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   "source": [
    "# Example-10: Dipole element factory (cylindrical multipole)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f744ba48-1780-4a06-a57b-ab549fbbf6c3",
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   "source": [
    "In this example dipole factory is illustrated for dipole with multipole cylindrical multipoles. \n",
    "\n",
    "The dipole hamiltonian is:\n",
    "\n",
    "$\n",
    "\\begin{align}\n",
    "& H(q_x, q_y, q_s, p_x, p_y, p_s; s) = \\frac{p_s}{\\beta} - t(s)(q_x p_y - q_y p_x) - (1 + h(s) q_x) \\left(\\sqrt{P_s^2 - P_x^2 - P_y^2 - \\frac{1}{\\beta^2 \\gamma^2}} + a_s(q_x, q_y, q_s; s)\\right)  \\\\\n",
    "& \\\\\n",
    "& P_s = p_s + 1/\\beta - \\varphi(q_x, q_y, q_s; s)  \\\\\n",
    "& P_x = p_x - a_x(q_x, q_y, q_s; s)  \\\\\n",
    "& P_y = p_y - a_y(q_x, q_y, q_s; s) \\\\\n",
    "\\\\\n",
    "& (a_x, a_y, a_s) = (0, 0, -\\frac{1}{1 + q_x/\\rho}\\left(\\frac{q_x}{ \\rho} + \\frac{q_x^2}{2 \\rho^2}   \\right) + U(q_x, q_y; \\rho) )\\\\\n",
    "& \\varphi = 0 \\\\\n",
    "& t = 0 \\\\\n",
    "& h = \\frac{1}{\\rho} = \\frac{\\alpha}{l}\n",
    "\\end{align}\n",
    "$\n",
    "\n",
    "The constructed element signature is:\n",
    "\n",
    "```python\n",
    "def dipole(qsps:Array, length:Array, angle:Array, kq_n:Array, kq_s:Array, ks_n:Array, ks_s:Array, ko_n:Array, ko_s:Array) -> Array:\n",
    "    ...\n",
    "```\n",
    "\n",
    "Note, no fringe effects are icluded.\n",
    "\n",
    "Cylindrical potential is precomputed for quadrupole, sextuipole and octupole components up to degree 10 in transverse coordinates."
   ]
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   "cell_type": "code",
   "execution_count": 1,
   "id": "206cff65-dbbe-4592-8927-fae0518a964a",
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   "outputs": [],
   "source": [
    "import jax\n",
    "from jax import jit\n",
    "from jax import jacrev\n",
    "\n",
    "from elementary.util import ptc\n",
    "from elementary.util import beta\n",
    "from elementary.dipole import dipole_factory\n",
    "\n",
    "jax.numpy.set_printoptions(linewidth=256, precision=12)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "a699f41b-1475-437e-93f1-b6e07400957d",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Set data type\n",
    "\n",
    "jax.config.update(\"jax_enable_x64\", True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "0ab982ad-6729-4515-8daa-07e15d6aaff3",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Set device\n",
    "\n",
    "device, *_ = jax.devices('cpu')\n",
    "jax.config.update('jax_default_device', device)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "7a28ae6d-52c1-4a0d-aeb4-ee10bba7cd13",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Set initial condition\n",
    "\n",
    "(q_x, q_y, q_s) = qs = jax.numpy.array([-0.01, 0.005, 0.001])\n",
    "(p_x, p_y, p_s) = ps = jax.numpy.array([0.001, 0.001, -0.0001])\n",
    "qsps = jax.numpy.hstack([qs, ps])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "06b4f75a-f769-4d5b-a5c6-897f1063abde",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Define generic dipole element\n",
    "# Note, exact should be set to false\n",
    "\n",
    "gamma = 10**3\n",
    "element = jit(dipole_factory(exact=False, multipole=True, beta=beta(gamma), gamma=gamma, order=2**1, iterations=1E3))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "86425b83-95ed-4e40-a227-e79d2fbfd9e2",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-0.017200420757 -0.003042779236  0.001519392779 -0.017822768071 -0.015620313086 -0.0001        ]\n",
      "[-0.017200420757 -0.003042779236  0.00151939277  -0.017822768072 -0.015620313086 -0.0001        ]\n",
      "True\n"
     ]
    }
   ],
   "source": [
    "# Compare with PTC\n",
    "\n",
    "length = jax.numpy.float64(1.0)\n",
    "angle = jax.numpy.float64(0.05)\n",
    "\n",
    "kq_n = jax.numpy.float64(-2.0)\n",
    "kq_s = jax.numpy.float64(+1.5)\n",
    "ks_n = jax.numpy.float64(-50.0)\n",
    "ks_s = jax.numpy.float64(+75.0)\n",
    "ko_n = jax.numpy.float64(-100.0)\n",
    "ko_s = jax.numpy.float64(+500.0)\n",
    "\n",
    "print(res := element(qsps, length, angle, kq_n, kq_s, ks_n, ks_s, ko_n, ko_s))\n",
    "print(ref := ptc(qsps, 'sbend', {'l': float(length), 'angle': float(angle), 'knl': f'{{0.0,{float(kq_n*length)}, {float(ks_n*length)}, {float(ko_n*length)}}}', 'ksl': f'{{0.0,{float(kq_s*length)}, {float(ks_s*length)}, {float(ko_s*length)}}}', 'kill_ent_fringe': 'true', 'kill_exi_fringe': 'true'}, gamma=gamma))\n",
    "print(jax.numpy.allclose(res, ref))"
   ]
  }
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