05-ilang-free-fall / 11-bound-pair
11bound pairverified
Determines the bound level of two bodies of different types with a binding, its curvature against the inertias and the cost, and the clock rates of the parts, with no field.
check_bound_pair.py17 KB · raw
"""Check for 05-ilang-free-fall/11-bound-pair, derivation v1 (role: check).
Numerical diagonalization of the fibre contract H(P) of (11.1) on the finite
range r = -L..L (open ends), for diagonal on-site bindings, and comparison with
the derivation:
- exact relations (11.5)-(11.9) of item 2 (all parameter sets),
- the bound-level equation (11.10) of item 3(a) (all sets),
- the closed forms (11.12), (11.14), (11.16) of items 3 and 4 (sets in the regime R),
- the scaling of the deviations with epsilon (scan).
Two solvers are used for the same matrix: a dense one (all eigenvectors, once per
set and P) and a banded shift-and-invert one (for the finite differences and the
doubled range); their bound levels are compared.
In the outputs, "Gamma" is the expectation value of the symmetry Xi of Step 6.
Outputs: output/check_bound_pair.json, output/check_bound_pair.txt
"""
import json
from pathlib import Path
import numpy as np
from scipy.linalg import solve_banded
OUT = Path(__file__).parent / "output"
OUT.mkdir(parents=True, exist_ok=True)
s0 = np.eye(2)
sx = np.array([[0.0, 1.0], [1.0, 0.0]])
sz = np.diag([1.0, -1.0])
SX1, SX2 = np.kron(sx, s0), np.kron(s0, sx)
SZ1, SZ2 = np.kron(sz, s0), np.kron(s0, sz)
def fibre_ops(P, p, L, eta):
"""Fibre operators on r=-L..L; index 4*(r+L)+2*(s1-1)+(s2-1)."""
n = 2 * L + 1
S = np.eye(n, k=-1) # (S psi)(r) = psi(r-1)
Sd = S.T
e1, e2 = np.exp(1j * eta * P), np.exp(1j * (1 - eta) * P)
A1 = 0.5j * p["tau1"] * np.kron(e1 * Sd - np.conj(e1) * S, SX1)
A2 = 0.5j * p["tau2"] * np.kron(e2 * S - np.conj(e2) * Sd, SX2)
V1 = -0.5 * p["tau1"] * np.kron(e1 * Sd + np.conj(e1) * S, SX1)
V2 = -0.5 * p["tau2"] * np.kron(e2 * S + np.conj(e2) * Sd, SX2)
Z1, Z2 = np.kron(np.eye(n), SZ1), np.kron(np.eye(n), SZ2)
U = np.zeros((4 * n, 4 * n))
U[4 * L:4 * L + 4, 4 * L:4 * L + 4] = np.diag(p["w"]) # w11, w12, w21, w22
return A1, A2, V1, V2, Z1, Z2, U
def bands(P, p, nq=4001):
"""The four two-body bands (11.2) at the total label P: (lower edge, upper edge)."""
q = np.linspace(-np.pi, np.pi, nq)
om1 = np.sqrt(p["M1"] ** 2 + p["tau1"] ** 2 * np.sin(0.5 * P + q) ** 2)
om2 = np.sqrt(p["M2"] ** 2 + p["tau2"] ** 2 * np.sin(0.5 * P - q) ** 2)
return {f"{a:+d}{b:+d}": (float((a * om1 + b * om2).min()), float((a * om1 + b * om2).max()))
for a in (1, -1) for b in (1, -1)}
def edge(P, p):
"""Edge of the (++) band next to the level: lower edge for w11<0, upper edge for w11>0."""
return bands(P, p)["+1+1"][0 if p["w"][0] < 0 else 1]
def solve_full(P, p, L, eta=0.5):
"""Dense diagonalization; bound level = eigenvector of largest weight at (r=0, s=11)."""
A1, A2, V1, V2, Z1, Z2, U = fibre_ops(P, p, L, eta)
H = A1 + A2 + p["M1"] * Z1 + p["M2"] * Z2 + U
E, X = np.linalg.eigh(H)
b = int(np.argmax(np.abs(X[4 * L, :]) ** 2))
psi = X[:, b]
ex = lambda O: float(np.real(np.vdot(psi, O @ psi)))
amp = X.conj().T @ ((eta * V1 + (1 - eta) * V2) @ psi)
mask = np.ones(len(E), bool)
mask[b] = False
resolvent_sum = float(np.sum(np.abs(amp[mask]) ** 2 / (E[mask] - E[b])))
d2H = -(eta ** 2) * ex(A1) - (1 - eta) ** 2 * ex(A2)
r = np.repeat(np.arange(-L, L + 1), 4)
Gam = ((-1.0) ** np.abs(r)) * np.tile(np.diag(SZ1 @ SZ2), 2 * L + 1) # symmetry Xi of Step 6
return dict(E=float(E[b]), s3_1=ex(Z1), s3_2=ex(Z2), A1=ex(A1), A2=ex(A2), U=ex(U),
V1=ex(V1), V2=ex(V2), Epp_formula=d2H - 2 * resolvent_sum,
weight0=float(np.abs(psi[4 * L]) ** 2),
tail=float(np.sum(np.abs(psi[np.abs(r) > L // 2]) ** 2)),
Gamma=float(np.real(np.sum(Gam * np.abs(psi) ** 2))))
def solve(P, p, L, ref, eta=0.5, theta=1.0):
"""Banded solver: the eigenvalue of H(P) nearest to ref, by inverse iteration from the
vector (r=0, s=11), four steps at the shift ref and then Rayleigh-quotient steps."""
