=== L = 8, N_f = 1, t_f = 1, x_s = 0, omega_min = 0.585786 (gap t_f*omega_min) === closed forms (7.4): max|G_r - (G_0 - u/2L)| = 5.0e-16; (G^2)_r - (-u/12L - u^2/24L) const to 1.8e-15; max|(7.3) - (7.5)| = 5.1e-21 grid: max_x |W_exact - W_(7.5)| and relative to max_x |W_exact|, two levels, both branches M_k sigma gamma g_s g_b/gap dev rel.dev dev of (4.4) alone 1.0 +1 0.002 0.001 +0.0034 3.116e-10 6.244e-04 9.366e-10 1.0 +1 0.002 0.004 +0.0034 4.986e-09 2.502e-03 7.486e-09 1.0 +1 0.010 0.001 +0.0171 1.505e-09 6.060e-04 1.713e-08 1.0 +1 0.010 0.004 +0.0171 2.461e-08 2.483e-03 8.711e-08 1.0 -1 0.002 0.001 -0.0034 3.125e-10 6.245e-04 3.126e-10 1.0 -1 0.002 0.004 -0.0034 5.000e-09 2.503e-03 2.500e-09 1.0 -1 0.010 0.001 -0.0171 1.527e-09 6.073e-04 1.411e-08 1.0 -1 0.010 0.004 -0.0171 2.496e-08 2.487e-03 3.758e-08 2.5 +1 0.002 0.001 +0.0085 7.725e-10 6.203e-04 4.679e-09 2.5 +1 0.002 0.004 +0.0085 1.242e-08 2.498e-03 2.804e-08 2.5 +1 0.010 0.001 +0.0427 3.104e-09 5.048e-04 1.008e-07 2.5 +1 0.010 0.004 +0.0427 5.841e-08 2.379e-03 4.490e-07 2.5 -1 0.002 0.001 -0.0085 7.779e-10 6.208e-04 3.129e-09 2.5 -1 0.002 0.004 -0.0085 1.251e-08 2.500e-03 3.128e-09 2.5 -1 0.010 0.001 -0.0427 3.257e-09 5.134e-04 9.450e-08 2.5 -1 0.010 0.004 -0.0427 6.065e-08 2.394e-03 3.304e-07 gamma scaling at g_s = 0.002 gap, g_b/gap = [0.02, 0.04, 0.08, 0.16]: g_s-odd deviation of (7.5): ['1.221e-10', '9.700e-10', '7.755e-09', '6.208e-08'], slope in g_b = 2.997 (expected 3) g_s-odd deviation of (4.4) alone: ['2.504e-08', '9.980e-08', '3.963e-07', '1.562e-06'], slope in g_b = 1.988 (expected 2) g_s scaling at g_b = 0.05 gap, g_s/gap = [0.002, 0.004, 0.008, 0.016]: g_s-even deviation of (7.5): ['6.164e-09', '2.466e-08', '9.863e-08', '3.945e-07'], slope in g_s = 2.000 (expected 2) local slope factor (s = 0 part of the geometric factor of (7.8)): [W'(x0)/W'_lead(x0) - 1]/g_b, odd part in g_s and in g_b (both branches), Richardson in g_b r0 = 1: numeric -0.791667, predicted -0.791667, diff +1.2e-07 r0 = 2: numeric -0.375000, predicted -0.375000, diff -3.9e-07 r0 = 3: numeric -0.124999, predicted -0.125000, diff +9.3e-07 r0 = 5: numeric -0.124999, predicted -0.125000, diff +7.3e-07 r0 = 6: numeric -0.374999, predicted -0.375000, diff +9.1e-07 r0 = 7: numeric -0.791666, predicted -0.791667, diff +1.9e-07 packet acceleration (7.7), (7.8): -<[H,[H,X]]>, H = (1/4)(1 - cos 2P) + W_exact (E'' = 1), triangular envelope of half-width h symmetric about x0, arc not containing x_s, cut 3 places beyond the arc h c_phi 12 s^2 max|<[K,[K,X]]>| max|lead ratio - 1| max|geom. factor num - pred| over r0 1 0.1667 0.000 0.0e+00 2.8e-06 9.3e-07 === L = 13, N_f = 1, t_f = 1, x_s = 0, omega_min = 0.229088 (gap t_f*omega_min) === closed forms (7.4): max|G_r - (G_0 - u/2L)| = 1.1e-15; (G^2)_r - (-u/12L - u^2/24L) const to 5.3e-15; max|(7.3) - (7.5)| = 3.4e-21 grid: max_x |W_exact - W_(7.5)| and