{
 "cells": [
  {
   "attachments": {},
   "cell_type": "markdown",
   "id": "ed1c26f4-6b1f-4333-8a47-e8964fb9b279",
   "metadata": {},
   "source": [
    "# Example-07: Sextupole (element)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "8a8f1fd1-7695-4af2-a56a-db6a1bcb121d",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Comparison of sextupole element with MADX-PTC and other features"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "be7f42de-9948-4efc-af35-701b14fda472",
   "metadata": {},
   "outputs": [],
   "source": [
    "from pathlib import Path\n",
    "from os import system\n",
    "\n",
    "import torch\n",
    "from model.library.drift import Drift\n",
    "from model.library.quadrupole import Quadrupole\n",
    "from model.library.sextupole import Sextupole"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "0f63654e-e265-4e86-86bd-3b342df72459",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.008745689261382875, -0.005080234821325765, -0.00476590910682766, 0.0008855562050471031]\n",
      "[0.008745689261382871, -0.00508023482132578, -0.00476590910682768, 0.0008855562050471008]\n",
      "[3.469446951953614e-18, 1.5612511283791264e-17, 1.9949319973733282e-17, 2.2768245622195593e-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",
    "ms = 10.0\n",
    "dp = 0.005\n",
    "length = 0.25\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: sextupole, l={length},k2={ms} ;\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",
    "S = Sextupole('S', length=length, ms=ms, dp=dp, exact=exact, order=5, ns=10)\n",
    "res = S(state, alignment=align, data={**S.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": "1770d4dc-1d8a-4ab9-ab99-fd95bd2ed371",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.008745672936611987, -0.005080234654232574, -0.004765906047078759, 0.000885556338827788]\n",
      "[0.00874567293661202, -0.0050802346542325495, -0.0047659060470788064, 0.0008855563388277869]\n",
      "[-3.2959746043559335e-17, -2.42861286636753e-17, 4.7704895589362195e-17, 1.0842021724855044e-18]\n"
     ]
    }
   ],
   "source": [
    "# Tracking (exact)\n",
    "\n",
    "ptc = Path('ptc')\n",
    "obs = Path('track.obs0001.p0001')\n",
    "\n",
    "exact = True\n",
    "align = False\n",
    "\n",
    "ms = 10.0\n",
    "dp = 0.005\n",
    "length = 0.25\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: sextupole, l={length},k2={ms} ;\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",
    "S = Sextupole('S', length=length, ms=ms, dp=dp, exact=exact, order=5, ns=10)\n",
    "res = S(state, alignment=align, data={**S.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": "2e0156cd-d879-4760-8e36-f9e7e89151b9",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.008663885569968804, -0.006258052120536049, -0.004896687680053297, -0.000915022709372755]\n",
      "[0.008663885569968922, -0.006258052120536033, -0.004896687680053253, -0.000915022709372733]\n",
      "[-1.1796119636642288e-16, -1.6479873021779667e-17, -4.423544863740858e-17, -2.200930410145574e-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",
    "ms = 10.0\n",
    "dp = 0.005\n",
    "length = 0.25\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: sextupole, l={length},k2={ms} ;\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",
    "S = Sextupole('S', length=length, ms=ms, dp=dp, exact=exact, order=5, ns=10)\n",
    "res = S(state, alignment=align, data={**S.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": "c797301f-2f9d-4424-a535-3c3ebddfa453",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tensor([ 0.0087, -0.0051, -0.0048,  0.0009], dtype=torch.float64)\n",
      "\n",
      "tensor([ 0.0100, -0.0050, -0.0050,  0.0010], dtype=torch.float64)\n",
      "\n",
      "tensor([ 0.0087, -0.0062, -0.0049, -0.0009], dtype=torch.float64)\n",
      "\n",
      "tensor([0., 0., 0., 0.], dtype=torch.float64)\n"
