r2_disc.py -- verification run 2 (OWR-4531-016), dps = 40

T1 Blaschke condition
  sum_k (1 - |a_k|), |k| <= 22, per orbit: 0.518752 (ratio 0.026579933); 0.621025 (ratio 0.026579933); 0.363399 (ratio 0.026579933); 0.56412 (ratio 0.026579933); 0.392606 (ratio 0.026579933); 0.794919 (ratio 0.026579933)
  [PASS] T1 geometric decay with ratio e^(-2 pi/sqrt 3) = 0.026579933; the smallest 1 - |a| is above 1e-36 (resolved at 40 digits)

T2 identities at random interior points
  [PASS] T2 F^3 + G^3 = H^3 and F + iH = B at 200 random points with 1 - |z| in (1e-3, 1)   max 7.41e-41, 1.28e-41

T3 the poles are cancelled
  [PASS] T3 |F|, |G|, |H| <= 1 on circles of radius 1e-3 around three points of Z (48 points)   max 0.272435

  Gauss-Legendre, 24 nodes: sum of weights - 2 = 0.0
  7776 nodes on T
  [PASS] T4a quadrature: total measure 1 - (omitted arcs); |B| = |B~| = 1 at the nodes   1 - mass = 2.7e-18
  [PASS] T4 theta_0 conj(z) conj(f) = f~ at all 7776 nodes (f = F, G, H)   max 8.89e-24
  [PASS] T5 Fourier coefficients of index -1..-8 of F, G, H, F~, G~, H~ vanish   max 4.92e-19
  [PASS] T6 |<f, theta_0 z^k>| for f = F, G, H, k = 0..8 (membership from the definition)   max 4.92e-19

T7 wrong inputs
  [PASS] T7a W1 theta = B B~ (no factor z): <f, theta> = conj(f~(0)) != 0 and <f, theta z^k> = 0 for k >= 1 (f = F, G, H)   |f~(0)| = ['0.0774134', '0.101559', '0.101559']; deviation 4.44e-19; k >= 1: 4.92e-19
  [PASS] T7b W2 theta = z B (no B~) is rejected   max 0.281911
  [PASS] T7c W5 zeros of B~ moved by 0.01 i in the strip is rejected   max 0.001237
  [PASS] T7d W8 B~ in place of B: B~ (q o psi) has negative Fourier coefficients (poles in D)   max 0.1195
  F(0), G(0), H(0) = ['(0.065687572 + 0.040963173j)', '(0.10147574 - 0.0041053672j)', '(-0.040963173 - 0.092931147j)'] ; F~(0), G~(0), H~(0) = ['(0.01814277 - 0.075257415j)', '(-0.035916084 - 0.094995866j)', '(-0.075257415 + 0.068194583j)']

T8 sup norms (Remark 5.4)
  [PASS] T8 sup over the nodes on T of |F|, |G|, |H|, |F~|, |G~|, |H~| is <= 1 and close to 1; interior values <= 1   0.99999997 1.0 1.0 0.99999997 1.0 1.0
  at zeta = -i (w = 0): |G| = 1.0  |H| = 1.0

T9 Lemma 2.2 for gamma
  [PASS] T9 B(gamma(z))/B(z) is a unimodular constant (23 points) and |F o gamma| = |F|   constant (1.0 + 9.470832869e-35j), spread 5.29e-34

T10 orbit of x = 0.3 under gamma
  1 - gamma^6(0.3) = 3.798e-10 ; 1 + gamma^-6(0.3) = 1.31e-9
  [PASS] T10 |F|, |G|, |H| constant along the orbit (k = -6..6), nonzero: no atom at +-1 can occur   |F|, |G|, |H| = ['0.10216746', '0.10092305', '0.077434233']; spread 1.54e-27

T11 Proposition 6.1 and Remark 6.2 for phi(z) = e^{-1.1i}(z - c)/(1 - conj(c) z), c = -0.35 + 0.45i
  boundary spectrum of theta_phi: angles 1.8874169 and 3.0458062 (difference 1.1583893; pi would be antipodal)
  [PASS] T11a <f o phi, theta_phi z^k> = 0 for f = F, G, H, k = 0..8 (7728 nodes on the transported circle)   max 7.17e-19
  [PASS] T11b naive theta_0 o phi: <F o phi, theta_0 o phi> = conj(phi(0) F~(phi(0))) != 0, and = 0 for k >= 1   0.1267709632 vs 0.1267709632

T12 the automorphism of Section 7 of the note
  [PASS] T12 |phi(0) F~(phi(0))| for phi(z) = e^{0.7i}(z - c)/(1 - conj(c) z), c = 0.3 + 0.2i, is 0.0572858 (as in the note)   0.05728580645

T13 linear independence
  [PASS] T13 rank 3 (smallest singular value of a 30 x 3 sample matrix bounded away from 0)   3.8839 3.2824 0.8255

TOTAL failures: 0
