1 files changed, 5 insertions, 176 deletions
diff --git a/buch/papers/laguerre/scripts/gamma_approx.py b/buch/papers/laguerre/scripts/gamma_approx.py
index 53ba76b..208f770 100644
--- a/buch/papers/laguerre/scripts/gamma_approx.py
+++ b/buch/papers/laguerre/scripts/gamma_approx.py
@@ -8,6 +8,7 @@ import scipy.special
 EPSILON = 1e-7
 root = str(Path(__file__).parent)
 img_path = f"{root}/../images"
+fontsize = "medium"
 
 
 def _prep_zeros_and_weights(x, w, n):
@@ -26,17 +27,6 @@ def pochhammer(z, n):
     return np.prod(z + np.arange(n))
 
 
-def find_shift(z, target):
-    factor = 1.0
-    steps = int(np.floor(target - np.real(z)))
-    zs = z + steps
-    if steps > 0:
-        factor = 1 / pochhammer(z, steps)
-    elif steps < 0:
-        factor = pochhammer(zs, -steps)
-    return zs, factor
-
-
 def find_optimal_shift(z, n):
     mhat = 1.34093 * n + 0.854093
     steps = int(np.floor(mhat - np.real(z)))
@@ -44,6 +34,7 @@ def find_optimal_shift(z, n):
 
 
 def get_shifting_factor(z, steps):
+    factor = 1.0
     if steps > 0:
         factor = 1 / pochhammer(z, steps)
     elif steps < 0:
@@ -56,7 +47,9 @@ def laguerre_gamma_shifted(z, x=None, w=None, n=8, target=11):
     n = len(x)
 
     z += 0j
-    z_shifted, correction_factor = find_shift(z, target)
+    steps = int(np.floor(target - np.real(z)))
+    z_shifted = z + steps
+    correction_factor = get_shifting_factor(z, steps)
 
     res = np.sum(x ** (z_shifted - 1) * w)
     res *= correction_factor
@@ -112,167 +105,3 @@ def eval_laguerre_gamma(z, x=None, w=None, n=8, func="simple", **kwargs):
 
