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-rw-r--r--buch/papers/laguerre/scripts/estimates.py49
-rw-r--r--buch/papers/laguerre/scripts/gamma_approx.ipynb395
-rw-r--r--buch/papers/laguerre/scripts/gamma_approx.py116
-rw-r--r--buch/papers/laguerre/scripts/integrand.py42
-rw-r--r--buch/papers/laguerre/scripts/integrand_exp.py46
-rw-r--r--buch/papers/laguerre/scripts/laguerre_plot.py100
-rw-r--r--buch/papers/laguerre/scripts/laguerre_poly.py108
-rw-r--r--buch/papers/laguerre/scripts/rel_error_complex.py43
-rw-r--r--buch/papers/laguerre/scripts/rel_error_mirror.py38
-rw-r--r--buch/papers/laguerre/scripts/rel_error_range.py41
-rw-r--r--buch/papers/laguerre/scripts/rel_error_shifted.py40
-rw-r--r--buch/papers/laguerre/scripts/rel_error_simple.py41
-rw-r--r--buch/papers/laguerre/scripts/targets.py58
13 files changed, 622 insertions, 495 deletions
diff --git a/buch/papers/laguerre/scripts/estimates.py b/buch/papers/laguerre/scripts/estimates.py
new file mode 100644
index 0000000..1acd7f7
--- /dev/null
+++ b/buch/papers/laguerre/scripts/estimates.py
@@ -0,0 +1,49 @@
+if __name__ == "__main__":
+ import matplotlib as mpl
+ import matplotlib.pyplot as plt
+ import numpy as np
+
+ import gamma_approx as ga
+ import targets
+
+ mpl.rcParams.update(
+ {
+ "mathtext.fontset": "stix",
+ "font.family": "serif",
+ "font.serif": "TeX Gyre Termes",
+ }
+ )
+
+ N = 200
+ ns = np.arange(1, 13)
+ step = 1 / (N - 1)
+ x = np.linspace(step, 1 - step, N + 1)
+
+ bests = targets.find_best_loc(N, ns=ns)
+ mean_m = np.mean(bests, -1)
+
+ intercept, bias = np.polyfit(ns, mean_m, 1)
+ fig, axs = plt.subplots(
+ 2, num=1, sharex=True, clear=True, constrained_layout=True, figsize=(4.5, 3.6)
+ )
+ xl = np.array([ns[0] - 0.5, ns[-1] + 0.5])
+ axs[0].plot(xl, intercept * xl + bias, label=r"$\hat{m}$")
+ axs[0].plot(ns, mean_m, "x", label=r"$\overline{m}$")
+ axs[1].plot(
+ ns, ((intercept * ns + bias) - mean_m), "-x", label=r"$\hat{m} - \overline{m}$"
+ )
+ axs[0].set_xlim(*xl)
+ axs[0].set_xticks(ns)
+ axs[0].set_yticks(np.arange(np.floor(mean_m[0]), np.ceil(mean_m[-1]) + 0.1, 2))
+ # axs[0].set_title("Schätzung von Mittelwert")
+ # axs[1].set_title("Fehler")
+ axs[-1].set_xlabel(r"$n$")
+ for ax in axs:
+ ax.grid(1)
+ ax.legend()
+ # fig.savefig(f"{ga.img_path}/estimates.pgf")
+ fig.savefig(f"{ga.img_path}/estimates.pdf")
+
+ print(f"Intercept={intercept:.6g}, Bias={bias:.6g}")
+ predicts = np.ceil(intercept * ns[:, None] + bias - np.real(x))
+ print(f"Error: {np.mean(np.abs(bests - predicts))}")
diff --git a/buch/papers/laguerre/scripts/gamma_approx.ipynb b/buch/papers/laguerre/scripts/gamma_approx.ipynb
deleted file mode 100644
index 44f3abd..0000000
--- a/buch/papers/laguerre/scripts/gamma_approx.ipynb
+++ /dev/null
@@ -1,395 +0,0 @@
-{
- "cells": [
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "# Gauss-Laguerre Quadratur für die Gamma-Funktion\n",
- "\n",
- "$$\n",
- " \\Gamma(z)\n",
- " = \n",
- " \\int_0^\\infty t^{z-1}e^{-t}dt\n",
- "$$\n",
- "\n",
- "$$\n",
- " \\int_0^\\infty f(x) e^{-x} dx \n",
- " \\approx \n",
- " \\sum_{i=1}^{N} f(x_i) w_i\n",
- " \\qquad\\text{ wobei }\n",
- " w_i = \\frac{x_i}{(n+1)^2 [L_{n+1}(x_i)]^2}\n",
- "$$\n",
- "und $x_i$ sind Nullstellen des Laguerre Polynoms $L_n(x)$\n",
- "\n",
- "Der Fehler ist gegeben als\n",
- "\n",
- "$$\n",
- " E \n",
- " =\n",
- " \\frac{(n!)^2}{(2n)!} f^{(2n)}(\\xi) \n",
- " = \n",
- " \\frac{(-2n + z)_{2n}}{(z-m)_m} \\frac{(n!)^2}{(2n)!} \\xi^{z + m - 2n - 1}\n",
- "$$"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": null,
- "metadata": {},
- "outputs": [],
- "source": [
- "import numpy as np\n",
- "import matplotlib.pyplot as plt\n",
- "from cmath import exp, pi, sin, sqrt\n",
- "import scipy.special\n",
- "\n",
- "EPSILON = 1e-07\n"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": null,
- "metadata": {},
- "outputs": [],
- "source": [
- "lanczos_p = [\n",
- " 676.5203681218851,\n",
- " -1259.1392167224028,\n",
- " 771.32342877765313,\n",
- " -176.61502916214059,\n",
- " 12.507343278686905,\n",
- " -0.13857109526572012,\n",
- " 9.9843695780195716e-6,\n",
- " 1.5056327351493116e-7,\n",
- "]\n",
- "\n",
- "\n",
- "def drop_imag(z):\n",
