{
 "cells": [
  {
   "attachments": {},
   "cell_type": "markdown",
   "id": "9a5af188-3962-4346-b9e7-69c0fd7913ab",
   "metadata": {},
   "source": [
    "# Example-08: Octupole (element)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "7e7c4591-356c-498f-a12d-b11234e962ad",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Comparison of octupole element with MADX-PTC and other features"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "6366eebc-548d-4631-8a28-a9e55b3baebb",
   "metadata": {},
   "outputs": [],
   "source": [
    "from pathlib import Path\n",
    "from os import system\n",
    "\n",
    "import torch\n",
    "from model.library.octupole import Octupole"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "d1e16585-2e20-4c2c-a98d-3115473dbeae",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.009004942132987458, -0.005000291994759593, -0.004801204788639501, 0.0009979801079392878]\n",
      "[0.009004942132987486, -0.005000291994759635, -0.004801204788639504, 0.0009979801079392843]\n",
      "[-2.7755575615628914e-17, 4.2500725161431774e-17, 3.469446951953614e-18, 3.469446951953614e-18]\n"
     ]
    }
   ],
   "source": [
    "# Tracking (paraxial)\n",
    "\n",
    "ptc = Path('ptc')\n",
    "obs = Path('track.obs0001.p0001')\n",
    "\n",
    "exact = False\n",
    "align = False\n",
    "\n",
    "mo = + 50.0\n",
    "dp = 0.005\n",
    "length = 0.2\n",
    "state = torch.tensor([0.01, -0.005, -0.005, 0.001], dtype=torch.float64)\n",
    "qx, px, qy, py = state.tolist()\n",
    "\n",
    "dx = align*torch.tensor(0.05, dtype=torch.float64)\n",
    "dy = align*torch.tensor(-0.02, dtype=torch.float64)\n",
    "dz = align*torch.tensor(0.05, dtype=torch.float64)\n",
    "\n",
    "wx = align*torch.tensor(0.005, dtype=torch.float64)\n",
    "wy = align*torch.tensor(-0.005, dtype=torch.float64)\n",
    "wz = align*torch.tensor(0.1, dtype=torch.float64)\n",
    "\n",
    "error = {'dx': dx, 'dy': dy, 'dz': dz, 'wx': wx, 'wy': wy, 'wz': wz}\n",
    "\n",
    "code = f\"\"\"\n",
    "mag: octupole, l={length},k3={mo};\n",
    "map:line=(mag) ;\n",
    "beam,energy=1.0E+6,particle=electron ;\n",
    "set,format=\"20.20f\",\"-20s\" ;\n",
    "use,period=map ;\n",
    "select,flag=error,pattern=\"mag\" ;\n",
    "ealign,dx={dx.item()},dy={dy.item()},ds={dz.item()},dphi={wx.item()},dtheta={wy.item()},dpsi={wz.item()} ;\n",
    "ptc_create_universe,sector_nmul_max=10,sector_nmul=10 ;\n",
    "ptc_create_layout,model=1,method=6,nst=1000,exact={str(exact).lower()} ;\n",
    "ptc_setswitch,fringe=false,time=true,totalpath=true,exact_mis=true ;\n",
    "ptc_align ;\n",
    "ptc_start,x={qx},px={px},y={qy},py={py},pt={dp},t=0.0 ;\n",
    "ptc_track,icase=5,deltap=0.,turns=1,file=track,maxaper={{1.,1.,1.,1.,1.,1.}} ;\n",
    "ptc_track_end ;\n",
    "ptc_end ;\n",
    "\"\"\" \n",
    "\n",
    "with ptc.open('w') as stream:\n",
    "    stream.write(code)\n",
    "    \n",
    "system(f'madx < {str(ptc)} > /dev/null')\n",
    "\n",
    "with obs.open('r') as stream:\n",
    "    for line in stream:\n",
    "        continue\n",
    "    _, _, qx, px, qy, py, *_ = line.split()\n",
