05-ilang-free-fall / 13-pair-fall
13pair fallverified
Determines how a bound pair of two bodies of different types falls in a sharp stationary field: its drift acceleration, the comparison with the free parts, and the tie to the clocks of the parts.
pair_fall_check.py45 KB · raw
"""Numerical check for 13-pair-fall, derivation v4 (role: check).
The script of v3 with the paths changed and one table added (second_derivative_table,
for Step 5.3: the second derivative at lambda = 0 of the function c_i(lambda; P) of
(13.29), against (13.29) and (13.30)); the evolution, its parameters and the other
tables are unchanged.
A packet of a bound pair of two Model D bodies is built from the numerical bound
level of the fibre contract H(P) (11.1), put on a finite chain |x_i| <= L and
evolved by C_bod (13.2) with small gradients (Chebyshev expansion of
exp(-i C_bod lambda)). The first order in the gradients is a symmetric
difference of two evolutions (+eps, -eps).
Compared with the derivation:
(13.7) slope of the first-order velocities of both parts = <alpha>_g, with
alpha(P) of (13.8) computed from the numerical bound level;
(13.10) first-order change of the mean label separation = lambda <beta>_g + bounded;
(13.13) closed form of the drift acceleration from rest in the regime R;
(13.22) derivative of the clock ratio Gamma_i = dE_b(0)/dM_i in the data M_1, M_2
(second derivatives of the level), from the numerical bound level at P = 0,
and with it the term of order eps^2 of (13.18);
Step 5.3 d<K>_P/dP and f_i = Re F_{i,Lambda}(P) for windows [0, Lambda], from the
numerical bound level at P = 0 and P = 0.2, and the identity (13.6);
(13.29), (13.30) the second derivative at lambda = 0 of
c_i(lambda; P) = Re <V_i psi| R exp(-i (H - E_b) lambda) |K psi>, from the
numerical bound level at P = 0.1 and P = 0.2.
Sets: S1-S3 in the regime R (tau_i/M_i <= 0.2; S2 with tau_1 != tau_2, M_1 != M_2;
S3 with w_11 > 0), S4 with tau_i/M_i of order one, S5 a control (binding of range 1
without symmetry, moving packet). Each with the common gradient, with g_1 only,
and with a plain gradient in the place of kappa<X>.
Run from the project root:
.venv/Scripts/python.exe 05-ilang-free-fall/13-pair-fall/code/v4/pair_fall_check.py
Optional arguments: names of sets or variants (default: all).
Outputs: output/results.json, output/summary.txt, output/first_order_motion.png
"""
import os
for _v in ("OMP_NUM_THREADS", "MKL_NUM_THREADS", "OPENBLAS_NUM_THREADS"):
os.environ.setdefault(_v, "1")
import json
import sys
import time
from multiprocessing import Pool
from pathlib import Path
import numpy as np
import scipy.sparse as sp
from scipy.special import jv
try: # in-place Y += A X; the public product is the fallback
from scipy.sparse._sparsetools import csr_matvecs
except ImportError:
csr_matvecs = None
OUT = Path(__file__).parent / "output"
ETA = 0.5 # phase convention of (11.1); results do not depend on it
S1 = np.array([[0, 1], [1, 0]], dtype=complex)
S3 = np.diag([1.0, -1.0]).astype(complex)
I2 = np.eye(2, dtype=complex)
S1A, S1B = np.kron(S1, I2), np.kron(I2, S1) # branch index = 2*s1 + s2
S3A, S3B = np.kron(S3, I2).real.diagonal(), np.kron(I2, S3).real.diagonal()
def diag_w(w11, w12, w21, w22):
return {0: np.diag([w11, w12, w21, w22]).astype(complex)}
def control_w():
"""Binding of the class W with range 1: not diagonal, no symmetry r -> -r or P -> -P."""
w = diag_w(-0.5, 0.3, -0.2, 0.4)
w1 = np.zeros((4, 4), dtype=complex)
w1[0, 0] = -0.6
w1[0, 1], w1[1, 0] = 0.3j, -0.3j
w1[0, 3], w1[3, 0] = 0.2 + 0.15j, 0.2 - 0.15j
w[1] = w1
return w
# tau_i, M_i, binding, packet (P0, half support Pc, centre label xc), chain L, fibre range rmax,
# range of lambda, sampling step (a number, or an interval from which the step with the largest
# alias margin is taken), gradient eps (g); the plain gradient is eps*(M1+M2)/2;
# cfgs: 0 common gradient, 1 g1 only, 2 plain gradient
BASE = {
"S1": dict(t1=0.20, M1=1.0, t2=0.20, M2=1.0, W=diag_w(-0.020, 0.012, 0.012, 0.016), R=True, lam_max=3000.0, dlam=(32.0, 44.0), eps=5e-7),
"S2": dict(t1=0.20, M1=1.0, t2=0.18, M2=1.25, W=diag_w(-0.020, 0.010, -0.006, 0.020), R=True, lam_max=3000.0, dlam=(32.0, 44.0), eps=5e-7),
"S3": dict(t1=0.20, M1=1.0, t2=0.17, M2=0.9, W=diag_w(0.022, -0.010, 0.007, -0.015), R=True, lam_max=3000.0, dlam=(32.0, 44.0), eps=5e-7),
"S4": dict(t1=1.00, M1=1.0, t2=0.80, M2=1.2, W=diag_w(-0.9, 0.3, -0.2, 0.4), R=False, lam_max=200.0, dlam=1.0, eps=2e-6),
"S5": dict(t1=1.00, M1=1.0, t2=0.80, M2=1.2, W=control_w(), R=False, lam_max=400.0, dlam=1.0, eps=2e-6,
P0=0.2, Pc=0.4, xc=-12),
}
DEFAULT = dict(P0=0.0, Pc=0.5, xc=0, L=60, rmax=48, cfgs=(0, 1, 2))
# convergence variants: one numerical parameter changed, only the configuration "g1 only"
VARIANTS = {
"S1_L45": ("S1", dict(L=45)),
"S1_eps2": ("S1", dict(eps=1e-6)),
"S1_Pc035": ("S1", dict(Pc=0.35)),
"S1_dlam": ("S1", dict(dlam=(20.0, 28.0))),
"S4_L45": ("S4", dict(L=45)),
"S5_L75": ("S5", dict(L=75)),
"S5_long": ("S5", dict(L=75, lam_max=600.0, xc=-15)),
}
LABELS = ("common gradient g1=g2", "g1 only (g2=0)", "plain gradient")
# ---------------------------------------------------------------- fibre H(P)
def bands(P, p):
q = np.linspace(-np.pi, np.pi, 4001)
o1 = np.sqrt(p["M1"] ** 2 + p["t1"] ** 2 * np.sin(q) ** 2)
o2 = np.sqrt(p["M2"] ** 2 + p["t2"] ** 2 * np.sin(P - q) ** 2)
out = []
for s1 in (1, -1):
for s2 in (1, -1):
e = s1 * o1 + s2 * o2
out.append((e.min(), e.max()))
return out
def fibre_ops(P, p, rmax):
nr = 2 * rmax + 1
S = np.eye(nr, k=-1).astype(complex) # (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["t1"] * np.kron(e1 * Sd - np.conj(e1) * S, S1A)
A2 = 0.5j * p["t2"] * np.kron(e2 * S - np.conj(e2) * Sd, S1B)
V1 = -0.5 * p["t1"] * np.kron(e1 * Sd + np.conj(e1) * S, S1A)
V2 = -0.5 * p["t2"] * np.kron(e2 * S + np.conj(e2) * Sd, S1B)
H = A1 + A2 + np.diag(np.kron(np.ones(nr), p["M1"] * S3A + p["M2"] * S3B))
for r, Wr in p["W"].items():
i = (r + rmax) * 4
H[i:i + 4, i:i + 4] += Wr
d2H = -ETA ** 2 * A1 - (1 - ETA) ** 2 * A2
return H, V1, V2, d2H
def level(P, p, rmax):
"""Bound level of H(P) attached to the band B_++ and the functions of item 2."""
nr = 2 * rmax + 1
H, V1, V2, d2H = fibre_ops(P, p, rmax)
vals, vecs = np.linalg.eigh(H)
b = int(np.argmax(np.abs(vecs[rmax * 4, :]) ** 2)) # weight on r = 0, branch |11>
psi = vecs[:, b] * np.exp(-1j * np.angle(vecs[rmax * 4, b]))
Eb = vals[b]
dE = vals - Eb
gap = np.min(np.abs(np.delete(dE, b)))
dE[b] = np.inf
def Rap(x): # Q (H - Eb)^-1 Q x
return vecs @ ((vecs.conj().T @ x) / dE)
rvec = np.kron(np.arange(-rmax, rmax + 1), np.ones(4))
s3 = [np.kron(np.ones(nr), S3A), np.kron(np.ones(nr), S3B)]
Mi = [p["M1"], p["M2"]]
vpsi = (ETA * V1 + (1 - ETA) * V2) @ psi
Rv = Rap(vpsi)
E1 = np.vdot(psi, vpsi).real
E2 = np.vdot(psi, d2H @ psi).real - 2 * np.vdot(vpsi, Rv).real
def ev(d):
return np.vdot(psi, d * psi).real
dPr = -2 * np.vdot(rvec * psi, Rv).real
alpha, beta, s3m = [], [], []
for i in range(2):
s3m.append(ev(s3[i]))
dPs3 = -2 * np.vdot(s3[i] * psi, Rv).real
rRs3 = 2 * np.vdot(rvec * psi, Rap(s3[i] * psi)).real
alpha.append(-Mi[i] * (E2 * s3m[i] - E1 * dPs3)) # coefficient of g_i (13.8)
beta.append(-Mi[i] * (s3m[i] * dPr + E1 * rRs3)) # coefficient of g_i (13.10)
alpha.append(-2 * E2) # coefficient of kappa<X>
beta.append(-2 * dPr)
bd = bands(P, p)
dist_band = min(min(abs(Eb - lo), abs(Eb - hi)) for lo, hi in bd)
inside = any(lo <= Eb <= hi for lo, hi in bd)
return dict(Eb=Eb, psi=psi, E1=E1, E2=E2, s3=s3m, alpha=alpha, beta=beta, gap=gap,
freqs=np.abs(np.delete(vals - Eb, b)),
v1=np.vdot(psi, V1 @ psi).real, v2=np.vdot(psi, V2 @ psi).real,
rmean=ev(rvec), dist_band=dist_band, inside=inside,
tail=float(np.abs(psi[:4]).max() + np.abs(psi[-4:]).max()))
def closed_form(p):
"""Leading order in R, (13.13): coefficients of g_1, g_2, kappa<X> at P0 = 0."""
t1, t2, w = p["t1"], p["t2"], p["W"][0][0, 0].real
a1, a2 = t1 ** 2 / (4 * p["M1"]), t2 ** 2 / (4 * p["M2"])
u = abs(w) / (a1 + a2)
pref = np.sign(w) / np.sqrt(1 + u * u) / (p["M1"] / t1 ** 2 + p["M2"] / t2 ** 2)
return [pref * p["M1"], pref * p["M2"], 2 * pref], u
def closed_form_packet(p, Pg, wg2):
"""(11.12) averaged over |g|^2: -<E_b''> (g1 M1 + g2 M2 + 2 kappa<X>)."""
w = p["W"][0][0, 0].real
a1, a2 = p["t1"] ** 2 / (4 * p["M1"]), p["t2"] ** 2 / (4 * p["M2"])
j2 = a1 * a1 + a2 * a2 + 2 * a1 * a2 * np.cos(2 * Pg)
s = np.sqrt(w * w + j2)
E2 = np.sign(w) * (-8 * a1 * a2 * np.cos(2 * Pg) / (2 * s) - (4 * a1 * a2 * np.sin(2 * Pg)) ** 2 / (4 * s ** 3))
m = -np.sum(wg2 * E2)
return [m * p["M1"], m * p["M2"], 2 * m]
# ---------------------------------------------------------------- real chain
def real_ops(p, L):
n = 2 * L + 1
x = np.arange(-L, L + 1).astype(float)
T = sp.eye(n, k=-1, format="csr", dtype=complex) # |x+1><x|
Xs = sp.diags(np.kron(x, np.ones(2))).tocsr()
Id = sp.identity(2 * n, format="csr", dtype=complex)
def body(t, M):
h = 0.5 * t * (sp.kron(T, -1j * S1) + sp.kron(T.T, 1j * S1)) + M * sp.kron(sp.identity(n), S3)
h = h.tocsr()
return h, (1j * (h @ Xs - Xs @ h)).tocsr() # V = i [C, x]
h1, v1 = body(p["t1"], p["M1"])
h2, v2 = body(p["t2"], p["M2"])
rows, cols, data = [], [], []
for r, Wr in p["W"].items():
i2 = np.arange(n)
i1 = i2 + r
ok = (i1 >= 0) & (i1 < n)
i1, i2 = i1[ok], i2[ok]
for a in range(4):
for b in range(4):
if Wr[a, b] != 0:
rows.append((i1 * 2 + a // 2) * 2 * n + i2 * 2 + a % 2)
cols.append((i1 * 2 + b // 2) * 2 * n + i2 * 2 + b % 2)
data.append(np.full(len(i1), Wr[a, b]))
dim = 4 * n * n
U = sp.csr_matrix((np.concatenate(data), (np.concatenate(rows), np.concatenate(cols))), shape=(dim, dim))
H0 = (sp.kron(h1, Id) + sp.kron(Id, h2) + U).tocsr()
one = np.ones(2 * n)
x1 = np.kron(np.kron(x, np.ones(2)), one)
x2 = np.kron(one, np.kron(x, np.ones(2)))
s31 = np.kron(np.kron(np.ones(n), [1.0, -1.0]), one)
s32 = np.kron(one, np.kron(np.ones(n), [1.0, -1.0]))
# gauge with real hopping: psi'(x1, x2) = (-i)^(x1 + x2) psi(x1, x2)
ph = np.array([1, -1j, -1, 1j])[np.rint(x1 + x2).astype(int) % 4]
G = sp.diags(ph)
def gauged(A):
B = (G @ A @ G.conj()).tocsr()
Br, Bi = B.real.tocsr(), B.imag.tocsr()
Br.eliminate_zeros()
Bi.eliminate_zeros()
return Br, Bi
Hr, Hi = gauged(H0)
K = []
for V in (sp.kron(v1, Id), sp.kron(Id, v2)):
Vr, Vi = gauged(V)
assert Vr.nnz == 0 # V' = i K with K real, antisymmetric
K.append(Vi)
return dict(H0=H0, Hr=Hr, Hi=Hi, K=K, ph=ph, x1=x1, x2=x2,
D=[p["M1"] * x1 * s31, p["M2"] * x2 * s32, x1 + x2], n=n, x=x)
def packet(p, L, rmax, Pg, coef, psis):
"""Phi(x1,s1,x2,s2) = sum_j coef_j (2 pi)^-1/2 e^{i P_j (eta x1 + (1-eta) x2)} psi_j(x1-x2)."""
n = 2 * L + 1
x = np.arange(-L, L + 1)
ridx = x[:, None] - x[None, :] + 2 * L
Phi = np.zeros((n, n, 4), dtype=complex)
pad = np.zeros((4 * L + 1, 4), dtype=complex)
lo = 2 * L - rmax
for Pj, cj, psi in zip(Pg, coef, psis):
if cj == 0:
continue
pad[:] = 0
ps = psi.reshape(2 * rmax + 1, 4)
if lo >= 0:
pad[lo:lo + 2 * rmax + 1] = ps
else:
pad[:] = ps[-lo:-lo + 4 * L + 1]
ph = np.exp(1j * Pj * ETA * x)[:, None] * np.exp(1j * Pj * (1 - ETA) * x)[None, :]
Phi += cj * ph[:, :, None] * pad[ridx]
Phi /= np.sqrt(2 * np.pi)
return Phi.reshape(n, n, 2, 2).transpose(0, 2, 1, 3).reshape(-1)
def cheb_coef(z):
"""exp(-i z h) = sum_k c_k T_k(h) with c_k = (2 - delta_k0) (-i)^k J_k(z)."""
kmax = int(z + 12 * z ** (1 / 3) + 30)
k = np.arange(kmax + 1)
c = jv(k, z) * np.array([1, -1j, -1, 1j])[k % 4]
c[1:] *= 2
keep = np.nonzero(np.abs(c) > 1e-16)[0].max() + 1
return c[:keep]
def slope(lam, v, lo, hi):
"""Smooth-window average of dv/dlambda: -int W' v with W = 8/(3T) sin^4 on [lo, hi]."""
