08-ilang-propagation / 06-mark-inertia
06mark inertiaverified
Determines the inertia of a moving body with its own mark against its cost in oscillator cells, as a function of the ratio of body speed to field speed, and whether weight over inertia is universal.
check_mark_inertia.py10 KB · raw
"""Check script of 08-ilang-propagation/06-mark-inertia, derivation v1.
Part A (check required by the question): ring of L = 32 cells. For a square grip
and one other grip of class (G2), two clock levels, both branches and
r = tau/c0 in {0.5, 1, 1.5}: the second-order coefficients (in delta F) of
Cost_{k,sigma} and of E''_{k,sigma}(0) from the formula (6.2), (6.3), (6.5) of the
derivation are compared with an independent computation: exact diagonalisation
of the total contract restricted to states with at most two field quanta, in the
sector of the joint label P (P continued continuously, E'' from the exact second
derivative of the eigenvalue), the second-order coefficient being extracted from
several small values of the scale s of delta F.
Part B: the formula (6.3) on long rings (L = 8 ell): rho = I^mark tau^2 / C^mark
of the one-quantum (F1^2) and two-quantum (F2^2) parts against 2 - r^2, and the
held mark against its leading form in ell, as ell grows.
Units: c0 = sqrt(kappa_h) = 1, F0 = 1, so eps_k = F0 M_k = M_k.
"""
from itertools import combinations_with_replacement
from pathlib import Path
import numpy as np
OUT = Path(__file__).parent / "output"
OUT.mkdir(exist_ok=True)
LOG = []
def log(s=""):
print(s, flush=True)
LOG.append(s)
S1 = np.array([[0.0, 1.0], [1.0, 0.0]])
S3 = np.diag([1.0, -1.0])
def modes(L, c0, mh):
q = 2 * np.pi * np.arange(L) / L
Om = np.sqrt(mh**2 + 2 * c0**2 * (1 - np.cos(q)))
return q, Om, 1.0 / (2 * L * Om) # c_q^2 = 1/(2 L Omega_q)
def omega(p, tau, eps):
return np.sqrt(eps**2 + (tau * np.sin(p))**2)
def theta(p, tau, eps):
return np.arctan(tau * np.sin(p) / eps)
# ------------------------------------------------------------------ formula (6.3)
def mark_sums(P, sigma, tau, eps, q, Om, c2, chunk=256):
"""(S1, S2) with E^(2)(P) = -M_k^2 (F1^2 S1 + F2^2 S2)."""
wP, tP = omega(P, tau, eps), theta(P, tau, eps)
def S(E, Q, w):
p = P - Q
th = 0.5 * (tP + theta(p, tau, eps))
wp = omega(p, tau, eps)
return np.sum(w * (np.cos(th)**2 / (E + sigma * (wp - wP))
+ np.sin(th)**2 / (E - sigma * (wp + wP))))
s1 = S(Om, q, c2)
s2 = 0.0
for i in range(0, len(q), chunk):
sl = slice(i, i + chunk)
s2 += S(Om[sl, None] + Om[None, :], q[sl, None] + q[None, :],
2 * c2[sl, None] * c2[None, :])
return s1, s2
def curv(f, h=1e-2):
"""Second derivative at 0 of an even function, Richardson in h."""
f0 = f(0.0)
c = lambda hh: 2 * (f(hh) - f0) / hh**2
return (4 * c(h) - c(2 * h)) / 3
def formula_second_order(sigma, tau, eps, Mk, F1, F2, q, Om, c2):
"""Second-order coefficients of Cost and E''(0): mark part (6.3) plus the
second-order part of the zero-quantum level sigma*omega-bar (6.2)."""