- " if abs(z.imag) <= EPSILON:\n",
- " z = z.real\n",
- " return z\n",
- "\n",
- "\n",
- "def lanczos_gamma(z):\n",
- " z = complex(z)\n",
- " if z.real < 0.5:\n",
- " y = pi / (sin(pi * z) * lanczos_gamma(1 - z)) # Reflection formula\n",
- " else:\n",
- " z -= 1\n",
- " x = 0.99999999999980993\n",
- " for (i, pval) in enumerate(lanczos_p):\n",
- " x += pval / (z + i + 1)\n",
- " t = z + len(lanczos_p) - 0.5\n",
- " y = sqrt(2 * pi) * t ** (z + 0.5) * exp(-t) * x\n",
- " return drop_imag(y)\n"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": null,
- "metadata": {},
- "outputs": [],
- "source": [
- "zeros, weights = np.polynomial.laguerre.laggauss(8)\n",
- "# zeros = np.array(\n",
- "# [\n",
- "# 1.70279632305101000e-1,\n",
- "# 9.03701776799379912e-1,\n",
- "# 2.25108662986613069e0,\n",
- "# 4.26670017028765879e0,\n",
- "# 7.04590540239346570e0,\n",
- "# 1.07585160101809952e1,\n",
- "# 1.57406786412780046e1,\n",
- "# 2.28631317368892641e1,\n",
- "# ]\n",
- "# )\n",
- "\n",
- "# weights = np.array(\n",
- "# [\n",
- "# 3.69188589341637530e-1,\n",
- "# 4.18786780814342956e-1,\n",
- "# 1.75794986637171806e-1,\n",
- "# 3.33434922612156515e-2,\n",
- "# 2.79453623522567252e-3,\n",
- "# 9.07650877335821310e-5,\n",
- "# 8.48574671627253154e-7,\n",
- "# 1.04800117487151038e-9,\n",
- "# ]\n",
- "# )\n",
- "\n",
- "\n",
- "def pochhammer(z, n):\n",
- " return np.prod(z + np.arange(n))\n",
- "\n",
- "\n",
- "def find_shift(z, target):\n",
- " factor = 1.0\n",
- " steps = int(np.floor(target - np.real(z)))\n",
- " zs = z + steps\n",
- " if steps > 0:\n",
- " factor = 1 / pochhammer(z, steps)\n",
- " elif steps < 0:\n",
- " factor = pochhammer(zs, -steps)\n",
- " return zs, factor\n",
- "\n",
- "\n",
- "def laguerre_gamma(z, x, w, target=11):\n",
- " # res = 0.0\n",
- " z = complex(z)\n",
- " if z.real < 1e-3:\n",
- " res = pi / (\n",
- " sin(pi * z) * laguerre_gamma(1 - z, x, w, target)\n",
- " ) # Reflection formula\n",
- " else:\n",
- " z_shifted, correction_factor = find_shift(z, target)\n",
- " res = np.sum(x ** (z_shifted - 1) * w)\n",
- " res *= correction_factor\n",
- " res = drop_imag(res)\n",
- " return res\n"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": null,
- "metadata": {},
- "outputs": [],
- "source": [
- "def eval_laguerre(x, target=12):\n",
- " return np.array([laguerre_gamma(xi, zeros, weights, target) for xi in x])\n",
- "\n",
- "\n",
- "def eval_lanczos(x):\n",
- " return np.array([lanczos_gamma(xi) for xi in x])\n",
- "\n",
- "\n",
- "def eval_mean_laguerre(x, targets):\n",
- " return np.mean([eval_laguerre(x, target) for target in targets], 0)\n",
- "\n",
- "\n",
- "def calc_rel_error(x, y):\n",
- " return (y - x) / x\n",
- "\n",
- "\n",
- "def evaluate(x, target=12):\n",
- " lanczos_gammas = eval_lanczos(x)\n",
- " laguerre_gammas = eval_laguerre(x, target)\n",
- " rel_error = calc_rel_error(lanczos_gammas, laguerre_gammas)\n",
- " return rel_error\n"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "### Test with real values"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "Empirische Tests zeigen:\n",
- "- $n=4 \\Rightarrow m=6$\n",
- "- $n=5 \\Rightarrow m=7$ oder $m=8$\n",
- "- $n=6 \\Rightarrow m=9$\n",
- "- $n=7 \\Rightarrow m=10$\n",
- "- $n=8 \\Rightarrow m=11$ oder $m=12$\n",
- "- $n=9 \\Rightarrow m=13$\n",
- "- $n=10 \\Rightarrow m=14$\n",
- "- $n=11 \\Rightarrow m=15$ oder $m=16$\n",
- "- $n=12 \\Rightarrow m=17$\n",
- "- $n=13 \\Rightarrow m=18 \\Rightarrow $ Beginnt numerisch instabil zu werden \n"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": null,
- "metadata": {},
- "outputs": [],
- "source": [
- "zeros, weights = np.polynomial.laguerre.laggauss(12)\n",
- "targets = np.arange(16, 21)\n",
- "mean_targets = ((16, 17),)\n",
- "x = np.linspace(EPSILON, 1 - EPSILON, 101)\n",
- "_, axs = plt.subplots(\n",
- " 2, sharex=True, clear=True, constrained_layout=True, figsize=(12, 12)\n",
- ")\n",
- "\n",
- "lanczos = eval_lanczos(x)\n",
- "for mean_target in mean_targets:\n",
- " vals = eval_mean_laguerre(x, mean_target)\n",
- " rel_error_mean = calc_rel_error(lanczos, vals)\n",
- " axs[0].plot(x, rel_error_mean, label=mean_target)\n",
