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-{
- "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
-}
generated by cgit v1.2.1 (git 2.18.0) at 2024-07-03 21:33:46 +0200