06-ilang-source / 06-own-mark
06own markverified
Determines what the field change made by a body itself does to the cost of its levels, its clock spacings and its inertia, and whether these change in proportion to the gaps of all levels.
check_own_mark.py10 KB · raw
"""Check for 06-own-mark (derivation v2).
Exact diagonalization of the total contract in the sectors (k, P) for one field
object (N_f = 1) on rings of two sizes, for the couplings (a) and (c), three clock
levels and both branches. Compared with the second-order results of the
derivation: dressed level E_{k,sigma}(0) (6.5)-(6.6), clock spacings (6.8), and
curvature E''_{k,sigma}(0) (6.10) with the leading-tau^2 forms (6.11), and the inertia-cost comparison (6.12).
The curvature of the exact level is the exact derivative of the eigenvalue of
the sector matrix C(P) (6.1) at P = 0 (second-order Hellmann-Feynman formula).
"""
from pathlib import Path
import numpy as np
OUT = Path(__file__).parent / "output"
OUT.mkdir(parents=True, exist_ok=True)
T_F, M_REST, D_C = 1.0, 3.0, 0.5 # t_f, m, clock spacing d_{k+1}-d_k = 2 pi/(N lambda_0)
LEVELS = [0, 1, 2] # clock levels k (N = 3)
RINGS = [6, 10]
TAUS = [1.0, 0.25]
GAMMAS = [0.01, 0.02, 0.04, 0.08]
N_F = 1
S1 = np.array([[0, 1], [1, 0]], dtype=complex)
S3 = np.array([[1, 0], [0, -1]], dtype=complex)
def shift(L, d): # |r> -> |r+d>, r = x_f - x_b
S = np.zeros((L, L))
for r in range(L):
S[(r + d) % L, r] = 1.0
return S
def half_proj(L, rs): # (1/2) sum_{r in rs} |r><r|
P = np.zeros((L, L))
for r in rs:
P[r % L, r % L] += 0.5
return P
def sector_parts(L, M, tau, g, coupling):
"""C(P) = C0 + e^{-iP} Am + e^{iP} Ap on branch (x) relative coordinate."""
I2, Ir = np.eye(2), np.eye(L)
Sp, Sm = shift(L, 1), shift(L, -1)
n0 = np.zeros((L, L)); n0[0, 0] = 1.0
C0 = M * np.kron(S3, Ir) - T_F * np.kron(I2, Sp + Sm) + g * M * np.kron(S3, n0)
Am = 0.5 * tau * np.kron(-1j * S1, Sm) # body x -> x+1
Ap = 0.5 * tau * np.kron(1j * S1, Sp) # body x+1 -> x
if coupling == "c": # bond factor (Phi_x + Phi_{x+1})/2
Am = Am + g * 0.5 * tau * np.kron(-1j * S1, Sm @ half_proj(L, [0, 1]))
Ap = Ap + g * 0.5 * tau * np.kron(1j * S1, Sp @ half_proj(L, [-1, 0]))
return C0, Am, Ap
def exact(L, M, tau, g, coupling, sigma):
C0, Am, Ap = sector_parts(L, M, tau, g, coupling)
C = C0 + Am + Ap
assert np.allclose(C, C.conj().T)
w, V = np.linalg.eigh(C)
beta = np.array([1.0, 0.0]) if sigma > 0 else np.array([0.0, 1.0])
v0 = np.kron(beta, np.ones(L) / np.sqrt(L)) # body (P=0, sigma), field lowest
i = int(np.argmax(np.abs(V.conj().T @ v0)))
psi, oth = V[:, i], np.arange(len(w)) != i
a = V.conj().T @ ((-1j * Am + 1j * Ap) @ psi)
curv = np.real(psi.conj() @ (-(Am + Ap)) @ psi) \
+ 2.0 * np.sum(np.abs(a[oth]) ** 2 / (w[i] - w[oth]))
# sanity: finite difference of the analytic continuation C(P)
h = 1e-3
ev = [np.linalg.eigvalsh(C0 + np.exp(-1j * P) * Am + np.exp(1j * P) * Ap) for P in (-h, h)]
fd = (ev[0][i] + ev[1][i] - 2 * w[i]) / h ** 2
assert abs(fd - curv) < 1e-4 * max(1.0, abs(curv)), (fd, curv)
return w[i], curv, float(np.min(np.abs(w[oth] - w[i])))
def G(P, q, M, tau, sigma, coupling):
"""Per-label second-order function G_{k,sigma}(P,q) of (6.5)."""