T = hi - lo
m = (lam >= lo - 1e-9) & (lam <= hi + 1e-9)
th = np.pi * (lam[m] - lo) / T
dW = 8 / (3 * T) * 4 * np.sin(th) ** 3 * np.cos(th) * np.pi / T
return float(-np.sum(dW * v[m]) * (lam[1] - lam[0]))
# ---------------------------------------------------------------- one task
def run(task):
name, p = task
t0 = time.time()
L, rmax, eps = p["L"], p["rmax"], p["eps"]
epsf = eps * 0.5 * (p["M1"] + p["M2"])
# packet weight g(P) = c cos^4(pi (P-P0) / (2 Pc)), class G(P_0)
nP = 161
Pg = np.linspace(p["P0"] - p["Pc"], p["P0"] + p["Pc"], nP)
dP = Pg[1] - Pg[0]
g = np.cos(np.pi * (Pg - p["P0"]) / (2 * p["Pc"])) ** 4
g /= np.sqrt(np.sum(g * g) * dP)
wg2 = g * g * dP
gc = g * np.exp(-1j * Pg * p["xc"]) # joint shift of the packet by xc labels
lev = [level(P, p, rmax) for P in Pg]
Eb = np.array([l["Eb"] for l in lev])
E1 = np.array([l["E1"] for l in lev])
E2 = np.array([l["E2"] for l in lev])
def avg(f):
return float(np.sum(wg2 * np.array(f)))
Pm = avg(Pg)
dPsp = np.sqrt(avg((Pg - Pm) ** 2))
alpha_g = [avg([l["alpha"][i] for l in lev]) for i in range(3)]
beta_g = [avg([l["beta"][i] for l in lev]) for i in range(3)]
l0 = level(p["P0"], p, rmax)
fd1 = np.gradient(Eb, dP)
fd2 = np.gradient(fd1, dP)
core = slice(8, nP - 8)
info = dict(
Eb_P0=l0["Eb"], dEb_P0=l0["E1"], d2Eb_P0=l0["E2"], s3_P0=l0["s3"], rmean_P0=l0["rmean"],
gap_min=float(min(l["gap"] for l in lev)), dist_band_min=float(min(l["dist_band"] for l in lev)),
inside_band=bool(any(l["inside"] for l in lev)), tail_max=float(max(l["tail"] for l in lev)),
V1_minus_V2_max=float(max(abs(l["v1"] - l["v2"]) for l in lev)),
fd_check_dE=float(np.max(np.abs(fd1 - E1)[core])), fd_check_d2E=float(np.max(np.abs(fd2 - E2)[core])),
P_mean=Pm, delta_P=dPsp, alpha_g=alpha_g, beta_g=beta_g, alpha_P0=l0["alpha"], beta_P0=l0["beta"],
mean_dE_packet=avg(E1))
# sampling step: the rest of the first-order motion oscillates with the frequencies |E_n(P) - E_b(P)|;
# a sum over samples at the step dlam aliases the multiples of 2 pi / dlam onto zero frequency
freqs = np.concatenate([l["freqs"] for l in lev])
def margin(dl):
fs = 2 * np.pi / dl
return float(np.min(np.abs(freqs - np.maximum(1, np.rint(freqs / fs)) * fs)))
if isinstance(p["dlam"], tuple):
cand = np.arange(p["dlam"][0], p["dlam"][1] + 1e-9, 0.25)
dlam = float(cand[int(np.argmax([margin(c) for c in cand]))])
else:
dlam = float(p["dlam"])
info["alias_margin"] = margin(dlam)
# start on the chain
ops = real_ops(p, L)
psis = [l["psi"] for l in lev]
Phi = packet(p, L, rmax, Pg, gc * dP, psis)
res = ops["H0"] @ Phi - packet(p, L, rmax, Pg, gc * Eb * dP, psis)
info["packet_norm"] = float(np.linalg.norm(Phi))
info["packet_residual"] = float(np.linalg.norm(res))
Phi /= np.linalg.norm(Phi)
# runs: common gradient, g1 only, plain gradient; each with + and -
sel = list(p["cfgs"])
nc = 2 * len(sel)
cfg = [(eps, eps, 0.0), (eps, 0.0, 0.0), (0.0, 0.0, epsf)]
par = [eps, eps, epsf]
pred = [alpha_g[0] + alpha_g[1], alpha_g[0], alpha_g[2]]
pred_b = [beta_g[0] + beta_g[1], beta_g[0], beta_g[2]]
pred0 = [l0["alpha"][0] + l0["alpha"][1], l0["alpha"][0], l0["alpha"][2]]
Dg = np.zeros((Phi.size, nc))
for c, ci in enumerate(sel):
g1, g2, f = cfg[ci]
d = g1 * ops["D"][0] + g2 * ops["D"][1] + f * ops["D"][2]
Dg[:, 2 * c], Dg[:, 2 * c + 1] = d, -d
wn = sum(np.linalg.norm(Wr, 2) for Wr in p["W"].values())
scale = 1.03 * (np.hypot(p["M1"], p["t1"]) + np.hypot(p["M2"], p["t2"]) + wn + np.abs(Dg).max())
# real arithmetic: Psi[0] real parts, Psi[1] imaginary parts of the runs
H2 = ((2 / scale) * ops["Hr"]).tocsr()
dim = H2.shape[0]
D2 = (2 / scale) * Dg
tmp = np.empty_like(D2)
rows = np.unique(ops["Hi"].nonzero()[0]) # complex entries of the binding only
Hi2 = (2 / scale) * ops["Hi"][rows] if len(rows) else None
def step2(F, Y): # Y <- 2 h F - Y, h = C_bod / scale
np.negative(Y, out=Y)
for a in range(2):
if csr_matvecs is None:
Y[a] += H2 @ F[a]
else:
csr_matvecs(dim, dim, nc, H2.indptr, H2.indices, H2.data, F[a].ravel(), Y[a].ravel())
np.multiply(D2, F[a], out=tmp)
Y[a] += tmp
if Hi2 is not None:
Y[0][rows] -= Hi2 @ F[1]
Y[1][rows] += Hi2 @ F[0]
return Y
coef = cheb_coef(scale * dlam)
nst = int(round(p["lam_max"] / dlam))
lam = dlam * np.arange(nst + 1)
Phig = ops["ph"] * Phi
Psi = np.stack([np.repeat(Phig.real[:, None], nc, axis=1), np.repeat(Phig.imag[:, None], nc, axis=1)])
obs = np.zeros((nst + 1, 4, nc))
def measure(k):
w2 = Psi[0] ** 2 + Psi[1] ** 2
for i in range(2): # <V_i> = -2 Re^T K_i Im
obs[k, i] = -2 * np.einsum("ij,ij->j", Psi[0], ops["K"][i] @ Psi[1])
obs[k, 2] = ops["x1"] @ w2
obs[k, 3] = ops["x2"] @ w2
return w2
measure(0)
for k in range(1, nst + 1):
f0 = Psi
f1 = step2(Psi, np.zeros_like(Psi))
f1 *= 0.5
acc = coef[0].real * f0
for j in range(1, len(coef)):
if j > 1:
f0, f1 = f1, step2(f1, f0) # T_j = 2 h T_{j-1} - T_{j-2}
if j % 2 == 0:
for a in range(2):
np.multiply(f1[a], coef[j].real, out=tmp)
acc[a] += tmp
else: # i c (re + i im) = c (-im + i re)
np.multiply(f1[1], coef[j].imag, out=tmp)
acc[0] -= tmp
np.multiply(f1[0], coef[j].imag, out=tmp)
acc[1] += tmp
Psi = acc
w2 = measure(k)
info["norm_drift"] = float(np.max(np.abs(np.sum(w2, axis=0) - 1)))
info["v0_start"] = [float(obs[0, 0, 0]), float(obs[0, 1, 0])]
edge = (np.abs(ops["x1"]) > L - 3) | (np.abs(ops["x2"]) > L - 3)
info["edge_weight_end"] = float(np.sum(w2[edge, 0]))
T = lam[-1]
out = dict(name=name, info=info, configs=[], lam=lam.tolist(), series={})
for c, ci in enumerate(sel):
lab = LABELS[ci]
fo = (obs[:, :, 2 * c] - obs[:, :, 2 * c + 1]) / (2 * par[ci]) # first order per unit gradient
v1, v2, sep = fo[:, 0], fo[:, 1], fo[:, 2] - fo[:, 3]
s1, s2 = slope(lam, v1, 0, T), slope(lam, v2, 0, T)
half = [slope(lam, v, T / 2, T) for v in (v1, v2)] + [slope(lam, v, 0, T / 2) for v in (v1, v2)]
ssep = slope(lam, sep, 0, T)
ssep_half = [slope(lam, sep, 0, T / 2), slope(lam, sep, T / 2, T)]
out["configs"].append(dict(
config=lab, predicted_slope=pred[ci], alpha_P0=pred0[ci], slope_v1=s1, slope_v2=s2,
rel_dev_v1=s1 / pred[ci] - 1, rel_dev_v2=s2 / pred[ci] - 1,
half_window_rel_dev=[h / pred[ci] - 1 for h in half],
two_point_rel_dev=[float(v1[-1] / T / pred[ci] - 1), float(v2[-1] / T / pred[ci] - 1)],
rest_sup=[float(np.max(np.abs(v1 - pred[ci] * lam))), float(np.max(np.abs(v2 - pred[ci] * lam)))],
linear_part_end=float(abs(pred[ci]) * T),
sep_predicted_slope=pred_b[ci], sep_slope=ssep, sep_slope_half_windows=ssep_half, sep_end=float(sep[-1]),
sep_max_abs=float(np.max(np.abs(sep))), sep_minus_linear_sup=float(np.max(np.abs(sep - pred_b[ci] * lam)))))
out["series"][lab] = dict(v1=v1.tolist(), v2=v2.tolist(), sep=sep.tolist())
if p["R"] and len(sel) == 3:
cf, u = closed_form(p)
cfp = closed_form_packet(p, Pg, wg2)
def comb(z):
return [z[0] + z[1], z[0], z[2]]
out["closed_form"] = dict(u=u, eps2=[(p["t1"] / p["M1"]) ** 2, (p["t2"] / p["M2"]) ** 2],
closed_P0=comb(cf), closed_packet=comb(cfp), level_P0=pred0,
slope_v1=[c["slope_v1"] for c in out["configs"]])
out["numerics"] = dict(L=L, chain_sites=2 * L + 1, dim=int(Phi.size), rmax=rmax, nP=nP, Pc=p["Pc"], P0=p["P0"], xc=p["xc"],
lam_max=float(T), dlam=dlam, window_half_width=6 * np.pi / T, eps=eps, eps_plain=epsf, cheb_terms=int(len(coef)),
seconds=round(time.time() - t0, 1))
return out
def scaling_table():
"""alpha(0) from the numerical level against (13.13) at tau -> s tau, w -> s^2 w."""
rows = []
for name in ("S1", "S2", "S3"):
for s in (1.0, 0.5, 0.25):
p = BASE[name]
q = dict(p, t1=p["t1"] * s, t2=p["t2"] * s, W={0: p["W"][0] * s * s})
l0 = level(0.0, q, 64)
cf, u = closed_form(q)
rows.append(dict(set=name, s=s, tau_over_M=[q["t1"] / q["M1"], q["t2"] / q["M2"]], u=u,
level=l0["alpha"], closed=cf,
rel_dev=[l0["alpha"][i] / cf[i] - 1 for i in range(3)]))
return rows
def clock_ratio(p, rmax):
"""Numerical level at P = 0: Gamma_i = <sigma_3^(i)>_0 = dE_b(0)/dM_i and its derivative in the data,
T[i][j] = M_j dGamma_i/dM_j = M_j d^2E_b(0)/dM_i dM_j = -2 M_j Re <sigma_3^(i) psi| R |sigma_3^(j) psi>."""
nr = 2 * rmax + 1
vals, vecs = np.linalg.eigh(fibre_ops(0.0, p, rmax)[0])
b = int(np.argmax(np.abs(vecs[rmax * 4, :]) ** 2)) # weight on r = 0, branch |11>
psi = vecs[:, b]
dE = vals - vals[b]
dE[b] = np.inf
s3 = [np.kron(np.ones(nr), S3A), np.kron(np.ones(nr), S3B)]
Mi = [p["M1"], p["M2"]]
c = [vecs.conj().T @ (s * psi) for s in s3]
G = [float(np.vdot(psi, s * psi).real) for s in s3]
T = [[float(-2 * Mi[j] * np.sum(c[i].conj() * c[j] / dE).real) for j in range(2)] for i in range(2)]
return G, T
def clock_ratio_closed(p):
"""Leading order in R: Gamma_i of (11.20) and M_j times the second derivatives (13.22)."""
t, M, w = [p["t1"], p["t2"]], [p["M1"], p["M2"]], p["W"][0][0, 0].real
a = [t[i] ** 2 / (4 * M[i]) for i in range(2)]
asum = a[0] + a[1]
u, sg = abs(w) / asum, np.sign(w)
lam = 1 + sg / np.sqrt(1 + u * u)
d2 = sg * u * u / (asum * (1 + u * u) ** 1.5) # second derivative of a + sgn(w) sqrt(w^2 + a^2) in a
G = [1 - a[i] / M[i] * lam for i in range(2)]
T = [[(2 * a[i] / M[i] * lam if i == j else 0.0) + a[i] * a[j] / M[i] * d2 for j in range(2)] for i in range(2)]
return G, T, u
def clock_ratio_table():
"""Derivative of the clock ratio in the data against (13.22), and rho_i - g_i against (13.18),
at tau -> s tau, w -> s^2 w (numerical level at P = 0, fibre |r| <= 64)."""