n = 4 * (2 * L + 1)
low = np.zeros((8, n), complex) # lower band storage: low[k, j] = H[j+k, j]
low[0, :] = np.tile(np.diag(p["M1"] * SZ1 + p["M2"] * SZ2), 2 * L + 1)
low[0, 4 * L:4 * L + 4] += theta * np.array(p["w"])
e1, e2 = np.exp(1j * eta * P), np.exp(1j * (1 - eta) * P)
B = -0.5j * p["tau1"] * np.conj(e1) * SX1 + 0.5j * p["tau2"] * e2 * SX2 # block (r+1, r): coefficient of S
for s_row in range(4):
for s_col in range(4):
low[4 + s_row - s_col, s_col:n - 4:4] = B[s_row, s_col]
full = np.zeros((15, n), complex) # storage of solve_banded: full[7+i-j, j] = H[i, j]
for k in range(8):
full[7 + k, :n - k] = low[k, :n - k]
if k:
full[7 - k, k:] = np.conj(low[k, :n - k])
def matvec(v):
out = low[0] * v
for k in range(1, 8):
out[k:] += low[k, :n - k] * v[:n - k]
out[:n - k] += np.conj(low[k, :n - k]) * v[k:]
return out
v = np.zeros(n, complex)
v[4 * L] = 1.0
shift, E = ref, ref
for it in range(12):
m = full.copy()
m[7] -= shift
try:
v = solve_banded((7, 7), m, v)
except np.linalg.LinAlgError:
break
v /= np.linalg.norm(v)
Hv = matvec(v)
E = float(np.real(np.vdot(v, Hv)))
if np.linalg.norm(Hv - E * v) < 1e-13:
break
if it >= 3:
shift = E
assert abs(v[4 * L]) ** 2 > 0.05, "banded solver left the bound level"
return E
def d1(f, x, h):
return (-f(x + 2 * h) + 8 * f(x + h) - 8 * f(x - h) + f(x - 2 * h)) / (12 * h)
def d2(f, x, h):
return (-f(x + 2 * h) + 16 * f(x + h) - 30 * f(x) + 16 * f(x - h) - f(x - 2 * h)) / (12 * h * h)
def green(E, P, p, nq=4096, eta=0.5):
"""Coefficients of (11.10): G_{s1 s2}(E,P) and g_x(E,P), as sums over q."""
q = -np.pi + 2 * np.pi * np.arange(nq) / nq
p1, p2 = eta * P + q, (1 - eta) * P - q
om1 = np.sqrt(p["M1"] ** 2 + p["tau1"] ** 2 * np.sin(p1) ** 2)
om2 = np.sqrt(p["M2"] ** 2 + p["tau2"] ** 2 * np.sin(p2) ** 2)
G, gx = {}, 0.0
for sg1 in (1, -1):
for sg2 in (1, -1):
den = 4 * (sg1 * om1 + sg2 * om2 - E)
gx += np.mean(sg1 * sg2 * p["tau1"] * p["tau2"] * np.sin(p1) * np.sin(p2) / (om1 * om2 * den))
for t1 in (1, 2):
for t2 in (1, 2):
c1, c2 = (1 if t1 == 1 else -1), (1 if t2 == 1 else -1)
G[(t1, t2)] = G.get((t1, t2), 0.0) + np.mean(
(1 + c1 * sg1 * p["M1"] / om1) * (1 + c2 * sg2 * p["M2"] / om2) / den)
return G, gx
def green_direct(E, P, p, nq=4096, eta=0.5):
"""<r=0|(H0(P)-E)^(-1)|r=0> as a 4x4 matrix, by inverting h_P(q)-E on the q grid."""
q = -np.pi + 2 * np.pi * np.arange(nq) / nq
p1, p2 = eta * P + q, (1 - eta) * P - q
h = (-p["tau1"] * np.sin(p1))[:, None, None] * SX1 + (-p["tau2"] * np.sin(p2))[:, None, None] * SX2 \
+ p["M1"] * SZ1 + p["M2"] * SZ2 - E * np.eye(4)
return np.linalg.inv(h).mean(axis=0)
def F_plus(E, P, p):
"""The factor F_+ of (11.10) (sector Xi=+1)."""
G, gx = green(E, P, p)
w11, _, _, w22 = p["w"]
return (1 + w11 * G[(1, 1)]) * (1 + w22 * G[(2, 2)]) - w11 * w22 * gx ** 2
def root_F_plus(Eb, P, p, gap):
lo, hi = Eb - 0.3 * gap, Eb + 0.3 * gap
flo, fhi = F_plus(lo, P, p), F_plus(hi, P, p)
if flo * fhi > 0:
return None
for _ in range(100):
mid = 0.5 * (lo + hi)
fm = F_plus(mid, P, p)
if flo * fm <= 0:
hi, fhi = mid, fm
else:
lo, flo = mid, fm
return 0.5 * (lo + hi)
def closed(P, p):
"""Closed forms of item 3 (leading order in R); expectation values at P=0."""
M1, M2, t1, t2 = p["M1"], p["M2"], p["tau1"], p["tau2"]
w = p["w"][0]
sg = np.sign(w)
a1, a2 = t1 ** 2 / (4 * M1), t2 ** 2 / (4 * M2)
a = a1 + a2
j = np.sqrt(a1 ** 2 + a2 ** 2 + 2 * a1 * a2 * np.cos(2 * P))
root0 = np.sqrt(w * w + a * a)
c = 1 + sg * a / root0
return dict(E=M1 + M2 + a + sg * np.sqrt(w * w + j * j),
Ep=-sg * 2 * a1 * a2 * np.sin(2 * P) / np.sqrt(w * w + j * j),
Epp0=-sg * 4 * a1 * a2 / root0, Ebind=a + sg * root0,
s3_1=1 - a1 / M1 * c, s3_2=1 - a2 / M2 * c, A1=2 * a1 * c, A2=2 * a2 * c,
U=sg * w * w / root0, a=a, u=abs(w) / a)
def with_(p, **kw):
q = dict(p)
q.update(kw)
return q
SETS = {
"A_equal_R": dict(M1=1.0, M2=1.0, tau1=0.10, tau2=0.10, w=[-0.0040, 0.0030, -0.0020, 0.0060]),
"B_unequal_R": dict(M1=1.0, M2=1.7, tau1=0.20, tau2=0.12, w=[-0.0100, 0.0050, -0.0040, 0.0080]),
"C_above_band_R": dict(M1=1.0, M2=0.8, tau1=0.10, tau2=0.15, w=[0.0200, -0.0050, 0.0030, -0.0100]),
"D_wide_R": dict(M1=1.3, M2=1.0, tau1=0.15, tau2=0.20, w=[-0.0040, 0.0020, 0.0010, 0.0030]),
"E_order_one": dict(M1=1.0, M2=1.3, tau1=1.00, tau2=0.80, w=[-0.6000, 0.2000, -0.1500, 0.3000]),
}
REGIME_R = {"A_equal_R", "B_unequal_R", "C_above_band_R", "D_wide_R"}
P_VALUES = [-0.30, -0.15, 0.0, 0.15, 0.30]
L = 120
HP, HX = 0.01, 1e-3
results, lines = {}, []
def say(s):
lines.append(s)
print(s, flush=True)
say(f"range of r: -L..L with L={L} (open ends); convergence: comparison with L={2 * L} and weight at |r|>L/2")
say(f"finite differences: 5-point, step {HP} in P, relative step {HX} in M_i, tau_i, theta; convention eta=1/2")
for name, p in SETS.items():
say(f"\n=== set {name}: {p} max tau_i/M_i = {max(p['tau1'] / p['M1'], p['tau2'] / p['M2']):.3f}")
res = dict(params=p, P={})
keys = ("dM", "dtau", "dtheta", "euler", "V1", "V2", "V12", "Epp", "eq310", "green", "conv", "tail", "eta",
"solvers", "closed_E", "closed_Ep")
worst = dict.fromkeys(keys, 0.0)
for P in P_VALUES:
o = solve_full(P, p, L)
bd = bands(P, p)
gap = min(abs(o["E"] - e) for lohi in bd.values() for e in lohi)
inside = any(lo <= o["E"] <= hi for lo, hi in bd.values())
e0 = edge(P, p)
def fE(PP=P, pp=p, **kw):
"""Banded bound level at changed P or data; the reference follows the band edge."""