    "    \n",
    "ref = torch.tensor([float(x) for x in (qx, px, qy, py)], dtype=torch.float64)\n",
    "\n",
    "O = Octupole('O', length=length, mo=mo, dp=dp, exact=exact, order=5, ns=10)\n",
    "res = O(state, alignment=align, data={**O.data(), **error})\n",
    "\n",
    "print(ref.tolist())\n",
    "print(res.tolist())\n",
    "print((ref - res).tolist())\n",
    "\n",
    "ptc.unlink()\n",
    "obs.unlink()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "7103c0ad-ebe8-4712-8124-9107d046db6c",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.00900492932570409, -0.005000291993307186, -0.004801202229721349, 0.0009979801112567327]\n",
      "[0.009004929325704207, -0.0050002919933071705, -0.004801202229721335, 0.00099798011125672]\n",
      "[-1.1622647289044608e-16, -1.5612511283791264e-17, -1.3877787807814457e-17, 1.2576745200831851e-17]\n"
     ]
    }
   ],
   "source": [
    "# Tracking (exact)\n",
    "\n",
    "ptc = Path('ptc')\n",
    "obs = Path('track.obs0001.p0001')\n",
    "\n",
    "exact = True\n",
    "align = False\n",
    "\n",
    "mo = + 50.0\n",
    "dp = 0.005\n",
    "length = 0.2\n",
    "state = torch.tensor([0.01, -0.005, -0.005, 0.001], dtype=torch.float64)\n",
    "qx, px, qy, py = state.tolist()\n",
    "\n",
    "dx = align*torch.tensor(0.05, dtype=torch.float64)\n",
    "dy = align*torch.tensor(-0.02, dtype=torch.float64)\n",
    "dz = align*torch.tensor(0.05, dtype=torch.float64)\n",
    "\n",
    "wx = align*torch.tensor(0.005, dtype=torch.float64)\n",
    "wy = align*torch.tensor(-0.005, dtype=torch.float64)\n",
    "wz = align*torch.tensor(0.1, dtype=torch.float64)\n",
    "\n",
    "error = {'dx': dx, 'dy': dy, 'dz': dz, 'wx': wx, 'wy': wy, 'wz': wz}\n",
    "\n",
    "code = f\"\"\"\n",
    "mag: octupole, l={length},k3={mo};\n",
    "map:line=(mag) ;\n",
    "beam,energy=1.0E+6,particle=electron ;\n",
    "set,format=\"20.20f\",\"-20s\" ;\n",
    "use,period=map ;\n",
    "select,flag=error,pattern=\"mag\" ;\n",
    "ealign,dx={dx.item()},dy={dy.item()},ds={dz.item()},dphi={wx.item()},dtheta={wy.item()},dpsi={wz.item()} ;\n",
    "ptc_create_universe,sector_nmul_max=10,sector_nmul=10 ;\n",
    "ptc_create_layout,model=1,method=6,nst=1000,exact={str(exact).lower()} ;\n",
    "ptc_setswitch,fringe=false,time=true,totalpath=true,exact_mis=true ;\n",
    "ptc_align ;\n",
    "ptc_start,x={qx},px={px},y={qy},py={py},pt={dp},t=0.0 ;\n",
    "ptc_track,icase=5,deltap=0.,turns=1,file=track,maxaper={{1.,1.,1.,1.,1.,1.}} ;\n",
    "ptc_track_end ;\n",
    "ptc_end ;\n",
    "\"\"\" \n",
    "\n",
    "with ptc.open('w') as stream:\n",
    "    stream.write(code)\n",
    "    \n",
    "system(f'madx < {str(ptc)} > /dev/null')\n",
    "\n",
    "with obs.open('r') as stream:\n",
    "    for line in stream:\n",
    "        continue\n",
    "    _, _, qx, px, qy, py, *_ = line.split()\n",
    "    \n",
    "ref = torch.tensor([float(x) for x in (qx, px, qy, py)], dtype=torch.float64)\n",
    "\n",
    "O = Octupole('O', length=length, mo=mo, dp=dp, exact=exact, order=5, ns=10)\n",
    "res = O(state, alignment=align, data={**O.data(), **error})\n",