rows, h = [], 1e-4
for name in ("S1", "S2", "S3"):
for s in (1.0, 0.5, 0.25):
p = BASE[name]
q = dict(p, t1=p["t1"] * s, t2=p["t2"] * s, W={0: p["W"][0] * s * s})
G, T = clock_ratio(q, 64)
Gc, Tc, u = clock_ratio_closed(q)
T48 = clock_ratio(q, 48)[1]
fd = [[0.0, 0.0], [0.0, 0.0]] # symmetric difference of Gamma_i in M_j
for j, key in enumerate(("M1", "M2")):
Gp = clock_ratio(dict(q, **{key: q[key] * (1 + h)}), 64)[0]
Gm = clock_ratio(dict(q, **{key: q[key] * (1 - h)}), 64)[0]
for i in range(2):
fd[i][j] = (Gp[i] - Gm[i]) / (2 * h)
ij = [(i, j) for i in range(2) for j in range(2)]
# rho_i - g_i per unit gradient, (13.21) against (13.18): common gradient, g1 only, g2 only
rho = [[(T[i][0] + T[i][1]) / G[i], T[i][0] / G[i], T[i][1] / G[i]] for i in range(2)]
rhoc = [[Tc[i][0] + Tc[i][1], Tc[i][0], Tc[i][1]] for i in range(2)]
rows.append(dict(
set=name, s=s, tau_over_M=[q["t1"] / q["M1"], q["t2"] / q["M2"]], u=u,
Gamma=G, Gamma_closed=Gc, one_minus_Gamma_rel_dev=[(1 - G[i]) / (1 - Gc[i]) - 1 for i in range(2)],
T=T, T_closed=Tc, T_rel_dev=[[T[i][j] / Tc[i][j] - 1 for j in range(2)] for i in range(2)],
T_abs_dev=[[T[i][j] - Tc[i][j] for j in range(2)] for i in range(2)],
fd_rel_dev=float(max(abs(fd[i][j] / T[i][j] - 1) for i, j in ij)),
rmax48_rel_change=float(max(abs(T48[i][j] / T[i][j] - 1) for i, j in ij)),
rho_minus_g=rho, rho_minus_g_closed=rhoc,
rho_rel_dev=[[rho[i][k] / rhoc[i][k] - 1 for k in range(3)] for i in range(2)]))
return rows
def other_order_table():
"""Step 5.3 of the derivation (the limits of item 2(c) in the other order). From the numerical level at
P = 0 and P = 0.2, per unit gradient (common gradient; g1 only):
d<sigma_3^(i)>_P/dP = -2 Re <sigma_3^(i) psi| R |V_eta psi>, dK = d<K>_P/dP = sum_i g_i M_i d<sigma_3^(i)>_P/dP,
f_i = Re F_{i,Lambda}(P) for the windows [0, Lambda], F_{i,lambda} = <V_i psi| R (1 - exp(-i (H - E_b) lambda)) |K psi>,
identity (13.6): 2 Re <V_i psi| R |K psi> = -dK for i = 1, 2.
Fibre |r| <= 96, against 64; d<sigma_3>/dP also by a symmetric difference in P (fibre |r| <= 64). For the sets
in the regime R the leading order from (11.12), differentiated once in M_i and once in P, is given for
comparison (a check of the table; no statement of the derivation rests on it)."""
rows, hP, wins = [], 1e-4, (5.0, 10.0, 20.0)
for name, p in BASE.items():
Mi, ti = [p["M1"], p["M2"]], [p["t1"], p["t2"]]
for P in (0.0, 0.2):
res = {}
for rmax in (64, 96):
nr = 2 * rmax + 1
H, V1, V2, _ = fibre_ops(P, p, rmax)
vals, vecs = np.linalg.eigh(H)
b = int(np.argmax(np.abs(vecs[rmax * 4, :]) ** 2)) # weight on r = 0, branch |11>
psi = vecs[:, b]
keep = np.arange(len(vals)) != b
dE = (vals - vals[b])[keep]
Uq = vecs[:, keep].conj().T # components in Q h
cV = [Uq @ (V @ psi) for V in (V1, V2)]
c3 = [Uq @ (np.kron(np.ones(nr), s) * psi) for s in (S3A, S3B)]
cVe = ETA * cV[0] + (1 - ETA) * cV[1]
ds3 = [float(-2 * np.sum(c3[i].conj() * cVe / dE).real) for i in range(2)]
cK = [Mi[0] * c3[0] + Mi[1] * c3[1], Mi[0] * c3[0]] # Q K psi per unit gradient: common, g1 only
dK = [Mi[0] * ds3[0] + Mi[1] * ds3[1], Mi[0] * ds3[0]]
id136 = [[float(2 * np.sum(cV[i].conj() * cK[c] / dE).real + dK[c]) for c in range(2)] for i in range(2)]
f = [[[float(np.sum(cV[i].conj() * cK[c] * (1 - np.exp(-1j * dE * lam)) / dE).real) for lam in wins]
for c in range(2)] for i in range(2)]
res[rmax] = dict(ds3=ds3, dK=dK, id136=id136, f=f)
lv = level(P, p, 64)
sp_, sm_ = level(P + hP, p, 64)["s3"], level(P - hP, p, 64)["s3"]
fd = [(sp_[i] - sm_[i]) / (2 * hP) for i in range(2)]
closed = None
if p["R"]:
w = p["W"][0][0, 0].real
a = [ti[i] ** 2 / (4 * Mi[i]) for i in range(2)]
j2 = a[0] ** 2 + a[1] ** 2 + 2 * a[0] * a[1] * np.cos(2 * P)
closed = [float(np.sign(w) * a[i] / Mi[i] * 2 * a[1 - i] * np.sin(2 * P)
* (w * w + a[1 - i] ** 2 + a[0] * a[1] * np.cos(2 * P)) / (w * w + j2) ** 1.5) for i in range(2)]
r64, r96 = res[64], res[96]
idx = [(i, c, k) for i in range(2) for c in range(2) for k in range(len(wins))]
rows.append(dict(
set=name, P=P, dEb=float(lv["E1"]), gap=float(lv["gap"]), dist_band=float(lv["dist_band"]),
inside_band=bool(lv["inside"]), ds3=r96["ds3"], ds3_closed=closed, dK=r96["dK"],
id136_abs_dev=float(max(abs(v) for row in r96["id136"] for v in row)),
ds3_fd_abs_dev=float(max(abs(fd[i] - r64["ds3"][i]) for i in range(2))),
ds3_change_64_96=float(max(abs(r64["ds3"][i] - r96["ds3"][i]) for i in range(2))),
windows=list(wins), f=r96["f"],
f_change_64_96=float(max(abs(r64["f"][i][c][k] - r96["f"][i][c][k]) for i, c, k in idx))))
return rows
def second_derivative_table():
"""Step 5.3 of the derivation, (13.29) and (13.30). With c_i(lambda; P) = Re <V_i psi| R exp(-i (H - E_b) lambda) |K psi>,
the second derivative at lambda = 0 is -Re <V_i psi| (H - E_b) |K psi>. From the numerical level at P = 0.1 and
P = 0.2 (fibre |r| <= 96, against 64), per unit gradient (the gradient of one part only, K = M_c sigma_3^(c)):
spectral sum -Re sum_n conj(<n|V_i psi>) <n|K psi> (E_n - E_b) (the levels n of the fibre other than the bound level),
double commutator -1/2 <[V_i, [H, K]]>_P (13.29), every binding,
(13.29) -M_i tau_i^2 Im <sigma_3^(i) T_i^2>_P for the gradient of the same part and 0 for the gradient of the
other part (a binding that commutes with K: the sets S1-S4, not the control S5), with
T_1 = e^{i eta P} S^dagger, T_2 = e^{i (1 - eta) P} S,
(13.30) sgn(w) M_i tau_i^2 a_j sin(2P) / sqrt(w^2 + j(P)^2), j != i: the leading order in R (sets S1-S3)."""
rows = []
for name, p in BASE.items():
Mi, ti = [p["M1"], p["M2"]], [p["t1"], p["t2"]]
for P in (0.1, 0.2):
res = {}
for rmax in (64, 96):
nr = 2 * rmax + 1
H, V1, V2, _ = fibre_ops(P, p, rmax)
vals, vecs = np.linalg.eigh(H)
b = int(np.argmax(np.abs(vecs[rmax * 4, :]) ** 2)) # weight on r = 0, branch |11>
psi = vecs[:, b]
keep = np.arange(len(vals)) != b
dE = (vals - vals[b])[keep]
Uq = vecs[:, keep].conj().T # components in Q h
S = np.eye(nr, k=-1).astype(complex) # (S psi)(r) = psi(r-1)
T2 = [np.exp(2j * ETA * P) * np.kron(S.T @ S.T, np.eye(4)),
np.exp(2j * (1 - ETA) * P) * np.kron(S @ S, np.eye(4))]
s3 = [np.kron(np.ones(nr), s) for s in (S3A, S3B)]
Vs = (V1, V2)
spec = [[0.0, 0.0], [0.0, 0.0]] # [part i][gradient of the part c]
comm = [[0.0, 0.0], [0.0, 0.0]]
for i in range(2):
vpsi = Vs[i] @ psi
cV = Uq @ vpsi
for c in range(2):
kd = Mi[c] * s3[c] # diagonal of K
spec[i][c] = float(-np.sum(cV.conj() * (Uq @ (kd * psi)) * dE).real)
HK = H * kd[None, :] - kd[:, None] * H # [H, K]
comm[i][c] = float(-0.5 * np.vdot(psi, Vs[i] @ (HK @ psi) - HK @ vpsi).real)
f29 = [float(-Mi[i] * ti[i] ** 2 * np.vdot(psi, s3[i] * (T2[i] @ psi)).imag) for i in range(2)]
res[rmax] = dict(spec=spec, comm=comm, f29=f29)
closed = None
if p["R"]:
w = p["W"][0][0, 0].real
a = [ti[i] ** 2 / (4 * Mi[i]) for i in range(2)]
j2 = a[0] ** 2 + a[1] ** 2 + 2 * a[0] * a[1] * np.cos(2 * P)
closed = [float(np.sign(w) * Mi[i] * ti[i] ** 2 * a[1 - i] * np.sin(2 * P) / np.sqrt(w * w + j2)) for i in range(2)]
r64, r96 = res[64], res[96]
ic = [(i, c) for i in range(2) for c in range(2)]
rows.append(dict(
set=name, P=P, own=[r96["spec"][0][0], r96["spec"][1][1]], cross=[r96["spec"][0][1], r96["spec"][1][0]],
commutator_abs_dev=float(max(abs(r96["spec"][i][c] - r96["comm"][i][c]) for i, c in ic)),
f1329=r96["f29"], f1329_abs_dev=[r96["spec"][i][i] - r96["f29"][i] for i in range(2)],
closed=closed,
closed_rel_dev=None if closed is None else [r96["spec"][i][i] / closed[i] - 1 for i in range(2)],
change_64_96=float(max(abs(r64["spec"][i][c] - r96["spec"][i][c]) for i, c in ic))))
return rows
def fibre_range_table():
"""Coefficients (13.8), (13.10) at P0 for the fibre ranges |r| <= 32, 48 against 64."""