return solve(PP, pp, L, o["E"] + edge(PP, pp) - e0, **kw)
solvers = abs(fE() - o["E"])
conv = abs(solve(P, p, 2 * L, o["E"]) - o["E"])
eta_dep = max(abs(fE(eta=e) - o["E"]) for e in (0.0, 1.0, 0.3))
# item 2(a): derivatives of E_b against expectation values
dM1 = d1(lambda x: fE(pp=with_(p, M1=x)), p["M1"], HX * p["M1"])
dM2 = d1(lambda x: fE(pp=with_(p, M2=x)), p["M2"], HX * p["M2"])
dt1 = p["tau1"] * d1(lambda x: fE(pp=with_(p, tau1=x)), p["tau1"], HX * p["tau1"])
dt2 = p["tau2"] * d1(lambda x: fE(pp=with_(p, tau2=x)), p["tau2"], HX * p["tau2"])
dth = d1(lambda x: fE(theta=x), 1.0, HX)
euler = p["M1"] * o["s3_1"] + p["M2"] * o["s3_2"] + o["A1"] + o["A2"] + o["U"] - o["E"]
# items 2(b), 2(c)
Ep = d1(lambda x: fE(PP=x), P, HP)
Epp = d2(lambda x: fE(PP=x), P, HP)
# item 3(a): coefficients against a direct inversion, root against E_b
G, gx = green(o["E"], P, p)
Gd = green_direct(o["E"], P, p)
gerr = max(abs(G[(1, 1)] - Gd[0, 0]), abs(G[(1, 2)] - Gd[1, 1]), abs(G[(2, 1)] - Gd[2, 2]),
abs(G[(2, 2)] - Gd[3, 3]), abs(gx - Gd[0, 3]), abs(gx - Gd[1, 2]),
abs(Gd[0, 1]), abs(Gd[0, 2]), abs(Gd[1, 3]), abs(Gd[2, 3])) / np.abs(Gd).max()
root = root_F_plus(o["E"], P, p, gap)
c = closed(P, p)
res["P"][f"{P:+.2f}"] = dict(
E=o["E"], isolation=gap, inside_band=inside, weight0=o["weight0"], Gamma=o["Gamma"],
tail=o["tail"], conv_2L=conv, eta_dependence=eta_dep, dense_minus_banded=solvers,
dM1=[dM1, o["s3_1"]], dM2=[dM2, o["s3_2"]], dtau1=[dt1, o["A1"]], dtau2=[dt2, o["A2"]],
dtheta=[dth, o["U"]], euler_residual=euler, Ep=[Ep, o["V1"], o["V2"]], Epp=[Epp, o["Epp_formula"]],
green_rel_error=float(gerr), eq_3a_root=root,
eq_3a_root_minus_E=(None if root is None else root - o["E"]),
closed_E=float(c["E"]), closed_Ep=float(c["Ep"]))
for k, v in (("dM", max(abs(dM1 - o["s3_1"]), abs(dM2 - o["s3_2"]))),
("dtau", max(abs(dt1 - o["A1"]), abs(dt2 - o["A2"]))), ("dtheta", abs(dth - o["U"])),
("euler", abs(euler)), ("V1", abs(Ep - o["V1"])), ("V2", abs(Ep - o["V2"])),
("V12", abs(o["V1"] - o["V2"])), ("Epp", abs(Epp - o["Epp_formula"])),
("eq310", np.inf if root is None else abs(root - o["E"])), ("green", float(gerr)),
("conv", conv), ("tail", o["tail"]), ("eta", eta_dep), ("solvers", solvers),
("closed_E", abs(c["E"] - o["E"])), ("closed_Ep", abs(c["Ep"] - Ep))):
worst[k] = max(worst[k], float(v))
say(f"P={P:+.2f} E_b={o['E']:.12f} isolation={gap:.3e} in a band: {inside} weight(r=0,11)={o['weight0']:.4f}"
f" Gamma={o['Gamma']:+.6f} E_b'={Ep:+.6e} <V1>={o['V1']:+.6e} <V2>={o['V2']:+.6e}"
f" E_b''={Epp:+.6e} (11.9)={o['Epp_formula']:+.6e}")
res["max_abs_deviation"] = worst
say("max |deviation| over the P values:")
say(f" 2(a): dE/dM_i - <sigma3_i> {worst['dM']:.2e}; tau_i dE/dtau_i - <A_i> {worst['dtau']:.2e};"
f" dE/dtheta - <U> {worst['dtheta']:.2e}; Euler residual {worst['euler']:.2e}")
say(f" 2(b): E_b' - <V1> {worst['V1']:.2e}; E_b' - <V2> {worst['V2']:.2e}; <V1> - <V2> {worst['V12']:.2e}")
say(f" 2(c): E_b'' (finite difference) - formula (11.9) {worst['Epp']:.2e}")
say(f" 3(a): coefficients of (11.10) against direct inversion (relative) {worst['green']:.2e};"
f" root of (11.10) - E_b {worst['eq310']:.2e}")
say(f" numerics: |E_b(L) - E_b(2L)| {worst['conv']:.2e}; weight at |r|>L/2 {worst['tail']:.2e};"
f" dependence on eta {worst['eta']:.2e}; dense - banded {worst['solvers']:.2e}")
# closed forms at P = 0
o0 = solve_full(0.0, p, L)
c0 = closed(0.0, p)
Epp0 = o0["Epp_formula"]
eps2 = max(p["tau1"] / p["M1"], p["tau2"] / p["M2"]) ** 2
cmp0 = {k: [o0[k], float(c0[k])] for k in ("s3_1", "s3_2", "A1", "A2", "U")}
cmp0["Ebind"] = [o0["E"] - p["M1"] - p["M2"], float(c0["Ebind"])]
cmp0["Epp0"] = [Epp0, float(c0["Epp0"])]
m1, m2 = p["M1"] / p["tau1"] ** 2, p["M2"] / p["tau2"] ** 2
mb = 1.0 / Epp0
item4 = dict(mb_over_m1_plus_m2=[mb / (m1 + m2), float(-np.sign(p["w"][0]) * np.sqrt(1 + c0["u"] ** 2))],
mb_minus_m1_m2=[mb - m1 - m2, -4 * m1 * m2 * (o0["E"] - p["M1"] - p["M2"])])
res["closed_forms_P0"] = dict(u=float(c0["u"]), a=float(c0["a"]), eps2=eps2, compare=cmp0, item4=item4,
max_dE_over_P=worst["closed_E"], max_dEp_over_P=worst["closed_Ep"],