    "\n",
    "print(ref.tolist())\n",
    "print(res.tolist())\n",
    "print((ref - res).tolist())\n",
    "\n",
    "ptc.unlink()\n",
    "obs.unlink()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "b1afc429-20bf-4780-9267-4e416b1c7a66",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.009005660276793346, -0.004983923494470567, -0.004794436076349048, 0.0011299481578555504]\n",
      "[0.009005660276792638, -0.004983923494470562, -0.004794436076348805, 0.0011299481578555233]\n",
      "[7.077671781985373e-16, -5.204170427930421e-18, -2.42861286636753e-16, 2.710505431213761e-17]\n"
     ]
    }
   ],
   "source": [
    "# Tracking (exact, alignment)\n",
    "\n",
    "ptc = Path('ptc')\n",
    "obs = Path('track.obs0001.p0001')\n",
    "\n",
    "exact = True\n",
    "align = True\n",
    "\n",
    "mo = + 50.0\n",
    "dp = 0.005\n",
    "length = 0.2\n",
    "state = torch.tensor([0.01, -0.005, -0.005, 0.001], dtype=torch.float64)\n",
    "qx, px, qy, py = state.tolist()\n",
    "\n",
    "dx = align*torch.tensor(0.05, dtype=torch.float64)\n",
    "dy = align*torch.tensor(-0.02, dtype=torch.float64)\n",
    "dz = align*torch.tensor(0.05, dtype=torch.float64)\n",
    "\n",
    "wx = align*torch.tensor(0.005, dtype=torch.float64)\n",
    "wy = align*torch.tensor(-0.005, dtype=torch.float64)\n",
    "wz = align*torch.tensor(0.1, dtype=torch.float64)\n",
    "\n",
    "error = {'dx': dx, 'dy': dy, 'dz': dz, 'wx': wx, 'wy': wy, 'wz': wz}\n",
    "\n",
    "code = f\"\"\"\n",
    "mag: octupole, l={length},k3={mo};\n",
    "map:line=(mag) ;\n",
    "beam,energy=1.0E+6,particle=electron ;\n",
    "set,format=\"20.20f\",\"-20s\" ;\n",
    "use,period=map ;\n",
    "select,flag=error,pattern=\"mag\" ;\n",
    "ealign,dx={dx.item()},dy={dy.item()},ds={dz.item()},dphi={wx.item()},dtheta={wy.item()},dpsi={wz.item()} ;\n",
    "ptc_create_universe,sector_nmul_max=10,sector_nmul=10 ;\n",
    "ptc_create_layout,model=1,method=6,nst=1000,exact={str(exact).lower()} ;\n",
    "ptc_setswitch,fringe=false,time=true,totalpath=true,exact_mis=true ;\n",
    "ptc_align ;\n",
    "ptc_start,x={qx},px={px},y={qy},py={py},pt={dp},t=0.0 ;\n",
    "ptc_track,icase=5,deltap=0.,turns=1,file=track,maxaper={{1.,1.,1.,1.,1.,1.}} ;\n",
    "ptc_track_end ;\n",
    "ptc_end ;\n",
    "\"\"\" \n",
    "\n",
    "with ptc.open('w') as stream:\n",
    "    stream.write(code)\n",
    "    \n",
    "system(f'madx < {str(ptc)} > /dev/null')\n",
    "\n",
    "with obs.open('r') as stream:\n",
    "    for line in stream:\n",
    "        continue\n",
    "    _, _, qx, px, qy, py, *_ = line.split()\n",
    "    \n",
    "ref = torch.tensor([float(x) for x in (qx, px, qy, py)], dtype=torch.float64)\n",
    "\n",
    "O = Octupole('O', length=length, mo=mo, dp=dp, exact=exact, order=5, ns=10)\n",
    "res = O(state, alignment=align, data={**O.data(), **error})\n",
    "\n",
    "print(ref.tolist())\n",
    "print(res.tolist())\n",
    "print((ref - res).tolist())\n",
    "\n",
    "ptc.unlink()\n",