 def calc_rel_error(x, y):
     return (y - x) / x
-
-
-# Simple / naive
-xmin = -5
-xmax = 30
-ns = np.arange(2, 12, 2)
-ylim = np.array([-11, 6])
-x = np.linspace(xmin + EPSILON, xmax - EPSILON, 400)
-gamma = scipy.special.gamma(x)
-fig, ax = plt.subplots(num=1, clear=True, constrained_layout=True, figsize=(5, 2.5))
-for n in ns:
-    gamma_lag = eval_laguerre_gamma(x, n=n)
-    rel_err = calc_rel_error(gamma, gamma_lag)
-    ax.semilogy(x, np.abs(rel_err), label=f"$n={n}$")
-ax.set_xlim(x[0], x[-1])
-ax.set_ylim(*(10.0 ** ylim))
-ax.set_xticks(np.arange(xmin, xmax + EPSILON, 5))
-ax.set_xticks(np.arange(xmin, xmax), minor=True)
-ax.set_yticks(10.0 ** np.arange(*ylim, 2))
-ax.set_yticks(10.0 ** np.arange(*ylim, 2))
-ax.set_xlabel(r"$z$")
-ax.set_ylabel("Relativer Fehler")
-ax.legend(ncol=3, fontsize="small")
-ax.grid(1, "both")
-fig.savefig(f"{img_path}/rel_error_simple.pgf")
-
-
-# Mirrored
-xmin = -15
-xmax = 15
-ylim = np.array([-11, 1])
-x = np.linspace(xmin + EPSILON, xmax - EPSILON, 400)
-gamma = scipy.special.gamma(x)
-fig2, ax2 = plt.subplots(num=2, clear=True, constrained_layout=True, figsize=(5, 2.5))
-for n in ns:
-    gamma_lag = eval_laguerre_gamma(x, n=n, func="mirror")
-    rel_err = calc_rel_error(gamma, gamma_lag)
-    ax2.semilogy(x, np.abs(rel_err), label=f"$n={n}$")
-ax2.set_xlim(x[0], x[-1])
-ax2.set_ylim(*(10.0 ** ylim))
-ax2.set_xticks(np.arange(xmin, xmax + EPSILON, 5))
-ax2.set_xticks(np.arange(xmin, xmax), minor=True)
-ax2.set_yticks(10.0 ** np.arange(*ylim, 2))
-# locmin = mpl.ticker.LogLocator(base=10.0,subs=0.1*np.arange(1,10),numticks=100)
-# ax2.yaxis.set_minor_locator(locmin)
-# ax2.yaxis.set_minor_formatter(mpl.ticker.NullFormatter())
-ax2.set_xlabel(r"$z$")
-ax2.set_ylabel("Relativer Fehler")
-ax2.legend(ncol=1, loc="upper left", fontsize="small")
-ax2.grid(1, "both")
-fig2.savefig(f"{img_path}/rel_error_mirror.pgf")
-
-
-# Move to target
-bests = []
-N = 200
-step = 1 / (N - 1)
-a = 11 / 8
-b = 1 / 2
-x = np.linspace(step, 1 - step, N + 1)
-gamma = scipy.special.gamma(x)[:, None]
-ns = np.arange(2, 13)
-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)
-    gamma_lag = np.stack(
-        [
-            eval_laguerre_gamma(x, target=target, x=zeros, w=weights, func="shifted")
-            for target in targets
-        ],
-        -1,
-    )
-    rel_error = np.abs(calc_rel_error(gamma, gamma_lag))
-    best = np.argmin(rel_error, -1) + targets[0]
-    bests.append(best)
-bests = np.stack(bests, 0)
-
-fig3, ax3 = plt.subplots(num=3, clear=True, constrained_layout=True, figsize=(5, 3))
-v = ax3.imshow(bests, cmap="inferno", aspect="auto", interpolation="nearest")
-plt.colorbar(v, ax=ax3, label=r"$m^*$")
-ticks = np.arange(0, N + 1, N // 5)
-ax3.set_xlim(0, 1)
-ax3.set_xticks(ticks)
-ax3.set_xticklabels([f"{v:.2f}" for v in ticks / N])
-ax3.set_xticks(np.arange(0, N + 1, N // 20), minor=True)
-ax3.set_yticks(np.arange(len(ns)))
-ax3.set_yticklabels(ns)
-ax3.set_xlabel(r"$z$")
-ax3.set_ylabel(r"$n$")
-fig3.savefig(f"{img_path}/targets.pdf")
-
-targets = np.mean(bests, -1)
-intercept, bias = np.polyfit(ns, targets, 1)
-fig4, axs4 = plt.subplots(
-    2, num=4, sharex=True, clear=True, constrained_layout=True, figsize=(5, 4)
-)
-xl = np.array([ns[0] - 0.5, ns[-1] + 0.5])
-axs4[0].plot(xl, intercept * xl + bias, label=r"$\hat{m}$")
-axs4[0].plot(ns, targets, "x", label=r"$\overline{m}$")
-axs4[1].plot(ns, ((intercept * ns + bias) - targets), "-x", label=r"$\hat{m} - \overline{m}$")
-axs4[0].set_xlim(*xl)
-# axs4[0].set_title("Schätzung von Mittelwert")
-# axs4[1].set_title("Fehler")
-axs4[-1].set_xlabel(r"$n$")
-for ax in axs4:
-    ax.grid(1)
-    ax.legend()
-fig4.savefig(f"{img_path}/estimate.pgf")
-
-print(f"Intercept={intercept:.6g}, Bias={bias:.6g}")
-predicts = np.ceil(intercept * ns[:, None] + bias - x)
-print(f"Error: {int(np.sum(np.abs(bests-predicts)))}")
-
-# Comparison relative error between methods
-N = 200
-step = 1 / (N - 1)
-x = np.linspace(step, 1 - step, N + 1)
-gamma = scipy.special.gamma(x)[:, None]
-n = 8
-targets = np.arange(10, 14)
-gamma = scipy.special.gamma(x)
-fig5, ax5 = plt.subplots(num=5, clear=True, constrained_layout=True)
-for target in targets:
-    gamma_lag = eval_laguerre_gamma(x, target=target, n=n, func="shifted")
-    rel_error = np.abs(calc_rel_error(gamma, gamma_lag))
-    ax5.semilogy(x, rel_error, label=f"$m={target}$", linewidth=3)
-gamma_lgo = eval_laguerre_gamma(x, n=n, func="optimal_shifted")
-rel_error = np.abs(calc_rel_error(gamma, gamma_lgo))
-ax5.semilogy(x, rel_error, "m", linestyle="dotted", label="$m^*$", linewidth=3)
-ax5.set_xlim(x[0], x[-1])
-ax5.set_ylim(5e-9, 5e-8)
-ax5.set_xlabel(r"$z$")
-ax5.grid(1, "both")
-ax5.legend()
-fig5.savefig(f"{img_path}/rel_error_shifted.pgf")
-
-N = 200
-x = np.linspace(-5+ EPSILON, 5-EPSILON, N)
-gamma = scipy.special.gamma(x)[:, None]
-n = 8
-gamma = scipy.special.gamma(x)
-fig6, ax6 = plt.subplots(num=6, clear=True, constrained_layout=True)
-gamma_lgo = eval_laguerre_gamma(x, n=n, func="optimal_shifted")
-rel_error = np.abs(calc_rel_error(gamma, gamma_lgo))
-ax6.semilogy(x, rel_error, label="$m^*$", linewidth=3)
-ax6.set_xlim(x[0], x[-1])
-ax6.set_ylim(5e-9, 5e-8)
-ax6.set_xlabel(r"$z$")
-ax6.grid(1, "both")
-ax6.legend()
-fig6.savefig(f"{img_path}/rel_error_range.pgf")
-
-N = 2001
-x = np.linspace(-5, 5, N)
-gamma = scipy.special.gamma(x)
-fig7, ax7 = plt.subplots(num=7, clear=True, constrained_layout=True)
-ax7.plot(x, gamma)
-ax7.set_xlim(x[0], x[-1])
-ax7.set_ylim(-7.5, 25)
-ax7.grid(1, "both")
-fig7.savefig(f"{img_path}/gamma.pgf")
-
-# plt.show()