w0 = np.sum(c2)
def E2(P):
a, b = mark_sums(P, sigma, tau, eps, q, Om, c2)
zq = sigma * (F2 * w0 * Mk)**2 * (tau * np.sin(P))**2 / (2 * omega(P, tau, eps)**3)
return -Mk**2 * (F1**2 * a + F2**2 * b) + zq
return E2(0.0), curv(E2)
# ------------------------------------------------------------------ exact diagonalisation
def field_ops(L, q, Om, c2):
states = [()] + [(i,) for i in range(L)] + list(combinations_with_replacement(range(L), 2))
idx = {s: i for i, s in enumerate(states)}
n = len(states)
A = np.zeros((n, n)) # sum_q c_q a_q on <= 2 quanta
cq = np.sqrt(c2)
for j, s in enumerate(states):
for m in set(s):
t = list(s)
t.remove(m)
A[idx[tuple(t)], j] += cq[m] * np.sqrt(s.count(m))
Ef = np.array([Om[list(s)].sum() if s else 0.0 for s in states])
Qn = np.array([q[list(s)].sum() if s else 0.0 for s in states])
q0 = A + A.T # q at the body's cell (x = 0)
nq2 = A @ A + A.T @ A.T + 2 * A.T @ A # :q0^2:, exact on <= 2 quanta
return Ef, Qn, q0, nq2, np.sum(c2)
def ed_levels(P, tau, eps, Mk, F1, F2, s, ops):
Ef, Qn, q0, nq2, w0 = ops
n = len(Ef)
sn, cs = np.sin(P - Qn), np.cos(P - Qn)
H = (np.kron(np.diag(Ef), np.eye(2)) + eps * np.kron(np.eye(n), S3)
- tau * np.kron(np.diag(sn), S1)
+ s * Mk * np.kron(F1 * q0 + F2 * (nq2 + w0 * np.eye(n)), S3))
H1 = -tau * np.kron(np.diag(cs), S1) # dH/dP
H2 = tau * np.kron(np.diag(sn), S1) # d2H/dP2
E, V = np.linalg.eigh(H)
_, u0 = np.linalg.eigh(-tau * np.sin(P) * S1 + eps * S3)
out = {}
for sigma, col in ((+1, 1), (-1, 0)):
ref = np.zeros(2 * n)
ref[:2] = u0[:, col] # field vacuum x band sigma
j = int(np.argmax(np.abs(V.T @ ref)))
psi = V[:, j]
d1 = V.T @ (H1 @ psi)
m = np.arange(len(E)) != j
Epp = psi @ H2 @ psi + 2 * np.sum(d1[m]**2 / (E[j] - E[m]))
out[sigma] = (E[j], Epp)
return out
def coeffs(svals, fpos, fneg, f0):
"""Fit even part s^2 a2 + s^4 a4 + s^6 a6 and odd part s a1 + s^3 a3 + s^5 a5."""
s = np.array(svals)
even = (np.array(fpos) + np.array(fneg)) / 2 - f0
odd = (np.array(fpos) - np.array(fneg)) / 2
a2 = np.linalg.solve(np.vstack([s**2, s**4, s**6]).T, even)[0]
a1 = np.linalg.solve(np.vstack([s, s**3, s**5]).T, odd)[0]
return a1, a2
def part_a():
L, c0, mh = 32, 1.0, 0.25
q, Om, c2 = modes(L, c0, mh)
w0 = np.sum(c2)
log(f"PART A: ring L={L}, c0={c0}, m_h={mh} (ell={c0/mh}), F0=1, "
f"G0={2*np.sum(c2/Om):.6f}, w0={w0:.6f}, Omega_max={Om.max():.4f}")
log("second-order coefficients: formula (6.2), (6.3), (6.5) vs ED restricted to <= 2 quanta")
ops = field_ops(L, q, Om, c2)