rel_dev_Epp0=float(abs(Epp0 - c0["Epp0"]) / abs(c0["Epp0"])))
tag = "regime R" if name in REGIME_R else "control, outside R"
say(f"closed forms (items 3, 4), {tag}: |w11|/a = {c0['u']:.3f}, (max tau/M)^2 = {eps2:.3e}")
say(f" max_P |E_b - (11.12)| = {worst['closed_E']:.3e} = {worst['closed_E'] / (eps2 ** 2):.3f} * (tau/M)^4;"
f" max_P |E_b' - d(11.12)/dP| = {worst['closed_Ep']:.3e}")
say(f" E_b''(0): numeric {Epp0:+.8e} closed {c0['Epp0']:+.8e} rel.dev {abs(Epp0 - c0['Epp0']) / abs(c0['Epp0']):.3e}"
f" = {abs(Epp0 - c0['Epp0']) / abs(c0['Epp0']) / eps2:.3f} * (tau/M)^2")
for k, (num, cl) in cmp0.items():
say(f" {k:6s} numeric {num:+.10e} closed {cl:+.10e} diff {num - cl:+.2e}")
say(f" m_b/(m_1+m_2): numeric {item4['mb_over_m1_plus_m2'][0]:+.6f} (11.16) {item4['mb_over_m1_plus_m2'][1]:+.6f};"
f" m_b-(m_1+m_2): numeric {item4['mb_minus_m1_m2'][0]:+.6e} -4 m_1 m_2 E_bind {item4['mb_minus_m1_m2'][1]:+.6e}")
results[name] = res
# epsilon scan: tau_i = eps*tauhat_i, w = eps^2*what, for the data of set B (eps = 1)
say("\n=== epsilon scan on set B_unequal_R (tau -> eps*tau, w -> eps^2*w)")
base = SETS["B_unequal_R"]
scan, prev = [], None
for eps in (1.0, 0.5, 0.25, 0.125):
p = with_(base, tau1=eps * base["tau1"], tau2=eps * base["tau2"], w=[eps * eps * x for x in base["w"]])
o0, c0 = solve_full(0.0, p, L), closed(0.0, p)
row = dict(eps=eps, dE_P0=o0["E"] - c0["E"], dE_P03=solve_full(0.3, p, L)["E"] - closed(0.3, p)["E"],
rel_dEpp0=(o0["Epp_formula"] - c0["Epp0"]) / c0["Epp0"], d_s3_1=o0["s3_1"] - c0["s3_1"],
d_s3_2=o0["s3_2"] - c0["s3_2"])
row = {k: float(v) for k, v in row.items()}
if prev:
row["ratio_to_previous"] = {k: prev[k] / row[k] for k in row if k != "eps"}
scan.append(row)
prev = {k: v for k, v in row.items() if k != "ratio_to_previous"}
say(f"eps={eps:6.3f} E_b-(11.12): P=0 {row['dE_P0']:+.4e}, P=0.3 {row['dE_P03']:+.4e}; rel.dev E_b''(0) {row['rel_dEpp0']:+.4e};"
f" <sigma3_i>-(11.14): {row['d_s3_1']:+.4e}, {row['d_s3_2']:+.4e}")
if "ratio_to_previous" in row:
say(" ratios to the previous eps: " + ", ".join(f"{k} {v:.2f}" for k, v in row["ratio_to_previous"].items()))
say("expected for halving eps: 16 for E_b and <sigma3_i> (next order eps^4), 4 for the relative deviation of E_b''(0)")
results["epsilon_scan_set_B"] = scan
results["settings"] = dict(L=L, L_check=2 * L, P_values=P_VALUES, eta=0.5, h_P=HP, h_rel_parameters=HX)
(OUT / "check_bound_pair.json").write_text(json.dumps(results, indent=1), encoding="utf-8")
(OUT / "check_bound_pair.txt").write_text("\n".join(lines) + "\n", encoding="utf-8")
output/check_bound_pair.json39 KB · raw
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0.1286413687625461
],
"dtheta": [
-0.416264047684541,
-0.41626404768504804
],
"euler_residual": -4.440892098500626e-16,
"Ep": [
0.027769969482642026,
0.027769969570427034,
0.027769969570427072
],
"Epp": [
0.18012699007659747,
0.1801269902164982
],
"green_rel_error": 3.1895603535462812e-15,
"eq_3a_root": 2.014120790385851,
"eq_3a_root_minus_E": -4.440892098500626e-16,
"closed_E": 1.9684934526588127,
"closed_Ep": 0.025810794086150356
},
"+0.30": {
"E": 2.0202430653650967,
"isolation": 0.29440305721726867,
"inside_band": false,
"weight0": 0.701707383851894,
"Gamma": 1.0000000000000004,
"tail": 1.7199875421734497e-30,
"conv_2L": 4.440892098500626e-16,
"eta_dependence": 4.440892098500626e-16,
"dense_minus_banded": 4.440892098500626e-16,
"dM1": [
0.8503318666591339,
0.8503318666583716
],
"dM2": [
0.9328219265201905,
0.9328219265196965
],
"dtau1": [
0.23778305062290467,
0.23778305062442884
],
"dtau2": [
0.13944747911157998,
0.13944747911219654
],
"dtheta": [
-0.41998783550600827,
-0.41998783550550445
],
"euler_residual": 1.3322676295501878e-15,
"Ep": [
0.0533071623368205,
0.05330716251086799,
0.05330716251086764
],
"Epp": [
0.15802105920041204,
0.15802105937262773
],
"green_rel_error": 3.9876928283906716e-16,
"eq_3a_root": 2.020243065365098,
"eq_3a_root_minus_E": 1.3322676295501878e-15,
"closed_E": 1.9741936617317075,
"closed_Ep": 0.049718216265647104
}
},
"max_abs_deviation": {
"dM": 7.622791287076325e-13,
"dtau": 1.5241696793566462e-12,
"dtheta": 1.1470269178914805e-12,