rows = []
for name, p in BASE.items():
q = dict(DEFAULT, **p)
ref = level(q["P0"], q, 64)
for rmax in (32, 48):
l = level(q["P0"], q, rmax)
da = max(abs(l["alpha"][i] / ref["alpha"][i] - 1) for i in range(3))
db = max(abs(l["beta"][i] - ref["beta"][i]) for i in range(3))
rows.append(dict(set=name, rmax=rmax, tail=l["tail"], alpha_rel_change=da, beta_abs_change=db))
return rows
def main():
t0 = time.time()
OUT.mkdir(exist_ok=True)
oth = other_order_table() # cheap; before the evolutions
sdt = second_derivative_table() # cheap; before the evolutions
names = sys.argv[1:] or list(BASE) + list(VARIANTS)
tasks = []
for nm in names:
if nm in BASE:
tasks.append((nm, dict(DEFAULT, **BASE[nm])))
else:
b, ch = VARIANTS[nm]
tasks.append((nm, dict(dict(DEFAULT, **BASE[b]), cfgs=(1,), **ch)))
with Pool(min(len(tasks), max(1, (os.cpu_count() or 2) - 2))) as pool:
results = pool.map(run, tasks, chunksize=1)
res = {r["name"]: r for r in results}
scal = scaling_table()
frng = fibre_range_table()
clk = clock_ratio_table()
pad = " " * 22
lines = []
for nm, r in res.items():
nu, inf = r["numerics"], r["info"]
lines.append(f"== {nm}: chain |x_i|<={nu['L']} ({nu['chain_sites']} sites, dim {nu['dim']}), fibre |r|<={nu['rmax']}, "
f"packet cos^4 on |P-{nu['P0']}|<={nu['Pc']} (delta P={inf['delta_P']:.4f}), centre label {nu['xc']}, "
f"lambda<={nu['lam_max']:.0f} step {nu['dlam']} (alias margin {inf['alias_margin']:.3f}, window half width "
f"{nu['window_half_width']:.4f}), eps={nu['eps']:.1e}, {nu['seconds']} s")
lines.append(f" level: E_b={inf['Eb_P0']:.8f} E_b'={inf['dEb_P0']:+.3e} E_b''={inf['d2Eb_P0']:+.6e} "
f"<s3>={inf['s3_P0'][0]:.6f},{inf['s3_P0'][1]:.6f} gap>={inf['gap_min']:.3e} "
f"dist to bands>={inf['dist_band_min']:.3e} inside band: {inf['inside_band']}")
lines.append(f" checks: |<V1>-<V2>|<={inf['V1_minus_V2_max']:.1e}, FD(E_b') {inf['fd_check_dE']:.1e}, "
f"FD(E_b'') {inf['fd_check_d2E']:.1e}, fibre tail {inf['tail_max']:.1e}, packet norm {inf['packet_norm']:.10f}, "
f"residual {inf['packet_residual']:.1e}, norm drift {inf['norm_drift']:.1e}, edge weight {inf['edge_weight_end']:.1e}")
for c in r["configs"]:
lines.append(f" {c['config']:<22} slope v1 {c['slope_v1']:+.8e} v2 {c['slope_v2']:+.8e} | (13.7) {c['predicted_slope']:+.8e} | "
f"rel dev {c['rel_dev_v1']:+.1e} {c['rel_dev_v2']:+.1e} | half windows {max(abs(h) for h in c['half_window_rel_dev']):.1e} | "
f"two-point {c['two_point_rel_dev'][0]:+.1e} {c['two_point_rel_dev'][1]:+.1e} | "
f"rest sup {c['rest_sup'][0]:.2e} {c['rest_sup'][1]:.2e} (linear part at end {c['linear_part_end']:.2e})")
lines.append(f" {pad} separation: slope {c['sep_slope']:+.4e} (half windows {c['sep_slope_half_windows'][0]:+.4e} "
f"{c['sep_slope_half_windows'][1]:+.4e}) | (13.10) <beta>_g {c['sep_predicted_slope']:+.4e} | "
f"max |sep| {c['sep_max_abs']:.3e} | sup |sep - lambda<beta>| {c['sep_minus_linear_sup']:.3e}")
if "closed_form" in r:
cf = r["closed_form"]
for i, lab in enumerate(("common", "g1 only", "plain")):
lines.append(f" closed form {lab:<8} (13.13) {cf['closed_P0'][i]:+.6e} | level at P0 {cf['level_P0'][i]:+.6e} "
f"(rel {cf['level_P0'][i] / cf['closed_P0'][i] - 1:+.2e}) | (11.12) packet average {cf['closed_packet'][i]:+.6e} "
f"vs evolution {cf['slope_v1'][i]:+.6e} (rel {cf['slope_v1'][i] / cf['closed_packet'][i] - 1:+.2e}); "
f"u={cf['u']:.3f}, (tau/M)^2={cf['eps2'][0]:.3f},{cf['eps2'][1]:.3f}")
conv = []
lines.append("== convergence (configuration g1 only): relative change of the slopes against the base run")
for nm in res:
if nm in VARIANTS and VARIANTS[nm][0] in res:
b, ch = VARIANTS[nm]
cb = [c for c in res[b]["configs"] if c["config"] == LABELS[1]][0]
cv = res[nm]["configs"][0]
row = dict(variant=nm, base=b, changed=ch, rel_dev_variant=[cv["rel_dev_v1"], cv["rel_dev_v2"]],
rel_dev_base=[cb["rel_dev_v1"], cb["rel_dev_v2"]],
slope_change=[cv["slope_v1"] / cb["slope_v1"] - 1, cv["slope_v2"] / cb["slope_v2"] - 1],
sep_slope=[cv["sep_slope"], cb["sep_slope"]])
conv.append(row)
lines.append(f" {nm:<10} {str(ch):<22} slope change v1 {row['slope_change'][0]:+.1e} v2 {row['slope_change'][1]:+.1e} | "
f"rel dev from (13.7): variant {cv['rel_dev_v1']:+.1e} {cv['rel_dev_v2']:+.1e}, base {cb['rel_dev_v1']:+.1e} {cb['rel_dev_v2']:+.1e} | "
f"separation slope: variant {cv['sep_slope']:+.4e}, base {cb['sep_slope']:+.4e}")
lines.append("== fibre range: change of the coefficients (13.8), (13.10) at P0 against |r|<=64")
for row in frng:
lines.append(f" {row['set']} |r|<={row['rmax']}: tail of psi {row['tail']:.1e}, alpha rel change {row['alpha_rel_change']:.1e}, "
f"beta abs change {row['beta_abs_change']:.1e}")
lines.append("== scaling of the deviation from (13.13): tau -> s tau, w -> s^2 w (numerical level at P0=0, fibre |r|<=64)")
for row in scal:
lines.append(f" {row['set']} s={row['s']:<5} tau/M={row['tau_over_M'][0]:.3f},{row['tau_over_M'][1]:.3f} u={row['u']:.3f} "
f"rel dev of coefficients (g1, g2, plain): {row['rel_dev'][0]:+.3e} {row['rel_dev'][1]:+.3e} {row['rel_dev'][2]:+.3e}")
lines.append("== clock ratio Gamma_i = dE_b(0)/dM_i and its derivative in the data, T_ij = M_j dGamma_i/dM_j, against (11.20), "
"(13.22): tau -> s tau, w -> s^2 w (numerical level at P0=0, fibre |r|<=64)")
for row in clk:
td, ta, rd = row["T_rel_dev"], row["T_abs_dev"], row["rho_rel_dev"]
lines.append(f" {row['set']} s={row['s']:<5} tau/M={row['tau_over_M'][0]:.3f},{row['tau_over_M'][1]:.3f} u={row['u']:.3f} "
f"1-Gamma_i rel dev {row['one_minus_Gamma_rel_dev'][0]:+.3e} {row['one_minus_Gamma_rel_dev'][1]:+.3e} | "
f"T_11 {row['T'][0][0]:+.6e} T_12 {row['T'][0][1]:+.6e} T_21 {row['T'][1][0]:+.6e} T_22 {row['T'][1][1]:+.6e} | "
f"rel dev from (13.22) {td[0][0]:+.3e} {td[0][1]:+.3e} {td[1][0]:+.3e} {td[1][1]:+.3e} | "
f"abs dev {ta[0][0]:+.2e} {ta[0][1]:+.2e} {ta[1][0]:+.2e} {ta[1][1]:+.2e}")
lines.append(f" {pad} rho_i - g_i, (13.21) against (13.18), rel dev (common, g1 only, g2 only): part 1 {rd[0][0]:+.3e} {rd[0][1]:+.3e} "
f"{rd[0][2]:+.3e} | part 2 {rd[1][0]:+.3e} {rd[1][1]:+.3e} {rd[1][2]:+.3e} | symmetric difference in M_j against "
f"the resolvent {row['fd_rel_dev']:.1e} | fibre |r|<=48 against 64 {row['rmax48_rel_change']:.1e}")
lines.append("== other order of the limits (Step 5.3): d<sigma_3^(i)>_P/dP, d<K>_P/dP and f_i = Re F_{i,Lambda}(P) for the windows "
"[0, Lambda], per unit gradient (numerical level, fibre |r|<=96; changes against |r|<=64)")
for row in oth:
cl = "" if row["ds3_closed"] is None else (f" | leading order from (11.12) {row['ds3_closed'][0]:+.6e} "
f"{row['ds3_closed'][1]:+.6e}")
lines.append(f" {row['set']} P={row['P']:<4} E_b'={row['dEb']:+.6e} gap {row['gap']:.3e} dist to bands {row['dist_band']:.3e} "
f"inside band: {row['inside_band']} | d<s3>/dP {row['ds3'][0]:+.6e} {row['ds3'][1]:+.6e}{cl} | "
f"d<K>/dP common {row['dK'][0]:+.6e} g1 only {row['dK'][1]:+.6e} | (13.6) abs dev {row['id136_abs_dev']:.1e} | "
f"symmetric difference in P abs dev {row['ds3_fd_abs_dev']:.1e} | |r|<=64 against 96: d<s3>/dP "
f"{row['ds3_change_64_96']:.1e}, f {row['f_change_64_96']:.1e}")
for c, lab in enumerate(("common", "g1 only")):
lines.append(f" {pad} f_i {lab:<8} " + " | ".join(
f"Lambda={lam:g}: f_1 {row['f'][0][c][k]:+.6e} f_2 {row['f'][1][c][k]:+.6e}" for k, lam in enumerate(row["windows"])))
lines.append("== second derivative at lambda = 0 of c_i(lambda; P) = Re <V_i psi| R exp(-i (H - E_b) lambda) |K psi> (Step 5.3, "
"(13.29), (13.30)), per unit gradient (numerical level, fibre |r|<=96; changes against |r|<=64)")
for row in sdt:
cl = "" if row["closed"] is None else (f" | (13.30) {row['closed'][0]:+.6e} {row['closed'][1]:+.6e} "
f"(rel dev {row['closed_rel_dev'][0]:+.3e} {row['closed_rel_dev'][1]:+.3e})")
lines.append(f" {row['set']} P={row['P']:<4} part 1 with g1 only {row['own'][0]:+.6e}, part 2 with g2 only {row['own'][1]:+.6e} | "
f"part 1 with g2 only {row['cross'][0]:+.2e}, part 2 with g1 only {row['cross'][1]:+.2e} | "
f"double commutator abs dev {row['commutator_abs_dev']:.1e} | (13.29) for a binding that commutes with K "
f"{row['f1329'][0]:+.6e} {row['f1329'][1]:+.6e} (abs dev {row['f1329_abs_dev'][0]:+.1e} "
f"{row['f1329_abs_dev'][1]:+.1e}){cl} | |r|<=64 against 96: {row['change_64_96']:.1e}")
try:
import matplotlib
matplotlib.use("Agg")
import matplotlib.pyplot as plt
show = [nm for nm in ("S1", "S2", "S3", "S4", "S5") if nm in res]
fig, ax = plt.subplots(2, len(show), figsize=(4.2 * len(show), 6.4), squeeze=False)
for j, nm in enumerate(show):
lam = np.array(res[nm]["lam"])
for c, col in zip(res[nm]["configs"], ("C0", "C1", "C2")):
s = res[nm]["series"][c["config"]]
ax[0, j].plot(lam, s["v1"], col, lw=1.2, label=c["config"])
ax[0, j].plot(lam, s["v2"], col, lw=0.8, ls=":")
ax[0, j].plot(lam, c["predicted_slope"] * lam, "k--", lw=0.6)
ax[1, j].plot(lam, s["sep"], col, lw=1.0)
ax[1, j].plot(lam, c["sep_predicted_slope"] * lam, "k--", lw=0.6)
ax[0, j].set_title(f"{nm}: first-order velocities (solid 1, dotted 2)", fontsize=9)
ax[1, j].set_title("first-order separation; dashed: linear parts of (13.7), (13.10)", fontsize=9)
ax[1, j].set_xlabel("lambda")
ax[0, 0].legend(fontsize=7)
fig.tight_layout()
fig.savefig(OUT / "first_order_motion.png", dpi=110)
except Exception as exc: # the figure is not needed for the check
lines.append(f"figure skipped: {exc}")
lines.append(f"total run time {time.time() - t0:.1f} s")
(OUT / "summary.txt").write_text("\n".join(lines) + "\n", encoding="utf-8")
slim = {nm: {k: v for k, v in r.items() if k not in ("series", "lam")} for nm, r in res.items()}
(OUT / "results.json").write_text(json.dumps(dict(sets=slim, convergence=conv, fibre_range=frng, scaling=scal, clock_ratio=clk,
other_order=oth, second_derivative=sdt), indent=1, default=float),
encoding="utf-8")
print("\n".join(lines))
if __name__ == "__main__":
main()
output/first_order_motion.png302 KB · raw
output/results.json85 KB · raw
{
"sets": {
"S1": {
"name": "S1",
"info": {
"Eb_P0": 1.991769390586346,
"dEb_P0": 6.678403045245842e-24,
"d2Eb_P0": 0.014176799535102808,
"s3_P0": [
0.9970498586401999,
0.9970498586401999
],
"rmean_P0": -1.8521084904794254e-13,
"gap_min": 0.008271701136514764,
"dist_band_min": 0.008230609413653944,
"inside_band": false,
"tail_max": 1.6879367253016974e-09,
"V1_minus_V2_max": 5.637851296924623e-18,
"fd_check_dE": 1.0823381577963753e-07,
"fd_check_d2E": 5.077184005355256e-07,
"P_mean": -6.938893903907228e-18,
"delta_P": 0.1058883937041857,
"alpha_g": [
-0.013935046029841205,
-0.013935046029841205,
-0.027950861446090602
],
"beta_g": [
1.5226982824844961e-15,
1.5227100793188133e-15,
3.065032769421403e-15
],
"alpha_P0": [
-0.014134975972444706,
-0.014134975972444706,
-0.028353599070205616
],
"beta_P0": [
2.097789295022752e-18,
2.097789295022752e-18,
4.20799276353896e-18
],
"mean_dE_packet": -2.3852447794681098e-18,
"alias_margin": 0.04872050667965899,
"packet_norm": 0.9999999999287391,
"packet_residual": 3.439989775100738e-07,
"norm_drift": 1.0647038806155251e-13,
"v0_start": [
-2.4345714377312935e-18,
-2.5396211165233316e-18
],
"edge_weight_end": 2.9986378102189095e-09
},
"configs": [
{
"config": "common gradient g1=g2",
"predicted_slope": -0.02787009205968241,
"alpha_P0": -0.02826995194488941,
"slope_v1": -0.027870011785811395,
"slope_v2": -0.027870011785812505,
"rel_dev_v1": -2.8802872571453975e-06,
"rel_dev_v2": -2.880287217288391e-06,
"half_window_rel_dev": [
-7.339928193683676e-06,
-7.339928216776315e-06,
3.1207085600559736e-05,
3.1207085359863385e-05