grips = {"square (F1=0.3, F2=F1^2/4=0.0225)": (0.3, 0.0225),
"other (F1=0.3, F2=-0.15)": (0.3, -0.15)}
svals = [0.01, 0.02, 0.04]
maxrel = 0.0
log(f"{'grip':36s} {'Mk':>3s} {'sig':>3s} {'r':>4s} | {'Cost2 form':>12s} {'Cost2 ED':>12s}"
f" | {'E2pp form':>12s} {'E2pp ED':>12s} | {'Cost1 ED-form':>13s} {'Epp1 ED-form':>13s}")
for gname, (F1, F2) in grips.items():
for Mk in (3.0, 4.0):
eps = Mk
for r in (0.5, 1.0, 1.5):
tau = r * c0
res = {sv: ed_levels(0.0, tau, eps, Mk, F1, F2, sv, ops)
for sv in [0.0] + svals + [-x for x in svals]}
for sigma in (+1, -1):
e0, p0 = res[0.0][sigma]
c1, c2c = coeffs(svals, [res[x][sigma][0] for x in svals],
[res[-x][sigma][0] for x in svals], e0)
d1, d2 = coeffs(svals, [res[x][sigma][1] for x in svals],
[res[-x][sigma][1] for x in svals], p0)
fc, fpp = formula_second_order(sigma, tau, eps, Mk, F1, F2, q, Om, c2)
c1f = sigma * Mk * F2 * w0
d1f = -sigma * tau**2 * F2 * w0 / Mk
rel = max(abs(c2c - fc) / abs(fc), abs(d2 - fpp) / abs(fpp))
maxrel = max(maxrel, rel)
log(f"{gname:36s} {Mk:3.0f} {sigma:+3d} {r:4.1f} | {fc:12.8f} {c2c:12.8f}"
f" | {fpp:12.8f} {d2:12.8f} | {c1-c1f:13.2e} {d1-d1f:13.2e}")
# zeroth order: bare level and curvature of (6.7) of 04-ilang-time/06-inertia-internal-energy
assert abs(e0 - sigma * eps) < 1e-10
assert abs(p0 - sigma * tau**2 / eps) < 1e-8
log(f"max relative deviation of second-order coefficients (formula vs ED): {maxrel:.2e}")
log("zeroth order: ED level = sigma*F0*M_k and E''(0) = sigma*tau^2/(F0*M_k) asserted to 1e-10 / 1e-8")
return maxrel
def part_b():
c0 = 1.0
log("")
log("PART B: formula (6.3) on rings with L = 8 ell, c0 = 1, F0 = 1;")
log("rho_1, rho_2 = I^mark tau^2/C^mark of the F1^2 and F2^2 parts; prediction (6.9): 2 - r^2")
log("held mark: g1 = G0*2c0^2/ell, g2 = J2*16c0^3/ell (prediction (6.7): -> 1);"
" Cratio = C^mark(tau)/C^mark(0) for the F1^2 part")
log("t_i = rho_i + 2 sigma eps_k R_i: rho with the term (iv) of Step 6 removed,"
" R_i = <<sin^2 Q/E^2>>/<<1/E>> of the part i")
log(f"{'ell':>5s} {'g1':>8s} {'g2':>8s} {'Mk':>3s} {'sig':>3s} {'r':>4s} {'2-r^2':>6s}"
f" {'rho_1':>9s} {'rho_2':>9s} {'t_1':>9s} {'t_2':>9s} {'Cratio':>8s}")
for ell in (16, 32, 64, 128, 256, 512):
mh, L = c0 / ell, 8 * ell
q, Om, c2 = modes(L, c0, mh)
G0 = 2 * np.sum(c2 / Om)
J2, N2 = 0.0, 0.0
for i in range(0, L, 256):
Es = Om[i:i+256, None] + Om[None, :]
ww = c2[i:i+256, None] * c2[None, :]
J2 += np.sum(ww / Es)
N2 += np.sum(ww * np.sin(q[i:i+256, None] + q[None, :])**2 / Es**2)