Om = sigma * np.sqrt(M ** 2 + tau ** 2 * np.sin(P) ** 2 + 0j)
sp, spp = np.sin(P), np.sin(P - q)
eps = 2 * T_F * (1 - np.cos(q))
z = Om - eps
wp2 = M ** 2 + tau ** 2 * spp ** 2
den = z ** 2 - wp2
if coupling == "a":
num = M ** 2 * (z + (M ** 2 - tau ** 2 * sp * spp) / Om)
else:
cp = (M ** 2 + tau ** 2 * sp * spp) / Om
S = Om ** 2 + wp2
num = (z * S + 2 * Om * wp2 + cp * (S + 2 * Om * z)) / 4
return num / den
def d2_at0(f, r=0.1, n=64): # exact second derivative of an analytic f at 0 (Cauchy)
th = 2 * np.pi * np.arange(n) / n
return float(np.real(2 * np.mean(f(r * np.exp(1j * th)) * np.exp(-2j * th)) / r ** 2))
def qs(L):
return 2 * np.pi * np.arange(1, L) / L
def c2_energy(L, M, tau, sigma, coupling): # gamma^2 coefficient of E(0), (6.5)
n = N_F / L
return n / L * sum(float(np.real(G(0.0, q, M, tau, sigma, coupling))) for q in qs(L))
def c2_energy_closed(L, M, tau, sigma, coupling): # closed forms (6.6)
n, Ms, tot = N_F / L, sigma * M, 0.0
for q in qs(L):
e, b2 = 2 * T_F * (1 - np.cos(q)), tau ** 2 * np.sin(q) ** 2
K = e * (2 * Ms - e) + b2
tot += -(M ** 2 * (2 * Ms - e) + (b2 * (Ms - e / 4) if coupling == "c" else 0.0)) / K
return n / L * tot
def c2_curv(L, M, tau, sigma, coupling): # gamma^2 coefficient of E''(0), (6.10)
n = N_F / L
return n / L * sum(d2_at0(lambda P: G(P, q, M, tau, sigma, coupling)) for q in qs(L))
def c2_curv_lead(L, M, tau, sigma, coupling): # leading tau^2 forms (6.11)
n, Ms, tot = N_F / L, sigma * M, 0.0
for q in qs(L):
c, e = np.cos(q), 2 * T_F * (1 - np.cos(q))
if coupling == "a":
tot += 2 * Ms * tau ** 2 * (1 + c) * (1 - Ms / T_F) / (e * (2 * Ms - e))
else:
tot += -tau ** 2 * (1 + c) * (2 * M ** 2 / T_F + 2 * Ms * (2 * c - 1) - e * c) / (e * (2 * Ms - e))
return n / L * tot
def mf_curv(M, tau, g, sigma, coupling, L): # mean-field part of (6.10), to gamma^2
gn = g * N_F / L
return sigma * tau ** 2 / M * ((1 - gn + gn ** 2) if coupling == "a" else (1 + gn))
def slope(xs, ys):
return float(np.polyfit(np.log(xs), np.log(np.abs(ys)), 1)[0])
lines, rows = [], ["coupling,L,tau,k,sigma,gamma,E_exact,E_pred,dE,curv_exact,curv_pred,dcurv,gap"]
lines.append("Check of 06-own-mark v2: N_f=1, t_f=%.2f, m=%.2f, clock spacing=%.2f, levels k=%s"
% (T_F, M_REST, D_C, LEVELS))
lines.append("Closed forms (6.6) vs general (6.5) at P=0, max abs difference of gamma^2 coefficients:")
mx = max(abs(c2_energy(L, M_REST + k * D_C, t, s, cp) - c2_energy_closed(L, M_REST + k * D_C, t, s, cp))
for L in RINGS for t in TAUS for k in LEVELS for s in (1, -1) for cp in "ac")
lines.append(" %.2e" % mx)
lines.append("")
lines.append("A. Dressed level E(0) and curvature E''(0): deviation exact - second order, slope in gamma")
lines.append(" coupling L tau k sigma | dE(g=0.01) dE(g=0.08) slope | dE''(g=0.01) dE''(g=0.08) slope | min gap")
spac = {}
for cp in "ac":
for L in RINGS:
for tau in TAUS:
for k in LEVELS:
M = M_REST + k * D_C
for s in (1, -1):
c2E, c2C = c2_energy(L, M, tau, s, cp), c2_curv(L, M, tau, s, cp)
dEs, dCs, gaps = [], [], []
for g in GAMMAS:
E, curv, gap = exact(L, M, tau, g, cp, s)
Ep = s * M * (1 + g * N_F / L) - 2 * T_F * N_F + g ** 2 * c2E
Cp = mf_curv(M, tau, g, s, cp, L) + g ** 2 * c2C
dEs.append(E - Ep); dCs.append(curv - Cp); gaps.append(gap)
