python hotrg_qr.py -cuda 0 -niter 6 -init hotrg -exact -0.9397153248292055 gets lower than exact ?

support/exact_ising_2d_periodic.py

+from __future__ import division
+
+from math import fsum
+from numpy import (
+    arccosh,
+    arcsinh,
+    arctanh,
+    cos,
+    cosh,
+    exp,
+    log,
+    pi,
+    sin,
+    sinh,
+    sqrt,
+    sum,
+    tan,
+    tanh,
+)
+from scipy.special import logsumexp
+
+def h(j, beta):
+    return beta * j
+
+
+def h_star(j, beta):
+    return arctanh(exp(-2 * beta * j))
+
+
+def gamma(n, j, beta, r):
+    return arccosh(
+        cosh(2 * h_star(j, beta)) * cosh(2 * h(j, beta)) -
+        sinh(2 * h_star(j, beta)) * sinh(2 * h(j, beta)) * cos(r * pi / n))
+
+
+def log_z(n, j, beta):
+    return (
+        -log(2) + 1 / 2 * n**2 * log(2 * sinh(2 * h(j, beta))) + logsumexp([
+            fsum([
+                log(2 * cosh(n / 2 * gamma(n, j, beta, 2 * r)))
+                for r in range(n)
+            ]),
+            fsum([
+                log(2 * sinh(n / 2 * gamma(n, j, beta, 2 * r)))
+                for r in range(n)
+            ]),
+            fsum([
+                log(2 * cosh(n / 2 * gamma(n, j, beta, 2 * r + 1)))
+                for r in range(n)
+            ]),
+            fsum([
+                log(2 * sinh(n / 2 * gamma(n, j, beta, 2 * r + 1)))
+                for r in range(n)
+            ]),
+        ]))
+
+
+def free_energy(n, j, beta):
+    return -1 / n**2 / beta * log_z(n, j, beta)
+
+
+if __name__ == '__main__':
+    import argparse
+    parser = argparse.ArgumentParser()
+    parser.add_argument('-L',type=int,default=4,help='L')
+    parser.add_argument('-beta',type=float,default=0.44068679350977147,help='beta')
+    args = parser.parse_args()
+
+    print(args.L, args.beta, -log_z(n=args.L, j=1, beta=args.beta)/args.L**2)

support/lnZ.dat

+2 1.09550665866847
+4 0.9701197161722052
+8 0.9397153248292055
+16 0.9321963149809878
+32 0.9303203901841237
+64 0.9298516316942892
+128 0.9297344557701893
+512 0.9296978394131481
+1024 0.9296960086092727
+2048 0.9296955509085117
+4096 0.9296954364833345