R1 = np.sum(c2 * np.sin(q)**2 / Om**2) / (G0 / 2)
R2 = N2 / J2
g1, g2 = G0 * 2 * c0**2 / ell, J2 * 16 * c0**3 / ell
for Mk in (3.0, 4.0):
eps = Mk
for sigma in (+1, -1):
for r in (0.5, 1.0, 1.5):
tau = r * c0
cache = {}
def ms(P):
if P not in cache:
cache[P] = mark_sums(P, sigma, tau, eps, q, Om, c2)
return cache[P]
C1, C2 = -Mk**2 * ms(0.0)[0], -Mk**2 * ms(0.0)[1]
E1 = -Mk**2 * curv(lambda P: ms(P)[0])
E2 = -Mk**2 * curv(lambda P: ms(P)[1])
I1, I2 = -(eps**2 / tau**4) * E1, -(eps**2 / tau**4) * E2
rho1, rho2 = I1 * tau**2 / C1, I2 * tau**2 / C2
cr = C1 / (-Mk**2 * G0 / 2)
t1, t2 = rho1 + 2 * sigma * eps * R1, rho2 + 2 * sigma * eps * R2
log(f"{ell:5d} {g1:8.5f} {g2:8.5f} {Mk:3.0f} {sigma:+3d} {r:4.1f} {2-r*r:6.2f}"
f" {rho1:9.5f} {rho2:9.5f} {t1:9.5f} {t2:9.5f} {cr:8.5f}")
if __name__ == "__main__":
part_a()
part_b()
(OUT / "check_mark_inertia.txt").write_text("\n".join(LOG) + "\n", encoding="utf-8")
output/check_mark_inertia.txt11 KB · raw
PART A: ring L=32, c0=1.0, m_h=0.25 (ell=4.0), F0=1, G0=1.985916, w0=0.550116, Omega_max=2.0156
second-order coefficients: formula (6.2), (6.3), (6.5) vs ED restricted to <= 2 quanta
grip Mk sig r | Cost2 form Cost2 ED | E2pp form E2pp ED | Cost1 ED-form Epp1 ED-form
square (F1=0.3, F2=F1^2/4=0.0225) 3 +1 0.5 | -0.79709146 -0.79709146 | 0.00272483 0.00272483 | -5.88e-11 -7.56e-13
square (F1=0.3, F2=F1^2/4=0.0225) 3 -1 0.5 | -0.81355379 -0.81355379 | 0.06469109 0.06469109 | 6.38e-11 -1.30e-11
square (F1=0.3, F2=F1^2/4=0.0225) 3 +1 1.0 | -0.77188596 -0.77188596 | -0.01790101 -0.01790101 | -5.00e-11 -1.63e-11
square (F1=0.3, F2=F1^2/4=0.0225) 3 -1 1.0 | -0.83695453 -0.83695453 | 0.24473099 0.24473099 | 6.85e-11 -5.10e-11
square (F1=0.3, F2=F1^2/4=0.0225) 3 +1 1.5 | -0.73621933 -0.73621933 | -0.12253651 -0.12253651 | -3.93e-11 -5.93e-11
square (F1=0.3, F2=F1^2/4=0.0225) 3 -1 1.5 | -0.88257932 -0.88257932 | 0.47513821 0.47513820 | 7.74e-11 -1.10e-10
square (F1=0.3, F2=F1^2/4=0.0225) 4 +1 0.5 | -1.42156754 -1.42156754 | -0.00839540 -0.00839540 | -4.41e-10 -1.88e-11
square (F1=0.3, F2=F1^2/4=0.0225) 4 -1 0.5 | -1.44333641 -1.44333642 | 0.07578325 0.07578325 | 4.69e-10 -6.91e-11
square (F1=0.3, F2=F1^2/4=0.0225) 4 +1 1.0 | -1.38830326 -1.38830326 | -0.06039710 -0.06039710 | -3.94e-10 -1.21e-10
square (F1=0.3, F2=F1^2/4=0.0225) 4 -1 1.0 | -1.47480499 -1.47480499 | 0.28739467 0.28739468 | 4.99e-10 -2.70e-10
square (F1=0.3, F2=F1^2/4=0.0225) 4 +1 1.5 | -1.33921483 -1.33921484 | -0.21748355 -0.21748356 | -3.33e-10 -3.65e-10