"euler": 1.3322676295501878e-15,
"V1": 1.7404749169669387e-10,
"V2": 1.7404713781310477e-10,
"V12": 3.5388358909926865e-16,
"Epp": 2.4022667188816627e-10,
"eq310": 1.3322676295501878e-15,
"green": 3.588255397739565e-15,
"conv": 8.881784197001252e-16,
"tail": 2.2610645562733344e-30,
"eta": 8.881784197001252e-16,
"solvers": 2.6645352591003757e-15,
"closed_E": 0.04604940363338916,
"closed_Ep": 0.003588946071173396
},
"closed_forms_P0": {
"u": 1.6082474226804122,
"a": 0.3730769230769231,
"eps2": 1.0,
"compare": {
"s3_1": [
0.8564696712688329,
0.8820100594684337
],
"s3_2": [
0.9385908911908176,
0.9553174189702944
],
"A1": [
0.22543811169631017,
0.23597988106313272
],
"A2": [
0.1249503925193654,
0.11617471067723459
],
"U": [
-0.4150023989441348,
-0.5095316109791912
],
"Ebind": [
-0.287976064911563,
-0.3334543151090077
],
"Epp0": [
0.18764361764368498,
0.1741988413604073
]
},
"item4": {
"mb_over_m1_plus_m2": [
1.7581035334915232,
1.893795071426206
],
"mb_minus_m1_m2": [
2.298001335896179,
2.339805527406449
]
},
"max_dE_over_P": 0.04604940363338916,
"max_dEp_over_P": 0.003588946071173396,
"rel_dev_Epp0": 0.07718062978077568
}
},
"epsilon_scan_set_B": [
{
"eps": 1.0,
"dE_P0": 8.151469865325112e-06,
"dE_P03": 8.267477692225356e-06,
"rel_dEpp0": 0.0005445581432701723,
"d_s3_1": -4.2426369883941106e-07,
"d_s3_2": -1.7673889233238427e-06
},
{
"eps": 0.5,
"dE_P0": 5.094345443268367e-07,
"dE_P03": 5.167414247608804e-07,
"rel_dEpp0": 0.00013707918516712943,
"d_s3_1": -2.720324154825704e-08,
"d_s3_2": -1.10197183156302e-07,
"ratio_to_previous": {
"dE_P0": 16.001015157102092,
"dE_P03": 15.99925474535178,
"rel_dEpp0": 3.972580830607048,
"d_s3_1": 15.596071449308377,
"d_s3_2": 16.038421969616095
}
},
{
"eps": 0.25,
"dE_P0": 3.183914820681366e-08,
"dE_P03": 3.2296711527379784e-08,
"rel_dEpp0": 3.432847031626075e-05,
"d_s3_1": -1.7109325067821146e-09,
"d_s3_2": -6.883195680096321e-09,
"ratio_to_previous": {
"dE_P0": 16.000256697125344,
"dE_P03": 15.999815471083144,
"rel_dEpp0": 3.9931632229530947,
"d_s3_1": 15.899657900252487,
"d_s3_2": 16.00959616402478
}
},
{
"eps": 0.125,
"dE_P0": 1.9899379921639593e-09,
"dE_P03": 2.018550659954599e-09,
"rel_dEpp0": 8.585780804080197e-06,
"d_s3_1": -1.0710143882874945e-10,
"d_s3_2": -4.3013559380966626e-10,
"ratio_to_previous": {
"dE_P0": 16.00007052088601,
"dE_P03": 15.999950939109054,
"rel_dEpp0": 3.9982933526496423,
"d_s3_1": 15.974878820422024,
"d_s3_2": 16.002385710823354
}
}
],
"settings": {
"L": 120,
"L_check": 240,
"P_values": [
-0.3,
-0.15,
0.0,
0.15,
0.3
],
"eta": 0.5,
"h_P": 0.01,
"h_rel_parameters": 0.001
}
}output/check_bound_pair.txt14 KB · raw
range of r: -L..L with L=120 (open ends); convergence: comparison with L=240 and weight at |r|>L/2
finite differences: 5-point, step 0.01 in P, relative step 0.001 in M_i, tau_i, theta; convention eta=1/2
=== set A_equal_R: {'M1': 1.0, 'M2': 1.0, 'tau1': 0.1, 'tau2': 0.1, 'w': [-0.004, 0.003, -0.002, 0.006]} max tau_i/M_i = 0.100
P=-0.30 E_b=1.998771748883 isolation=1.452e-03 in a band: False weight(r=0,11)=0.6408 Gamma=+1.000000 E_b'=-1.133090e-03 <V1>=-1.133090e-03 <V2>=-1.133090e-03 E_b''=+3.517649e-03 (11.9)=+3.517649e-03
P=-0.15 E_b=1.998642641792 isolation=1.414e-03 in a band: False weight(r=0,11)=0.6278 Gamma=+1.000000 E_b'=-5.810445e-04 <V1>=-5.810445e-04 <V2>=-5.810445e-04 E_b''=+3.809572e-03 (11.9)=+3.809572e-03
P=+0.00 E_b=1.998598883527 isolation=1.401e-03 in a band: False weight(r=0,11)=0.6235 Gamma=+1.000000 E_b'=+1.850372e-15 <V1>=+7.893878e-21 <V2>=-2.053590e-20 E_b''=+3.905596e-03 (11.9)=+3.905596e-03
P=+0.15 E_b=1.998642641792 isolation=1.414e-03 in a band: False weight(r=0,11)=0.6278 Gamma=+1.000000 E_b'=+5.810445e-04 <V1>=+5.810445e-04 <V2>=+5.810445e-04 E_b''=+3.809572e-03 (11.9)=+3.809572e-03
P=+0.30 E_b=1.998771748883 isolation=1.452e-03 in a band: False weight(r=0,11)=0.6408 Gamma=+1.000000 E_b'=+1.133090e-03 <V1>=+1.133090e-03 <V2>=+1.133090e-03 E_b''=+3.517649e-03 (11.9)=+3.517649e-03
max |deviation| over the P values:
2(a): dE/dM_i - <sigma3_i> 1.25e-12; tau_i dE/dtau_i - <A_i> 1.37e-12; dE/dtheta - <U> 8.96e-13; Euler residual 4.22e-15