- " axs[1].semilogy(x, np.abs(rel_error_mean), label=mean_target)\n",
- "\n",
- "mins = []\n",
- "maxs = []\n",
- "for target in targets:\n",
- " rel_error = evaluate(x, target)\n",
- " mins.append(np.min(np.abs(rel_error[(0.1 <= x) & (x <= 0.9)])))\n",
- " maxs.append(np.max(np.abs(rel_error)))\n",
- " axs[0].plot(x, rel_error, label=target)\n",
- " axs[1].semilogy(x, np.abs(rel_error), label=target)\n",
- "# axs[0].set_ylim(*(np.array([-1, 1]) * 3.5e-8))\n",
- "\n",
- "axs[0].set_xlim(x[0], x[-1])\n",
- "axs[1].set_ylim(np.min(mins), 1.04*np.max(maxs))\n",
- "for ax in axs:\n",
- " ax.legend()\n",
- " ax.grid(which=\"both\")\n"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": null,
- "metadata": {},
- "outputs": [],
- "source": [
- "targets = (16, 17)\n",
- "xmax = 15\n",
- "x = np.linspace(-xmax + EPSILON, xmax - EPSILON, 1000)\n",
- "\n",
- "mean_lag = eval_mean_laguerre(x, targets)\n",
- "lanczos = eval_lanczos(x)\n",
- "rel_error = calc_rel_error(lanczos, mean_lag)\n",
- "rel_error_simple = evaluate(x, targets[-1])\n",
- "# rel_error = evaluate(x, target)\n",
- "\n",
- "_, axs = plt.subplots(\n",
- " 2, sharex=True, clear=True, constrained_layout=True, figsize=(12, 12)\n",
- ")\n",
- "axs[0].plot(x, rel_error, label=targets)\n",
- "axs[1].semilogy(x, np.abs(rel_error), label=targets)\n",
- "axs[0].plot(x, rel_error_simple, label=targets[-1])\n",
- "axs[1].semilogy(x, np.abs(rel_error_simple), label=targets[-1])\n",
- "axs[0].set_xlim(x[0], x[-1])\n",
- "# axs[0].set_ylim(*(np.array([-1, 1]) * 4.2e-8))\n",
- "# axs[1].set_ylim(1e-10, 5e-8)\n",
- "for ax in axs:\n",
- " ax.legend()\n",
- "\n",
- "x2 = np.linspace(-5 + EPSILON, 5, 4001)\n",
- "_, ax = plt.subplots(constrained_layout=True, figsize=(8, 6))\n",
- "ax.plot(x2, eval_mean_laguerre(x2, targets))\n",
- "ax.set_xlim(x2[0], x2[-1])\n",
- "ax.set_ylim(-7.5, 25)\n"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "### Test with complex values"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": null,
- "metadata": {},
- "outputs": [],
- "source": [
- "targets = (16, 17)\n",
- "vals = np.linspace(-5 + EPSILON, 5, 100)\n",
- "x, y = np.meshgrid(vals, vals)\n",
- "mesh = x + 1j * y\n",
- "input = mesh.flatten()\n",
- "\n",
- "mean_lag = eval_mean_laguerre(input, targets).reshape(mesh.shape)\n",
- "lanczos = eval_lanczos(input).reshape(mesh.shape)\n",
- "rel_error = np.abs(calc_rel_error(lanczos, mean_lag))\n",
- "\n",
- "lag = eval_laguerre(input, targets[-1]).reshape(mesh.shape)\n",
- "rel_error_simple = np.abs(calc_rel_error(lanczos, lag))\n",
- "# rel_error = evaluate(x, target)\n",
- "\n",
- "fig, axs = plt.subplots(\n",
- " 2,\n",
- " 2,\n",
- " sharex=True,\n",
- " sharey=True,\n",
- " clear=True,\n",
- " constrained_layout=True,\n",
- " figsize=(12, 10),\n",
- ")\n",
- "_c = axs[0, 1].pcolormesh(x, y, np.log10(np.abs(lanczos - mean_lag)), shading=\"gouraud\")\n",
- "_c = axs[0, 0].pcolormesh(x, y, np.log10(np.abs(lanczos - lag)), shading=\"gouraud\")\n",
- "fig.colorbar(_c, ax=axs[0, :])\n",
- "_c = axs[1, 1].pcolormesh(x, y, np.log10(rel_error), shading=\"gouraud\")\n",
- "_c = axs[1, 0].pcolormesh(x, y, np.log10(rel_error_simple), shading=\"gouraud\")\n",
- "fig.colorbar(_c, ax=axs[1, :])\n",
- "_ = axs[0, 0].set_title(\"Absolute Error\")\n",
- "_ = axs[1, 0].set_title(\"Relative Error\")\n"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": null,
- "metadata": {},
- "outputs": [],
- "source": [
- "z = 0.5\n",
- "ns = [4, 5, 5, 6, 7, 8, 8, 9, 10, 11, 11, 12] # np.arange(4, 13)\n",
- "ms = np.arange(6, 18)\n",
- "xi = np.logspace(0, 2, 201)[:, None]\n",
- "lanczos = eval_lanczos([z])[0]\n",
- "\n",
- "_, ax = plt.subplots(clear=True, constrained_layout=True, figsize=(12, 8))\n",
- "ax.grid(1)\n",
- "for n, m in zip(ns, ms):\n",
- " zeros, weights = np.polynomial.laguerre.laggauss(n)\n",
- " c = scipy.special.factorial(n) ** 2 / scipy.special.factorial(2 * n)\n",
- " e = np.abs(\n",
- " scipy.special.poch(z - 2 * n, 2 * n)\n",
- " / scipy.special.poch(z - m, m)\n",
- " * c\n",
- " * xi ** (z - 2 * n + m - 1)\n",
- " )\n",
- " ez = np.sum(\n",
- " scipy.special.poch(z - 2 * n, 2 * n)\n",
- " / scipy.special.poch(z - m, m)\n",
- " * c\n",