square (F1=0.3, F2=F1^2/4=0.0225) 4 -1 1.5 | -1.53381757 -1.53381758 | 0.57028552 0.57028551 | 5.55e-10 -5.68e-10
other (F1=0.3, F2=-0.15) 3 +1 0.5 | -0.88255089 -0.88255088 | 0.00244496 0.00244496 | 1.97e-09 1.22e-10
other (F1=0.3, F2=-0.15) 3 -1 0.5 | -0.90108819 -0.90108819 | 0.07033359 0.07033359 | -2.17e-09 6.41e-10
other (F1=0.3, F2=-0.15) 3 +1 1.0 | -0.85386846 -0.85386846 | -0.01963079 -0.01963079 | 1.67e-09 8.37e-10
other (F1=0.3, F2=-0.15) 3 -1 1.0 | -0.92705583 -0.92705583 | 0.26807672 0.26807672 | -2.47e-09 2.53e-09
other (F1=0.3, F2=-0.15) 3 +1 1.5 | -0.81331009 -0.81331009 | -0.12802719 -0.12802719 | 1.32e-09 2.48e-09
other (F1=0.3, F2=-0.15) 3 -1 1.5 | -0.97758359 -0.97758359 | 0.53083127 0.53083125 | -3.16e-09 4.96e-09
other (F1=0.3, F2=-0.15) 4 +1 0.5 | -1.57419026 -1.57419024 | -0.01025575 -0.01025575 | 1.49e-08 1.09e-09
other (F1=0.3, F2=-0.15) 4 -1 0.5 | -1.59864379 -1.59864377 | 0.08292655 0.08292654 | -1.60e-08 3.27e-09
other (F1=0.3, F2=-0.15) 4 +1 1.0 | -1.53652382 -1.53652380 | -0.06819105 -0.06819104 | 1.32e-08 5.66e-09
other (F1=0.3, F2=-0.15) 4 -1 1.0 | -1.63363731 -1.63363729 | 0.31662572 0.31662569 | -1.77e-08 1.28e-08
other (F1=0.3, F2=-0.15) 4 +1 1.5 | -1.48093983 -1.48093982 | -0.23595428 -0.23595426 | 1.11e-08 1.53e-08
other (F1=0.3, F2=-0.15) 4 -1 1.5 | -1.69916986 -1.69916983 | 0.63864993 0.63864987 | -2.13e-08 2.58e-08
max relative deviation of second-order coefficients (formula vs ED): 2.26e-07
zeroth order: ED level = sigma*F0*M_k and E''(0) = sigma*tau^2/(F0*M_k) asserted to 1e-10 / 1e-8
PART B: formula (6.3) on rings with L = 8 ell, c0 = 1, F0 = 1;
rho_1, rho_2 = I^mark tau^2/C^mark of the F1^2 and F2^2 parts; prediction (6.9): 2 - r^2
held mark: g1 = G0*2c0^2/ell, g2 = J2*16c0^3/ell (prediction (6.7): -> 1); Cratio = C^mark(tau)/C^mark(0) for the F1^2 part
t_i = rho_i + 2 sigma eps_k R_i: rho with the term (iv) of Step 6 removed, R_i = <<sin^2 Q/E^2>>/<<1/E>> of the part i
ell g1 g2 Mk sig r 2-r^2 rho_1 rho_2 t_1 t_2 Cratio
16 1.00018 0.96912 3 +1 0.5 1.75 1.02947 0.27353 1.71048 1.55478 0.99488
16 1.00018 0.96912 3 +1 1.0 1.00 0.42042 -0.09180 1.10143 1.18945 0.98053
16 1.00018 0.96912 3 +1 1.5 -0.25 -0.49901 -0.61591 0.18200 0.66535 0.95935
16 1.00018 0.96912 3 -1 0.5 1.75 2.34232 2.68110 1.66131 1.39984 1.00443
16 1.00018 0.96912 3 -1 1.0 1.00 1.77840 2.40227 1.09739 1.12101 1.01848
16 1.00018 0.96912 3 -1 1.5 -0.25 0.72788 1.84891 0.04687 0.56766 1.04497