relative to max_x |W_exact|, two levels, both branches M_k sigma gamma g_s g_b/gap dev rel.dev dev of (4.4) alone 1.0 +1 0.002 0.001 +0.0087 5.260e-10 1.062e-03 1.584e-09 1.0 +1 0.002 0.004 +0.0087 8.417e-09 4.260e-03 1.265e-08 1.0 +1 0.010 0.001 +0.0437 2.453e-09 9.988e-04 2.890e-08 1.0 +1 0.010 0.004 +0.0437 4.107e-08 4.193e-03 1.468e-07 1.0 -1 0.002 0.001 -0.0087 5.286e-10 1.062e-03 5.292e-10 1.0 -1 0.002 0.004 -0.0087 8.458e-09 4.263e-03 4.227e-09 1.0 -1 0.010 0.001 -0.0437 2.519e-09 1.004e-03 2.393e-08 1.0 -1 0.010 0.004 -0.0437 4.211e-08 4.209e-03 6.368e-08 2.5 +1 0.002 0.001 +0.0218 1.294e-09 1.048e-03 7.905e-09 2.5 +1 0.002 0.004 +0.0218 2.090e-08 4.245e-03 4.734e-08 2.5 +1 0.010 0.001 +0.1091 3.941e-09 6.521e-04 1.692e-07 2.5 +1 0.010 0.004 +0.1091 9.246e-08 3.837e-03 7.536e-07 2.5 -1 0.002 0.001 -0.0218 1.310e-09 1.050e-03 5.301e-09 2.5 -1 0.002 0.004 -0.0218 2.116e-08 4.252e-03 5.294e-09 2.5 -1 0.010 0.001 -0.1091 4.379e-09 6.870e-04 1.609e-07 2.5 -1 0.010 0.004 -0.1091 9.906e-08 3.898e-03 5.621e-07 gamma scaling at g_s = 0.002 gap, g_b/gap = [0.02, 0.04, 0.08, 0.16]: g_s-odd deviation of (7.5): ['8.885e-12', '7.049e-11', '5.618e-10', '4.474e-09'], slope in g_b = 2.992 (expected 3) g_s-odd deviation of (4.4) alone: ['2.537e-09', '1.012e-08', '4.025e-08', '1.592e-07'], slope in g_b = 1.991 (expected 2) g_s scaling at g_b = 0.05 gap, g_s/gap = [0.002, 0.004, 0.008, 0.016]: g_s-even deviation of (7.5): ['6.270e-10', '2.508e-09', '1.003e-08', '4.013e-08'], slope in g_s = 2.000 (expected 2) local slope factor (s = 0 part of the geometric factor of (7.8)): [W'(x0)/W'_lead(x0) - 1]/g_b, odd part in g_s and in g_b (both branches), Richardson in g_b r0 = 1: numeric -1.576923, predicted -1.576923, diff -2.1e-07 r0 = 2: numeric -1.064097, predicted -1.064103, diff +6.0e-06 r0 = 3: numeric -0.653849, predicted -0.653846, diff -2.6e-06 r0 = 4: numeric -0.346152, predicted -0.346154, diff +2.2e-06 r0 = 5: numeric -0.141038, predicted -0.141026, diff -1.2e-05 r0 = 6: numeric -0.038337, predicted -0.038462, diff +1.2e-04 r0 = 7: numeric -0.038469, predicted -0.038462, diff -7.8e-06 r0 = 8: numeric -0.140966, predicted -0.141026, diff +6.0e-05 r0 = 9: numeric -0.346157, predicted -0.346154, diff -2.9e-06 r0 = 10: numeric -0.653856, predicted -0.653846, diff -1.0e-05 r0 = 11: numeric -1.064105, predicted -1.064103, diff -2.3e-06 r0 = 12: numeric -1.576922, predicted -1.576923, diff +6.7e-07 packet acceleration (7.7), (7.8): -<[H,[H,X]]>, H = (1/4)(1 - cos 2P) + W_exact (E'' = 1), triangular envelope of half-width h symmetric about x0, arc not containing x_s, cut 3 places beyond the arc h c_phi 12 s^2 max|<[K,[K,X]]>| max|lead ratio - 1| max|geom. factor num - pred| over r0 1 0.1667 0.000 0.0e+00 1.4e-06 1.2e-04 2 0.5263 7.200 0.0e+00 1.2e-06 4.1e-05 3 0.7045 15.484 0.0e+00 1.0e-06 1.4e-05 4 0.8000 26.118 0.0e+00 7.6e-07 9.2e-06