    "obs.unlink()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "b5d27e5b-0f6c-4d62-91f4-afd2eabf0872",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tensor([ 0.0090, -0.0050, -0.0048,  0.0010], dtype=torch.float64)\n",
      "\n",
      "tensor([ 0.0100, -0.0050, -0.0050,  0.0010], dtype=torch.float64)\n",
      "\n",
      "tensor([ 0.0090, -0.0050, -0.0048,  0.0011], dtype=torch.float64)\n",
      "\n",
      "tensor([0., 0., 0., 0.], dtype=torch.float64)\n"
     ]
    }
   ],
   "source": [
    "# Deviation/error variables\n",
    "\n",
    "mo = 50.0\n",
    "dp = 0.005\n",
    "length = 0.2\n",
    "state = torch.tensor([0.01, -0.005, -0.005, 0.001], dtype=torch.float64)\n",
    "\n",
    "dx = align*torch.tensor(0.05, dtype=torch.float64)\n",
    "dy = align*torch.tensor(-0.02, dtype=torch.float64)\n",
    "dz = align*torch.tensor(0.05, dtype=torch.float64)\n",
    "\n",
    "wx = align*torch.tensor(0.005, dtype=torch.float64)\n",
    "wy = align*torch.tensor(-0.005, dtype=torch.float64)\n",
    "wz = align*torch.tensor(0.1, dtype=torch.float64)\n",
    "\n",
    "error = {'dx': dx, 'dy': dy, 'dz': dz, 'wx': wx, 'wy': wy, 'wz': wz}\n",
    "\n",
    "O = Octupole('O', length, mo, dp)\n",
    "\n",
    "# Each element has two variant of a call method\n",
    "# In the first case only state is passed, it is transformed using parameters specified on initializaton\n",
    "\n",
    "print(O(state))\n",
    "print()\n",
    "\n",
    "# Deviation errors can be also passed to call method\n",
    "# These variables are added to corresponding parameters specified on initializaton\n",
    "# For example, element lenght can changed\n",
    "\n",
    "print(O(state, data={**O.data(), **{'dl': -O.length}}))\n",
    "print()\n",
    "\n",
    "# In the above O.data() creates default deviation dictionary (with zero values for each deviaton)\n",
    "# {**O.data(), **{'dl': -O.length}} replaces the 'dl' key value \n",
    "\n",
    "# Additionaly, alignment errors are passed as deivation variables\n",
    "# They are used if alignment flag is raised\n",
    "\n",
    "print(O(state, data={**O.data(), **error}, alignment=True))\n",
    "print()\n",
    "\n",
    "# The following elements can be made equivalent using deviation variables\n",
    "\n",
    "OA = Octupole('OA', length, mo, dp)\n",
    "OB = Octupole('OB', length - 0.1, mo, dp)\n",
    "\n",
    "print(OA(state) - OB(state, data={**OB.data(), **{'dl': torch.tensor(+0.1, dtype=OB.dtype)}}))\n",
    "\n",
    "# Note, while in some cases float values can be passed as values to deviation variables\n",
    "# The correct behaviour in guaranteed only for tensors"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "ca7cf2e5-590b-41eb-b08c-61b20de24ad6",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tensor([ 0.0000e+00, -4.1667e-07,  0.0000e+00, -2.2917e-06],\n",
      "       dtype=torch.float64)\n",
      "tensor([0., 0., 0., 0.], dtype=torch.float64)\n",
      "tensor([-4.9754e-04, -5.2588e-07,  9.9342e-05, -3.2270e-06],\n",
      "       dtype=torch.float64)\n",
      "tensor([-1.7351e-08, -4.1976e-07, -6.9314e-09, -2.2953e-06],\n",
      "       dtype=torch.float64)\n"
     ]
    }
   ],