],
"two_point_rel_dev": [
5.959485893702876e-05,
5.959485893058947e-05
],
"rest_sup": [
0.022638003105342364,
0.022638003107900317
],
"linear_part_end": 83.98652242185294,
"sep_predicted_slope": 3.0454083618033094e-15,
"sep_slope": 6.960385936653441e-13,
"sep_slope_half_windows": [
1.3883980096187664e-12,
-6.9118297973402096e-12
],
"sep_end": -9.764335118234158e-09,
"sep_max_abs": 1.5890691429376602e-08,
"sep_minus_linear_sup": 1.5898637661144638e-08
},
{
"config": "g1 only (g2=0)",
"predicted_slope": -0.013935046029841205,
"alpha_P0": -0.014134975972444706,
"slope_v1": -0.013935321093479476,
"slope_v2": -0.01393475581775897,
"rel_dev_v1": 1.973898311358191e-05,
"rel_dev_v2": -2.0826058386491653e-05,
"half_window_rel_dev": [
-0.00029380969959280634,
0.00028895345465795863,
0.012094909204611204,
-0.01203132211314828
],
"two_point_rel_dev": [
0.0010310957653001385,
-0.0009061494480220311
],
"rest_sup": [
0.8339878244819787,
0.8346920694678415
],
"linear_part_end": 41.99326121092647,
"sep_predicted_slope": 1.5226982824844961e-15,
"sep_slope": -0.00038828532974404694,
"sep_slope_half_windows": [
-0.039187898654764616,
-0.019537570710251553
],
"sep_end": -162.04222183697857,
"sep_max_abs": 270.0405195839799,
"sep_minus_linear_sup": 270.0405195839803
},
{
"config": "plain gradient",
"predicted_slope": -0.027950861446090602,
"alpha_P0": -0.028353599070205616,
"slope_v1": -0.027950779459121094,
"slope_v2": -0.02795077945911782,
"rel_dev_v1": -2.933253762771315e-06,
"rel_dev_v2": -2.933253879899844e-06,
"half_window_rel_dev": [
-6.693219174680465e-06,
-6.693218834397108e-06,
2.6372748688885395e-05,
2.6372748561209747e-05
],
"two_point_rel_dev": [
-6.528241241376964e-05,
-6.528241252101719e-05
],
"rest_sup": [
0.018989099877418036,
0.01898909988048625
],
"linear_part_end": 84.22992096779403,
"sep_predicted_slope": 3.065032769421403e-15,
"sep_slope": -2.4351369338467862e-12,
"sep_slope_half_windows": [
4.152185660134693e-12,
2.203395697241323e-12
],
"sep_end": 7.566995918750763e-10,
"sep_max_abs": 2.7168425731360912e-08,
"sep_minus_linear_sup": 2.7160203014698746e-08
}
],
"closed_form": {
"u": 0.9999999999999998,
"eps2": [
0.04000000000000001,
0.04000000000000001
],
"closed_P0": [
-0.02828427124746191,
-0.014142135623730954,
-0.02828427124746191
],
"closed_packet": [
-0.027886726991595077,
-0.013943363495797539,
-0.027886726991595077
],
"level_P0": [
-0.02826995194488941,
-0.014134975972444706,
-0.028353599070205616
],
"slope_v1": [
-0.027870011785811395,
-0.013935321093479476,
-0.027950779459121094
]
},
"numerics": {
"L": 60,
"chain_sites": 121,
"dim": 58564,
"rmax": 48,
"nP": 161,
"Pc": 0.5,
"P0": 0.0,
"xc": 0,
"lam_max": 3013.5,
"dlam": 36.75,
"window_half_width": 0.006255037637809444,
"eps": 5e-07,
"eps_plain": 5e-07,
"cheb_terms": 126,
"seconds": 350.7
}
},
"S2": {
"name": "S2",
"info": {
"Eb_P0": 2.240614014293727,
"dEb_P0": -3.18574091085535e-19,
"d2Eb_P0": 0.01003066752863435,
"s3_P0": [
0.996335156921266,
0.998095630527932
],
"rmean_P0": 2.1015849628986644e-14,
"gap_min": 0.0094198453517782,
"dist_band_min": 0.009385985706273026,
"inside_band": false,
"tail_max": 5.442634781938319e-11,
"V1_minus_V2_max": 1.3964523981613297e-16,
"fd_check_dE": 8.380500898189785e-08,
"fd_check_d2E": 3.737024656200877e-07,
"P_mean": -6.938893903907228e-18,
"delta_P": 0.1058883937041857,
"alpha_g": [
-0.009834120537409495,
-0.012314314370034045,
-0.019739735737077473
],
"beta_g": [
1.2644793269433296e-15,
1.5882292958010834e-15,
2.5482382477718345e-15
],
"alpha_P0": [
-0.009993906706166952,
-0.012514456789510443,
-0.0200613350572687
],
"beta_P0": [
-2.827870092308001e-17,
-3.541083493875294e-17,
-5.676543826971411e-17
],
"mean_dE_packet": -3.2526065174565133e-19,
"alias_margin": 0.03598128065087458,
"packet_norm": 0.9999999999305118,
"packet_residual": 2.873561629312379e-07,
"norm_drift": 5.080380560684716e-13,
"v0_start": [
2.3378439123695995e-19,
-1.29102031151915e-18
],
"edge_weight_end": 8.107926510263556e-10
},
"configs": [
{
"config": "common gradient g1=g2",
"predicted_slope": -0.02214843490744354,
"alpha_P0": -0.022508363495677395,
"slope_v1": -0.022148315920302926,
"slope_v2": -0.022148355357563106,
"rel_dev_v1": -5.372259534786039e-06,
"rel_dev_v2": -3.59167050700826e-06,
"half_window_rel_dev": [
-0.00020905812590665818,
0.00011983645765223372,
6.072275087931622e-05,
-3.577112540442684e-05
],
"two_point_rel_dev": [
-0.0001811624395272604,
7.916547814401653e-05
],
"rest_sup": [
0.17472053565036783,
0.10033215580765686
],
"linear_part_end": 66.74430859358111,
"sep_predicted_slope": 2.852708622744413e-15,
"sep_slope": -3.0080171597785477e-05,
"sep_slope_half_windows": [
-0.003565634056944688,
0.0011616043803721516
],
"sep_end": -22.20763365998573,
"sep_max_abs": 38.96600546249135,
"sep_minus_linear_sup": 38.966005462491985
},
{
"config": "g1 only (g2=0)",
"predicted_slope": -0.009834120537409495,
"alpha_P0": -0.009993906706166952,
"slope_v1": -0.009833990415379714,
"slope_v2": -0.00983419302048956,
"rel_dev_v1": -1.323168953304421e-05,
"rel_dev_v2": 7.37057063604496e-06,
"half_window_rel_dev": [
-0.002315103879232727,
0.0014907764889497965,
0.0006833488370772134,
-0.0004363427340573489
],
"two_point_rel_dev": [
-0.0011354662903279023,
0.0005903457960099168
],
"rest_sup": [
0.84314854589507,
0.5439792508747452
],
"linear_part_end": 29.635122239483515,
"sep_predicted_slope": 1.2644793269433296e-15,
"sep_slope": -0.00015424419793151801,
"sep_slope_half_windows": [
-0.018304614885373036,
0.005948830710825901
],
"sep_end": -114.55400701647159,
"sep_max_abs": 200.7028476803575,
"sep_minus_linear_sup": 200.7028476803578
},
{
"config": "plain gradient",
"predicted_slope": -0.019739735737077473,
"alpha_P0": -0.0200613350572687,
"slope_v1": -0.019739606629620193,
"slope_v2": -0.019739678324936626,
"rel_dev_v1": -6.540485597228418e-06,
"rel_dev_v2": -2.908455392303111e-06,
"half_window_rel_dev": [
-0.00041622495725635034,
0.0002540605485907843,
0.00012160439517217014,
-7.50704477437969e-05
],
"two_point_rel_dev": [
-8.077033907416542e-05,
0.00010025622098619458
],
"rest_sup": [
0.31027436838855404,
0.1903778951299231
],
"linear_part_end": 59.485693643682964,
"sep_predicted_slope": 2.5482382477718345e-15,
"sep_slope": -5.46005694325876e-05,
"sep_slope_half_windows": [
-0.006482615592151719,
0.002108662504530895
],
"sep_end": -40.55957139421662,
"sep_max_abs": 71.08718965993592,
"sep_minus_linear_sup": 71.08718965993648
}
],
"closed_form": {
"u": 1.2135922330097086,
"eps2": [
0.04000000000000001,
0.020735999999999997
],
"closed_P0": [
-0.022504287919540393,
-0.010001905742017952,
-0.020003811484035905
],
"closed_packet": [
-0.022145586656948147,
-0.00984248295864362,
-0.01968496591728724
],
"level_P0": [
-0.022508363495677395,
-0.009993906706166952,
-0.0200613350572687
],
"slope_v1": [
-0.022148315920302926,
-0.009833990415379714,
-0.019739606629620193
]
},
"numerics": {
"L": 60,
"chain_sites": 121,
"dim": 58564,
"rmax": 48,
"nP": 161,
"Pc": 0.5,
"P0": 0.0,
"xc": 0,
"lam_max": 3013.5,
"dlam": 36.75,
"window_half_width": 0.006255037637809444,
"eps": 5e-07,
"eps_plain": 5.625e-07,
"cheb_terms": 137,
"seconds": 368.5
}
},
"S3": {
"name": "S3",
"info": {
"Eb_P0": 1.9460122838986929,
"dEb_P0": -3.3538541632865066e-20,
"d2Eb_P0": -0.011038448387855348,
"s3_P0": [
0.9843268447697007,
0.9860022906747499
],
"rmean_P0": 8.670598743224439e-14,
"gap_min": 0.010330621748721125,
"dist_band_min": 0.010293537976904465,
"inside_band": false,
"tail_max": 1.895045545372569e-11,
"V1_minus_V2_max": 1.465841337200402e-16,
"fd_check_dE": 9.075780518872506e-08,
"fd_check_d2E": 4.056227565045478e-07,
"P_mean": -6.938893903907228e-18,
"delta_P": 0.1058883937041857,
"alpha_g": [
0.010694956507430117,
0.009641763759717484,
0.021731553110541375
],
"beta_g": [
1.1676400444095362e-15,
1.0522131084957993e-15,
2.35518827091307e-15
],
"alpha_P0": [
0.010865441072770843,
0.009795541856328334,
0.022076896775710696
],
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"change_64_96": 4.2500725161431774e-17
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{
"set": "S2",
"P": 0.1,
"own": [
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"cross": [
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3.681575765681613e-18
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"commutator_abs_dev": 1.5178830414797062e-17,
"f1329": [
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"f1329_abs_dev": [
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6.071532165918825e-18
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"closed": [
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],
"change_64_96": 2.727852665973529e-16
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{
"set": "S2",
"P": 0.2,
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"commutator_abs_dev": 1.9081958235744878e-17,
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"change_64_96": 1.8474805019152996e-16
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{
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"cross": [
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"commutator_abs_dev": 4.552787841074399e-18,
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"change_64_96": 1.5265566588595902e-16
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{
"set": "S3",
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"own": [
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0.003528876970532609
],
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{
"set": "S4",
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"change_64_96": 5.620504062164855e-16
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{
"set": "S4",
"P": 0.2,
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"commutator_abs_dev": 2.5673907444456745e-16,
"f1329": [
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],
"f1329_abs_dev": [
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8.326672684688674e-17
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"closed": null,
"closed_rel_dev": null,
"change_64_96": 3.0531133177191805e-16
},
{
"set": "S5",
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],
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"change_64_96": 3.3306690738754696e-16
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{
"set": "S5",
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}
]
}output/summary.txt37 KB · raw
== S1: chain |x_i|<=60 (121 sites, dim 58564), fibre |r|<=48, packet cos^4 on |P-0.0|<=0.5 (delta P=0.1059), centre label 0, lambda<=3014 step 36.75 (alias margin 0.049, window half width 0.0063), eps=5.0e-07, 350.7 s
level: E_b=1.99176939 E_b'=+6.678e-24 E_b''=+1.417680e-02 <s3>=0.997050,0.997050 gap>=8.272e-03 dist to bands>=8.231e-03 inside band: False
checks: |<V1>-<V2>|<=5.6e-18, FD(E_b') 1.1e-07, FD(E_b'') 5.1e-07, fibre tail 1.7e-09, packet norm 0.9999999999, residual 3.4e-07, norm drift 1.1e-13, edge weight 3.0e-09
common gradient g1=g2 slope v1 -2.78700118e-02 v2 -2.78700118e-02 | (13.7) -2.78700921e-02 | rel dev -2.9e-06 -2.9e-06 | half windows 3.1e-05 | two-point +6.0e-05 +6.0e-05 | rest sup 2.26e-02 2.26e-02 (linear part at end 8.40e+01)
separation: slope +6.9604e-13 (half windows +1.3884e-12 -6.9118e-12) | (13.10) <beta>_g +3.0454e-15 | max |sep| 1.589e-08 | sup |sep - lambda<beta>| 1.590e-08
g1 only (g2=0) slope v1 -1.39353211e-02 v2 -1.39347558e-02 | (13.7) -1.39350460e-02 | rel dev +2.0e-05 -2.1e-05 | half windows 1.2e-02 | two-point +1.0e-03 -9.1e-04 | rest sup 8.34e-01 8.35e-01 (linear part at end 4.20e+01)