2(b): E_b' - <V1> 1.59e-12; E_b' - <V2> 1.59e-12; <V1> - <V2> 6.51e-19
2(c): E_b'' (finite difference) - formula (11.9) 1.59e-11
3(a): coefficients of (11.10) against direct inversion (relative) 1.19e-14; root of (11.10) - E_b 1.55e-15
numerics: |E_b(L) - E_b(2L)| 2.00e-15; weight at |r|>L/2 3.17e-20; dependence on eta 2.44e-15; dense - banded 2.44e-15
closed forms (items 3, 4), regime R: |w11|/a = 0.800, (max tau/M)^2 = 1.000e-02
max_P |E_b - (11.12)| = 2.053e-06 = 0.021 * (tau/M)^4; max_P |E_b' - d(11.12)/dP| = 2.354e-07
E_b''(0): numeric +3.90559613e-03 closed +3.90434405e-03 rel.dev 3.207e-04 = 0.032 * (tau/M)^2
s3_1 numeric +9.9945154586e-01 closed +9.9945217202e-01 diff -6.26e-07
s3_2 numeric +9.9945154586e-01 closed +9.9945217202e-01 diff -6.26e-07
A1 numeric +1.0948993397e-03 closed +1.0956559528e-03 diff -7.57e-07
A2 numeric +1.0948993397e-03 closed +1.0956559528e-03 diff -7.57e-07
U numeric -2.4940068754e-03 closed -2.4987801902e-03 diff +4.77e-06
Ebind numeric -1.4011164735e-03 closed -1.4031242374e-03 diff +2.01e-06
Epp0 numeric +3.9055961250e-03 closed +3.9043440472e-03 diff +1.25e-06
m_b/(m_1+m_2): numeric +1.280214 (11.16) +1.280625; m_b-(m_1+m_2): numeric +5.604286e+01 -4 m_1 m_2 E_bind +5.604466e+01
=== set B_unequal_R: {'M1': 1.0, 'M2': 1.7, 'tau1': 0.2, 'tau2': 0.12, 'w': [-0.01, 0.005, -0.004, 0.008]} max tau_i/M_i = 0.200
P=-0.30 E_b=2.696652071452 isolation=3.657e-03 in a band: False weight(r=0,11)=0.6441 Gamma=+1.000000 E_b'=-1.546134e-03 <V1>=-1.546134e-03 <V2>=-1.546134e-03 E_b''=+4.673251e-03 (11.9)=+4.673251e-03
P=-0.15 E_b=2.696475088077 isolation=3.603e-03 in a band: False weight(r=0,11)=0.6368 Gamma=+1.000000 E_b'=-8.001290e-04 <V1>=-8.001290e-04 <V2>=-8.001290e-04 E_b''=+5.213785e-03 (11.9)=+5.213785e-03
P=+0.00 E_b=2.696414739677 isolation=3.585e-03 in a band: False weight(r=0,11)=0.6344 Gamma=+1.000000 E_b'=+1.110223e-14 <V1>=-1.752348e-19 <V2>=+3.687192e-20 E_b''=+5.394417e-03 (11.9)=+5.394417e-03
P=+0.15 E_b=2.696475088077 isolation=3.603e-03 in a band: False weight(r=0,11)=0.6368 Gamma=+1.000000 E_b'=+8.001290e-04 <V1>=+8.001290e-04 <V2>=+8.001290e-04 E_b''=+5.213785e-03 (11.9)=+5.213785e-03
P=+0.30 E_b=2.696652071452 isolation=3.657e-03 in a band: False weight(r=0,11)=0.6441 Gamma=+1.000000 E_b'=+1.546134e-03 <V1>=+1.546134e-03 <V2>=+1.546134e-03 E_b''=+4.673251e-03 (11.9)=+4.673251e-03
max |deviation| over the P values:
2(a): dE/dM_i - <sigma3_i> 1.49e-12; tau_i dE/dtau_i - <A_i> 1.56e-12; dE/dtheta - <U> 6.98e-13; Euler residual 4.44e-15
2(b): E_b' - <V1> 8.50e-13; E_b' - <V2> 8.50e-13; <V1> - <V2> 7.35e-17
2(c): E_b'' (finite difference) - formula (11.9) 1.83e-11
3(a): coefficients of (11.10) against direct inversion (relative) 4.28e-14; root of (11.10) - E_b 4.88e-15
numerics: |E_b(L) - E_b(2L)| 3.11e-15; weight at |r|>L/2 9.86e-21; dependence on eta 6.22e-15; dense - banded 3.55e-15
closed forms (items 3, 4), regime R: |w11|/a = 0.825, (max tau/M)^2 = 4.000e-02
max_P |E_b - (11.12)| = 8.267e-06 = 0.005 * (tau/M)^4; max_P |E_b' - d(11.12)/dP| = 6.697e-07
E_b''(0): numeric +5.39441742e-03 closed +5.39148145e-03 rel.dev 5.446e-04 = 0.014 * (tau/M)^2
s3_1 numeric +9.9771238948e-01 closed +9.9771281374e-01 diff -4.24e-07
s3_2 numeric +9.9971332360e-01 closed +9.9971509098e-01 diff -1.77e-06
A1 numeric +4.5625732894e-03 closed +4.5743725191e-03 diff -1.18e-05
A2 numeric +9.7104322032e-04 closed +9.6869065111e-04 diff +2.35e-06
U numeric -6.3439164227e-03 closed -6.3649433780e-03 diff +2.10e-05
Ebind numeric -3.5852603230e-03 closed -3.5934117929e-03 diff +8.15e-06
Epp0 numeric +5.3944174247e-03 closed +5.3914814496e-03 diff +2.94e-06
m_b/(m_1+m_2): numeric +1.295838 (11.16) +1.296544; m_b-(m_1+m_2): numeric +4.232127e+01 -4 m_1 m_2 E_bind +4.232599e+01
=== set C_above_band_R: {'M1': 1.0, 'M2': 0.8, 'tau1': 0.1, 'tau2': 0.15, 'w': [0.02, -0.005, 0.003, -0.01]} max tau_i/M_i = 0.187
P=-0.30 E_b=1.831346531561 isolation=1.274e-02 in a band: False weight(r=0,11)=0.9037 Gamma=+1.000000 E_b'=+8.867732e-04 <V1>=+8.867732e-04 <V2>=+8.867732e-04 E_b''=-2.629234e-03 (11.9)=-2.629234e-03
