r2_tables.py (verification run 2); mpmath 1.3.0, 50 digits
T1   ok    Table 1: S = x - 1/100: True; signs: True; psi = 1.000e+00, -2.271e-05, 1.476e-05, -1.662e-05, 2.190e-05, -1.000e+00 rounds to the printed values: True; bit lengths 21892 to 24265 (printed: 21892 to 24265)
F1   ok    Figure 1: 132 plotted points, x from 0.9815 to 0.9946; largest difference between the plotted and the exact value of 10^5 psi: 5.0e-04; filled dots at the four interior test points: True; circles within 10^-5 of the five roots: True
T2a  ok    Table 2: 14 rows; N, b, nu_1 equal to the data file and beta, S_T round to the printed values, alpha = gamma = phi = L1 = R = 1, kappa = 2 nu_1: True; column 'found by' agrees with the data file: True; exact values printed for N = 651 and N = 2000: True; the point (5) of Theorem 1.2 equals the entry N1000 of the data file: True
T2a  ok    N^2 nu for the twelve points with N <= 700: from 3.5182 to 3.5858 (Section 8.2 prints 3.518 and 3.586)
T2b  ok    relative length of the window of X = b + S_T with five roots (numerical, 50 digits); the point lies inside its windows: N1000: window of X 6.637e-07 (printed 6.6e-7), of rho 6.381e-05 (printed 6.4e-5); N651-B: 1.002e-13 (printed 1.0e-13); N652-B: 1.426e-12 (printed 1.4e-12); N652-A: 3.538e-12 (printed 3.5e-12); N653-A: 1.237e-11 (printed 1.2e-11); N655-B: 3.516e-11 (printed 3.5e-11); N655-A: 3.469e-11 (printed 3.5e-11); N660-B: 2.476e-10 (printed 2.5e-10); N660-A: 2.749e-10 (printed 2.7e-10); N680-A: 4.560e-09 (printed 4.6e-9); N690-A: 9.330e-09 (printed 9.3e-9); N700-B: 7.432e-09 (printed 7.4e-9); N700-A: 1.512e-08 (printed 1.5e-8); N2000: 2.558e-06 (printed 2.6e-6); N4000-A: 3.987e-06 (printed 4.0e-6)
R    ok    Remark 5.1, chain length N' with the reduced parameters of the point (5), EXACT counts of the roots of P in [x0, X]: 850: 1, 900: 3, 950: 3, 975: 3, 990: 3, 995: 3, 998: 3, 999: 5, 1000: 5, 1001: 3, 1002: 3, 1005: 3, 1010: 3, 1025: 1, 1050: 1
S5   ok    Remark 5.1 (110 digits): S = 0.973078759015, 0.976976484125, 0.980131721344, 0.982433429148, 0.983715309104 (printed: 0.973078759, 0.976976484, 0.980131721, 0.982433429, 0.983715309); smallest C_j at the five states: 8.759e-5, 0.0001127, 0.0001474, 0.0001901, 0.0002259 (printed: larger than 8e-5); largest residual of the 1002 equations (1)-(4) as printed: 1.3e-113, 2.6e-113, 1.3e-113, 1.3e-113, 6.5e-114
T3   ok    Table 3: the 18 printed pairs (eta_c, lambda_c) are the rounded values of the recorded outputs of run B: True; in the recorded list for N = 5..660: eta_c < 0 for N = 5..650 (646 values), > 0 for N = 651..660; minimum -13.02 at N = 17; value at N = 650: -0.00069, at N = 651: +0.00080
T3   ok    eta = x + 2G'/G'' evaluated here (30 digits) on xi = N(1 - x) in [9.55, 9.67] for N^2 nu = lambda_c - 10^-5 (lambda_c of run B): G' < 0 < G'' there; N = 650 (N^2 nu = 3.58637432): local min -0.000703671 at xi = 9.6003; local max -0.000703625 at xi = 9.6174; N = 651 (N^2 nu = 3.58636564): local min +0.000790225 at xi = 9.6004; local max +0.000790271 at xi = 9.6175; one fold each, the fold maximum rounds to the values -0.000704 and +0.000790 of Section 8.2: True
failures: 0 