- " * zeros[:, None] ** (z - 2 * n + m - 1),\n",
- " 0,\n",
- " )\n",
- " lag = eval_laguerre([z], m)[0]\n",
- " err = np.abs(lanczos - lag)\n",
- " # print(m+z,ez)\n",
- " # for zi,ezi in zip(z[0], ez):\n",
- " # print(f\"{m+zi}: {ezi}\")\n",
- " # ax.semilogy(xi, e, color=color)\n",
- " lines = ax.loglog(xi, e, label=str(n))\n",
- " ax.axhline(err, color=lines[0].get_color())\n",
- " # ax.set_xticks(np.arange(xi[-1] + 1))\n",
- " # ax.set_ylim(1e-8, 1e5)\n",
- "_ = ax.legend()\n",
- "# _ = ax.legend([f\"z={zi}\" for zi in z[0]])\n",
- "# _ = [ax.axvline(x) for x in zeros]\n"
- ]
- }
- ],
- "metadata": {
- "interpreter": {
- "hash": "767d51c1340bd893661ea55ea3124f6de3c7a262a8b4abca0554b478b1e2ff90"
- },
- "kernelspec": {
- "display_name": "Python 3.8.10 64-bit",
- "language": "python",
- "name": "python3"
- },
- "language_info": {
- "codemirror_mode": {
- "name": "ipython",
- "version": 3
- },
- "file_extension": ".py",
- "mimetype": "text/x-python",
- "name": "python",
- "nbconvert_exporter": "python",
- "pygments_lexer": "ipython3",
- "version": "3.8.10"
- },
- "orig_nbformat": 4
- },
- "nbformat": 4,
- "nbformat_minor": 2
-}
diff --git a/buch/papers/laguerre/scripts/gamma_approx.py b/buch/papers/laguerre/scripts/gamma_approx.py
new file mode 100644
index 0000000..5b09e59
--- /dev/null
+++ b/buch/papers/laguerre/scripts/gamma_approx.py
@@ -0,0 +1,116 @@
+from pathlib import Path
+
+import numpy as np
+import scipy.special
+
+EPSILON = 1e-7
+root = str(Path(__file__).parent)
+img_path = f"{root}/../images"
+fontsize = "medium"
+cmap = "plasma"
+
+
+def _prep_zeros_and_weights(x, w, n):
+ if x is None or w is None:
+ return np.polynomial.laguerre.laggauss(n)
+ return x, w
+
+
+def drop_imag(z):
+ if abs(z.imag) <= EPSILON:
+ z = z.real
+ return z
+
+
+def pochhammer(z, n):
+ return np.prod(z + np.arange(n))
+
+
+def find_optimal_shift(z, n):
+ mhat = 1.34093 * n + 0.854093
+ steps = int(np.floor(mhat - np.real(z)))
+ return steps
+
+
+def get_shifting_factor(z, steps):
+ factor = 1.0
+ if steps > 0:
+ factor = 1 / pochhammer(z, steps)
+ elif steps < 0:
+ factor = pochhammer(z + steps, -steps)
+ return factor
+
+
+def laguerre_gamma_shifted(z, x=None, w=None, n=8, target=11):
+ x, w = _prep_zeros_and_weights(x, w, n)
+ n = len(x)
+
+ z += 0j
+ 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
+ res = drop_imag(res)
+ return res
+
+
+def laguerre_gamma_opt_shifted(z, x=None, w=None, n=8):
+ if z == 0.0:
+ return np.infty
+ x, w = _prep_zeros_and_weights(x, w, n)
+ n = len(x)
+
+ z += 0j
+ opt_shift = find_optimal_shift(z, n)
+ correction_factor = get_shifting_factor(z, opt_shift)
+ z_shifted = z + opt_shift
+
+ res = np.sum(x ** (z_shifted - 1) * w)
+ res *= correction_factor
+ res = drop_imag(res)
+ return res
+
+
+def laguerre_gamma_simple(z, x=None, w=None, n=8):
+ if z == 0.0:
+ return np.infty
+ x, w = _prep_zeros_and_weights(x, w, n)
+ z += 0j
+ res = np.sum(x ** (z - 1) * w)
+ res = drop_imag(res)
+ return res
+
+
+def laguerre_gamma_mirror(z, x=None, w=None, n=8):
+ if z == 0.0:
+ return np.infty
+ x, w = _prep_zeros_and_weights(x, w, n)
+ z += 0j
+ if z.real < 1e-3:
+ return np.pi / (
+ np.sin(np.pi * z) * laguerre_gamma_simple(1 - z, x, w)
+ ) # Reflection formula
+ return laguerre_gamma_simple(z, x, w)
+
+
+def eval_laguerre_gamma(z, x=None, w=None, n=8, func="simple", **kwargs):
+ x, w = _prep_zeros_and_weights(x, w, n)
+ if func == "simple":
+ f = laguerre_gamma_simple
+ elif func == "mirror":
+ f = laguerre_gamma_mirror
+ elif func == "optimal_shifted":
+ f = laguerre_gamma_opt_shifted
+ else:
+ f = laguerre_gamma_shifted
+ return np.array([f(zi, x, w, n, **kwargs) for zi in z])
+
+
+def calc_rel_error(x, y):
+ mask = np.abs(x) != np.infty
+ rel_error = np.zeros_like(y)
+ rel_error[mask] = (y[mask] - x[mask]) / x[mask]
+ rel_error[~mask] = 0.0
+ return rel_error
diff --git a/buch/papers/laguerre/scripts/integrand.py b/buch/papers/laguerre/scripts/integrand.py
new file mode 100644
index 0000000..e970721
--- /dev/null
+++ b/buch/papers/laguerre/scripts/integrand.py
@@ -0,0 +1,42 @@
+#!/usr/bin/env python3
+# -*- coding:utf-8 -*-
+"""Plot for integrand of gamma function with shifting terms."""
