Minimum Norm (MN)
Interpolation points on the Sphere S2

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Summary of data for Minimum Norm (MN) 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 norm (MN) points, Degree n = 1, Number of points dn = 4
  • mn01_0004pts.jpg
    mn01_0004msh.jpg mn01_0004int.jpg
    MN 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
  • Minimum norm (MN) points, Degree n = 2, Number of points dn = 9
  • mn02_0009pts.jpg
    mn02_0009msh.jpg mn02_0009int.jpg
    MN points, 
    n = 2, dn = 9
    Mesh norm function,
    h = 0.848
    Interpolation norm function
    ||Ln|| = 3.06
    Eigenvalues of G
    lmin = 0.387, lavg = 0.716, lmax = 0.978
  • Minimum norm (MN) points, Degree n = 3, Number of points dn = 16
  • mn03_0016pts.jpg
    mn03_0016msh.jpg mn03_0016int.jpg
    MN points,
    n = 3, dn = 16
    Mesh norm function,
    h = 0.595
    Interpolation norm function,
    ||Ln|| = 3.56
    Eigenvalues of G
    lmin = 0.825, lavg = 1.273, lmax = 2.278
  • Minimum norm (MN) points, Degree n = 4, Number of points dn = 25
  • mn04_0025pts.jpg
    mn04_0025msh.jpg
    mn04_0025int.jpg
    MN points
    n = 4, dn = 25
    Mesh norm function
    h = 0.500
    Interpolation norm function
    ||Ln|| = 4.53
     Eigenvalues of G
    lmin = 0.856, lavg = 1.989, lmax = 3.293
  • Minimum norm (MN) points, Degree n = 5, Number of points dn = 36
  • mn05_0036pts.jpg
    mn05_0036msh.jpg
    mn05_0036int.jpg
    mn05_0036ev.jpg
    MN points
    n = 5, dn = 36
    Mesh norm function
    h = 0.417
    Interpolation norm function
    ||Ln|| = 5.23
     Eigenvalues of G
    lmin = 1.175, lavg = 2.865, lmax = 4.704
  • Minimum norm (MN) points, Degree n = 6, Number of points dn = 49
  • mn06_0049pts.jpg
    mn06_0049msh.jpg
    mn06_0049int.jpg
    MN points
    n = 6, dn = 49
    Mesh norm function
    h = 0.366
    Interpolation norm function
    ||Ln|| = 6.13
     Eigenvalues of G
    lmin = 1.453, lavg = 3.899, lmax = 6.825
  • Minimum norm (MN) points, Degree n = 7, Number of points dn = 64
  • mn07_0064pts.jpg
    mn07_0064msh.jpg
    mn07_0064int.jpg
    MN points
    n = 7, dn = 64
    Mesh norm function
    h = 0.305
    Interpolation norm function
    ||Ln|| = 6.59
     Eigenvalues of G
    lmin = 1.610, lavg = 5.093, lmax = 8.734
  • Minimum norm (MN) points, Degree n = 8, Number of points dn = 81
  • mn08_0081pts0.jpg mn08_0081msh.jpg mn08_0081int.jpg
    MN points
    n = 8, dn = 81
    Mesh norm function
    h = 0.285
    Interpolation norm function
    ||Ln|| = 7.45
     Eigenvalues of G
    lmin = 1.869, lavg = 6.446, lmax = 11.97
  • Minimum norm (MN) points, Degree n = 9, Number of points dn = 100
  • mn09_0100pts.jpg mn09_0100msh.jpg mn09_0100int.jpg
    MN points
    n = 9, dn = 100
    Mesh norm function
    h =0.253
    Interpolation norm function
    ||Ln|| = 8.18
     Eigenvalues of G
    lmin = 2.112, lavg = 7.958, lmax = 14.66
  • Minimum norm (MN) points, Degree n = 10, Number of points dn = 121
  • mn10_0121pts.jpg mn10_0121msh.jpg mn10_0121int.jpg mn10_0121ev.jpg
    MN points
    n = 10, dn = 121
    Mesh norm function
    h =0.231
    Interpolation norm function
    ||Ln|| = 8.85
     Eigenvalues of G