rows.append("%s,%d,%.3f,%d,%d,%.3f,%.12f,%.12f,%.3e,%.12f,%.12f,%.3e,%.4f"
% (cp, L, tau, k, s, g, E, Ep, E - Ep, curv, Cp, curv - Cp, gap))
if s > 0:
spac[(cp, L, tau, k, g)] = E
lines.append(" (%s) %2d %.2f %d %+d | %10.2e %10.2e %4.2f | %10.2e %10.2e %4.2f | %.3f"
% (cp, L, tau, k, s, dEs[0], dEs[-1], slope(GAMMAS, dEs),
dCs[0], dCs[-1], slope(GAMMAS, dCs), min(gaps)))
lines.append("")
lines.append("B. Clock spacings E_{k+1,+}(0)-E_{k,+}(0): exact - [(1+gamma n) spacing + gamma^2 (c2_{k+1}-c2_k)],")
lines.append(" slope in gamma; and the second-order change per unit spacing, (c2_{k+1}-c2_k)/spacing, for k=0,1")
for cp in "ac":
for L in RINGS:
for tau in TAUS:
c2 = [c2_energy(L, M_REST + k * D_C, tau, 1, cp) for k in LEVELS]
rel = [(c2[k + 1] - c2[k]) / D_C for k in LEVELS[:-1]]
out = []
for k in LEVELS[:-1]:
devs = [spac[(cp, L, tau, k + 1, g)] - spac[(cp, L, tau, k, g)]
- ((1 + g * N_F / L) * D_C + g ** 2 * (c2[k + 1] - c2[k])) for g in GAMMAS]
out.append("k=%d: dev(g=0.08)=%.2e slope %.2f" % (k, devs[-1], slope(GAMMAS, devs)))
lines.append(" (%s) L=%2d tau=%.2f | %s | %s | change/spacing: k=0 %.5f, k=1 %.5f"
% (cp, L, tau, out[0], out[1], rel[0], rel[1]))
lines.append("")
lines.append("C. Leading-tau^2 curvature (6.11) vs full second-order curvature (6.10), relative deviation")
for cp in "ac":
for L in RINGS:
for s in (1, -1):
devs = []
for tau in (0.2, 0.1, 0.05):
full, lead = c2_curv(L, M_REST, tau, s, cp), c2_curv_lead(L, M_REST, tau, s, cp)
devs.append(abs(lead - full) / abs(full))
lines.append(" (%s) L=%2d sigma=%+d k=0 | tau=0.2: %.2e tau=0.1: %.2e tau=0.05: %.2e"
% (cp, L, s, *devs))
lines.append("")
lines.append("D. Own mark (6.12): gamma^2 coefficients at P=0, sigma=+: cost c2E, curvature c2C,")
lines.append(" inertia change -(M^2/tau^4) c2C vs cost change / tau^2 = c2E/tau^2")
for cp in "ac":
for L in RINGS:
for tau in TAUS:
for k in LEVELS:
M = M_REST + k * D_C
c2E, c2C = c2_energy(L, M, tau, 1, cp), c2_curv(L, M, tau, 1, cp)
lines.append(" (%s) L=%2d tau=%.2f k=%d M=%.2f | c2E=%+.5f c2C=%+.6f | inertia %+.4f cost/tau^2 %+.4f"
% (cp, L, tau, k, M, c2E, c2C, -M ** 2 / tau ** 4 * c2C, c2E / tau ** 2))
(OUT / "check_own_mark.log").write_text("\n".join(lines) + "\n", encoding="utf-8")
(OUT / "check_own_mark.csv").write_text("\n".join(rows) + "\n", encoding="utf-8")
print("\n".join(lines))
output/check_own_mark.csv21 KB · raw
coupling,L,tau,k,sigma,gamma,E_exact,E_pred,dE,curv_exact,curv_pred,dcurv,gap
a,6,1.000,0,1,0.010,1.004935503833,1.004934887124,6.167e-07,0.332758391296,0.332757963128,4.282e-07,1.1174
a,6,1.000,0,1,0.020,1.009744457287,1.009739548495,4.909e-06,0.332146362674,0.332142963625,3.399e-06,1.1124
a,6,1.000,0,1,0.040,1.018997058831,1.018958193980,3.886e-05,0.330820837535,0.330794076722,2.676e-05,1.1027
a,6,1.000,0,1,0.080,1.036137024834,1.035832775920,3.042e-04,0.327827677258,0.327620751332,2.069e-04,1.0849
a,6,1.000,0,-1,0.010,-5.005080302507,-5.005079392857,-9.096e-07,-0.332741465109,-0.332742137317,6.722e-07,0.8728
a,6,1.000,0,-1,0.020,-5.010324886570,-5.010317571429,-7.315e-06,-0.332074264181,-0.332079660380,5.396e-06,0.8682