+
+if __name__ == "__main__":
+ import os
+ from pathlib import Path
+
+ import matplotlib as mpl
+ import matplotlib.pyplot as plt
+ import numpy as np
+
+ mpl.rcParams.update(
+ {
+ "mathtext.fontset": "stix",
+ "font.family": "serif",
+ "font.serif": "TeX Gyre Termes",
+ }
+ )
+
+ EPSILON = 1e-12
+ xlims = np.array([-3, 3])
+
+ root = str(Path(__file__).parent)
+ img_path = f"{root}/../images"
+ os.makedirs(img_path, exist_ok=True)
+
+ t = np.logspace(*xlims, 1001)[:, None]
+
+ z = np.array([-4.5, -2, -1, -0.5, 0.0, 0.5, 1, 2, 4.5])
+ r = t ** z
+
+ fig, ax = plt.subplots(num=1, clear=True, constrained_layout=True, figsize=(4, 2.4))
+ ax.semilogx(t, r)
+ ax.set_xlim(*(10.0 ** xlims))
+ ax.set_ylim(1e-3, 40)
+ ax.set_xlabel(r"$x$")
+ # ax.set_ylabel(r"$x^z$")
+ ax.grid(1, "both")
+ labels = [f"$z={zi: 3.1f}$" for zi in np.squeeze(z)]
+ ax.legend(labels, ncol=2, loc="upper left", fontsize="small")
+ fig.savefig(f"{img_path}/integrand.pdf")
diff --git a/buch/papers/laguerre/scripts/integrand_exp.py b/buch/papers/laguerre/scripts/integrand_exp.py
new file mode 100644
index 0000000..e649b26
--- /dev/null
+++ b/buch/papers/laguerre/scripts/integrand_exp.py
@@ -0,0 +1,46 @@
+#!/usr/bin/env python3
+# -*- coding:utf-8 -*-
+"""Plot for integrand of gamma function with shifting terms."""
+
+if __name__ == "__main__":
+ import os
+ from pathlib import Path
+
+ import matplotlib as mpl
+ import matplotlib.pyplot as plt
+ import numpy as np
+
+ mpl.rcParams.update(
+ {
+ "mathtext.fontset": "stix",
+ "font.family": "serif",
+ "font.serif": "TeX Gyre Termes",
+ }
+ )
+
+ EPSILON = 1e-12
+ xlims = np.array([-3, 3])
+
+ root = str(Path(__file__).parent)
+ img_path = f"{root}/../images"
+ os.makedirs(img_path, exist_ok=True)
+
+ t = np.logspace(*xlims, 1001)[:, None]
+
+ z = np.array([-1, -0.5, 0.0, 0.5, 1, 2, 3, 4, 4.5])
+ e = np.exp(-t)
+ r = t ** z * e
+
+ fig, ax = plt.subplots(num=2, clear=True, constrained_layout=True, figsize=(4, 2.4))
+ ax.semilogx(t, r)
+ # ax.plot(t,np.exp(-t))
+ ax.set_xlim(10 ** (-2), 20)
+ ax.set_ylim(1e-3, 10)
+ ax.set_xlabel(r"$x$")
+ # ax.set_ylabel(r"$x^z e^{-x}$")
+ ax.grid(1, "both")
+ labels = [f"$z={zi: 3.1f}$" for zi in np.squeeze(z)]
+ ax.legend(labels, ncol=2, loc="upper left", fontsize="small")
+ # fig.savefig(f"{img_path}/integrand_exp.pgf")
+ fig.savefig(f"{img_path}/integrand_exp.pdf")
+ # plt.show()
diff --git a/buch/papers/laguerre/scripts/laguerre_plot.py b/buch/papers/laguerre/scripts/laguerre_plot.py
deleted file mode 100644
index b9088d0..0000000
--- a/buch/papers/laguerre/scripts/laguerre_plot.py
+++ /dev/null
@@ -1,100 +0,0 @@
-#!/usr/bin/env python3
-# -*- coding:utf-8 -*-
-"""Some plots for Laguerre Polynomials."""