    lmin = 2.367, lavg = 9.629, lmax = 16.73
  • Minimum norm (MN) points, Degree n = 11, Number of points dn = 144
  • mn11_0144pts.jpg mn11_0144msh.jpg mn11_0144int.jpg mn11_0144ev.jpg
    MN points
    n = 11, dn = 144
    Mesh norm function
    h = 0.218
    Interpolation norm function
    ||Ln|| = 9.61
     Eigenvalues of G
    lmin = 2.768, lavg = 11.46, lmax = 20.79
  • Minimum norm (MN) points, Degree n = 12, Number of points dn = 169
  • mn12_0169pts.jpg mn12_0169msh.jpg mn12_0169int.jpg mn12_0169ev.jpg
    MN points
    n = 12, dn = 169
    Mesh norm function
    h = 0.204
    Interpolation norm function
    ||Ln|| = 10.28
     Eigenvalues of G
    lmin = 3.165, lavg = 13.45, lmax = 24.55
  • Minimum norm (MN) points, Degree n = 13, Number of points dn = 196
  • mn13_0196pts.jpg mn13_0196msh.jpg mn13_0196int.jpg mn13_0196ev.jpg
    MN points
    n = 13, dn = 196
    Mesh norm function
    h = 0.187
    Interpolation norm function
    ||Ln|| = 11.03
     Eigenvalues of G
    lmin = 3.261, lavg = 15.60, lmax = 28.37
  • Minimum norm (MN) points, Degree n = 14, Number of points dn = 225
  • mn14_0225pts.jpg mn14_0225msh.jpg mn14_0225int.jpg mn14_0225ev.jpg
    MN points
    n = 14, dn = 225
    Mesh norm function
    h = 0.178
    Interpolation norm function
    ||Ln|| = 11.50
     Eigenvalues of G
    lmin = 3.758, lavg = 17.90, lmax = 36.12
  • Minimum norm (MN) points, Degree n = 15, Number of points dn = 256
  • mn15_0256pts.jpg mn15_0256msh.jpg mn15_0256int.jpg mn15_0256ev.jpg
    MN points
    n = 15, dn = 256
    Mesh norm function
    h = 0.171
    Interpolation norm function
    ||Ln|| = 12.35
     Eigenvalues of G
    lmin = 3.995, lavg = 20.37, lmax = 38.23
  • Minimum norm (MN) points, Degree n = 16, Number of points dn = 289
  • mn16_0289pts.jpg mn16_0289msh.jpg mn16_0289int.jpg mn16_0289ev.jpg
    MN points
    n = 16, dn = 289
    Mesh norm function
    h = 0.162
    Interpolation norm function
    ||Ln|| = 13.04
     Eigenvalues of G
    lmin = 4.340, lavg = 23.000, lmax = 43.55
  • Minimum norm (MN) points, Degree n = 17, Number of points dn = 324
  • mn17_0324pts.jpg mn17_0324msh.jpg mn17_0324int.jpg mn17_0324ev.jpg
    MN points
    n = 17, dn = 324
    Mesh norm function
    h = 0.153
    Interpolation norm function
    ||Ln|| = 13.74
     Eigenvalues of G
    lmin = 4.956, lavg = 25.78, lmax = 48.43
  • Minimum norm (MN) points, Degree n = 18, Number of points dn = 361
  • mn18_0361pts.jpg mn18_0361msh.jpg mn18_0361int.jpg mn18_0361ev.jpg
    MN points
    n = 18, dn = 361
    Mesh norm function
    h = 0.146
    Interpolation norm function
    ||Ln|| = 14.34
     Eigenvalues of G
    lmin = 5.271, lavg = 28.73, lmax = 59.66
  • Minimum norm (MN) points, Degree n = 19, Number of points dn = 400
  • mn19_0400pts.jpg mn19_0400msh.jpg mn19_0400int.jpg mn19_0400ev.jpg
    MN points
    n = 19, dn = 400
    Mesh norm function
    h = 0.140
    Interpolation norm function
    ||Ln|| = 14.98
     Eigenvalues of G
    lmin = 5.659, lavg = 31.83, lmax = 68.21
  • Minimum norm (MN) points, Degree n = 20, Number of points dn = 441
  • mn20_0441pts.jpg mn20_0441msh.jpg mn20_0441int.jpg mn20_0441ev.jpg
    MN points
    n = 20, dn = 441
