Merge remote-tracking branch 'origin/master'

2 files changed, 495 insertions, 0 deletions

diff --git a/buch/papers/laguerre/scripts/gamma_approx.ipynb b/buch/papers/laguerre/scripts/gamma_approx.ipynb
new file mode 100644
index 0000000..44f3abd
--- /dev/null
+++ b/buch/papers/laguerre/scripts/gamma_approx.ipynb
@@ -0,0 +1,395 @@
+{
+ "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/laguerre_plot.py b/buch/papers/laguerre/scripts/laguerre_plot.py
new file mode 100644
index 0000000..b9088d0
--- /dev/null
+++ b/buch/papers/laguerre/scripts/laguerre_plot.py
@@ -0,0 +1,100 @@
+#!/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")