-
-import os
-from pathlib import Path
-
-import matplotlib.pyplot as plt
-import numpy as np
-import scipy.special as ss
-
-
-def get_ticks(start, end, step=1):
- ticks = np.arange(start, end, step)
- return ticks[ticks != 0]
-
-
-N = 1000
-step = 5
-t = np.linspace(-1.05, 10.5, N)[:, None]
-root = str(Path(__file__).parent)
-img_path = f"{root}/../images"
-os.makedirs(img_path, exist_ok=True)
-
-
-# fig = plt.figure(num=1, clear=True, tight_layout=True, figsize=(5.5, 3.7))
-# ax = fig.add_subplot(axes_class=AxesZero)
-fig, ax = plt.subplots(num=1, clear=True, constrained_layout=True, figsize=(6, 4))
-for n in np.arange(0, 8):
- k = np.arange(0, n + 1)[None]
- L = np.sum((-1) ** k * ss.binom(n, k) / ss.factorial(k) * t ** k, -1)
- ax.plot(t, L, label=f"n={n}")
-
-ax.set_xticks(get_ticks(int(t[0]), t[-1]), minor=True)
-ax.set_xticks(get_ticks(0, t[-1], step))
-ax.set_xlim(t[0], t[-1] + 0.1 * (t[1] - t[0]))
-ax.set_xlabel(r"$x$", x=1.0, labelpad=-10, rotation=0, fontsize="large")
-
-ylim = 13
-ax.set_yticks(np.arange(-ylim, ylim), minor=True)
-ax.set_yticks(np.arange(-step * (ylim // step), ylim, step))
-ax.set_ylim(-ylim, ylim)
-ax.set_ylabel(r"$y$", y=0.95, labelpad=-18, rotation=0, fontsize="large")
-
-ax.legend(ncol=2, loc=(0.125, 0.01), fontsize="large")
-
-# set the x-spine
-ax.spines[["left", "bottom"]].set_position("zero")
-ax.spines[["right", "top"]].set_visible(False)
-ax.xaxis.set_ticks_position("bottom")
-hlx = 0.4
-dx = t[-1, 0] - t[0, 0]
-dy = 2 * ylim
-hly = dy / dx * hlx
-dps = fig.dpi_scale_trans.inverted()
-bbox = ax.get_window_extent().transformed(dps)
-width, height = bbox.width, bbox.height
-
-# manual arrowhead width and length
-hw = 1.0 / 60.0 * dy
-hl = 1.0 / 30.0 * dx
-lw = 0.5 # axis line width
-ohg = 0.0 # arrow overhang
-
-# compute matching arrowhead length and width
-yhw = hw / dy * dx * height / width
-yhl = hl / dx * dy * width / height
-
-# draw x and y axis
-ax.arrow(
- t[-1, 0] - hl,
- 0,
- hl,
- 0.0,
- fc="k",
- ec="k",
- lw=lw,
- head_width=hw,
- head_length=hl,
- overhang=ohg,
- length_includes_head=True,
- clip_on=False,
-)
-
-ax.arrow(
- 0,
- ylim - yhl,
- 0.0,
- yhl,
- fc="k",
- ec="k",
- lw=lw,
- head_width=yhw,
- head_length=yhl,
- overhang=ohg,
- length_includes_head=True,
- clip_on=False,
-)
-
-fig.savefig(f"{img_path}/laguerre_polynomes.pdf")
diff --git a/buch/papers/laguerre/scripts/laguerre_poly.py b/buch/papers/laguerre/scripts/laguerre_poly.py
new file mode 100644
index 0000000..05db5d3
--- /dev/null
+++ b/buch/papers/laguerre/scripts/laguerre_poly.py
@@ -0,0 +1,108 @@
+import numpy as np
+
+
+def get_ticks(start, end, step=1):
+ ticks = np.arange(start, end, step)
+ return ticks[ticks != 0]
+
+
+if __name__ == "__main__":
+ import os
+ from pathlib import Path
+
+ import matplotlib as mpl
+ import matplotlib.pyplot as plt
+ import scipy.special as ss
+
+ mpl.rcParams.update(
+ {
+ "mathtext.fontset": "stix",
+ "font.family": "serif",
+ "font.serif": "TeX Gyre Termes",
+ }
+ )
+
+ N = 1000
+ step = 5
+ t = np.linspace(-1.05, 10.5, N)[:, None]
+ root = str(Path(__file__).parent)
+ img_path = f"{root}/../images"
+ os.makedirs(img_path, exist_ok=True)
+
+ # fig = plt.figure(num=1, clear=True, tight_layout=True, figsize=(5.5, 3.7))
+ # ax = fig.add_subplot(axes_class=AxesZero)
+ fig, ax = plt.subplots(num=1, clear=True, constrained_layout=True, figsize=(6, 4))
+ for n in np.arange(0, 8):
+ k = np.arange(0, n + 1)[None]
+ L = np.sum((-1) ** k * ss.binom(n, k) / ss.factorial(k) * t ** k, -1)
+ ax.plot(t, L, label=f"$n={n}$")
+
+ ax.set_xticks(get_ticks(int(t[0]), t[-1]), minor=True)
+ ax.set_xticks(get_ticks(0, t[-1], step))
+ ax.set_xlim(t[0], t[-1] + 0.1 * (t[1] - t[0]))
+ ax.set_xlabel(r"$x$", x=1.0, labelpad=-10, rotation=0, fontsize="large")
+
+ ylim = 13
+ ax.set_yticks(get_ticks(-ylim, ylim), minor=True)
+ ax.set_yticks(get_ticks(-step * (ylim // step), ylim, step))
+ ax.set_ylim(-ylim, ylim)
+ ax.set_ylabel(r"$y$", y=0.95, labelpad=-14, rotation=0, fontsize="large")
+
+ ax.legend(ncol=2, loc=(0.125, 0.01), fontsize="large")
+
+ # set the x-spine
+ ax.spines[["left", "bottom"]].set_position("zero")
+ ax.spines[["right", "top"]].set_visible(False)
+ ax.xaxis.set_ticks_position("bottom")
+ hlx = 0.4
+ dx = t[-1, 0] - t[0, 0]
+ dy = 2 * ylim
+ hly = dy / dx * hlx
+ dps = fig.dpi_scale_trans.inverted()
+ bbox = ax.get_window_extent().transformed(dps)
+ width, height = bbox.width, bbox.height
+
+ # manual arrowhead width and length
+ hw = 1.0 / 60.0 * dy
