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@olivierverdier
Created April 25, 2024 15:02
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Polynomial roots exploration
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"source": [
"#from polynomial import *\n",
"%run polynomial.py\n",
"import numpy as np\n",
"%matplotlib inline\n",
"import matplotlib_inline\n",
"matplotlib_inline.backend_inline.set_matplotlib_formats('svg')\n",
"import matplotlib.pyplot as plt\n",
"import math"
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"metadata": {},
"source": [
"Using polynomials and shifts"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The following function defines the list\n",
"\\\\[f(n) = [∞, 1!, 2!,\\ldots, (n-1)!]\\\\]"
]
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{
"cell_type": "code",
"execution_count": 18,
"metadata": {},
"outputs": [],
"source": [
"def texp(k):\n",
" return np.hstack([np.inf, np.cumprod(np.arange(k-1)+1)])"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now define an approximation of \\\\(\\exp(x)-1 = \\sum_{k=1}^{∞}x^k/k!\\\\) by\n",
"\\\\[\n",
"p_n(x) := \\sum_{k=1}^{n-1} \\frac{x^k}{k!}\n",
"\\\\]"
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {},
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"source": [
"def poly_expm1(k):\n",
" return Polynomial(1./texp(k))"
]
},
{
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"metadata": {},
"source": [
"Compare the approximation \\\\(p_{20}\\\\) with the true function on the imaginary axis."
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"def plot_both_imaginary(k=20, x=.9*3*np.pi):\n",
" y = 1\n",
" #plt.figure(figsize=(zoom*2*x,zoom*y))\n",
" xs = np.linspace(-x,x,200)\n",
" zs = 1j*xs\n",
" expected = np.expm1(zs)\n",
" computed = poly_expm1(k)(zs)\n",
" plt.subplot(1,2,1)\n",
" for vals in [computed, expected]:\n",
" plt.plot(np.real(vals), np.imag(vals))\n",
" plt.axis('scaled')\n",
" plt.subplot(1,2,2)\n",
" plt.plot(xs, np.imag(expected))\n",
" plt.plot(xs, np.imag(computed))\n",
" plt.axis('scaled')\n",
" plt.grid(ls='-', alpha=.2)"
]
},
{
"cell_type": "code",
"execution_count": 21,
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"for s,k in enumerate(range(15,40,5)):\n",
" p = poly_expm1(k)\n",
" zeros = p.zeros()\n",
" plt.plot(np.real(zeros), np.imag(zeros), ls='', marker='o', label=\"{}\".format(k), markersize=s+3)\n",
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{
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"source": [
"def plot_abs(k):\n",
" p = poly_expm1(k)\n",
" zeros = p.zeros()\n",
" #def get_grid(x0,x1,y0,y1)\n",
" margin = 1\n",
" xmin = np.min(np.real(zeros)) - margin\n",
" xmax = np.max(np.real(zeros)) + margin\n",
" ymin = np.min(np.imag(zeros)) - margin\n",
" ymax = np.max(np.imag(zeros)) + margin\n",
" res = 100\n",
" xs,ys = np.meshgrid(np.linspace(xmin,xmax,res), np.linspace(ymin,ymax,res))\n",
" zs = 10*np.log10(np.abs(p(xs + 1j*ys))+np.finfo(float).eps)\n",
" \n",
" CS=plt.contour(xs,ys,zs)\n",
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{
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"Using a Taylor basis"
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"The basis is $ω^k X^k/k!$. A vector with coordinates $x_i$ represents the polynomial\n",
"\\\\[\n",
"x_0 + x_1 ωX + x_2 ω^2X^2/2! + \\cdots = ∑_i x_i \\frac{ω^i X^i}{i!}\n",
"\\\\]\n",
"The shift operation is thus\n",
"\\begin{align}\n",
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"\\end{align}\n",
"\n",
"We look for the roots of the polynomial of degree $n$\n",
"\\begin{align}\n",
"P = ∑_i^{n} p_i e_i\n",
"\\end{align}\n",
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"\\begin{align}\n",
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"\\begin{align}\n",
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"def get_matrix(coeffs):\n",
" ohm = 2*np.pi\n",
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" n = len(coeffs) + 1\n",
" shift = np.arange(n-2, dtype=complex)+1\n",
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"metadata": {},
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"text/plain": [
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},
"metadata": {},
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],
"source": [
"for size in range(50,350,50):\n",
" plot_eigvals(np.linalg.eigvals(get_expm1_matrix(size)))\n",
"plt.grid(ls='-', alpha=.2)"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
}
],
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