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import numpy as np
import scipy.special
import gamma_approx as ga
def find_best_loc(N=200, a=1.375, b=0.5, ns=None):
if ns is None:
ns = np.arange(2, 13)
bests = []
step = 1 / (N - 1)
x = np.linspace(step, 1 - step, N + 1)
gamma = scipy.special.gamma(x)[:, None]
for n in ns:
zeros, weights = np.polynomial.laguerre.laggauss(n)
est = np.ceil(b + a * n)
targets = np.arange(max(est - 2, 0), est + 3)
glag = [
ga.eval_laguerre_gamma(x, target=target, x=zeros, w=weights, func="shifted")
for target in targets
]
gamma_lag = np.stack(glag, -1)
rel_error = np.abs(ga.calc_rel_error(gamma, gamma_lag))
best = np.argmin(rel_error, -1) + targets[0]
bests.append(best)
return np.stack(bests, 0)
if __name__ == "__main__":
import matplotlib.pyplot as plt
N = 200
ns = np.arange(2, 13)
bests = find_best_loc(N, ns=ns)
fig, ax = plt.subplots(num=1, clear=True, constrained_layout=True, figsize=(4, 2.4))
v = ax.imshow(bests, cmap="inferno", aspect="auto", interpolation="nearest")
plt.colorbar(v, ax=ax, label=r"$m^*$")
ticks = np.arange(0, N + 1, N // 5)
ax.set_xlim(0, 1)
ax.set_xticks(ticks)
ax.set_xticklabels([f"{v:.2f}" for v in ticks / N])
ax.set_xticks(np.arange(0, N + 1, N // 20), minor=True)
ax.set_yticks(np.arange(len(ns)))
ax.set_yticklabels(ns)
ax.set_xlabel(r"$z$")
ax.set_ylabel(r"$n$")
fig.savefig(f"{ga.img_path}/targets.pgf")
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