separation: slope -3.8829e-04 (half windows -3.9188e-02 -1.9538e-02) | (13.10) <beta>_g +1.5227e-15 | max |sep| 2.700e+02 | sup |sep - lambda<beta>| 2.700e+02
plain gradient slope v1 -2.79507795e-02 v2 -2.79507795e-02 | (13.7) -2.79508614e-02 | rel dev -2.9e-06 -2.9e-06 | half windows 2.6e-05 | two-point -6.5e-05 -6.5e-05 | rest sup 1.90e-02 1.90e-02 (linear part at end 8.42e+01)
separation: slope -2.4351e-12 (half windows +4.1522e-12 +2.2034e-12) | (13.10) <beta>_g +3.0650e-15 | max |sep| 2.717e-08 | sup |sep - lambda<beta>| 2.716e-08
closed form common (13.13) -2.828427e-02 | level at P0 -2.826995e-02 (rel -5.06e-04) | (11.12) packet average -2.788673e-02 vs evolution -2.787001e-02 (rel -5.99e-04); u=1.000, (tau/M)^2=0.040,0.040
closed form g1 only (13.13) -1.414214e-02 | level at P0 -1.413498e-02 (rel -5.06e-04) | (11.12) packet average -1.394336e-02 vs evolution -1.393532e-02 (rel -5.77e-04); u=1.000, (tau/M)^2=0.040,0.040
closed form plain (13.13) -2.828427e-02 | level at P0 -2.835360e-02 (rel +2.45e-03) | (11.12) packet average -2.788673e-02 vs evolution -2.795078e-02 (rel +2.30e-03); u=1.000, (tau/M)^2=0.040,0.040
== S2: chain |x_i|<=60 (121 sites, dim 58564), fibre |r|<=48, packet cos^4 on |P-0.0|<=0.5 (delta P=0.1059), centre label 0, lambda<=3014 step 36.75 (alias margin 0.036, window half width 0.0063), eps=5.0e-07, 368.5 s
level: E_b=2.24061401 E_b'=-3.186e-19 E_b''=+1.003067e-02 <s3>=0.996335,0.998096 gap>=9.420e-03 dist to bands>=9.386e-03 inside band: False
checks: |<V1>-<V2>|<=1.4e-16, FD(E_b') 8.4e-08, FD(E_b'') 3.7e-07, fibre tail 5.4e-11, packet norm 0.9999999999, residual 2.9e-07, norm drift 5.1e-13, edge weight 8.1e-10
common gradient g1=g2 slope v1 -2.21483159e-02 v2 -2.21483554e-02 | (13.7) -2.21484349e-02 | rel dev -5.4e-06 -3.6e-06 | half windows 2.1e-04 | two-point -1.8e-04 +7.9e-05 | rest sup 1.75e-01 1.00e-01 (linear part at end 6.67e+01)
separation: slope -3.0080e-05 (half windows -3.5656e-03 +1.1616e-03) | (13.10) <beta>_g +2.8527e-15 | max |sep| 3.897e+01 | sup |sep - lambda<beta>| 3.897e+01
g1 only (g2=0) slope v1 -9.83399042e-03 v2 -9.83419302e-03 | (13.7) -9.83412054e-03 | rel dev -1.3e-05 +7.4e-06 | half windows 2.3e-03 | two-point -1.1e-03 +5.9e-04 | rest sup 8.43e-01 5.44e-01 (linear part at end 2.96e+01)
separation: slope -1.5424e-04 (half windows -1.8305e-02 +5.9488e-03) | (13.10) <beta>_g +1.2645e-15 | max |sep| 2.007e+02 | sup |sep - lambda<beta>| 2.007e+02
plain gradient slope v1 -1.97396066e-02 v2 -1.97396783e-02 | (13.7) -1.97397357e-02 | rel dev -6.5e-06 -2.9e-06 | half windows 4.2e-04 | two-point -8.1e-05 +1.0e-04 | rest sup 3.10e-01 1.90e-01 (linear part at end 5.95e+01)
separation: slope -5.4601e-05 (half windows -6.4826e-03 +2.1087e-03) | (13.10) <beta>_g +2.5482e-15 | max |sep| 7.109e+01 | sup |sep - lambda<beta>| 7.109e+01
closed form common (13.13) -2.250429e-02 | level at P0 -2.250836e-02 (rel +1.81e-04) | (11.12) packet average -2.214559e-02 vs evolution -2.214832e-02 (rel +1.23e-04); u=1.214, (tau/M)^2=0.040,0.021
closed form g1 only (13.13) -1.000191e-02 | level at P0 -9.993907e-03 (rel -8.00e-04) | (11.12) packet average -9.842483e-03 vs evolution -9.833990e-03 (rel -8.63e-04); u=1.214, (tau/M)^2=0.040,0.021
closed form plain (13.13) -2.000381e-02 | level at P0 -2.006134e-02 (rel +2.88e-03) | (11.12) packet average -1.968497e-02 vs evolution -1.973961e-02 (rel +2.78e-03); u=1.214, (tau/M)^2=0.040,0.021
== S3: chain |x_i|<=60 (121 sites, dim 58564), fibre |r|<=48, packet cos^4 on |P-0.0|<=0.5 (delta P=0.1059), centre label 0, lambda<=3008 step 32.0 (alias margin 0.040, window half width 0.0063), eps=5.0e-07, 357.2 s
level: E_b=1.94601228 E_b'=-3.354e-20 E_b''=-1.103845e-02 <s3>=0.984327,0.986002 gap>=1.033e-02 dist to bands>=1.029e-02 inside band: False
checks: |<V1>-<V2>|<=1.5e-16, FD(E_b') 9.1e-08, FD(E_b'') 4.1e-07, fibre tail 1.9e-11, packet norm 0.9999999999, residual 2.9e-07, norm drift 1.7e-13, edge weight 1.0e-09
common gradient g1=g2 slope v1 +2.03366779e-02 v2 +2.03366466e-02 | (13.7) +2.03367203e-02 | rel dev -2.1e-06 -3.6e-06 | half windows 2.2e-04 | two-point +7.4e-05 -4.6e-05 | rest sup 1.95e-01 1.56e-01 (linear part at end 6.12e+01)
separation: slope +2.3910e-05 (half windows +1.6482e-03 -2.0806e-04) | (13.10) <beta>_g +2.2199e-15 | max |sep| 4.354e+01 | sup |sep - lambda<beta>| 4.354e+01
g1 only (g2=0) slope v1 +1.06950117e-02 v2 +1.06948982e-02 | (13.7) +1.06949565e-02 | rel dev +5.2e-06 -5.5e-06 | half windows 1.5e-03 | two-point +6.7e-04 -5.8e-04 | rest sup 6.65e-01 5.44e-01 (linear part at end 3.22e+01)
separation: slope +8.6585e-05 (half windows +5.9659e-03 -7.5495e-04) | (13.10) <beta>_g +1.1676e-15 | max |sep| 1.574e+02 | sup |sep - lambda<beta>| 1.574e+02
plain gradient slope v1 +2.17315005e-02 v2 +2.17314776e-02 | (13.7) +2.17315531e-02 | rel dev -2.4e-06 -3.5e-06 | half windows 1.5e-04 | two-point +6.9e-05 -6.3e-05 | rest sup 1.45e-01 1.03e-01 (linear part at end 6.54e+01)
separation: slope +1.7507e-05 (half windows +1.2057e-03 -1.5206e-04) | (13.10) <beta>_g +2.3552e-15 | max |sep| 3.183e+01 | sup |sep - lambda<beta>| 3.183e+01
closed form common (13.13) +2.145035e-02 | level at P0 +2.066098e-02 (rel -3.68e-02) | (11.12) packet average +2.111232e-02 vs evolution +2.033668e-02 (rel -3.67e-02); u=1.220, (tau/M)^2=0.040,0.036
closed form g1 only (13.13) +1.128966e-02 | level at P0 +1.086544e-02 (rel -3.76e-02) | (11.12) packet average +1.111175e-02 vs evolution +1.069501e-02 (rel -3.75e-02); u=1.220, (tau/M)^2=0.040,0.036
closed form plain (13.13) +2.257932e-02 | level at P0 +2.207690e-02 (rel -2.23e-02) | (11.12) packet average +2.222350e-02 vs evolution +2.173150e-02 (rel -2.21e-02); u=1.220, (tau/M)^2=0.040,0.036
== S4: chain |x_i|<=60 (121 sites, dim 58564), fibre |r|<=48, packet cos^4 on |P-0.0|<=0.5 (delta P=0.1059), centre label 0, lambda<=200 step 1.0 (alias margin 1.716, window half width 0.0942), eps=2.0e-06, 222.5 s
level: E_b=1.71210419 E_b'=-2.020e-17 E_b''=+1.539205e-01 <s3>=0.768512,0.881314 gap>=4.887e-01 dist to bands>=4.879e-01 inside band: False
checks: |<V1>-<V2>|<=6.5e-16, FD(E_b') 1.5e-06, FD(E_b'') 7.7e-06, fibre tail 1.2e-14, packet norm 0.9999999999, residual 1.8e-06, norm drift 1.5e-14, edge weight 8.6e-10
common gradient g1=g2 slope v1 -2.75262112e-01 v2 -2.75262112e-01 | (13.7) -2.75262183e-01 | rel dev -2.6e-07 -2.6e-07 | half windows 7.0e-07 | two-point +1.5e-03 -9.4e-04 | rest sup 5.77e-01 5.65e-01 (linear part at end 5.51e+01)
separation: slope -7.3986e-08 (half windows -2.3618e-06 -1.0784e-07) | (13.10) <beta>_g +1.1255e-17 | max |sep| 1.211e+00 | sup |sep - lambda<beta>| 1.211e+00
g1 only (g2=0) slope v1 -1.15854383e-01 v2 -1.15854384e-01 | (13.7) -1.15854388e-01 | rel dev -4.7e-08 -3.8e-08 | half windows 1.1e-06 | two-point +3.9e-03 +6.8e-04 | rest sup 6.94e-01 2.59e-01 (linear part at end 2.32e+01)
separation: slope -1.8394e-07 (half windows -5.9357e-06 -2.3784e-07) | (13.10) <beta>_g +5.0628e-18 | max |sep| 2.411e+00 | sup |sep - lambda<beta>| 2.411e+00
plain gradient slope v1 -3.01229645e-01 v2 -3.01229646e-01 | (13.7) -3.01229766e-01 | rel dev -4.0e-07 -4.0e-07 | half windows 9.3e-07 | two-point -1.5e-03 +4.7e-04 | rest sup 6.75e-01 2.78e-01 (linear part at end 6.02e+01)
separation: slope -7.3840e-08 (half windows -2.4097e-06 -8.0318e-08) | (13.10) <beta>_g +1.0811e-17 | max |sep| 9.639e-01 | sup |sep - lambda<beta>| 9.639e-01
== S5: chain |x_i|<=60 (121 sites, dim 58564), fibre |r|<=48, packet cos^4 on |P-0.2|<=0.4 (delta P=0.0847), centre label -12, lambda<=400 step 1.0 (alias margin 1.427, window half width 0.0471), eps=2.0e-06, 362.2 s
level: E_b=2.00101648 E_b'=+5.016e-02 E_b''=+2.405046e-01 <s3>=0.896692,0.925514 gap>=7.444e-02 dist to bands>=2.040e-01 inside band: False
checks: |<V1>-<V2>|<=1.2e-15, FD(E_b') 1.7e-06, FD(E_b'') 6.8e-06, fibre tail 2.1e-10, packet norm 0.9999999973, residual 5.2e-06, norm drift 8.1e-14, edge weight 3.2e-06
common gradient g1=g2 slope v1 -4.81459652e-01 v2 -4.81467652e-01 | (13.7) -4.81456283e-01 | rel dev +7.0e-06 +2.4e-05 | half windows 5.4e-03 | two-point -3.9e-03 -2.1e-03 | rest sup 2.10e+00 3.11e+00 (linear part at end 1.93e+02)
separation: slope -8.6620e-02 (half windows -1.1118e-01 -7.2999e-02) | (13.10) <beta>_g -8.6508e-02 | max |sep| 4.690e+01 | sup |sep - lambda<beta>| 1.495e+01
g1 only (g2=0) slope v1 -2.14925383e-01 v2 -2.14928547e-01 | (13.7) -2.14923838e-01 | rel dev +7.2e-06 +2.2e-05 | half windows 5.4e-03 | two-point -3.5e-03 -1.7e-03 | rest sup 1.94e+00 1.42e+00 (linear part at end 8.60e+01)
separation: slope -2.8194e-02 (half windows -3.9456e-02 -2.2817e-02) | (13.10) <beta>_g -2.8133e-02 | max |sep| 2.319e+01 | sup |sep - lambda<beta>| 1.483e+01
plain gradient slope v1 -4.75406552e-01 v2 -4.75417030e-01 | (13.7) -4.75402144e-01 | rel dev +9.3e-06 +3.1e-05 | half windows 6.0e-03 | two-point +1.1e-03 +3.7e-03 | rest sup 8.27e-01 9.60e-01 (linear part at end 1.90e+02)
separation: slope -7.4571e-02 (half windows -1.0636e-01 -5.5695e-02) | (13.10) <beta>_g -7.4440e-02 | max |sep| 4.637e+01 | sup |sep - lambda<beta>| 1.927e+01
== S1_L45: chain |x_i|<=45 (91 sites, dim 33124), fibre |r|<=48, packet cos^4 on |P-0.0|<=0.5 (delta P=0.1059), centre label 0, lambda<=3014 step 36.75 (alias margin 0.049, window half width 0.0063), eps=5.0e-07, 123.5 s
level: E_b=1.99176939 E_b'=+6.678e-24 E_b''=+1.417680e-02 <s3>=0.997050,0.997050 gap>=8.272e-03 dist to bands>=8.231e-03 inside band: False
checks: |<V1>-<V2>|<=5.6e-18, FD(E_b') 1.1e-07, FD(E_b'') 5.1e-07, fibre tail 1.7e-09, packet norm 0.9999999988, residual 7.0e-07, norm drift 2.4e-14, edge weight 3.0e-07
g1 only (g2=0) slope v1 -1.39353174e-02 v2 -1.39347522e-02 | (13.7) -1.39350460e-02 | rel dev +1.9e-05 -2.1e-05 | half windows 1.2e-02 | two-point +1.0e-03 -9.1e-04 | rest sup 8.34e-01 8.35e-01 (linear part at end 4.20e+01)
separation: slope -3.8828e-04 (half windows -3.9188e-02 -1.9538e-02) | (13.10) <beta>_g +1.5227e-15 | max |sep| 2.700e+02 | sup |sep - lambda<beta>| 2.700e+02
== S1_eps2: chain |x_i|<=60 (121 sites, dim 58564), fibre |r|<=48, packet cos^4 on |P-0.0|<=0.5 (delta P=0.1059), centre label 0, lambda<=3014 step 36.75 (alias margin 0.049, window half width 0.0063), eps=1.0e-06, 206.6 s
level: E_b=1.99176939 E_b'=+6.678e-24 E_b''=+1.417680e-02 <s3>=0.997050,0.997050 gap>=8.272e-03 dist to bands>=8.231e-03 inside band: False
checks: |<V1>-<V2>|<=5.6e-18, FD(E_b') 1.1e-07, FD(E_b'') 5.1e-07, fibre tail 1.7e-09, packet norm 0.9999999999, residual 3.4e-07, norm drift 9.6e-14, edge weight 3.1e-09
g1 only (g2=0) slope v1 -1.39352885e-02 v2 -1.39347235e-02 | (13.7) -1.39350460e-02 | rel dev +1.7e-05 -2.3e-05 | half windows 1.2e-02 | two-point +1.0e-03 -9.1e-04 | rest sup 8.34e-01 8.35e-01 (linear part at end 4.20e+01)
separation: slope -3.8798e-04 (half windows -3.9187e-02 -1.9536e-02) | (13.10) <beta>_g +1.5227e-15 | max |sep| 2.700e+02 | sup |sep - lambda<beta>| 2.700e+02
== S1_Pc035: chain |x_i|<=60 (121 sites, dim 58564), fibre |r|<=48, packet cos^4 on |P-0.0|<=0.35 (delta P=0.0741), centre label 0, lambda<=3014 step 36.75 (alias margin 0.052, window half width 0.0063), eps=5.0e-07, 207.2 s
level: E_b=1.99176939 E_b'=+6.678e-24 E_b''=+1.417680e-02 <s3>=0.997050,0.997050 gap>=8.272e-03 dist to bands>=8.231e-03 inside band: False