16 1.00018 0.96912 4 +1 0.5 1.75 0.79609 -0.17873 1.70411 1.52961 0.99624
16 1.00018 0.96912 4 +1 1.0 1.00 0.19283 -0.53165 1.10085 1.17669 0.98551
16 1.00018 0.96912 4 +1 1.5 -0.25 -0.73980 -1.05627 0.16821 0.65207 0.96919
16 1.00018 0.96912 4 -1 0.5 1.75 2.57525 3.12404 1.66724 1.41570 1.00337
16 1.00018 0.96912 4 -1 1.0 1.00 2.00585 2.83561 1.09784 1.12727 1.01394
16 1.00018 0.96912 4 -1 1.5 -0.25 0.97571 2.28919 0.06769 0.58085 1.03324
32 1.00055 0.98473 3 +1 0.5 1.75 1.30991 0.66603 1.73271 1.63703 0.99686
32 1.00055 0.98473 3 +1 1.0 1.00 0.63602 0.17247 1.05882 1.14347 0.98799
32 1.00055 0.98473 3 +1 1.5 -0.25 -0.41564 -0.56204 0.00716 0.40895 0.97466
32 1.00055 0.98473 3 -1 0.5 1.75 2.12805 2.50760 1.70526 1.53660 1.00278
32 1.00055 0.98473 3 -1 1.0 1.00 1.48043 2.07982 1.05764 1.10882 1.01154
32 1.00055 0.98473 3 -1 1.5 -0.25 0.31874 1.26875 -0.10406 0.29775 1.02776
32 1.00055 0.98473 4 +1 0.5 1.75 1.16546 0.32687 1.72918 1.62153 0.99769
32 1.00055 0.98473 4 +1 1.0 1.00 0.49491 -0.15757 1.05863 1.13709 0.99105
32 1.00055 0.98473 4 +1 1.5 -0.25 -0.56826 -0.89831 -0.00454 0.39635 0.98081
32 1.00055 0.98473 4 -1 0.5 1.75 2.27233 2.84200 1.70860 1.54734 1.00211
32 1.00055 0.98473 4 -1 1.0 1.00 1.62149 2.40670 1.05776 1.11204 1.00869
32 1.00055 0.98473 4 -1 1.5 -0.25 0.47630 1.60897 -0.08742 0.31431 1.02055
64 1.00064 0.99254 3 +1 0.5 1.75 1.49021 0.99884 1.74289 1.68674 0.99814
64 1.00064 0.99254 3 +1 1.0 1.00 0.78115 0.41067 1.03383 1.09857 0.99285
64 1.00064 0.99254 3 +1 1.5 -0.25 -0.35207 -0.49218 -0.09939 0.19572 0.98478
64 1.00064 0.99254 3 -1 0.5 1.75 1.98048 2.31156 1.72780 1.62366 1.00168
64 1.00064 0.99254 3 -1 1.0 1.00 1.28617 1.76929 1.03349 1.08139 1.00692
64 1.00064 0.99254 3 -1 1.5 -0.25 0.07381 0.77769 -0.17887 0.08979 1.01650
64 1.00064 0.99254 4 +1 0.5 1.75 1.40406 0.76016 1.74097 1.67736 0.99863
64 1.00064 0.99254 4 +1 1.0 1.00 0.69686 0.17823 1.03377 1.09543 0.99467
64 1.00064 0.99254 4 +1 1.5 -0.25 -0.44489 -0.73290 -0.10798 0.18430 0.98849
64 1.00064 0.99254 4 -1 0.5 1.75 2.06655 2.54781 1.72965 1.63061 1.00127
64 1.00064 0.99254 4 -1 1.0 1.00 1.37043 2.00020 1.03352 1.08300 1.00521
64 1.00064 0.99254 4 -1 1.5 -0.25 0.16961 1.02296 -0.16730 0.10576 1.01223
128 1.00066 0.99645 3 +1 0.5 1.75 1.60052 1.25165 1.74753 1.71538 0.99893
128 1.00066 0.99645 3 +1 1.0 1.00 0.87314 0.59979 1.02014 1.06352 0.99585
128 1.00066 0.99645 3 +1 1.5 -0.25 -0.30818 -0.42551 -0.16118 0.03822 0.99111
128 1.00066 0.99645 3 -1 0.5 1.75 1.88631 2.14051 1.73931 1.67678 1.00098