   "source": [
    "# Insertion element\n",
    "\n",
    "# In this mode elements are treated as thin insertions (at the center)\n",
    "# Using parameters specified on initialization, transport two matrices are computed\n",
    "# These matrices are used to insert the element\n",
    "# Input state is transformed from the element center to its entrance\n",
    "# Next, transformation from the entrance frame to the exit frame is performed\n",
    "# This transformation can contain errors\n",
    "# The final step is to transform state from the exit frame back to the element center\n",
    "# Without errors, this results in identity transformation for linear elements\n",
    "\n",
    "mo = 50.0\n",
    "dp = 0.005\n",
    "length = 0.2\n",
    "state = torch.tensor([0.01, -0.005, -0.005, 0.001], dtype=torch.float64)\n",
    "\n",
    "dx = torch.tensor(0.05, dtype=torch.float64)\n",
    "dy = torch.tensor(-0.02, dtype=torch.float64)\n",
    "dz = torch.tensor(0.05, dtype=torch.float64)\n",
    "\n",
    "wx = torch.tensor(0.005, dtype=torch.float64)\n",
    "wy = torch.tensor(-0.005, dtype=torch.float64)\n",
    "wz = torch.tensor(0.1, dtype=torch.float64)\n",
    "\n",
    "error = {'dx': dx, 'dy': dy, 'dz': dz, 'wx': wx, 'wy': wy, 'wz': wz}\n",
    "\n",
    "O = Octupole('O', length, mo, dp, exact=False, insertion=True)\n",
    "\n",
    "# Since octupole is a nonlinear element, insertion is an identity transformation only for zero strenght\n",
    "\n",
    "print(O(state) - state)\n",
    "print(O(state, data={**O.data(), **{'mo': -mo}}) - state)\n",
    "\n",
    "# Represents effect of an error (any nonzero value of strengh or a change in other parameter)\n",
    "\n",
    "print(O(state, data={**O.data(), **{'dl': 0.1}}) - state)\n",
    "\n",
    "# Exact tracking corresponds to inclusion of kinematic term as errors\n",
    "\n",
    "O = Octupole('O', length, mo, dp, exact=True, insertion=True, ns=100, order=1)\n",
    "\n",
    "print(O(state) - state)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "58995f82-17dc-400b-b7b2-0b14ac5e0296",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "torch.Size([512, 4])\n",
      "torch.Size([512, 4])\n"
     ]
    }
   ],
   "source": [
    "# Mapping over a set of initial conditions\n",
    "\n",
    "# Call method can be used to map over a set of initial conditions\n",
    "# Note, device can be set to cpu or gpu via base element classvariables\n",
    "\n",
    "mo = 50.0\n",
    "dp = 0.005\n",
    "length = 0.2\n",
    "\n",
    "dx = torch.tensor(0.05, dtype=torch.float64)\n",
    "dy = torch.tensor(-0.02, dtype=torch.float64)\n",
    "dz = torch.tensor(0.05, dtype=torch.float64)\n",
    "\n",
    "wx = torch.tensor(0.005, dtype=torch.float64)\n",
    "wy = torch.tensor(-0.005, dtype=torch.float64)\n",
    "wz = torch.tensor(0.1, dtype=torch.float64)\n",
    "\n",
    "error = {'dx': dx, 'dy': dy, 'dz': dz, 'wx': wx, 'wy': wy, 'wz': wz}\n",
    "\n",
    "O = Octupole('O', length, mo, dp, exact=True)\n",
    "\n",
    "state = 1.0E-3*torch.randn((512, 4), dtype=O.dtype, device=O.device)\n",
    "\n",
