln|| = 15.16 Eigenvalues of G lmin = 2.045, lavg = 15.597, lmax = 27.608
• Maximum determinant (MD) points, Degree n = 14, Number of points dn = 225
•  MD points n = 14, dn = 225 Mesh norm function h = 0.176 Interpolation norm function ||Ln|| = 17.81 Eigenvalues of G lmin = 2.288, lavg = 17.905, lmax = 32.119
• Maximum determinant (MD) points, Degree n = 15, Number of points dn = 256
•  MD points n = 15, dn = 256 Mesh norm function h = 0.161 Interpolation norm function ||Ln|| = 18.08 Eigenvalues of G lmin = 2.321, lavg = 20.372, lmax = 36.687
• Maximum determinant (MD) points, Degree n = 16, Number of points dn = 289
•  MD points n = 16, dn = 289 Mesh norm function h = 0.157 Interpolation norm function ||Ln|| = 19.92 Eigenvalues of G lmin = 2.724, lavg = 22.998, lmax = 40.970
• Maximum determinant (MD) points, Degree n = 17, Number of points dn = 324
•  MD points n = 17, dn = 324 Mesh norm function h = 0.147 Interpolation norm function ||Ln|| = 21.55 Eigenvalues of G lmin = 2.562, lavg = 25.783, lmax = 45.959
• Maximum determinant (MD) points, Degree n = 18, Number of points dn = 361
•  MD points n = 18, dn = 361 Mesh norm function h =0.140 Interpolation norm function ||Ln|| = 21.08 Eigenvalues of G lmin = 3.074, lavg = 28.727, lmax = 52.078
• Maximum determinant (MD) points, Degree n = 19, Number of points dn = 400
•  MD points n = 19, dn = 400 Mesh norm function h = 0.132 Interpolation norm function ||Ln|| = 23.64 Eigenvalues of G lmin = 2.973, lavg = 31.831, lmax = 58.300
• Maximum determinant (MD) points, Degree n = 20, Number of points dn = 441
•  MD points n = 20, dn = 441 Mesh norm function h = 0.125 Interpolation norm function ||Ln|| = 24.66 Eigenvalues of G lmin = 2.924, lavg = 35.094, lmax = 63.669
• Maximum determinant (MD) points, Degree n = 21, Number of points dn = 484
•  MD points n = 21, dn = 484 Mesh norm function h = 0.120 Interpolation norm function ||Ln|| = 23.83 Eigenvalues of G lmin = 3.597, lavg =  38.515, lmax = 70.130
• Maximum determinant (MD) points, Degree n = 22, Number of points dn = 529
•  MD points n = 22, dn = 529 Mesh norm function h = 0.114 Interpolation norm function ||Ln|| = 25.25 Eigenvalues of G lmin = 3.460, lavg = 42.096, lmax = 76.522
• Maximum determinant (MD) points, Degree n = 23, Number of points dn = 576
•  MD points n = 23, dn = 576 Mesh norm function h = 0.112 Interpolation norm function ||Ln|| = 28.70 Eigenvalues of G lmin = 3.211, lavg = 45.837, lmax = 83.703
• Maximum determinant (MD) points, Degree n = 24, Number of points dn = 625
•  MD points n = 24, dn = 625 Mesh norm function h = 0.103 Interpolation norm function ||Ln|| = 28.41 Eigenvalues of G lmin = 3.324, lavg = 49.736, lmax = 91.704
• Maximum determinant (MD) points, Degree n = 25, Number of points dn = 676
•  MD points n = 25, dn = 676 Mesh norm function h = 0.101 Interpolation norm function ||Ln|| = 31.15 Eigenvalues of G lmin = 3.631, lavg = 53.794 , lmax = 98.977
• Maximum determinant (MD) points, Degree n = 26, Number of points dn = 729
•  MD points n = 26, dn = 729 Mesh norm function h = 0.095 Interpolation norm function ||Ln|| = 31.21 Eigenvalues of G lmin = 3.430, lavg = 58.012, lmax = 106.770
• Maximum determinant (MD) points, Degree n = 27, Number of points dn = 784
•  MD points n = 27, dn = 784 Mesh norm function h = 0.095 Interpolation norm function ||Ln|| = 32.31 Eigenvalues of G lmin = 4.007, lavg = 62.389, lmax = 114.840
• Maximum determinant (MD) points, Degree n = 28, Number of points dn = 841
•  MD points n = 28, dn = 841 Mesh norm function h = 0.092 Interpolation norm function ||Ln|| = 33.92 Eigenvalues of G lmin = 3.283, lavg = 66.925, lmax = 125.928
• Maximum determinant (MD) points, Degree n = 29, Number of points dn = 900
•  MD points n = 29, dn = 900 Mesh norm function h = 0.089 Interpolation norm function ||Ln|| = 36.98 Eigenvalues of G lmin = 4.033, lavg = 71.620, lmax = 132.464

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