128 1.00066 0.99645 3 -1 1.0 1.00 1.16705 1.51888 1.02005 1.05514 1.00403
128 1.00066 0.99645 3 -1 1.5 -0.25 -0.06643 0.41326 -0.21343 -0.05048 1.00955
128 1.00066 0.99645 4 +1 0.5 1.75 1.55048 1.09149 1.74648 1.70980 0.99921
128 1.00066 0.99645 4 +1 1.0 1.00 0.82412 0.44367 1.02012 1.06198 0.99690
128 1.00066 0.99645 4 +1 1.5 -0.25 -0.36296 -0.58958 -0.16696 0.02873 0.99328
128 1.00066 0.99645 4 -1 0.5 1.75 1.93632 2.29943 1.74032 1.68112 1.00074
128 1.00066 0.99645 4 -1 1.0 1.00 1.21606 1.67424 1.02005 1.05592 1.00303
128 1.00066 0.99645 4 -1 1.5 -0.25 -0.00998 0.58111 -0.20599 -0.03720 1.00709
256 1.00067 0.99840 3 +1 0.5 1.75 1.66579 1.43035 1.74963 1.73137 0.99939
256 1.00067 0.99840 3 +1 1.0 1.00 0.92909 0.73812 1.01293 1.03914 0.99764
256 1.00067 0.99840 3 +1 1.5 -0.25 -0.27981 -0.37076 -0.19597 -0.06974 0.99491
256 1.00067 0.99840 3 -1 0.5 1.75 1.82902 2.00926 1.74519 1.70824 1.00056
256 1.00067 0.99840 3 -1 1.0 1.00 1.09673 1.33612 1.01290 1.03510 1.00230
256 1.00067 0.99840 3 -1 1.5 -0.25 -0.14469 0.16335 -0.22853 -0.13767 1.00542
256 1.00067 0.99840 4 +1 0.5 1.75 1.63728 1.32673 1.74906 1.72810 0.99955
256 1.00067 0.99840 4 +1 1.0 1.00 0.90114 0.63703 1.01292 1.03839 0.99824
256 1.00067 0.99840 4 +1 1.5 -0.25 -0.31142 -0.47842 -0.19964 -0.07706 0.99616
256 1.00067 0.99840 4 -1 0.5 1.75 1.85751 2.11225 1.74573 1.71089 1.00043
256 1.00067 0.99840 4 -1 1.0 1.00 1.12468 1.43684 1.01290 1.03548 1.00173
256 1.00067 0.99840 4 -1 1.5 -0.25 -0.11219 0.27373 -0.22397 -0.12763 1.00403
512 1.00067 0.99938 3 +1 0.5 1.75 1.70349 1.55041 1.75057 1.74011 0.99966
512 1.00067 0.99938 3 +1 1.0 1.00 0.96211 0.83367 1.00920 1.02337 0.99868
512 1.00067 0.99938 3 +1 1.5 -0.25 -0.26229 -0.32977 -0.21521 -0.14007 0.99713
512 1.00067 0.99938 3 -1 0.5 1.75 1.79528 1.91619 1.74819 1.72649 1.00032
512 1.00067 0.99938 3 -1 1.0 1.00 1.05627 1.21113 1.00919 1.02143 1.00130
512 1.00067 0.99938 3 -1 1.5 -0.25 -0.18766 0.00087 -0.23475 -0.18883 1.00303
512 1.00067 0.99938 4 +1 0.5 1.75 1.68749 1.48528 1.75027 1.73821 0.99975
512 1.00067 0.99938 4 +1 1.0 1.00 0.94642 0.77007 1.00920 1.02300 0.99901
512 1.00067 0.99938 4 +1 1.5 -0.25 -0.28022 -0.39831 -0.21744 -0.14537 0.99784
512 1.00067 0.99938 4 -1 0.5 1.75 1.81127 1.98101 1.74849 1.72807 1.00024
512 1.00067 0.99938 4 -1 1.0 1.00 1.07197 1.27454 1.00919 1.02160 1.00097
512 1.00067 0.99938 4 -1 1.5 -0.25 -0.16927 0.07122 -0.23205 -0.18171 1.00226
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