checks: |<V1>-<V2>|<=5.2e-18, FD(E_b') 3.7e-08, FD(E_b'') 2.4e-07, fibre tail 1.7e-09, packet norm 0.9999999983, residual 1.7e-06, norm drift 2.8e-13, edge weight 1.8e-08
g1 only (g2=0) slope v1 -1.40374166e-02 v2 -1.40368376e-02 | (13.7) -1.40371355e-02 | rel dev +2.0e-05 -2.1e-05 | half windows 1.2e-02 | two-point +1.0e-03 -8.8e-04 | rest sup 8.37e-01 8.41e-01 (linear part at end 4.23e+01)
separation: slope -3.9129e-04 (half windows -3.9477e-02 -1.9888e-02) | (13.10) <beta>_g -1.4341e-15 | max |sep| 2.720e+02 | sup |sep - lambda<beta>| 2.720e+02
== S1_dlam: chain |x_i|<=60 (121 sites, dim 58564), fibre |r|<=48, packet cos^4 on |P-0.0|<=0.5 (delta P=0.1059), centre label 0, lambda<=3003 step 21.0 (alias margin 0.091, window half width 0.0063), eps=5.0e-07, 234.5 s
level: E_b=1.99176939 E_b'=+6.678e-24 E_b''=+1.417680e-02 <s3>=0.997050,0.997050 gap>=8.272e-03 dist to bands>=8.231e-03 inside band: False
checks: |<V1>-<V2>|<=5.6e-18, FD(E_b') 1.1e-07, FD(E_b'') 5.1e-07, fibre tail 1.7e-09, packet norm 0.9999999999, residual 3.4e-07, norm drift 1.3e-13, edge weight 3.0e-09
g1 only (g2=0) slope v1 -1.39353260e-02 v2 -1.39347463e-02 | (13.7) -1.39350460e-02 | rel dev +2.0e-05 -2.2e-05 | half windows 1.3e-02 | two-point +8.3e-04 -9.3e-04 | rest sup 8.50e-01 8.39e-01 (linear part at end 4.18e+01)
separation: slope -3.8979e-04 (half windows -3.9284e-02 -1.9855e-02) | (13.10) <beta>_g +1.5227e-15 | max |sep| 2.696e+02 | sup |sep - lambda<beta>| 2.696e+02
== S4_L45: chain |x_i|<=45 (91 sites, dim 33124), fibre |r|<=48, packet cos^4 on |P-0.0|<=0.5 (delta P=0.1059), centre label 0, lambda<=200 step 1.0 (alias margin 1.716, window half width 0.0942), eps=2.0e-06, 84.3 s
level: E_b=1.71210419 E_b'=-2.020e-17 E_b''=+1.539205e-01 <s3>=0.768512,0.881314 gap>=4.887e-01 dist to bands>=4.879e-01 inside band: False
checks: |<V1>-<V2>|<=6.5e-16, FD(E_b') 1.5e-06, FD(E_b'') 7.7e-06, fibre tail 1.2e-14, packet norm 0.9999999988, residual 5.3e-06, norm drift 9.1e-14, edge weight 4.9e-08
g1 only (g2=0) slope v1 -1.15854376e-01 v2 -1.15854377e-01 | (13.7) -1.15854388e-01 | rel dev -1.0e-07 -9.2e-08 | half windows 1.3e-06 | two-point +3.9e-03 +6.5e-04 | rest sup 6.94e-01 2.59e-01 (linear part at end 2.32e+01)
separation: slope -1.8301e-07 (half windows -5.9357e-06 -2.3421e-07) | (13.10) <beta>_g +5.0628e-18 | max |sep| 2.411e+00 | sup |sep - lambda<beta>| 2.411e+00
== S5_L75: chain |x_i|<=75 (151 sites, dim 91204), fibre |r|<=48, packet cos^4 on |P-0.2|<=0.4 (delta P=0.0847), centre label -12, lambda<=400 step 1.0 (alias margin 1.427, window half width 0.0471), eps=2.0e-06, 297.3 s
level: E_b=2.00101648 E_b'=+5.016e-02 E_b''=+2.405046e-01 <s3>=0.896692,0.925514 gap>=7.444e-02 dist to bands>=2.040e-01 inside band: False
checks: |<V1>-<V2>|<=1.2e-15, FD(E_b') 1.7e-06, FD(E_b'') 6.8e-06, fibre tail 2.1e-10, packet norm 0.9999999998, residual 1.1e-06, norm drift 1.4e-13, edge weight 4.2e-08
g1 only (g2=0) slope v1 -2.14925708e-01 v2 -2.14928869e-01 | (13.7) -2.14923838e-01 | rel dev +8.7e-06 +2.3e-05 | half windows 5.5e-03 | two-point -3.4e-03 -1.8e-03 | rest sup 1.94e+00 1.42e+00 (linear part at end 8.60e+01)
separation: slope -2.8194e-02 (half windows -3.9456e-02 -2.2817e-02) | (13.10) <beta>_g -2.8133e-02 | max |sep| 2.320e+01 | sup |sep - lambda<beta>| 1.483e+01
== S5_long: chain |x_i|<=75 (151 sites, dim 91204), fibre |r|<=48, packet cos^4 on |P-0.2|<=0.4 (delta P=0.0847), centre label -15, lambda<=600 step 1.0 (alias margin 1.427, window half width 0.0314), eps=2.0e-06, 369.2 s
level: E_b=2.00101648 E_b'=+5.016e-02 E_b''=+2.405046e-01 <s3>=0.896692,0.925514 gap>=7.444e-02 dist to bands>=2.040e-01 inside band: False
checks: |<V1>-<V2>|<=1.2e-15, FD(E_b') 1.7e-06, FD(E_b'') 6.8e-06, fibre tail 2.1e-10, packet norm 0.9999999997, residual 4.1e-06, norm drift 1.1e-13, edge weight 8.7e-06
g1 only (g2=0) slope v1 -2.14923024e-01 v2 -2.14922278e-01 | (13.7) -2.14923838e-01 | rel dev -3.8e-06 -7.3e-06 | half windows 3.0e-04 | two-point -4.3e-03 -6.4e-03 | rest sup 2.31e+00 1.65e+00 (linear part at end 1.29e+02)
separation: slope -2.8139e-02 (half windows -2.8900e-02 -2.7635e-02) | (13.10) <beta>_g -2.8133e-02 | max |sep| 2.828e+01 | sup |sep - lambda<beta>| 1.561e+01
== convergence (configuration g1 only): relative change of the slopes against the base run
S1_L45 {'L': 45} slope change v1 -2.6e-07 v2 -2.6e-07 | rel dev from (13.7): variant +1.9e-05 -2.1e-05, base +2.0e-05 -2.1e-05 | separation slope: variant -3.8828e-04, base -3.8829e-04
S1_eps2 {'eps': 1e-06} slope change v1 -2.3e-06 v2 -2.3e-06 | rel dev from (13.7): variant +1.7e-05 -2.3e-05, base +2.0e-05 -2.1e-05 | separation slope: variant -3.8798e-04, base -3.8829e-04
S1_Pc035 {'Pc': 0.35} slope change v1 +7.3e-03 v2 +7.3e-03 | rel dev from (13.7): variant +2.0e-05 -2.1e-05, base +2.0e-05 -2.1e-05 | separation slope: variant -3.9129e-04, base -3.8829e-04
S1_dlam {'dlam': (20.0, 28.0)} slope change v1 +3.5e-07 v2 -6.8e-07 | rel dev from (13.7): variant +2.0e-05 -2.2e-05, base +2.0e-05 -2.1e-05 | separation slope: variant -3.8979e-04, base -3.8829e-04
S4_L45 {'L': 45} slope change v1 -5.4e-08 v2 -5.4e-08 | rel dev from (13.7): variant -1.0e-07 -9.2e-08, base -4.7e-08 -3.8e-08 | separation slope: variant -1.8301e-07, base -1.8394e-07
S5_L75 {'L': 75} slope change v1 +1.5e-06 v2 +1.5e-06 | rel dev from (13.7): variant +8.7e-06 +2.3e-05, base +7.2e-06 +2.2e-05 | separation slope: variant -2.8194e-02, base -2.8194e-02
S5_long {'L': 75, 'lam_max': 600.0, 'xc': -15} slope change v1 -1.1e-05 v2 -2.9e-05 | rel dev from (13.7): variant -3.8e-06 -7.3e-06, base +7.2e-06 +2.2e-05 | separation slope: variant -2.8139e-02, base -2.8194e-02
== fibre range: change of the coefficients (13.8), (13.10) at P0 against |r|<=64
S1 |r|<=32: tail of psi 1.9e-06, alpha rel change 1.1e-11, beta abs change 1.8e-17
S1 |r|<=48: tail of psi 1.7e-09, alpha rel change 9.4e-14, beta abs change 2.7e-17
S2 |r|<=32: tail of psi 1.9e-07, alpha rel change 1.8e-13, beta abs change 2.1e-17
S2 |r|<=48: tail of psi 5.4e-11, alpha rel change 1.1e-14, beta abs change 6.6e-17
S3 |r|<=32: tail of psi 7.4e-08, alpha rel change 1.4e-13, beta abs change 3.0e-17
S3 |r|<=48: tail of psi 1.9e-11, alpha rel change 6.1e-14, beta abs change 4.2e-18
S4 |r|<=32: tail of psi 7.3e-10, alpha rel change 1.1e-14, beta abs change 3.2e-17
S4 |r|<=48: tail of psi 1.2e-14, alpha rel change 9.1e-15, beta abs change 1.1e-16
S5 |r|<=32: tail of psi 3.6e-07, alpha rel change 1.8e-13, beta abs change 3.7e-12
S5 |r|<=48: tail of psi 1.4e-10, alpha rel change 4.0e-15, beta abs change 7.6e-16
== scaling of the deviation from (13.13): tau -> s tau, w -> s^2 w (numerical level at P0=0, fibre |r|<=64)
S1 s=1.0 tau/M=0.200,0.200 u=1.000 rel dev of coefficients (g1, g2, plain): -5.063e-04 -5.063e-04 +2.451e-03
S1 s=0.5 tau/M=0.100,0.100 u=1.000 rel dev of coefficients (g1, g2, plain): -1.121e-04 -1.121e-04 +6.220e-04
S1 s=0.25 tau/M=0.050,0.050 u=1.000 rel dev of coefficients (g1, g2, plain): -2.711e-05 -2.711e-05 +1.561e-04
S2 s=1.0 tau/M=0.200,0.144 u=1.214 rel dev of coefficients (g1, g2, plain): -7.998e-04 +9.658e-04 +2.876e-03
S2 s=0.5 tau/M=0.100,0.072 u=1.214 rel dev of coefficients (g1, g2, plain): -1.851e-04 +2.540e-04 +7.272e-04
S2 s=0.25 tau/M=0.050,0.036 u=1.214 rel dev of coefficients (g1, g2, plain): -4.536e-05 +6.428e-05 +1.823e-04
S3 s=1.0 tau/M=0.200,0.189 u=1.220 rel dev of coefficients (g1, g2, plain): -3.758e-02 -3.594e-02 -2.225e-02
S3 s=0.5 tau/M=0.100,0.094 u=1.220 rel dev of coefficients (g1, g2, plain): -9.687e-03 -9.254e-03 -5.668e-03
S3 s=0.25 tau/M=0.050,0.047 u=1.220 rel dev of coefficients (g1, g2, plain): -2.441e-03 -2.331e-03 -1.424e-03
== clock ratio Gamma_i = dE_b(0)/dM_i and its derivative in the data, T_ij = M_j dGamma_i/dM_j, against (11.20), (13.22): tau -> s tau, w -> s^2 w (numerical level at P0=0, fibre |r|<=64)
S1 s=1.0 tau/M=0.200,0.200 u=1.000 1-Gamma_i rel dev +7.241e-03 +7.241e-03 | T_11 +4.116978e-03 T_12 -1.777954e-03 T_21 -1.777954e-03 T_22 +4.116978e-03 | rel dev from (13.22) +6.572e-03 +5.763e-03 +5.763e-03 +6.572e-03 | abs dev +2.69e-05 -1.02e-05 -1.02e-05 +2.69e-05
rho_i - g_i, (13.21) against (13.18), rel dev (common, g1 only, g2 only): part 1 +1.017e-02 +9.550e-03 +8.739e-03 | part 2 +1.017e-02 +8.739e-03 +9.550e-03 | symmetric difference in M_j against the resolvent 8.5e-09 | fibre |r|<=48 against 64 2.6e-13
S1 s=0.5 tau/M=0.100,0.100 u=1.000 1-Gamma_i rel dev +1.812e-03 +1.812e-03 | T_11 +1.024220e-03 T_12 -4.425802e-04 T_21 -4.425802e-04 T_22 +1.024220e-03 | rel dev from (13.22) +1.658e-03 +1.445e-03 +1.445e-03 +1.658e-03 | abs dev +1.70e-06 -6.38e-07 -6.38e-07 +1.70e-06
rho_i - g_i, (13.21) against (13.18), rel dev (common, g1 only, g2 only): part 1 +2.556e-03 +2.394e-03 +2.180e-03 | part 2 +2.556e-03 +2.180e-03 +2.394e-03 | symmetric difference in M_j against the resolvent 1.2e-08 | fibre |r|<=48 against 64 5.5e-13
S1 s=0.25 tau/M=0.050,0.050 u=1.000 1-Gamma_i rel dev +4.530e-04 +4.530e-04 | T_11 +2.557373e-04 T_12 -1.105254e-04 T_21 -1.105254e-04 T_22 +2.557373e-04 | rel dev from (13.22) +4.155e-04 +3.614e-04 +3.614e-04 +4.155e-04 | abs dev +1.06e-07 -3.99e-08 -3.99e-08 +1.06e-07
rho_i - g_i, (13.21) against (13.18), rel dev (common, g1 only, g2 only): part 1 +6.399e-04 +5.988e-04 +5.447e-04 | part 2 +6.399e-04 +5.447e-04 +5.988e-04 | symmetric difference in M_j against the resolvent 3.6e-08 | fibre |r|<=48 against 64 2.5e-12
S2 s=1.0 tau/M=0.200,0.144 u=1.214 1-Gamma_i rel dev +6.614e-03 +9.006e-03 | T_11 +5.010141e-03 T_12 -1.499704e-03 T_21 -1.199764e-03 T_22 +3.035993e-03 | rel dev from (13.22) +5.397e-03 +6.995e-03 +6.995e-03 +1.109e-02 | abs dev +2.69e-05 -1.04e-05 -8.33e-06 +3.33e-05
rho_i - g_i, (13.21) against (13.18), rel dev (common, g1 only, g2 only): part 1 +8.412e-03 +9.095e-03 +1.070e-02 | part 2 +1.572e-02 +8.917e-03 +1.302e-02 | symmetric difference in M_j against the resolvent 1.6e-08 | fibre |r|<=48 against 64 7.5e-13
S2 s=0.5 tau/M=0.100,0.072 u=1.214 1-Gamma_i rel dev +1.655e-03 +2.243e-03 | T_11 +1.247511e-03 T_12 -3.729741e-04 T_21 -2.983792e-04 T_22 +7.527511e-04 | rel dev from (13.22) +1.364e-03 +1.752e-03 +1.752e-03 +2.766e-03 | abs dev +1.70e-06 -6.52e-07 -5.22e-07 +2.08e-06
rho_i - g_i, (13.21) against (13.18), rel dev (common, g1 only, g2 only): part 1 +2.112e-03 +2.278e-03 +2.666e-03 | part 2 +3.908e-03 +2.226e-03 +3.241e-03 | symmetric difference in M_j against the resolvent 5.1e-08 | fibre |r|<=48 against 64 6.0e-13
S2 s=0.25 tau/M=0.050,0.036 u=1.214 1-Gamma_i rel dev +4.138e-04 +5.604e-04 | T_11 +3.115593e-04 T_12 -9.312121e-05 T_21 -7.449697e-05 T_22 +1.877983e-04 | rel dev from (13.22) +3.419e-04 +4.383e-04 +4.383e-04 +6.912e-04 | abs dev +1.06e-07 -4.08e-08 -3.26e-08 +1.30e-07
rho_i - g_i, (13.21) against (13.18), rel dev (common, g1 only, g2 only): part 1 +5.286e-04 +5.697e-04 +6.661e-04 | part 2 +9.758e-04 +5.564e-04 +8.094e-04 | symmetric difference in M_j against the resolvent 7.1e-08 | fibre |r|<=48 against 64 4.8e-12
S3 s=1.0 tau/M=0.200,0.189 u=1.220 1-Gamma_i rel dev -4.071e-02 -3.950e-02 | T_11 +3.236098e-02 T_12 +1.643041e-03 T_21 +1.825601e-03 T_22 +2.861149e-02 | rel dev from (13.22) -6.955e-02 -2.696e-02 -2.696e-02 -6.659e-02 | abs dev -2.42e-03 -4.55e-05 -5.06e-05 -2.04e-03
rho_i - g_i, (13.21) against (13.18), rel dev (common, g1 only, g2 only): part 1 -5.273e-02 -5.473e-02 -1.147e-02 | part 2 -5.102e-02 -1.315e-02 -5.334e-02 | symmetric difference in M_j against the resolvent 2.0e-08 | fibre |r|<=48 against 64 1.1e-13