P=-0.15 E_b=1.831448371991 isolation=1.260e-02 in a band: False weight(r=0,11)=0.8994 Gamma=+1.000000 E_b'=+4.619112e-04 <V1>=+4.619112e-04 <V2>=+4.619112e-04 E_b''=-2.996418e-03 (11.9)=-2.996418e-03
P=+0.00 E_b=1.831483249143 isolation=1.255e-02 in a band: False weight(r=0,11)=0.8980 Gamma=+1.000000 E_b'=+1.850372e-15 <V1>=-1.604183e-19 <V2>=+5.735736e-19 E_b''=-3.121011e-03 (11.9)=-3.121011e-03
P=+0.15 E_b=1.831448371991 isolation=1.260e-02 in a band: False weight(r=0,11)=0.8994 Gamma=+1.000000 E_b'=-4.619112e-04 <V1>=-4.619112e-04 <V2>=-4.619112e-04 E_b''=-2.996418e-03 (11.9)=-2.996418e-03
P=+0.30 E_b=1.831346531561 isolation=1.274e-02 in a band: False weight(r=0,11)=0.9037 Gamma=+1.000000 E_b'=-8.867732e-04 <V1>=-8.867732e-04 <V2>=-8.867732e-04 E_b''=-2.629234e-03 (11.9)=-2.629234e-03
max |deviation| over the P values:
2(a): dE/dM_i - <sigma3_i> 1.04e-12; tau_i dE/dtau_i - <A_i> 1.30e-12; dE/dtheta - <U> 1.11e-12; Euler residual 1.78e-15
2(b): E_b' - <V1> 2.74e-12; E_b' - <V2> 2.74e-12; <V1> - <V2> 8.55e-17
2(c): E_b'' (finite difference) - formula (11.9) 1.26e-11
3(a): coefficients of (11.10) against direct inversion (relative) 4.97e-15; root of (11.10) - E_b 8.88e-16
numerics: |E_b(L) - E_b(2L)| 2.22e-16; weight at |r|>L/2 5.56e-28; dependence on eta 1.33e-15; dense - banded 4.44e-16
closed forms (items 3, 4), regime R: |w11|/a = 2.098, (max tau/M)^2 = 3.516e-02
max_P |E_b - (11.12)| = 2.030e-04 = 0.164 * (tau/M)^4; max_P |E_b' - d(11.12)/dP| = 1.488e-05
E_b''(0): numeric -3.12101105e-03 closed -3.17366056e-03 rel.dev 1.659e-02 = 0.472 * (tau/M)^2
s3_1 numeric +9.9650760383e-01 closed +9.9642448170e-01 diff +8.31e-05
s3_2 numeric +9.8795706438e-01 closed +9.8742981847e-01 diff +5.27e-04
A1 numeric +7.0534612009e-03 closed +7.1510366031e-03 diff -9.76e-05
A2 numeric +1.9596946814e-02 closed +2.0112290446e-02 diff -5.15e-04
U numeric +1.7959585791e-02 closed +1.8054602308e-02 diff -9.50e-05
Ebind numeric +3.1483249143e-02 closed +3.1686265833e-02 diff -2.03e-04
Epp0 numeric -3.1210110472e-03 closed -3.1736605620e-03 diff +5.26e-05
m_b/(m_1+m_2): numeric -2.363673 (11.16) -2.324461; m_b-(m_1+m_2): numeric -4.559645e+02 -4 m_1 m_2 E_bind -4.477618e+02
=== set D_wide_R: {'M1': 1.3, 'M2': 1.0, 'tau1': 0.15, 'tau2': 0.2, 'w': [-0.004, 0.002, 0.001, 0.003]} max tau_i/M_i = 0.200
P=-0.30 E_b=2.299969840671 isolation=5.677e-04 in a band: False weight(r=0,11)=0.2780 Gamma=+1.000000 E_b'=-3.402251e-03 <V1>=-3.402251e-03 <V2>=-3.402251e-03 E_b''=+1.074662e-02 (11.9)=+1.074662e-02
P=-0.15 E_b=2.299583345287 isolation=5.522e-04 in a band: False weight(r=0,11)=0.2707 Gamma=+1.000000 E_b'=-1.734310e-03 <V1>=-1.734310e-03 <V2>=-1.734310e-03 E_b''=+1.141561e-02 (11.9)=+1.141561e-02
P=+0.00 E_b=2.299452860723 isolation=5.471e-04 in a band: False weight(r=0,11)=0.2684 Gamma=+1.000000 E_b'=+1.110223e-14 <V1>=-2.235218e-19 <V2>=+5.295572e-19 E_b''=+1.163513e-02 (11.9)=+1.163513e-02
P=+0.15 E_b=2.299583345287 isolation=5.522e-04 in a band: False weight(r=0,11)=0.2707 Gamma=+1.000000 E_b'=+1.734310e-03 <V1>=+1.734310e-03 <V2>=+1.734310e-03 E_b''=+1.141561e-02 (11.9)=+1.141561e-02
P=+0.30 E_b=2.299969840671 isolation=5.677e-04 in a band: False weight(r=0,11)=0.2780 Gamma=+1.000000 E_b'=+3.402251e-03 <V1>=+3.402251e-03 <V2>=+3.402251e-03 E_b''=+1.074662e-02 (11.9)=+1.074662e-02
max |deviation| over the P values:
2(a): dE/dM_i - <sigma3_i> 2.05e-12; tau_i dE/dtau_i - <A_i> 1.96e-12; dE/dtheta - <U> 2.07e-12; Euler residual 5.33e-15
2(b): E_b' - <V1> 5.04e-12; E_b' - <V2> 5.04e-12; <V1> - <V2> 8.46e-17
2(c): E_b'' (finite difference) - formula (11.9) 9.92e-11
3(a): coefficients of (11.10) against direct inversion (relative) 2.34e-13; root of (11.10) - E_b 1.78e-15
numerics: |E_b(L) - E_b(2L)| 4.44e-15; weight at |r|>L/2 4.85e-08; dependence on eta 4.44e-15; dense - banded 2.22e-15
closed forms (items 3, 4), regime R: |w11|/a = 0.279, (max tau/M)^2 = 4.000e-02
max_P |E_b - (11.12)| = 7.729e-07 = 0.000 * (tau/M)^4; max_P |E_b' - d(11.12)/dP| = 1.014e-06
E_b''(0): numeric +1.16351317e-02 closed +1.16355522e-02 rel.dev 3.615e-05 = 0.001 * (tau/M)^2
s3_1 numeric +9.9987712967e-01 closed +9.9987739882e-01 diff -2.69e-07