a,6,1.000,0,-1,0.040,-5.021329385137,-5.021270285714,-5.910e-05,-0.330497467091,-0.330540863742,4.340e-05,0.8595
a,6,1.000,0,-1,0.080,-5.045562087090,-5.045081142857,-4.809e-04,-0.326259676707,-0.326607899411,3.482e-04,0.8446
a,6,1.000,1,1,0.010,1.505744001569,1.505742981799,1.020e-06,0.285214281682,0.285213677197,6.045e-07,1.0996
a,6,1.000,1,1,0.020,1.511313368712,1.511305260530,8.108e-06,0.284669025211,0.284664232599,4.793e-06,1.0939
a,6,1.000,1,1,0.040,1.521951759250,1.521887708787,6.405e-05,0.283456462379,0.283418835156,3.763e-05,1.0829
a,6,1.000,1,1,0.080,1.541383178403,1.540884168482,4.990e-04,0.280631137094,0.280342007292,2.891e-04,1.0629
a,6,1.000,1,-1,0.010,-5.505941621543,-5.505940202920,-1.419e-06,-0.285196898300,-0.285197791727,8.934e-07,0.8889
a,6,1.000,1,-1,0.020,-5.512105563122,-5.512094145013,-1.142e-05,-0.284593511996,-0.284600690718,7.179e-06,0.8835
a,6,1.000,1,-1,0.040,-5.525135637662,-5.525043246717,-9.239e-05,-0.283106858473,-0.283164667633,5.781e-05,0.8735
a,6,1.000,1,-1,0.080,-5.554259826341,-5.553506320202,-7.535e-04,-0.278861424840,-0.279325337199,4.639e-04,0.8567
a,6,1.000,2,1,0.010,2.006548614423,2.006547050122,1.564e-06,0.249555112197,0.249554301467,8.107e-07,1.0860
a,6,1.000,2,1,0.020,2.012867291780,2.012854867156,1.242e-05,0.249056958776,0.249050539201,6.420e-06,1.0795
a,6,1.000,2,1,0.040,2.024850733101,2.024752801957,9.793e-05,0.247919082077,0.247868823469,5.026e-05,1.0672
a,6,1.000,2,1,0.080,2.046437248110,2.045677874495,7.594e-04,0.245192354886,0.244808627209,3.837e-04,1.0451
a,6,1.000,2,-1,0.010,-6.006807154704,-6.006805066166,-2.089e-06,-0.249537231323,-0.249538377165,1.146e-06,0.9009
a,6,1.000,2,-1,0.020,-6.013903755762,-6.013886931329,-1.682e-05,-0.248977627045,-0.248986841993,9.215e-06,0.8948
a,6,1.000,2,-1,0.040,-6.029017397782,-6.028881058649,-1.363e-04,-0.247539743300,-0.247614034637,7.429e-05,0.8834
a,6,1.000,2,-1,0.080,-6.063304899532,-6.062190901261,-1.114e-03,-0.243193776827,-0.243789471881,5.957e-04,0.8651
a,6,0.250,0,1,0.010,1.004928395912,1.004927634085,7.618e-07,0.020797233432,0.020797199102,3.433e-08,1.0029
a,6,0.250,0,1,0.020,1.009716596419,1.009710536342,6.060e-06,0.020758513072,0.020758240852,2.722e-07,0.9981
a,6,0.250,0,1,0.040,1.018890063003,1.018842145366,4.792e-05,0.020673990303,0.020671852297,2.138e-06,0.9889
a,6,0.250,0,1,0.080,1.035742666879,1.035368581466,3.741e-04,0.020481633881,0.020465186965,1.645e-05,0.9720
a,6,0.250,0,-1,0.010,-5.005074071792,-5.005073282731,-7.891e-07,-0.020796675857,-0.020796709543,3.369e-08,0.9873
a,6,0.250,0,-1,0.020,-5.010299477291,-5.010293130923,-6.346e-06,-0.020756012089,-0.020756282618,2.705e-07,0.9825
a,6,0.250,0,-1,0.040,-5.021223819025,-5.021172523693,-5.130e-05,-0.020661841248,-0.020664019360,2.178e-06,0.9733
a,6,0.250,0,-1,0.080,-5.045108159397,-5.044690094773,-4.181e-04,-0.020416309326,-0.020433855219,1.755e-05,0.9568
a,6,0.250,1,1,0.010,1.505735913575,1.505734701480,1.212e-06,0.017825717475,0.017825670804,4.667e-08,1.0009
a,6,0.250,1,1,0.020,1.511281771478,1.511272139253,9.632e-06,0.017791148101,0.017790778453,3.696e-07,0.9954
a,6,0.250,1,1,0.040,1.521831227425,1.521755223677,7.600e-05,0.017713628585,0.017710732858,2.896e-06,0.9848