    "print(torch.vmap(O)(state).shape)\n",
    "\n",
    "# To map over deviations parameters a wrapper function (or a lambda expression) can be used\n",
    "\n",
    "def wrapper(state, dp):\n",
    "    return O(state, data={**O.data(), **{'dp': dp}})\n",
    "\n",
    "dp = 1.0E-3*torch.randn(512, dtype=O.dtype, device=O.device)\n",
    "\n",
    "print(torch.vmap(wrapper)(state, dp).shape)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "bbb81635-b354-4392-9735-a0a758ef89f7",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tensor([[ 9.9997e-01,  1.9900e-01, -4.6335e-05, -4.6105e-06],\n",
      "        [-3.3141e-04,  9.9997e-01, -4.6567e-04, -4.6335e-05],\n",
      "        [-4.6335e-05, -4.6105e-06,  1.0000e+00,  1.9901e-01],\n",
      "        [-4.6567e-04, -4.6335e-05,  3.3141e-04,  1.0000e+00]],\n",
      "       dtype=torch.float64)\n",
      "\n",
      "tensor([-5.7528e-10, -5.7815e-09, -4.0127e-09, -4.0327e-08],\n",
      "       dtype=torch.float64)\n",
      "\n"
     ]
    }
   ],
   "source": [
    "# Differentiability\n",
    "\n",
    "# Both call methods are differentiable\n",
    "# Derivative with respect to state can be computed directly\n",
    "# For deviation variables, wrapping is required\n",
    "\n",
    "mo = 50.0\n",
    "dp = 0.005\n",
    "length = 0.2\n",
    "state = torch.tensor([0.01, -0.005, -0.005, 0.001], dtype=torch.float64)\n",
    "\n",
    "dx = torch.tensor(0.05, dtype=torch.float64)\n",
    "dy = torch.tensor(-0.02, dtype=torch.float64)\n",
    "dz = torch.tensor(0.05, dtype=torch.float64)\n",
    "\n",
    "wx = torch.tensor(0.005, dtype=torch.float64)\n",
    "wy = torch.tensor(-0.005, dtype=torch.float64)\n",
    "wz = torch.tensor(0.1, dtype=torch.float64)\n",
    "\n",
    "error = {'dx': dx, 'dy': dy, 'dz': dz, 'wx': wx, 'wy': wy, 'wz': wz}\n",
    "\n",
    "O = Octupole('O', length, mo, dp, exact=False)\n",
    "\n",
    "# Compute derivative with respect to state\n",
    "\n",
    "print(torch.func.jacrev(O)(state))\n",
    "print()\n",
    "\n",
    "# Compute derivative with respect to a deviation variable\n",
    "\n",
    "mo = torch.tensor(0.0, dtype=torch.float64)\n",
    "\n",
    "def wrapper(state, mo):\n",
    "    return O(state, data={**O.data(), **{'mo': mo}})\n",
    "\n",
    "print(torch.func.jacrev(wrapper, 1)(state, mo))\n",
    "print()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "90577ad9-b390-4bc4-9c4b-21e7feb03a5e",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tensor([ 0.0090, -0.0050, -0.0048,  0.0010], dtype=torch.float64)\n",
      "torch.Size([10, 4])\n",
      "torch.Size([10, 4, 4])\n",
      "torch.Size([100, 4])\n",
      "torch.Size([100, 4, 4])\n"
     ]
    }
   ],
   "source": [
    "# Output at each step\n",
    "\n",
    "# It is possible to collect output of state or tangent matrix at each integration step\n",
    "# Number of integratin steps is controlled by ns parameter on initialization\n",
    "# Alternatively, desired integration step length can be passed\n",
    "# Number of integration steps is computed as ceil(length/ds)\n",
    "\n",
    "mo = 50.0\n",
    "dp = 0.005\n",