    Mesh norm function
    h = 0.134
    Interpolation norm function
    ||Ln|| = 15.78
     Eigenvalues of G
    lmin = 6.263, lavg = 35.09, lmax = 69.42
  • Minimum norm (MN) points, Degree n = 21, Number of points dn = 484
  • mn21_0484pts.jpg mn21_0484msh.jpg mn21_0484int.jpg mn21_0484ev.jpg
    MN points
    n = 21, dn = 484
    Mesh norm function
    h = 0.126
    Interpolation norm function
    ||Ln|| = 16.25
     Eigenvalues of G
    lmin = 6.562, lavg = 38.52 , lmax = 76.78
  • Minimum norm (MN) points, Degree n = 22, Number of points dn = 529
  • mn22_0529pts.jpg mn22_0529msh.jpg mn22_0529int.jpg mn22_0529ev.jpg
    MN points
    n = 22, dn = 529
    Mesh norm function
    h = 0.123
    Interpolation norm function
    ||Ln|| = 17.08
     Eigenvalues of G
    lmin = 6.867, lavg = 42.10, lmax = 84.21
  • Minimum norm (MN) points, Degree n = 23, Number of points dn = 576
  • mn23_0576pts.jpg mn23_0576msh.jpg mn23_0576int.jpg mn23_0576ev.jpg
    MN points
    n = 23, dn = 576
    Mesh norm function
    h = 0.115
    Interpolation norm function
    ||Ln|| = 17.62
     Eigenvalues of G
    lmin = 7.323, lavg = 45.84, lmax = 87.44
  • Minimum norm (MN) points, Degree n = 24, Number of points dn = 625
  • mn24_0625pts.jpg mn24_0625msh.jpg mn24_0625int.jpg mn24_0625ev.jpg
    MN points
    n = 24, dn = 625
    Mesh norm function
    h = 0.113
    Interpolation norm function
    ||Ln|| = 18.50
     Eigenvalues of G
    lmin = 7.566, lavg = 49.74, lmax = 106.45
  • Minimum norm (MN) points, Degree n = 25, Number of points dn = 676
  • mn25_0676pts.jpg mn25_0676msh.jpg mn25_0676int.jpg mn25_0676ev.jpg
    MN points
    n = 25, dn = 676
    Mesh norm function
    h = 0.106
    Interpolation norm function
    ||Ln|| = 18.72
     Eigenvalues of G
    lmin = 7.625, lavg = 53.79, lmax = 103.28
  • Minimum norm (MN) points, Degree n = 26, Number of points dn = 729
  • mn26_0729pts.jpg mn26_0729msh.jpg mn26_0729int.jpg mn26_0729ev.jpg
    MN points
    n = 26, dn = 729
    Mesh norm function
    h = 0.107
    Interpolation norm function
    ||Ln|| = 19.73
     Eigenvalues of G
    lmin = 7.420, lavg =58.01 , lmax = 118.27
  • Minimum norm (MN) points, Degree n = 27, Number of points dn = 784
  • mn27_0784pts.jpg mn27_0784msh.jpg mn27_0784int.jpg mn27_0784ev.jpg
    MN points
    n = 27, dn = 784
    Mesh norm function
    h = 0.098
    Interpolation norm function
    ||Ln|| = 20.42
     Eigenvalues of G
    lmin = 8.780, lavg = 62.39, lmax = 121.42
  • Minimum norm (MN) points, Degree n = 28, Number of points dn = 841
  • mn28_0841pts.jpg mn28_0841msh.jpg mn28_0841int.jpg mn28_0841ev.jpg
    MN points
    n = 28, dn = 841
    Mesh norm function
    h = 0.095
    Interpolation norm function
    ||Ln|| = 21.12
     Eigenvalues of G
    lmin = 8.223, lavg = 66.93, lmax = 136.24
  • Minimum norm (MN) points, Degree n = 29, Number of points dn = 900
  • mn29_0900pts.jpg mn29_0900msh.jpg mn29_0900int.jpg mn29_0900ev.jpg
    MN points
    n = 29, dn = 900
    Mesh norm function
    h = 0.094
    Interpolation norm function
    ||Ln|| = 21.40
     Eigenvalues of G
    lmin = 9.31, lavg = 71.62, lmax = 145.35

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