a,6,0.250,1,1,0.080,1.540944913623,1.540354228040,5.907e-04,0.017531742413,0.017509598099,2.214e-05,0.9657
a,6,0.250,1,-1,0.010,-5.505934270437,-5.505933017633,-1.253e-06,-0.017825132225,-0.017825178319,4.609e-08,0.9876
a,6,0.250,1,-1,0.020,-5.512075488707,-5.512065403865,-1.008e-05,-0.017788438018,-0.017788808514,3.705e-07,0.9820
a,6,0.250,1,-1,0.040,-5.525009919925,-5.524928282126,-8.164e-05,-0.017699866308,-0.017702853103,2.987e-06,0.9714
a,6,0.250,1,-1,0.080,-5.553713335954,-5.553046461838,-6.669e-04,-0.017454012656,-0.017478079077,2.407e-05,0.9531
a,6,0.250,2,1,0.010,2.006539531717,2.006537720620,1.811e-06,0.015597012343,0.015596951443,6.090e-08,0.9993
a,6,0.250,2,1,0.020,2.012831927109,2.012817549146,1.438e-05,0.015565370861,0.015564889107,4.818e-07,0.9930
a,6,0.250,2,1,0.040,2.024716740532,2.024603529917,1.132e-04,0.015492486864,0.015488723093,3.764e-06,0.9811
a,6,0.250,2,1,0.080,2.045956631811,2.045080786335,8.758e-04,0.015316830516,0.015288225706,2.860e-05,0.9598
a,6,0.250,2,-1,0.010,-6.006798674030,-6.006796803831,-1.870e-06,-0.015596397065,-0.015596457522,6.046e-08,0.9876
a,6,0.250,2,-1,0.020,-6.013868949424,-6.013853881989,-1.507e-05,-0.015562427070,-0.015562913420,4.863e-07,0.9813
a,6,0.250,2,-1,0.040,-6.028871015023,-6.028748861289,-1.222e-04,-0.015476895187,-0.015480820345,3.925e-06,0.9694
a,6,0.250,2,-1,0.080,-6.062661930141,-6.061662111822,-9.998e-04,-0.015225004586,-0.015256614713,3.161e-05,0.9494
a,10,1.000,0,1,0.010,1.002935825522,1.002934730849,1.095e-06,0.332967945291,0.332966855728,1.090e-06,0.4360
a,10,1.000,0,1,0.020,1.005747581725,1.005738923396,8.658e-06,0.332542683553,0.332534089577,8.594e-06,0.4331
a,10,1.000,0,1,0.040,1.011023386073,1.010955693583,6.769e-05,0.331536456665,0.331469691642,6.677e-05,0.4278
a,10,1.000,0,1,0.080,1.020339679536,1.019822774332,5.169e-04,0.329047637905,0.328545433234,5.022e-04,0.4182
a,10,1.000,0,-1,0.010,-5.003085435001,-5.003083636039,-1.799e-06,-0.332963237100,-0.332964725440,1.488e-06,0.3221
a,10,1.000,0,-1,0.020,-5.006349119292,-5.006334544155,-1.458e-05,-0.332513410681,-0.332525568428,1.216e-05,0.3193
a,10,1.000,0,-1,0.040,-5.013457637262,-5.013338176619,-1.195e-04,-0.331334423996,-0.331435607046,1.012e-04,0.3144
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c,10,0.250,0,-1,0.040,-5.013295461516,-5.013198078781,-9.738e-05,-0.020884077758,-0.020889212721,5.135e-06,0.3674
c,10,0.250,0,-1,0.080,-5.029606589128,-5.028792315123,-8.143e-04,-0.020846572113,-0.020890184217,4.361e-05,0.3592
c,10,0.250,1,1,0.010,1.503401733299,1.503399526407,2.207e-06,0.017871974873,0.017871857757,1.171e-07,0.3817
c,10,0.250,1,1,0.020,1.506615500249,1.506598105630,1.739e-05,0.017881208429,0.017880288169,9.203e-07,0.3784
c,10,0.250,1,1,0.040,1.512527449901,1.512392422519,1.350e-04,0.017885386924,0.017878295533,7.091e-06,0.3725
c,10,0.250,1,1,0.080,1.522585952181,1.521569690076,1.016e-03,0.017851323169,0.017798896419,5.243e-05,0.3625
c,10,0.250,1,-1,0.010,-5.503604115058,-5.503601786168,-2.329e-06,-0.017872802588,-0.017872907820,1.052e-07,0.3755
c,10,0.250,1,-1,0.020,-5.507426044310,-5.507407144673,-1.890e-05,-0.017883628956,-0.017884488423,8.595e-07,0.3722
c,10,0.250,1,-1,0.040,-5.515784010312,-5.515628578692,-1.554e-04,-0.017887949303,-0.017895096548,7.147e-06,0.3663