    "length = 0.2\n",
    "state = torch.tensor([0.01, -0.005, -0.005, 0.001], dtype=torch.float64)\n",
    "\n",
    "dx = torch.tensor(0.05, dtype=torch.float64)\n",
    "dy = torch.tensor(-0.02, dtype=torch.float64)\n",
    "dz = torch.tensor(0.05, dtype=torch.float64)\n",
    "\n",
    "wx = torch.tensor(0.005, dtype=torch.float64)\n",
    "wy = torch.tensor(-0.005, dtype=torch.float64)\n",
    "wz = torch.tensor(0.1, dtype=torch.float64)\n",
    "\n",
    "error = {'dx': dx, 'dy': dy, 'dz': dz, 'wx': wx, 'wy': wy, 'wz': wz}\n",
    "\n",
    "O = Octupole('O', length, mo, dp, exact=False, ns=10, output=True, matrix=True)\n",
    "\n",
    "# Final state is still returned\n",
    "\n",
    "print(O(state))\n",
    "\n",
    "# Data is added to special attributes (state and tangent matrix)\n",
    "\n",
    "print(O.container_output.shape)\n",
    "print(O.container_matrix.shape)\n",
    "\n",
    "# Number of integration steps can be changed\n",
    "\n",
    "O.ns = 100\n",
    "\n",
    "O(state)\n",
    "print(O.container_output.shape)\n",
    "print(O.container_matrix.shape)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "5591a125-3e10-489d-94ce-79c60e0a15e3",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.00900492936806216, -0.005000291964120008, -0.004801202138358545, 0.0009979801465156844]\n",
      "[0.00900492932612774, -0.005000291993015304, -0.004801202228807697, 0.0009979801116093128]\n",
      "[4.193442121325219e-11, 2.889529587823958e-11, 9.044915164069245e-11, 3.4906371629978006e-11]\n",
      "\n",
      "[0, 1, 2, 1, 0]\n",
      "[0.5, 0.5, 1.0, 0.5, 0.5]\n"
     ]
    }
   ],
   "source": [
    "# Integration order is set on initialization (default value is zero)\n",
    "# This order is related to difference order as 2n + 2\n",
    "# Thus, zero corresponds to second order difference method\n",
    "\n",
    "mo = 50.0\n",
    "dp = 0.005\n",
    "length = 0.2\n",
    "state = torch.tensor([0.01, -0.005, -0.005, 0.001], dtype=torch.float64)\n",
    "\n",
    "dx = torch.tensor(0.05, dtype=torch.float64)\n",
    "dy = torch.tensor(-0.02, dtype=torch.float64)\n",
    "dz = torch.tensor(0.05, dtype=torch.float64)\n",
    "\n",
    "wx = torch.tensor(0.005, dtype=torch.float64)\n",
    "wy = torch.tensor(-0.005, dtype=torch.float64)\n",
    "wz = torch.tensor(0.1, dtype=torch.float64)\n",
    "\n",
    "error = {'dx': dx, 'dy': dy, 'dz': dz, 'wx': wx, 'wy': wy, 'wz': wz}\n",
    "\n",
    "O = Octupole('O', length, mo, dp, order=0, exact=True)\n",
    "\n",
    "# For octupole integration is always performed\n",
    "# In exact case, kinematic term error is added\n",
    "\n",
    "O.ns = 10\n",
    "ref = O(state)\n",
    "\n",
    "O.ns = 100\n",
    "res = O(state)\n",
    "\n",
    "print(ref.tolist())\n",
    "print(res.tolist())\n",
    "print((ref - res).tolist())\n",
    "print()\n",
    "\n",
    "# Integrator parameters are stored in data attribute (if integration is actually performed)\n",
    "\n",
    "maps, weights = O._data\n",
    "print(maps)\n",
    "print(weights)"
   ]
  }
 ],
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