+ hl = 1.0 / 30.0 * dx
+ lw = 0.5 # axis line width
+ ohg = 0.0 # arrow overhang
+
+ # compute matching arrowhead length and width
+ yhw = hw / dy * dx * height / width
+ yhl = hl / dx * dy * width / height
+
+ # draw x and y axis
+ ax.arrow(
+ t[-1, 0] - hl,
+ 0,
+ hl,
+ 0.0,
+ fc="k",
+ ec="k",
+ lw=lw,
+ head_width=hw,
+ head_length=hl,
+ overhang=ohg,
+ length_includes_head=True,
+ clip_on=False,
+ )
+
+ ax.arrow(
+ 0,
+ ylim - yhl,
+ 0.0,
+ yhl,
+ fc="k",
+ ec="k",
+ lw=lw,
+ head_width=yhw,
+ head_length=yhl,
+ overhang=ohg,
+ length_includes_head=True,
+ clip_on=False,
+ )
+
+ # fig.savefig(f"{img_path}/laguerre_poly.pgf")
+ fig.savefig(f"{img_path}/laguerre_poly.pdf")
+ # plt.show()
diff --git a/buch/papers/laguerre/scripts/rel_error_complex.py b/buch/papers/laguerre/scripts/rel_error_complex.py
new file mode 100644
index 0000000..4a714fa
--- /dev/null
+++ b/buch/papers/laguerre/scripts/rel_error_complex.py
@@ -0,0 +1,43 @@
+if __name__ == "__main__":
+ import matplotlib as mpl
+ import matplotlib.pyplot as plt
+ import numpy as np
+ import scipy.special
+
+ import gamma_approx as ga
+
+ mpl.rcParams.update(
+ {
+ "mathtext.fontset": "stix",
+ "font.family": "serif",
+ "font.serif": "TeX Gyre Termes",
+ }
+ )
+
+ xmax = 4
+ vals = np.linspace(-xmax + ga.EPSILON, xmax, 100)
+ x, y = np.meshgrid(vals, vals)
+ mesh = x + 1j * y
+ input = mesh.flatten()
+
+ lanczos = scipy.special.gamma(mesh)
+ lag = ga.eval_laguerre_gamma(input, n=8, func="optimal_shifted").reshape(mesh.shape)
+ rel_error = np.abs(ga.calc_rel_error(lanczos, lag))
+
+ fig, ax = plt.subplots(clear=True, constrained_layout=True, figsize=(3.5, 2.1))
+ _c = ax.pcolormesh(
+ x, y, rel_error, shading="gouraud", cmap=ga.cmap, norm=mpl.colors.LogNorm()
+ )
+ cbr = fig.colorbar(_c, ax=ax)
+ cbr.minorticks_off()
+ # ax.set_title("Relative Error")
+ ax.set_xlabel("Re($z$)")
+ ax.set_ylabel("Im($z$)")
+ minor_ticks = np.arange(-xmax, xmax + ga.EPSILON)
+ ticks = np.arange(-xmax, xmax + ga.EPSILON, 2)
+ ax.set_xticks(ticks)
+ ax.set_xticks(minor_ticks, minor=True)
+ ax.set_yticks(ticks)
+ ax.set_yticks(minor_ticks, minor=True)
+ fig.savefig(f"{ga.img_path}/rel_error_complex.pdf")
+ # plt.show()
diff --git a/buch/papers/laguerre/scripts/rel_error_mirror.py b/buch/papers/laguerre/scripts/rel_error_mirror.py
new file mode 100644
index 0000000..7348d5e
--- /dev/null
+++ b/buch/papers/laguerre/scripts/rel_error_mirror.py
@@ -0,0 +1,38 @@
+if __name__ == "__main__":
+ import matplotlib as mpl
+ import matplotlib.pyplot as plt
+ import numpy as np
+ import scipy.special
+
+ import gamma_approx as ga
+
+ mpl.rcParams.update(
+ {
+ "mathtext.fontset": "stix",
+ "font.family": "serif",
+ "font.serif": "TeX Gyre Termes",
+ }
+ )
+
+ xmin = -15
+ xmax = 15
+ ns = np.arange(2, 12, 2)
+ ylim = np.array([-11, 1])
+ x = np.linspace(xmin + ga.EPSILON, xmax - ga.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 = ga.eval_laguerre_gamma(x, n=n, func="mirror")
+ rel_err = ga.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 + ga.EPSILON, 5))
+ ax.set_xticks(np.arange(xmin, xmax), minor=True)
+ ax.set_yticks(10.0 ** np.arange(*ylim, 2))
+ ax.set_xlabel(r"$z$")
+ # ax.set_ylabel("Relativer Fehler")
+ ax.legend(ncol=1, loc="upper left", fontsize=ga.fontsize)
+ ax.grid(1, "both")
+ # fig.savefig(f"{ga.img_path}/rel_error_mirror.pgf")
+ fig.savefig(f"{ga.img_path}/rel_error_mirror.pdf")
diff --git a/buch/papers/laguerre/scripts/rel_error_range.py b/buch/papers/laguerre/scripts/rel_error_range.py
new file mode 100644
index 0000000..ece3b6d
--- /dev/null
+++ b/buch/papers/laguerre/scripts/rel_error_range.py
@@ -0,0 +1,41 @@
+if __name__ == "__main__":
+ import matplotlib as mpl
+ import matplotlib.pyplot as plt
+ import numpy as np
+ import scipy.special
+
+ import gamma_approx as ga
+
+ mpl.rcParams.update(
+ {
+ "mathtext.fontset": "stix",
+ "font.family": "serif",
+ "font.serif": "TeX Gyre Termes",
+ }
+ )
+ N = 1201
+ xmax = 6
+ xmin = -xmax
+ ns = np.arange(2, 12, 2)
+ ylim = np.array([-11, -1.2])
+
+ x = np.linspace(xmin + ga.EPSILON, xmax - ga.EPSILON, N)
+ gamma = scipy.special.gamma(x)
+ fig, ax = plt.subplots(num=1, clear=True, constrained_layout=True, figsize=(5, 2))
+ for n in ns:
+ gamma_lag = ga.eval_laguerre_gamma(x, n=n, func="optimal_shifted")
+ rel_err = ga.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 + ga.EPSILON, 2))
+ ax.set_xticks(np.arange(xmin, xmax + ga.EPSILON), minor=True)
+ ax.set_yticks(10.0 ** np.arange(*ylim, 2))