c,10,0.250,1,-1,0.080,-5.535821183833,-5.534514314769,-1.307e-03,-0.017805222923,-0.017866100478,6.088e-05,0.3578
c,10,0.250,2,1,0.010,2.003871973825,2.003868682652,3.291e-06,0.015637258063,0.015637106791,1.513e-07,0.3808
c,10,0.250,2,1,0.020,2.007500615389,2.007474730608,2.588e-05,0.015643362280,0.015642177164,1.185e-06,0.3772
c,10,0.250,2,1,0.040,2.014098971120,2.013898922433,2.000e-04,0.015640283928,0.015631208657,9.075e-06,0.3706
c,10,0.250,2,1,0.080,2.025088108598,2.023595689733,1.492e-03,0.015591064037,0.015524834630,6.623e-05,0.3596
c,10,0.250,2,-1,0.010,-6.004136294349,-6.004132817414,-3.477e-06,-0.015638017658,-0.015638156840,1.392e-07,0.3754
c,10,0.250,2,-1,0.020,-6.008559541343,-6.008531269656,-2.827e-05,-0.015645237966,-0.015646377358,1.139e-06,0.3717
c,10,0.250,2,-1,0.040,-6.018358413267,-6.018125078625,-2.333e-04,-0.015638500120,-0.015648009433,9.509e-06,0.3652
c,10,0.250,2,-1,0.080,-6.042471688103,-6.040500314499,-1.971e-03,-0.015510965778,-0.015592037731,8.107e-05,0.3568
output/check_own_mark.log10 KB · raw
Check of 06-own-mark v2: N_f=1, t_f=1.00, m=3.00, clock spacing=0.50, levels k=[0, 1, 2]
Closed forms (6.6) vs general (6.5) at P=0, max abs difference of gamma^2 coefficients:
6.66e-16
A. Dressed level E(0) and curvature E''(0): deviation exact - second order, slope in gamma
coupling L tau k sigma | dE(g=0.01) dE(g=0.08) slope | dE''(g=0.01) dE''(g=0.08) slope | min gap
(a) 6 1.00 0 +1 | 6.17e-07 3.04e-04 2.98 | 4.28e-07 2.07e-04 2.97 | 1.085
(a) 6 1.00 0 -1 | -9.10e-07 -4.81e-04 3.02 | 6.72e-07 3.48e-04 3.01 | 0.845
(a) 6 1.00 1 +1 | 1.02e-06 4.99e-04 2.98 | 6.04e-07 2.89e-04 2.97 | 1.063
(a) 6 1.00 1 -1 | -1.42e-06 -7.54e-04 3.02 | 8.93e-07 4.64e-04 3.01 | 0.857
(a) 6 1.00 2 +1 | 1.56e-06 7.59e-04 2.97 | 8.11e-07 3.84e-04 2.96 | 1.045
(a) 6 1.00 2 -1 | -2.09e-06 -1.11e-03 3.02 | 1.15e-06 5.96e-04 3.01 | 0.865
(a) 6 0.25 0 +1 | 7.62e-07 3.74e-04 2.98 | 3.43e-08 1.64e-05 2.97 | 0.972
(a) 6 0.25 0 -1 | -7.89e-07 -4.18e-04 3.02 | 3.37e-08 1.75e-05 3.01 | 0.957
(a) 6 0.25 1 +1 | 1.21e-06 5.91e-04 2.98 | 4.67e-08 2.21e-05 2.96 | 0.966
(a) 6 0.25 1 -1 | -1.25e-06 -6.67e-04 3.02 | 4.61e-08 2.41e-05 3.01 | 0.953
(a) 6 0.25 2 +1 | 1.81e-06 8.76e-04 2.97 | 6.09e-08 2.86e-05 2.96 | 0.960
(a) 6 0.25 2 -1 | -1.87e-06 -1.00e-03 3.02 | 6.05e-08 3.16e-05 3.01 | 0.949
(a) 10 1.00 0 +1 | 1.09e-06 5.17e-04 2.96 | 1.09e-06 5.02e-04 2.95 | 0.418
(a) 10 1.00 0 -1 | -1.80e-06 -9.97e-04 3.04 | 1.49e-06 8.65e-04 3.06 | 0.307
(a) 10 1.00 1 +1 | 1.81e-06 8.40e-04 2.95 | 1.52e-06 6.85e-04 2.94 | 0.408
(a) 10 1.00 1 -1 | -2.78e-06 -1.56e-03 3.04 | 2.01e-06 1.17e-03 3.06 | 0.314
(a) 10 1.00 2 +1 | 2.77e-06 1.27e-03 2.95 | 2.02e-06 8.90e-04 2.93 | 0.399
(a) 10 1.00 2 -1 | -4.06e-06 -2.30e-03 3.05 | 2.61e-06 1.53e-03 3.07 | 0.319
(a) 10 0.25 0 +1 | 1.38e-06 6.48e-04 2.96 | 7.22e-08 3.23e-05 2.94 | 0.366
(a) 10 0.25 0 -1 | -1.46e-06 -8.11e-04 3.04 | 8.96e-08 4.96e-05 3.04 | 0.359
(a) 10 0.25 1 +1 | 2.20e-06 1.01e-03 2.95 | 9.96e-08 4.35e-05 2.93 | 0.362
(a) 10 0.25 1 -1 | -2.32e-06 -1.30e-03 3.04 | 1.21e-07 6.76e-05 3.04 | 0.358
(a) 10 0.25 2 +1 | 3.28e-06 1.49e-03 2.94 | 1.31e-07 5.61e-05 2.91 | 0.360
