hankel.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "%reload_ext autoreload\n",
    "%autoreload 2\n",
    "\n",
    "import sympy as sp\n",
    "sp.init_printing(use_unicode=False, wrap_line=False)\n",
    "\n",
    "from hankel import *"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "matrix = sp.IndexedBase('h')\n",
    "i, j, k, l = sp.symbols('i j k l', cls=sp.Idx)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAskAAAAVBAMAAABGeG6jAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMrvvmVREiWZ2Iqtdv6urAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAFjklEQVRYCbVYPYhcVRQ+s5v9ycTdHUVBkGQWLFKIGNg0sZBptLLYZtIkguBPIRarIohYLLE04CpaRXDUTrZYrQUHRIKgsrY2prELuisLahHwnnPuPd957533dnbBW8z97nfOd75zz0zeS0KUV+/qpQIfeqGgsrtgofzuwqZ1nE9tYJdnWktyQeMIaVE44qAFivIiztlBDGTNQAuENENzWwYvGirABQvldxc2reN8agO7PNNakgsaR0iLwhEHLVCUF3HODmIgawZaIKQVtLZdEP1jqAAXLJTst/TkwqZ1XEVihw6t5YRFzIKicMRZPQZNXwuHWthZngMWhRbI5WX46aBwZw8Lsh1BowTc1CPC0IKrSuzUobWcqAgsKApHnNVj0PBFNNI6OyQaQhRaIEsz8IWhvj2hjULQKAG5Y4ShBVeV2KlDazlREVhQFI44q8eg4YtopHV2SDSEKLRAlmbg4nvf7+th7q1vnjNaAYLVQO4YYWjBVSV26tBaTlQEFhSFI87qMWj4IhppnR0SDSEKLZClFdD7a3/1jh6Gj9MfhdbdBasB7diFTeu4qsRO7VpLCYuYBUXhiLN6Ahq+Fg61sLM8BywKLZDLy3D5iFY3Fb8/ot+JevmXzVwJjjQBn9pxCSfetIVrSEzcrrWUsIhacE4On7d8cDTyZAVXfI/Xwq5SJR+C+5amffoDG7wu09I6LU6pvzshep3oVaL+lHRdOMhBosuZokfWE3pjY+PljY2nKNRqPUh6f3Zpe1+naKdv6bOkcQfZY9dXPpkvdWutK3QvnfYelamNSjNpVvBVlKaWl8wq4/kJ/TZKph8TPUkrR5ndSvvyOmlQuaW0nfnxkh5uyhZp6xK6kVJbtc9+l6LdvmpY0qQD75ECDY7tun05QVZDyzd3XXH3ZJxI5MN6Ri+K0tRq9+X84Q6/sV+ktf3ev2Rvzk1OneQgpxEt8MdyZcqRVjnOzZK3C4i09G56RHX7SqX0kdO4ivfgcJ2TVrt9S9WGlm/uuuIqVDjGutAzelGUplabFSu+JXrl3Pl7tLh97oAWJ1+d/ZlZrju/o8FB7/Z2OgZTjrTCecmVDu2A73OMb5LL0jSZinp8cidHHOda7fRt15aJZjuuEkwZPeO+itLUarPiCul9d9Q/c0jze6ubtDaa9IeDXHeYg7QkfwqCKUda4Zykd1B31aeN5vGUj/HlJnlpmkxUtZ/ZP6L4t3wy33btJntZV9J9MGX0DF9FaWrBlK8RfTlNU16cLuyllyD9khz020uPJAnS/f2tulInFWnrkrZvSPP4Psf4cju8NE2mLNqVh9NsdfGUT+TboS1TVjvpPpgyeoavInmQL3Bf3FVlrdzDfw1cp6Xx+KXxNZJHkqTN7dGb46vj8Z4pdcoSrGpLWZOk51S7Vn41WRL7lnq2o/eFKT09TmvfukpJM/q2aPPN+bvXlbpvcCWW9uuGk6+uNLXafS1H3n566h2OEuBv9MpAGaK1/YQq38+tEko7vzn1qFrBJpn/IJ1btbhPq6+Wtk9MefhgIcGh1W7fdi3fHF1JleC3rM71+zIrU6vcV3P5c5c+zIfl9UlCyat3MM0UPcOgRdnUisokw/u6tLhPq6+UwwcmeqNfWHBotdu3XVudslRpnbL2LF3IfRPSqbXMqr/7Wm65d5d/mOx1dydTJH/tYOXK539PC2l7XSsBk6z+kM6t2l+feCfXafU1HwGugws/5ZDj0Gq3b7uWb46upIpMAxwa0p7lLPdlJFNrvS+0jNgL6zGG8v2AOwZVJTNrA99jjGrhU/pKFdFWO9DqERf5GjfjfbdMkF4oHz3PJ/kHlaO7YF0yszbw7fKpx07tmwplre+glI+4EpM9a42b+b6moKXbAxxmQ6eQNAufosgpJOb7f2j/AxylUkhja9NFAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$- {h}_{0} {h}_{i + l} {h}_{j + 1} + {h}_{0} {h}_{i} {h}_{j + l + 1} + {h}_{1} {h}_{i + l} {h}_{j} - {h}_{1} {h}_{i - 1} {h}_{j + l + 1} + {h}_{i - 1} {h}_{j + 1} {h}_{l + 1} - {h}_{i} {h}_{j} {h}_{l + 1}$$"
      ],
      "text/plain": [
       "-h[0]*h[i + l]*h[j + 1] + h[0]*h[i]*h[j + l + 1] + h[1]*h[i + l]*h[j] - h[1]*h[i - 1]*h[j + l + 1] + h[i - 1]*h[j + 1]*h[l + 1] - h[i]*h[j]*h[l + 1]"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "hankel_submatrix(matrix, [0, 1, i], [0, l, j]).det() - hankel_submatrix(matrix, [0, l, i - 1], [0, 1, j + 1]).det()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- {h}_{0} {h}_{i + l} {h}_{j + 1} + {h}_{0} {h}_{i} {h}_{j + l + 1} + {h}_{1} {h}_{i + l} {h}_{j} - {h}_{1} {h}_{i - 1} {h}_{j + l + 1} + {h}_{i - 1} {h}_{j + 1} {h}_{l + 1} - {h}_{i} {h}_{j} {h}_{l + 1}$$"
      ],
      "text/plain": [
       "-h[0]*h[i + l]*h[j + 1] + h[0]*h[i]*h[j + l + 1] + h[1]*h[i + l]*h[j] - h[1]*h[i - 1]*h[j + l + 1] + h[i - 1]*h[j + 1]*h[l + 1] - h[i]*h[j]*h[l + 1]"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "hankel_submatrix(matrix, [0, l + 1, 1], [0, j, i - 1]).det()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "def doit(a, b, c):\n",
    "    expr = hankel_submatrix(matrix, *a).det() + hankel_submatrix(matrix, *b).det() - hankel_submatrix(matrix, *c).det()\n",
    "    return expr.equals(0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "doit(([0, 1, 2, 3], [0, 1, j, i - 1]), ([0, 1, 2, i], [0, 1, 2, j]), ([0, 1, 2, i - 1], [0, 1, 2, j + 1]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
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   "nbconvert_exporter": "python",
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 },
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