Exact mutual cost on a ring (one field object, t_f = 1), gamma = g L / t_f, w = d(L-d)/L^2 rel2 = (M_exact - M2)/M2, pred3 = M3/M2 from (4.8), rel3 = (M_exact - M2 - M3)/M2, pred3_L = -(gamma/5)(1-10w+20w^2)/(1-6w) (leading in 1/L, (4.9)); |closed - eigh| = max deviation of (4.11) roots -- L = 64 gamma d w M_exact M2 rel2 pred3 pred3_L rel3 |closed-eigh| 0.03 1 0.0154 -3.30464e-08 -3.32326e-08 -5.602e-03 -5.619e-03 -5.625e-03 1.63e-05 2.2e-16 0.03 8 0.1094 -1.25477e-08 -1.25796e-08 -2.535e-03 -2.530e-03 -2.540e-03 -5.34e-06 2.2e-16 0.03 16 0.1875 4.54887e-09 4.58658e-09 -8.221e-03 -8.249e-03 -8.250e-03 2.78e-05 2.2e-16 0.03 32 0.2500 1.82646e-08 1.83195e-08 -2.994e-03 -2.999e-03 -3.000e-03 5.02e-06 4.4e-16 0.10 1 0.0154 -3.62402e-07 -3.69251e-07 -1.855e-02 -1.873e-02 -1.875e-02 1.81e-04 4.4e-16 0.10 8 0.1094 -1.38586e-07 -1.39773e-07 -8.489e-03 -8.432e-03 -8.466e-03 -5.74e-05 4.4e-16 0.10 16 0.1875 4.95764e-08 5.09620e-08 -2.719e-02 -2.750e-02 -2.750e-02 3.06e-04 4.4e-16 0.10 32 0.2500 2.01527e-07 2.03550e-07 -9.940e-03 -9.995e-03 -1.000e-02 5.53e-05 4.4e-16 0.30 1 0.0154 -3.14194e-06 -3.32326e-06 -5.456e-02 -5.619e-02 -5.625e-02 1.63e-03 2.2e-16 0.30 8 0.1094 -1.22553e-06 -1.25796e-06 -2.578e-02 -2.530e-02 -2.540e-02 -4.80e-04 2.2e-16 0.30 16 0.1875 4.22078e-07 4.58658e-07 -7.975e-02 -8.249e-02 -8.250e-02 2.73e-03 2.2e-16 0.30 32 0.2500 1.77793e-06 1.83195e-06 -2.949e-02 -2.999e-02 -3.000e-02 4.96e-04 4.4e-16 1.00 1 0.0154 -3.06651e-05 -3.69251e-05 -1.695e-01 -1.873e-01 -1.875e-01 1.78e-02 4.4e-16 1.00 8 0.1094 -1.27422e-05 -1.39773e-05 -8.836e-02 -8.432e-02 -8.466e-02 -4.04e-03 4.4e-16 1.00 16 0.1875 3.84439e-06 5.09620e-06 -2.456e-01 -2.750e-01 -2.750e-01 2.93e-02 4.4e-16 1.00 32 0.2500 1.84310e-05 2.03550e-05 -9.452e-02 -9.995e-02 -1.000e-01 5.43e-03 4.4e-16 3.00 1 0.0154 -1.94134e-04 -3.32326e-04 -4.158e-01 -5.619e-01 -5.625e-01 1.46e-01 0.0e+00 3.00 8 0.1094 -9.24888e-05 -1.25796e-04 -2.648e-01 -2.530e-01 -2.540e-01 -1.18e-02 2.2e-16 3.00 16 0.1875 1.88729e-05 4.58658e-05 -5.885e-01 -8.249e-01 -8.250e-01 2.36e-01 2.2e-16 3.00 32 0.2500 1.36755e-04 1.83195e-04 -2.535e-01 -2.999e-01 -3.000e-01 4.64e-02 2.2e-16 -0.30 1 0.0154 -3.51540e-06 -3.32326e-06 5.782e-02 5.619e-02 5.625e-02 1.63e-03 4.4e-16 -0.30 8 0.1094 -1.28903e-06 -1.25796e-06 2.470e-02 2.530e-02 2.540e-02 -5.93e-04 0.0e+00 -0.30 16 0.1875 4.97778e-07 4.58658e-07 8.529e-02 8.249e-02 8.250e-02 2.81e-03 0.0e+00 -0.30 32 0.2500 1.88780e-06 1.83195e-06 3.049e-02 2.999e-02 3.000e-02 5.01e-04 4.4e-16 -1.00 1 0.0154 -4.45014e-05 -3.69251e-05 2.052e-01 1.873e-01 1.875e-01 1.79e-02 4.4e-16 -1.00 8 0.1094 -1.50412e-05 -1.39773e-05 7.612e-02 8.432e-02 8.466e-02 -8.20e-03 4.4e-16 -1.00 16 0.1875 6.66095e-06 5.09620e-06 3.070e-01 2.750e-01 2.750e-01 3.21e-02 4.4e-16 -1.00 32 0.2500 2.25039e-05 2.03550e-05 1.056e-01 9.995e-02 1.000e-01 5.62e-03 4.4e-16 -- L = 256 gamma d w M_exact M2 rel2 pred3 pred3_L rel3 |closed-eigh| 0.03 1 0.0039 -2.22219e-09 -2.23535e-09 -5.886e-03 -5.906e-03 -5.906e-03 1.94e-05 4.4e-16 0.03 32 0.1094 -7.84745e-10 -7.86746e-10 -2.543e-03 -2.539e-03 -2.540e-03 -4.26e-06 4.4e-16 0.03 64 0.1875 2.83784e-10 2.86137e-10 -8.225e-03 -8.250e-03 -8.250e-03 2.46e-05 4.4e-16 0.03 128 0.2500 1.14102e-09 1.14444e-09 -2.996e-03 -3.000e-03 -3.000e-03 4.24e-06 4.4e-16 0.10 1 0.0039 -2.43535e-08 -2.48372e-08 -1.948e-02 -1.969e-02 -1.969e-02 2.11e-04 2.2e-16 0.10 32 0.1094 -8.66715e-09 -8.74163e-09 -8.520e-03 -8.464e-03 -8.466e-03 -5.65e-05 2.2e-16 0.10 64 0.1875 3.09285e-09 3.17930e-09 -2.719e-02 -2.750e-02 -2.750e-02 3.07e-04 2.2e-16 0.10 128 0.2500 1.25896e-08 1.27160e-08 -9.944e-03 -1.000e-02 -1.000e-02 5.54e-05 2.2e-16 0.30 1 0.0039 -2.10755e-07 -2.23535e-07 -5.717e-02 -5.906e-02 -5.906e-02 1.89e-03 2.2e-16 0.30 32 0.1094 -7.66398e-08 -7.86746e-08 -2.586e-02 -2.539e-02 -2.540e-02 -4.73e-04 2.2e-16 0.30 64 0.1875 2.63314e-08 2.86137e-08 -7.976e-02 -8.250e-02 -8.250e-02 2.73e-03 2.2e-16 0.30 128 0.2500 1.11068e-07 1.14444e-07 -2.950e-02 -3.000e-02 -3.000e-02 4.97e-04 2.2e-16 1.00 1 0.0039 -2.04534e-06 -2.48372e-06 -1.765e-01 -1.969e-01 -1.969e-01 2.04e-02 2.2e-16 1.00 32 0.1094 -7.96704e-07 -8.74163e-07 -8.861e-02 -8.464e-02 -8.466e-02 -3.97e-03 4.4e-16 1.00 64 0.1875 2.39830e-07 3.17930e-07 -2.457e-01 -2.750e-01 -2.750e-01 2.93e-02 2.2e-16 1.00 128 0.2500 1.15137e-06 1.27160e-06 -9.456e-02 -1.000e-01 -1.000e-01 5.44e-03 2.2e-16 3.00 1 0.0039 -1.28013e-05 -2.23535e-05 -4.273e-01 -5.906e-01 -5.906e-01 1.63e-01 4.4e-16 3.00 32 0.1094 -5.78102e-06 -7.86746e-06 -2.652e-01 -2.539e-01 -2.540e-01 -1.13e-02 2.2e-16 3.00 64 0.1875 1.17749e-06 2.86137e-06 -5.885e-01 -8.250e-01 -8.250e-01 2.37e-01 0.0e+00 3.00 128 0.2500 8.54277e-06 1.14444e-05 -2.535e-01 -3.000e-01 -3.000e-01 4.64e-02 2.2e-16 -0.30 1 0.0039 -2.37166e-07 -2.23535e-07 6.098e-02 5.906e-02 5.906e-02 1.92e-03 0.0e+00 -0.30 32 0.1094 -8.06263e-08 -7.86746e-08 2.481e-02 2.539e-02 2.540e-02 -5.85e-04 4.4e-16 -0.30 64 0.1875 3.10548e-08 2.86137e-08 8.531e-02 8.250e-02 8.250e-02 2.81e-03 0.0e+00 -0.30 128 0.2500 1.17935e-07 1.14444e-07 3.050e-02 3.000e-02 3.000e-02 5.02e-04 4.4e-16 -1.00 1 0.0039 -3.02602e-06 -2.48372e-06 2.183e-01 1.969e-01 1.969e-01 2.15e-02 4.4e-16 -1.00 32 0.1094 -9.41059e-07 -8.74163e-07 7.653e-02 8.464e-02 8.466e-02 -8.11e-03 4.4e-16 -1.00 64 0.1875 4.15579e-07 3.17930e-07 3.071e-01 2.750e-01 2.750e-01 3.21e-02 4.4e-16 -1.00 128 0.2500 1.40593e-06 1.27160e-06 1.056e-01 1.000e-01 1.000e-01 5.64e-03 4.4e-16 Unequal sources g2 = 2 g1 (determinant equation (4.11)), L = 64, d = 8: gamma1 = 0.30: E_eigh = -1.999791523936965, E_(4.11) = -1.999791523936965, (M_exact - M2)/M2 = -3.896e-02 gamma1 = 1.00: E_eigh = -1.999382359521463, E_(4.11) = -1.999382359521462, (M_exact - M2)/M2 = -1.335e-01 gamma1 = -1.00: E_eigh = -2.000878297773575, E_(4.11) = -2.000878297773575, (M_exact - M2)/M2 = 1.042e-01