+ ax.set_yticks(10.0 ** np.arange(*ylim, 1), "", minor=True)
+ ax.set_xlabel(r"$z$")
+ # ax.set_ylabel("Relativer Fehler")
+ ax.legend(ncol=1, loc="upper left", fontsize=ga.fontsize)
+ ax.grid(1, "both")
+ # fig.savefig(f"{ga.img_path}/rel_error_range.pgf")
+ fig.savefig(f"{ga.img_path}/rel_error_range.pdf")
+ # plt.show()
diff --git a/buch/papers/laguerre/scripts/rel_error_shifted.py b/buch/papers/laguerre/scripts/rel_error_shifted.py
new file mode 100644
index 0000000..f53c89b
--- /dev/null
+++ b/buch/papers/laguerre/scripts/rel_error_shifted.py
@@ -0,0 +1,40 @@
+if __name__ == "__main__":
+ import matplotlib as mpl
+ import matplotlib.pyplot as plt
+ import numpy as np
+ import scipy.special
+
+ import gamma_approx as ga
+
+ mpl.rcParams.update(
+ {
+ "mathtext.fontset": "stix",
+ "font.family": "serif",
+ "font.serif": "TeX Gyre Termes",
+ }
+ )
+ n = 8 # order of Laguerre polynomial
+ N = 200 # number of points in interval
+
+ step = 1 / (N - 1)
+ x = np.linspace(step, 1 - step, N + 1)
+ targets = np.arange(10, 14)
+ gamma = scipy.special.gamma(x)
+ fig, ax = plt.subplots(num=1, clear=True, constrained_layout=True, figsize=(5, 2.1))
+ for target in targets:
+ gamma_lag = ga.eval_laguerre_gamma(x, target=target, n=n, func="shifted")
+ rel_error = np.abs(ga.calc_rel_error(gamma, gamma_lag))
+ ax.semilogy(x, rel_error, label=f"$m={target}$", linewidth=3)
+ gamma_lgo = ga.eval_laguerre_gamma(x, n=n, func="optimal_shifted")
+ rel_error = np.abs(ga.calc_rel_error(gamma, gamma_lgo))
+ ax.semilogy(x, rel_error, "m", linestyle=":", label="$m^*$", linewidth=3)
+ ax.set_xlim(x[0], x[-1])
+ ax.set_ylim(5e-9, 5e-8)
+ ax.set_xlabel(r"$z$")
+ ax.set_xticks(np.linspace(0, 1, 6))
+ ax.set_xticks(np.linspace(0, 1, 11), minor=True)
+ ax.grid(1, "both")
+ ax.legend(ncol=1, fontsize=ga.fontsize)
+ # fig.savefig(f"{ga.img_path}/rel_error_shifted.pgf")
+ fig.savefig(f"{ga.img_path}/rel_error_shifted.pdf")
+ # plt.show()
diff --git a/buch/papers/laguerre/scripts/rel_error_simple.py b/buch/papers/laguerre/scripts/rel_error_simple.py
new file mode 100644
index 0000000..e1ea36a
--- /dev/null
+++ b/buch/papers/laguerre/scripts/rel_error_simple.py
@@ -0,0 +1,41 @@
+if __name__ == "__main__":
+ import matplotlib as mpl
+ import matplotlib.pyplot as plt
+ import numpy as np
+ import scipy.special
+
+ import gamma_approx as ga
+
+ # mpl.rc("text", usetex=True)
+ mpl.rcParams.update(
+ {
+ "mathtext.fontset": "stix",
+ "font.family": "serif",
+ "font.serif": "TeX Gyre Termes",
+ }
+ )
+ # mpl.rcParams.update({"font.family": "serif", "font.serif": "TeX Gyre Termes"})
+
+ # Simple / naive
+ xmin = -5
+ xmax = 25
+ ns = np.arange(2, 12, 2)
+ ylim = np.array([-11, 6])
+ x = np.linspace(xmin + ga.EPSILON, xmax - ga.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 = ga.eval_laguerre_gamma(x, n=n)
+ rel_err = ga.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 + ga.EPSILON, 5))
+ ax.set_xticks(np.arange(xmin, xmax), minor=True)
+ ax.set_yticks(10.0 ** np.arange(*ylim, 2))
+ ax.set_xlabel(r"$z$")
+ # ax.set_ylabel("Relativer Fehler")
+ ax.legend(ncol=3, fontsize=ga.fontsize)
+ ax.grid(1, "both")
+ # fig.savefig(f"{ga.img_path}/rel_error_simple.pgf")
+ fig.savefig(f"{ga.img_path}/rel_error_simple.pdf")
diff --git a/buch/papers/laguerre/scripts/targets.py b/buch/papers/laguerre/scripts/targets.py
new file mode 100644
index 0000000..69f94ba
--- /dev/null
+++ b/buch/papers/laguerre/scripts/targets.py
@@ -0,0 +1,58 @@
+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)
+ 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)
+ rel_error = []
+ for target in targets:
+ gamma_lag = ga.eval_laguerre_gamma(x, target=target, x=zeros, w=weights, func="shifted")
+ rel_error.append(np.abs(ga.calc_rel_error(gamma, gamma_lag)))
+ rel_error = np.stack(rel_error, -1)
+ best = np.argmin(rel_error, -1) + targets[0]
+ bests.append(best)
+ return np.stack(bests, 0)
+
+
+if __name__ == "__main__":
+ import matplotlib as mpl
+ import matplotlib.pyplot as plt
+
+ mpl.rcParams.update(
+ {
+ "mathtext.fontset": "stix",
+ "font.family": "serif",
+ "font.serif": "TeX Gyre Termes",
+ }
+ )
+
+ N = 200
+ ns = np.arange(1, 13)
+
+ bests = find_best_loc(N, ns=ns)
+
+ fig, ax = plt.subplots(num=1, clear=True, constrained_layout=True, figsize=(3.5, 2.1))
+ v = ax.imshow(bests, cmap=ga.cmap, 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")
+ fig.savefig(f"{ga.img_path}/targets.pdf")
generated by cgit v1.2.1 (git 2.18.0) at 2024-07-06 11:03:27 +0200