r2_dixon.py -- verification run 2 (OWR-4531-016), mpmath dps = 40

D1 real period
  B(1/3,1/3)          = 5.2999162508563498719410684989453
  3 int_0^1 (1-t^3)^(-2/3) dt = 5.2999162508563498719410684989453
  4K(m)/kappa (route C) = 5.2999162508563498719410684989453   m = 0.066987298107780676618
  h = (sqrt 3/2) p0   = 4.5898621111715786247395825169265
  [PASS] D1 the three values of p0 agree   0.0, 0.0

D2 route C is the Weierstrass function of (0, 1/27) and gives Dixon's functions
  [PASS] D2a P'^2 = 4P^3 - 1/27 at 10 random points   max rel. residual 5.55e-40
  [PASS] D2b P(u) - u^-2 = O(u^4) at u = 0.01 + 0.007i (Laurent normalization)   |P - u^-2| = 2.94e-11
  [PASS] D2c route C satisfies sm' = cm^2, cm' = -sm^2, sm^3 + cm^3 = 1; sm(u) ~ u, cm(0) = 1   ODE 2.12e-41, first integral 2.45e-41
  [PASS] D2d route A (Taylor integration, chart at the poles) = route C at 8 points   max rel. diff 4.11e-31 (119 Taylor steps)

D3 periods and symmetries (route C, controls)
  [PASS] D3a p0, P = omega p0, 2ih, P + p0, -2P are periods of sm and cm; p0/2, P/2, ih are not   max 4.09e-40; smallest control difference 0.5021
  [PASS] D3b sm(omega u) = omega sm(u), cm(omega u) = cm(u), sm(conj u) = conj sm(u)   max 2.17e-40
  2ih - p0(1 + 2 omega) = 1.22e-40

D4 the real line
  [PASS] D4a sm(p0/3) = 1, cm(p0/3) = 0 (u = int_0^1 dt/(1-t^3)^(2/3) = p0/3)
  [PASS] D4b int_1^oo dx/(x^3-1)^(2/3) = p0/3, so the real pole is at p0/3 + p0/3 = 2p0/3   diff 2.3e-41
  [PASS] D4c route A: sm(2p0/3 - 1e-10) ~ 1e10 (pole), cm/sm -> -1 there   sm * 1e-10 = 1.0, cm/sm = -1.0
  [PASS] D4d sm is real and increasing on (0, 2p0/3) (132 points), with sm' = cm^2 >= 0

D5 zeros of sm -/+ i: Abel integrals and the automorphism u -> omega (u - p0/3) + p0/3
  sm = -i:
    u = (0.7497704548828558397944785 + 3.6469972581954693547947543j)   cm = (-0.290514555507 + 1.08421508149j)   cm^3 = (1.0 - 1.0j)
    u = (1.7666387502854499573136895 + 1.8857297059522185398896565j)   cm = (-0.793700525984 - 0.793700525984j)   cm^3 = (1.0 - 1.0j)
    u = (2.7835070456880440748329005 + 3.6469972581954693547947543j)   cm = (1.08421508149 - 0.290514555507j)   cm^3 = (1.0 - 1.0j)
  [PASS] D5a the three points satisfy sm = -i and cm = omega^k (1 - i)^(1/3) (routes A and C)   max 1.54e-31
  [PASS] D5c agreement with Table 1 (rounding 5e-16) and with the recorded 22-digit values   Table 1: 4.62e-16, recorded: 4.63e-22
  sm = +i:
    u = (0.13354892025986913886236625 + 0.94286485297610926994482824j)   cm = (1.08421508149 + 0.290514555507j)   cm^3 = (1.0 + 1.0j)
    u = (3.3997285803110307757650128 + 0.94286485297610926994482824j)   cm = (-0.290514555507 - 1.08421508149j)   cm^3 = (1.0 + 1.0j)
    u = (4.4165968757136248932842237 + 2.704132405219360084849926j)   cm = (-0.793700525984 + 0.793700525984j)   cm^3 = (1.0 + 1.0j)
  [PASS] D5b the three points satisfy sm = +i and cm = omega^k (1 + i)^(1/3) (routes A and C)   max 9.65e-32
  [PASS] D5d agreement with Table 1 (rounding 5e-16) and with the recorded 22-digit values   Table 1: 3.51e-16, recorded: 2.36e-22

D6 exactly three solutions per period (Newton from a 14 x 10 grid, route C)
  [PASS] D6a sm = -i: 3 distinct solutions in [0,p0) x [0,h), the three of D5; cm^3 = 1 - i != 0
  [PASS] D6b sm = +i: 3 distinct solutions in [0,p0) x [0,h), the three of D5; cm^3 = 1 + i != 0

D7 poles of sm in [0, p0) x [0, h): omega^k * 2p0/3 reduced
     (0.88331937514272497866 + 1.5299540370571928749j)   |1 - 3P'| there: 2.0e-30
     (3.5332775005708999146 + 3.0599080741143857498j)   |1 - 3P'| there: 2.0e-30
     (3.5332775005708999146 + 0.0j)   |1 - 3P'| there: 2.0e-30
  [PASS] D7 three poles p0/6 + ih/3, 2p0/3, 2p0/3 + 2ih/3; sm has degree 3, so these are all

D8 geometry of the zero sets
  side lengths ['2.03373659081', '2.03373659081', '2.03373659081']  centroid (1.76663875028545 + 3.05990807411439j)  distance to the lines 0.942864852976109
  [PASS] D8 equilateral, centroid p0/3 + 2ih/3, Z~_S = conj(Z_S) + P mod p0, distance 0.9428... > 0

D9 reality on the boundary lines
  [PASS] D9 sm, cm real on Im u = 0 and Im u = h (32 random points)   max rel. |Im| 5.28e-41

D10 the point psi(0) = ih/2
  sm(ih/2) = (-0.62996052494743658238 + 0.42916063195784049027j)   cm(ih/2) = (-0.36602540378443864676 + 0.93060485910209959894j)
  [PASS] D10 sm(ih/2) != -i, +i (so B(0) B~(0) != 0); routes A and C agree   min |sm(ih/2) -/+ i| = 0.850122

TOTAL failures: 0