(a) 10 0.25 2 -1 | -3.47e-06 -1.97e-03 3.05 | 1.57e-07 8.85e-05 3.05 | 0.357
(c) 6 1.00 0 +1 | 6.86e-07 3.37e-04 2.98 | 4.29e-07 2.11e-04 2.98 | 1.088
(c) 6 1.00 0 -1 | -9.85e-07 -5.23e-04 3.02 | 7.67e-07 4.14e-04 3.03 | 0.842
(c) 6 1.00 1 +1 | 1.10e-06 5.36e-04 2.98 | 6.01e-07 2.92e-04 2.98 | 1.066
(c) 6 1.00 1 -1 | -1.51e-06 -8.02e-04 3.02 | 9.95e-07 5.38e-04 3.03 | 0.854
(c) 6 1.00 2 +1 | 1.65e-06 8.00e-04 2.97 | 8.02e-07 3.86e-04 2.97 | 1.048
(c) 6 1.00 2 -1 | -2.19e-06 -1.17e-03 3.02 | 1.25e-06 6.78e-04 3.03 | 0.863
(c) 6 0.25 0 +1 | 7.67e-07 3.77e-04 2.98 | 3.27e-08 1.61e-05 2.98 | 0.972
(c) 6 0.25 0 -1 | -7.93e-07 -4.20e-04 3.02 | 3.65e-08 1.96e-05 3.02 | 0.957
(c) 6 0.25 1 +1 | 1.22e-06 5.93e-04 2.98 | 4.47e-08 2.17e-05 2.98 | 0.966
(c) 6 0.25 1 -1 | -1.26e-06 -6.69e-04 3.02 | 4.94e-08 2.66e-05 3.02 | 0.953
(c) 6 0.25 2 +1 | 1.82e-06 8.79e-04 2.97 | 5.86e-08 2.82e-05 2.97 | 0.960
(c) 6 0.25 2 -1 | -1.88e-06 -1.00e-03 3.02 | 6.42e-08 3.47e-05 3.03 | 0.949
(c) 10 1.00 0 +1 | 1.18e-06 5.53e-04 2.96 | 1.26e-06 5.87e-04 2.95 | 0.419
(c) 10 1.00 0 -1 | -1.91e-06 -1.06e-03 3.04 | 1.10e-06 7.05e-04 3.11 | 0.307
(c) 10 1.00 1 +1 | 1.90e-06 8.80e-04 2.95 | 1.73e-06 7.87e-04 2.94 | 0.408
(c) 10 1.00 1 -1 | -2.90e-06 -1.63e-03 3.05 | 1.60e-06 1.01e-03 3.10 | 0.313
(c) 10 1.00 2 +1 | 2.88e-06 1.31e-03 2.94 | 2.26e-06 1.01e-03 2.93 | 0.400
(c) 10 1.00 2 -1 | -4.20e-06 -2.38e-03 3.05 | 2.17e-06 1.37e-03 3.10 | 0.318
(c) 10 0.25 0 +1 | 1.39e-06 6.50e-04 2.96 | 8.72e-08 3.99e-05 2.95 | 0.366
(c) 10 0.25 0 -1 | -1.47e-06 -8.14e-04 3.04 | 7.61e-08 4.36e-05 3.05 | 0.359
(c) 10 0.25 1 +1 | 2.21e-06 1.02e-03 2.95 | 1.17e-07 5.24e-05 2.94 | 0.362
(c) 10 0.25 1 -1 | -2.33e-06 -1.31e-03 3.04 | 1.05e-07 6.09e-05 3.06 | 0.358
(c) 10 0.25 2 +1 | 3.29e-06 1.49e-03 2.94 | 1.51e-07 6.62e-05 2.93 | 0.360
(c) 10 0.25 2 -1 | -3.48e-06 -1.97e-03 3.05 | 1.39e-07 8.11e-05 3.06 | 0.357
B. Clock spacings E_{k+1,+}(0)-E_{k,+}(0): exact - [(1+gamma n) spacing + gamma^2 (c2_{k+1}-c2_k)],
slope in gamma; and the second-order change per unit spacing, (c2_{k+1}-c2_k)/spacing, for k=0,1
(a) L= 6 tau=1.00 | k=0: dev(g=0.08)=1.95e-04 slope 2.97 | k=1: dev(g=0.08)=2.60e-04 slope 2.97 | change/spacing: k=0 -0.50477, k=1 -0.58530
(a) L= 6 tau=0.25 | k=0: dev(g=0.08)=2.17e-04 slope 2.97 | k=1: dev(g=0.08)=2.85e-04 slope 2.97 | change/spacing: k=0 -0.52532, k=1 -0.60628
(a) L=10 tau=1.00 | k=0: dev(g=0.08)=3.23e-04 slope 2.94 | k=1: dev(g=0.08)=4.28e-04 slope 2.93 | change/spacing: k=0 -0.50722, k=1 -0.58937
(a) L=10 tau=0.25 | k=0: dev(g=0.08)=3.66e-04 slope 2.94 | k=1: dev(g=0.08)=4.76e-04 slope 2.93 | change/spacing: k=0 -0.53446, k=1 -0.61693
(c) L= 6 tau=1.00 | k=0: dev(g=0.08)=1.99e-04 slope 2.97 | k=1: dev(g=0.08)=2.65e-04 slope 2.97 | change/spacing: k=0 -0.50378, k=1 -0.58472
(c) L= 6 tau=0.25 | k=0: dev(g=0.08)=2.17e-04 slope 2.97 | k=1: dev(g=0.08)=2.85e-04 slope 2.97 | change/spacing: k=0 -0.52517, k=1 -0.60619
(c) L=10 tau=1.00 | k=0: dev(g=0.08)=3.27e-04 slope 2.94 | k=1: dev(g=0.08)=4.32e-04 slope 2.93 | change/spacing: k=0 -0.50672, k=1 -0.58910
(c) L=10 tau=0.25 | k=0: dev(g=0.08)=3.66e-04 slope 2.94 | k=1: dev(g=0.08)=4.76e-04 slope 2.93 | change/spacing: k=0 -0.53438, k=1 -0.61688