S3 s=0.5 tau/M=0.100,0.094 u=1.220 1-Gamma_i rel dev -1.049e-02 -1.016e-02 | T_11 +8.536987e-03 T_12 +4.192139e-04 T_21 +4.657932e-04 T_22 +7.530158e-03 | rel dev from (13.22) -1.817e-02 -6.933e-03 -6.933e-03 -1.736e-02 | abs dev -1.58e-04 -2.93e-06 -3.25e-06 -1.33e-04
rho_i - g_i, (13.21) against (13.18), rel dev (common, g1 only, g2 only): part 1 -1.366e-02 -1.418e-02 -2.903e-03 | part 2 -1.320e-02 -3.339e-03 -1.380e-02 | symmetric difference in M_j against the resolvent 2.2e-08 | fibre |r|<=48 against 64 8.1e-13
S3 s=0.25 tau/M=0.050,0.047 u=1.220 1-Gamma_i rel dev -2.643e-03 -2.560e-03 | T_11 +2.163755e-03 T_12 +1.053509e-04 T_21 +1.170566e-04 T_22 +1.907393e-03 | rel dev from (13.22) -4.594e-03 -1.746e-03 -1.746e-03 -4.387e-03 | abs dev -9.99e-06 -1.84e-07 -2.05e-07 -8.40e-06
rho_i - g_i, (13.21) against (13.18), rel dev (common, g1 only, g2 only): part 1 -3.447e-03 -3.579e-03 -7.286e-04 | part 2 -3.329e-03 -8.386e-04 -3.481e-03 | symmetric difference in M_j against the resolvent 4.2e-08 | fibre |r|<=48 against 64 4.3e-12
== other order of the limits (Step 5.3): d<sigma_3^(i)>_P/dP, d<K>_P/dP and f_i = Re F_{i,Lambda}(P) for the windows [0, Lambda], per unit gradient (numerical level, fibre |r|<=96; changes against |r|<=64)
S1 P=0.0 E_b'=+1.622880e-20 gap 8.254e-03 dist to bands 8.231e-03 inside band: False | d<s3>/dP +4.743385e-19 -1.246832e-18 | leading order from (11.12) -0.000000e+00 -0.000000e+00 | d<K>/dP common -7.724940e-19 g1 only +4.743385e-19 | (13.6) abs dev 2.7e-18 | symmetric difference in P abs dev 1.7e-18 | |r|<=64 against 96: d<s3>/dP 2.2e-18, f 3.6e-17
f_i common Lambda=5: f_1 +2.500441e-17 f_2 -2.368982e-17 | Lambda=10: f_1 +5.235341e-17 f_2 -5.067629e-17 | Lambda=20: f_1 +9.484059e-17 f_2 -9.025983e-17
f_i g1 only Lambda=5: f_1 +1.111307e-17 f_2 -1.192792e-17 | Lambda=10: f_1 +2.156885e-17 f_2 -2.535678e-17 | Lambda=20: f_1 +3.987153e-17 f_2 -5.533497e-17
S1 P=0.2 E_b'=+2.787495e-03 gap 8.357e-03 dist to bands 8.348e-03 inside band: False | d<s3>/dP -2.101506e-03 -2.101506e-03 | leading order from (11.12) -2.099885e-03 -2.099885e-03 | d<K>/dP common -4.203011e-03 g1 only -2.101506e-03 | (13.6) abs dev 2.9e-17 | symmetric difference in P abs dev 2.2e-13 | |r|<=64 against 96: d<s3>/dP 7.8e-17, f 1.5e-16
f_i common Lambda=5: f_1 +2.556160e-03 f_2 +2.556160e-03 | Lambda=10: f_1 +8.476831e-04 f_2 +8.476831e-04 | Lambda=20: f_1 +2.421435e-03 f_2 +2.421435e-03
f_i g1 only Lambda=5: f_1 +2.551134e-03 f_2 +5.026191e-06 | Lambda=10: f_1 +8.243607e-04 f_2 +2.332240e-05 | Lambda=20: f_1 +2.331810e-03 f_2 +8.962494e-05
S2 P=0.0 E_b'=+7.001012e-20 gap 9.405e-03 dist to bands 9.386e-03 inside band: False | d<s3>/dP +1.893174e-19 -1.231586e-19 | leading order from (11.12) -0.000000e+00 -0.000000e+00 | d<K>/dP common +3.536912e-20 g1 only +1.893174e-19 | (13.6) abs dev 1.0e-19 | symmetric difference in P abs dev 3.1e-19 | |r|<=64 against 96: d<s3>/dP 5.0e-19, f 4.3e-16
f_i common Lambda=5: f_1 +8.882792e-18 f_2 -3.188961e-17 | Lambda=10: f_1 +2.530548e-17 f_2 -4.431778e-17 | Lambda=20: f_1 +9.472226e-18 f_2 -9.124934e-17
f_i g1 only Lambda=5: f_1 -3.521432e-18 f_2 -8.401853e-18 | Lambda=10: f_1 +2.337345e-17 f_2 +1.818821e-18 | Lambda=20: f_1 -5.347580e-18 f_2 -5.592341e-18
S2 P=0.2 E_b'=+1.967891e-03 gap 9.509e-03 dist to bands 9.501e-03 inside band: False | d<s3>/dP -1.489668e-03 -1.332555e-03 | leading order from (11.12) -1.488631e-03 -1.328612e-03 | d<K>/dP common -3.155362e-03 g1 only -1.489668e-03 | (13.6) abs dev 1.5e-16 | symmetric difference in P abs dev 9.0e-12 | |r|<=64 against 96: d<s3>/dP 1.7e-16, f 5.8e-16
f_i common Lambda=5: f_1 +1.800514e-03 f_2 +5.635486e-06 | Lambda=10: f_1 +6.061171e-04 f_2 +2.426129e-05 | Lambda=20: f_1 +1.722972e-03 f_2 +9.666734e-05
f_i g1 only Lambda=5: f_1 +1.796676e-03 f_2 +3.746826e-06 | Lambda=10: f_1 +5.893425e-04 f_2 +1.666130e-05 | Lambda=20: f_1 +1.656913e-03 f_2 +6.555673e-05
S3 P=0.0 E_b'=-5.572527e-20 gap 1.031e-02 dist to bands 1.029e-02 inside band: False | d<s3>/dP +1.279541e-19 -1.780332e-19 | leading order from (11.12) +0.000000e+00 +0.000000e+00 | d<K>/dP common -3.227579e-20 g1 only +1.279541e-19 | (13.6) abs dev 9.5e-20 | symmetric difference in P abs dev 1.7e-19 | |r|<=64 against 96: d<s3>/dP 7.0e-20, f 5.4e-17
f_i common Lambda=5: f_1 -4.646274e-17 f_2 +3.697469e-17 | Lambda=10: f_1 -8.050759e-17 f_2 +6.928136e-17 | Lambda=20: f_1 -1.513519e-16 f_2 +1.409803e-16
f_i g1 only Lambda=5: f_1 -1.855558e-17 f_2 +2.417477e-17 | Lambda=10: f_1 -4.938903e-17 f_2 +2.803243e-17 | Lambda=20: f_1 -7.579718e-17 f_2 +6.582171e-17
S3 P=0.2 E_b'=-2.166630e-03 gap 1.044e-02 dist to bands 1.042e-02 inside band: False | d<s3>/dP +1.656606e-03 +1.945895e-03 | leading order from (11.12) +1.731668e-03 +2.033991e-03 | d<K>/dP common +3.407911e-03 g1 only +1.656606e-03 | (13.6) abs dev 1.7e-16 | symmetric difference in P abs dev 1.0e-11 | |r|<=64 against 96: d<s3>/dP 1.4e-17, f 2.4e-16
f_i common Lambda=5: f_1 -1.803497e-03 f_2 -2.071415e-03 | Lambda=10: f_1 -1.040826e-03 f_2 -1.545052e-04 | Lambda=20: f_1 -2.196446e-03 f_2 -5.852403e-04
f_i g1 only Lambda=5: f_1 -1.798404e-03 f_2 -5.145762e-06 | Lambda=10: f_1 -1.018591e-03 f_2 -2.216892e-05 | Lambda=20: f_1 -2.109923e-03 f_2 -8.637356e-05
S4 P=0.0 E_b'=+2.850665e-17 gap 4.883e-01 dist to bands 4.879e-01 inside band: False | d<s3>/dP +2.370494e-17 -5.816220e-18 | d<K>/dP common +1.672548e-17 g1 only +2.370494e-17 | (13.6) abs dev 5.3e-17 | symmetric difference in P abs dev 1.1e-17 | |r|<=64 against 96: d<s3>/dP 3.5e-17, f 1.6e-15
f_i common Lambda=5: f_1 +8.663541e-17 f_2 -2.374172e-17 | Lambda=10: f_1 -3.171902e-16 f_2 -1.455102e-16 | Lambda=20: f_1 +1.414619e-15 f_2 -7.318586e-16
f_i g1 only Lambda=5: f_1 +1.597223e-17 f_2 -8.807931e-17 | Lambda=10: f_1 -3.539973e-16 f_2 +4.138728e-17 | Lambda=20: f_1 +1.134676e-15 f_2 -2.074143e-16
S4 P=0.2 E_b'=+2.999623e-02 gap 4.920e-01 dist to bands 4.918e-01 inside band: False | d<s3>/dP -2.397833e-02 -2.300145e-02 | d<K>/dP common -5.158007e-02 g1 only -2.397833e-02 | (13.6) abs dev 1.1e-16 | symmetric difference in P abs dev 1.7e-10 | |r|<=64 against 96: d<s3>/dP 1.4e-16, f 1.5e-15
f_i common Lambda=5: f_1 +7.004686e-02 f_2 +4.778681e-02 | Lambda=10: f_1 +3.126625e-03 f_2 -1.889792e-03 | Lambda=20: f_1 +3.147125e-02 f_2 -3.561915e-02
f_i g1 only Lambda=5: f_1 +4.338467e-02 f_2 +1.597516e-02 | Lambda=10: f_1 +3.454884e-03 f_2 +1.478966e-02 | Lambda=20: f_1 +2.103920e-02 f_2 -2.228818e-03
S5 P=0.0 E_b'=-1.170449e-04 gap 7.445e-02 dist to bands 2.040e-01 inside band: False | d<s3>/dP -2.571495e-05 +3.012537e-04 | d<K>/dP common +3.357895e-04 g1 only -2.571495e-05 | (13.6) abs dev 2.7e-16 | symmetric difference in P abs dev 2.4e-12 | |r|<=64 against 96: d<s3>/dP 2.0e-16, f 1.0e-15
f_i common Lambda=5: f_1 +6.127794e-03 f_2 +1.273197e-02 | Lambda=10: f_1 +1.717838e-03 f_2 +4.221817e-02 | Lambda=20: f_1 +3.556962e-04 f_2 -9.320966e-02
f_i g1 only Lambda=5: f_1 +7.784976e-03 f_2 +3.430463e-03 | Lambda=10: f_1 -7.898672e-03 f_2 +1.415525e-02 | Lambda=20: f_1 -5.183665e-03 f_2 -1.387237e-02
S5 P=0.2 E_b'=+5.015569e-02 gap 7.573e-02 dist to bands 2.059e-01 inside band: False | d<s3>/dP -3.277265e-02 -3.887610e-02 | d<K>/dP common -7.942397e-02 g1 only -3.277265e-02 | (13.6) abs dev 1.7e-16 | symmetric difference in P abs dev 1.9e-10 | |r|<=64 against 96: d<s3>/dP 1.6e-16, f 2.4e-15
f_i common Lambda=5: f_1 +7.065473e-02 f_2 +2.827414e-02 | Lambda=10: f_1 +2.337343e-02 f_2 +1.113954e-01 | Lambda=20: f_1 +1.198083e-02 f_2 -6.281217e-02
f_i g1 only Lambda=5: f_1 +4.820649e-02 f_2 +1.840201e-02 | Lambda=10: f_1 -1.932053e-02 f_2 +3.799413e-02 | Lambda=20: f_1 -1.425763e-02 f_2 -2.852367e-03
== second derivative at lambda = 0 of c_i(lambda; P) = Re <V_i psi| R exp(-i (H - E_b) lambda) |K psi> (Step 5.3, (13.29), (13.30)), per unit gradient (numerical level, fibre |r|<=96; changes against |r|<=64)
S1 P=0.1 part 1 with g1 only -2.811827e-03, part 2 with g2 only -2.811827e-03 | part 1 with g2 only -1.76e-18, part 2 with g1 only +3.79e-19 | double commutator abs dev 2.2e-18 | (13.29) for a binding that commutes with K -2.811827e-03 -2.811827e-03 (abs dev +2.6e-18 +2.2e-18) | (13.30) -2.816636e-03 -2.816636e-03 (rel dev -1.707e-03 -1.707e-03) | |r|<=64 against 96: 2.0e-17
S1 P=0.2 part 1 with g1 only -5.552015e-03, part 2 with g2 only -5.552015e-03 | part 1 with g2 only +1.06e-17, part 2 with g1 only +1.54e-17 | double commutator abs dev 1.5e-17 | (13.29) for a binding that commutes with K -5.552015e-03 -5.552015e-03 (abs dev +0.0e+00 -5.2e-18) | (13.30) -5.562366e-03 -5.562366e-03 (rel dev -1.861e-03 -1.861e-03) | |r|<=64 against 96: 4.3e-17
S2 P=0.1 part 1 with g1 only -1.987960e-03, part 2 with g2 only -3.108879e-03 | part 1 with g2 only +1.87e-18, part 2 with g1 only +3.68e-18 | double commutator abs dev 1.5e-17 | (13.29) for a binding that commutes with K -1.987960e-03 -3.108879e-03 (abs dev +4.3e-18 +6.1e-18) | (13.30) -1.990905e-03 -3.110789e-03 (rel dev -1.479e-03 -6.139e-04) | |r|<=64 against 96: 2.7e-16
S2 P=0.2 part 1 with g1 only -3.918745e-03, part 2 with g2 only -6.128363e-03 | part 1 with g2 only -3.56e-18, part 2 with g1 only +2.27e-18 | double commutator abs dev 1.9e-17 | (13.29) for a binding that commutes with K -3.918745e-03 -6.128363e-03 (abs dev +3.1e-17 -4.3e-18) | (13.30) -3.924935e-03 -6.132711e-03 (rel dev -1.577e-03 -7.090e-04) | |r|<=64 against 96: 1.8e-16
S3 P=0.1 part 1 with g1 only +2.208063e-03, part 2 with g2 only +1.789490e-03 | part 1 with g2 only +8.95e-19, part 2 with g1 only +4.55e-18 | double commutator abs dev 4.6e-18 | (13.29) for a binding that commutes with K +2.208063e-03 +1.789490e-03 (abs dev -4.3e-18 +1.1e-18) | (13.30) +2.247359e-03 +1.820361e-03 (rel dev -1.749e-02 -1.696e-02) | |r|<=64 against 96: 1.5e-16
S3 P=0.2 part 1 with g1 only +4.354297e-03, part 2 with g2 only +3.528877e-03 | part 1 with g2 only +3.76e-18, part 2 with g1 only +4.41e-18 | double commutator abs dev 1.4e-17 | (13.29) for a binding that commutes with K +4.354297e-03 +3.528877e-03 (abs dev -1.2e-17 +4.3e-18) | (13.30) +4.431248e-03 +3.589311e-03 (rel dev -1.737e-02 -1.684e-02) | |r|<=64 against 96: 4.7e-16
S4 P=0.1 part 1 with g1 only -2.660877e-02, part 2 with g2 only -3.918639e-02 | part 1 with g2 only +3.64e-16, part 2 with g1 only +8.33e-17 | double commutator abs dev 3.7e-16 | (13.29) for a binding that commutes with K -2.660877e-02 -3.918639e-02 (abs dev +1.7e-16 -2.0e-16) | |r|<=64 against 96: 5.6e-16
S4 P=0.2 part 1 with g1 only -5.216475e-02, part 2 with g2 only -7.683019e-02 | part 1 with g2 only -3.47e-18, part 2 with g1 only +7.63e-17 | double commutator abs dev 2.6e-16 | (13.29) for a binding that commutes with K -5.216475e-02 -7.683019e-02 (abs dev -1.1e-16 +8.3e-17) | |r|<=64 against 96: 3.1e-16
S5 P=0.1 part 1 with g1 only -4.607152e-02, part 2 with g2 only -6.751468e-02 | part 1 with g2 only +4.09e-04, part 2 with g1 only -5.31e-04 | double commutator abs dev 2.9e-16 | (13.29) for a binding that commutes with K -4.552282e-02 -5.751605e-02 (abs dev -5.5e-04 -1.0e-02) | |r|<=64 against 96: 3.3e-16
S5 P=0.2 part 1 with g1 only -9.094493e-02, part 2 with g2 only -1.329232e-01 | part 1 with g2 only +4.02e-04, part 2 with g1 only -1.29e-03 | double commutator abs dev 4.3e-16 | (13.29) for a binding that commutes with K -8.953282e-02 -1.130641e-01 (abs dev -1.4e-03 -2.0e-02) | |r|<=64 against 96: 3.8e-16
total run time 418.0 s