s3_2 numeric +9.9963191441e-01 closed +9.9963165158e-01 diff +2.63e-07
A1 numeric +3.1883906936e-04 closed +3.1876305792e-04 diff +7.60e-08
A2 numeric +7.3524924751e-04 closed +7.3669684496e-04 diff -1.45e-06
U numeric -1.0734105787e-03 closed -1.0756421628e-03 diff +2.23e-06
Ebind numeric -5.4713927724e-04 closed -5.4791221131e-04 diff +7.73e-07
Epp0 numeric +1.1635131656e-02 closed +1.1635552241e-02 diff -4.21e-07
m_b/(m_1+m_2): numeric +1.038281 (11.16) +1.038244; m_b-(m_1+m_2): numeric +3.168822e+00 -4 m_1 m_2 E_bind +3.161249e+00
=== set E_order_one: {'M1': 1.0, 'M2': 1.3, 'tau1': 1.0, 'tau2': 0.8, 'w': [-0.6, 0.2, -0.15, 0.3]} max tau_i/M_i = 1.000
P=-0.30 E_b=2.020243065365 isolation=2.944e-01 in a band: False weight(r=0,11)=0.7017 Gamma=+1.000000 E_b'=-5.330716e-02 <V1>=-5.330716e-02 <V2>=-5.330716e-02 E_b''=+1.580211e-01 (11.9)=+1.580211e-01
P=-0.15 E_b=2.014120790386 isolation=2.896e-01 in a band: False weight(r=0,11)=0.6956 Gamma=+1.000000 E_b'=-2.776997e-02 <V1>=-2.776997e-02 <V2>=-2.776997e-02 E_b''=+1.801270e-01 (11.9)=+1.801270e-01
P=+0.00 E_b=2.012023935088 isolation=2.880e-01 in a band: False weight(r=0,11)=0.6936 Gamma=+1.000000 E_b'=-7.401487e-15 <V1>=-3.140157e-17 <V2>=+1.914190e-17 E_b''=+1.876436e-01 (11.9)=+1.876436e-01
P=+0.15 E_b=2.014120790386 isolation=2.896e-01 in a band: False weight(r=0,11)=0.6956 Gamma=+1.000000 E_b'=+2.776997e-02 <V1>=+2.776997e-02 <V2>=+2.776997e-02 E_b''=+1.801270e-01 (11.9)=+1.801270e-01
P=+0.30 E_b=2.020243065365 isolation=2.944e-01 in a band: False weight(r=0,11)=0.7017 Gamma=+1.000000 E_b'=+5.330716e-02 <V1>=+5.330716e-02 <V2>=+5.330716e-02 E_b''=+1.580211e-01 (11.9)=+1.580211e-01
max |deviation| over the P values:
2(a): dE/dM_i - <sigma3_i> 7.62e-13; tau_i dE/dtau_i - <A_i> 1.52e-12; dE/dtheta - <U> 1.15e-12; Euler residual 1.33e-15
2(b): E_b' - <V1> 1.74e-10; E_b' - <V2> 1.74e-10; <V1> - <V2> 3.54e-16
2(c): E_b'' (finite difference) - formula (11.9) 2.40e-10
3(a): coefficients of (11.10) against direct inversion (relative) 3.59e-15; root of (11.10) - E_b 1.33e-15
numerics: |E_b(L) - E_b(2L)| 8.88e-16; weight at |r|>L/2 2.26e-30; dependence on eta 8.88e-16; dense - banded 2.66e-15
closed forms (items 3, 4), control, outside R: |w11|/a = 1.608, (max tau/M)^2 = 1.000e+00
max_P |E_b - (11.12)| = 4.605e-02 = 0.046 * (tau/M)^4; max_P |E_b' - d(11.12)/dP| = 3.589e-03
E_b''(0): numeric +1.87643618e-01 closed +1.74198841e-01 rel.dev 7.718e-02 = 0.077 * (tau/M)^2
s3_1 numeric +8.5646967127e-01 closed +8.8201005947e-01 diff -2.55e-02
s3_2 numeric +9.3859089119e-01 closed +9.5531741897e-01 diff -1.67e-02
A1 numeric +2.2543811170e-01 closed +2.3597988106e-01 diff -1.05e-02
A2 numeric +1.2495039252e-01 closed +1.1617471068e-01 diff +8.78e-03
U numeric -4.1500239894e-01 closed -5.0953161098e-01 diff +9.45e-02
Ebind numeric -2.8797606491e-01 closed -3.3345431511e-01 diff +4.55e-02
Epp0 numeric +1.8764361764e-01 closed +1.7419884136e-01 diff +1.34e-02
m_b/(m_1+m_2): numeric +1.758104 (11.16) +1.893795; m_b-(m_1+m_2): numeric +2.298001e+00 -4 m_1 m_2 E_bind +2.339806e+00
=== epsilon scan on set B_unequal_R (tau -> eps*tau, w -> eps^2*w)
eps= 1.000 E_b-(11.12): P=0 +8.1515e-06, P=0.3 +8.2675e-06; rel.dev E_b''(0) +5.4456e-04; <sigma3_i>-(11.14): -4.2426e-07, -1.7674e-06
eps= 0.500 E_b-(11.12): P=0 +5.0943e-07, P=0.3 +5.1674e-07; rel.dev E_b''(0) +1.3708e-04; <sigma3_i>-(11.14): -2.7203e-08, -1.1020e-07
ratios to the previous eps: dE_P0 16.00, dE_P03 16.00, rel_dEpp0 3.97, d_s3_1 15.60, d_s3_2 16.04
eps= 0.250 E_b-(11.12): P=0 +3.1839e-08, P=0.3 +3.2297e-08; rel.dev E_b''(0) +3.4328e-05; <sigma3_i>-(11.14): -1.7109e-09, -6.8832e-09
ratios to the previous eps: dE_P0 16.00, dE_P03 16.00, rel_dEpp0 3.99, d_s3_1 15.90, d_s3_2 16.01
eps= 0.125 E_b-(11.12): P=0 +1.9899e-09, P=0.3 +2.0186e-09; rel.dev E_b''(0) +8.5858e-06; <sigma3_i>-(11.14): -1.0710e-10, -4.3014e-10
ratios to the previous eps: dE_P0 16.00, dE_P03 16.00, rel_dEpp0 4.00, d_s3_1 15.97, d_s3_2 16.00
expected for halving eps: 16 for E_b and <sigma3_i> (next order eps^4), 4 for the relative deviation of E_b''(0)