C. Leading-tau^2 curvature (6.11) vs full second-order curvature (6.10), relative deviation
(a) L= 6 sigma=+1 k=0 | tau=0.2: 5.03e-03 tau=0.1: 1.25e-03 tau=0.05: 3.13e-04
(a) L= 6 sigma=-1 k=0 | tau=0.2: 6.34e-03 tau=0.1: 1.59e-03 tau=0.05: 3.97e-04
(a) L=10 sigma=+1 k=0 | tau=0.2: 1.57e-02 tau=0.1: 4.06e-03 tau=0.05: 1.02e-03
(a) L=10 sigma=-1 k=0 | tau=0.2: 2.49e-03 tau=0.1: 5.95e-04 tau=0.05: 1.47e-04
(c) L= 6 sigma=+1 k=0 | tau=0.2: 6.68e-03 tau=0.1: 1.67e-03 tau=0.05: 4.17e-04
(c) L= 6 sigma=-1 k=0 | tau=0.2: 5.62e-03 tau=0.1: 1.41e-03 tau=0.05: 3.53e-04
(c) L=10 sigma=+1 k=0 | tau=0.2: 3.80e-03 tau=0.1: 9.87e-04 tau=0.05: 2.49e-04
(c) L=10 sigma=-1 k=0 | tau=0.2: 1.25e-02 tau=0.1: 3.03e-03 tau=0.05: 7.52e-04
D. Own mark (6.12): gamma^2 coefficients at P=0, sigma=+: cost c2E, curvature c2C,
inertia change -(M^2/tau^4) c2C vs cost change / tau^2 = c2E/tau^2
(a) L= 6 tau=1.00 k=0 M=3.00 | c2E=-0.65113 c2C=-0.207406 | inertia +1.8667 cost/tau^2 -0.6511
(a) L= 6 tau=1.00 k=1 M=3.50 | c2E=-0.90352 c2C=-0.252117 | inertia +3.0884 cost/tau^2 -0.9035
(a) L= 6 tau=1.00 k=2 M=4.00 | c2E=-1.19617 c2C=-0.297263 | inertia +4.7562 cost/tau^2 -1.1962
(a) L= 6 tau=0.25 k=0 M=3.00 | c2E=-0.72366 c2C=-0.014699 | inertia +33.8660 cost/tau^2 -11.5785
(a) L= 6 tau=0.25 k=1 M=3.50 | c2E=-0.98632 c2C=-0.017598 | inertia +55.1858 cost/tau^2 -15.7811
(a) L= 6 tau=0.25 k=2 M=4.00 | c2E=-1.28946 c2C=-0.020503 | inertia +83.9800 cost/tau^2 -20.6314
(a) L=10 tau=1.00 k=0 M=3.00 | c2E=-0.65269 c2C=-0.334776 | inertia +3.0130 cost/tau^2 -0.6527
(a) L=10 tau=1.00 k=1 M=3.50 | c2E=-0.90630 c2C=-0.395481 | inertia +4.8446 cost/tau^2 -0.9063
(a) L=10 tau=1.00 k=2 M=4.00 | c2E=-1.20099 c2C=-0.456167 | inertia +7.2987 cost/tau^2 -1.2010
(a) L=10 tau=0.25 k=0 M=3.00 | c2E=-0.73610 c2C=-0.017649 | inertia +40.6627 cost/tau^2 -11.7776
(a) L=10 tau=0.25 k=1 M=3.50 | c2E=-1.00333 c2C=-0.021398 | inertia +67.1055 cost/tau^2 -16.0533
(a) L=10 tau=0.25 k=2 M=4.00 | c2E=-1.31180 c2C=-0.025154 | inertia +103.0315 cost/tau^2 -20.9888
(c) L= 6 tau=1.00 k=0 M=3.00 | c2E=-0.68067 c2C=-0.268163 | inertia +2.4135 cost/tau^2 -0.6807
(c) L= 6 tau=1.00 k=1 M=3.50 | c2E=-0.93256 c2C=-0.314727 | inertia +3.8554 cost/tau^2 -0.9326
(c) L= 6 tau=1.00 k=2 M=4.00 | c2E=-1.22492 c2C=-0.361271 | inertia +5.7803 cost/tau^2 -1.2249
(c) L= 6 tau=0.25 k=0 M=3.00 | c2E=-0.72573 c2C=-0.019472 | inertia +44.8645 cost/tau^2 -11.6116
(c) L= 6 tau=0.25 k=1 M=3.50 | c2E=-0.98831 c2C=-0.022371 | inertia +70.1550 cost/tau^2 -15.8130
(c) L= 6 tau=0.25 k=2 M=4.00 | c2E=-1.29141 c2C=-0.025268 | inertia +103.4980 cost/tau^2 -20.6625
(c) L=10 tau=1.00 k=0 M=3.00 | c2E=-0.67330 c2C=-0.447612 | inertia +4.0285 cost/tau^2 -0.6733
(c) L=10 tau=1.00 k=1 M=3.50 | c2E=-0.92666 c2C=-0.513972 | inertia +6.2962 cost/tau^2 -0.9267
(c) L=10 tau=1.00 k=2 M=4.00 | c2E=-1.22121 c2C=-0.579069 | inertia +9.2651 cost/tau^2 -1.2212
(c) L=10 tau=0.25 k=0 M=3.00 | c2E=-0.73755 c2C=-0.027660 | inertia +63.7277 cost/tau^2 -11.8008
(c) L=10 tau=0.25 k=1 M=3.50 | c2E=-1.00474 c2C=-0.031422 | inertia +98.5408 cost/tau^2 -16.0758
(c) L=10 tau=0.25 k=2 M=4.00 | c2E=-1.31317 c2C=-0.035182 | inertia +144.1058 cost/tau^2 -21.0108