     ]
    }
   ],
   "source": [
    "# Deviation/error variables\n",
    "\n",
    "ms = 10.0\n",
    "dp = 0.005\n",
    "length = 0.25\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",
    "S = Sextupole('S', length, ms, 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(S(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(S(state, data={**S.data(), **{'dl': -S.length}}))\n",
    "print()\n",
    "\n",
    "# In the above S.data() creates default deviation dictionary (with zero values for each deviaton)\n",
    "# {**S.data(), **{'dl': -S.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(S(state, data={**S.data(), **error}, alignment=True))\n",
    "print()\n",
    "\n",
    "# The following elements can be made equivalent using deviation variables\n",
    "\n",
    "SA = Sextupole('SA', length, ms, dp)\n",
    "SB = Sextupole('SB', length - 0.1, ms, dp)\n",
    "\n",
    "print(SA(state) - SB(state, data={**SB.data(), **{'dl': torch.tensor(+0.1, dtype=SB.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": "5f4340f0-1082-4f86-a22c-e022c0575934",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tensor([ 0.0000, -0.0006,  0.0000, -0.0008], dtype=torch.float64)\n",
      "tensor([0., 0., 0., 0.], dtype=torch.float64)\n",
      "tensor([-5.2560e-04, -5.6465e-04,  6.1078e-05, -7.7234e-04],\n",
      "       dtype=torch.float64)\n",
      "tensor([-1.3875e-04, -5.5306e-04, -9.3764e-05, -7.7778e-04],\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",
    "ms = 10.0\n",
    "dp = 0.005\n",
    "length = 1.5\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",
    "S = Sextupole('S', length, ms, dp, exact=False, insertion=True)\n",
    "\n",
    "# Since sextupole is a nonlinear element, insertion is an identity transformation only for zero strenght\n",
    "\n",
    "print(S(state) - state)\n",
    "print(S(state, data={**S.data(), **{'ms': -ms}}) - state)\n",
    "\n",
    "# Represents effect of an error (any nonzero value of strengh or a change in other parameter)\n",
    "\n",
    "print(S(state, data={**S.data(), **{'dl': 0.1}}) - state)\n",
    "\n",
    "# Exact tracking corresponds to inclusion of kinematic term as errors\n",
    "\n",
    "S = Sextupole('S', length, ms, dp, exact=True, insertion=True, ns=100, order=1)\n",
    "\n",
    "print(S(state) - state)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "ab74f465-c887-40c6-bb5c-ddfd553d73d9",
   "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",
    "ms = 10.0\n",
    "dp = 0.0\n",
    "length = 1.5\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",
    "S = Sextupole('S', length, ms, dp, exact=True)\n",
    "\n",
    "state = 1.0E-3*torch.randn((512, 4), dtype=S.dtype, device=S.device)\n",
    "\n",
    "print(torch.vmap(S)(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 S(state, data={**S.data(), **{'dp': dp}})\n",
    "\n",
    "dp = 1.0E-3*torch.randn(512, dtype=S.dtype, device=S.device)\n",
    "\n",
    "print(torch.vmap(wrapper)(state, dp).shape)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "997dae49-c873-4e95-9b42-f0160eb9bee5",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tensor([[ 0.9297,  1.4473, -0.0478, -0.0359],\n",
      "        [-0.0938,  0.9297, -0.0638, -0.0478],\n",
      "        [-0.0478, -0.0359,  1.0703,  1.5527],\n",
      "        [-0.0638, -0.0478,  0.0938,  1.0703]], dtype=torch.float64)\n",
      "\n",
      "tensor([-1.1813e-05, -1.5750e-05, -2.9883e-05, -3.9844e-05],\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",
    "ms = 10.0\n",
    "dp = 0.0\n",
    "length = 1.5\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",
    "S = Sextupole('S', length, ms, dp, exact=False)\n",
    "\n",
    "# Compute derivative with respect to state\n",
    "\n",
    "print(torch.func.jacrev(S)(state))\n",
    "print()\n",
    "\n",
    "# Compute derivative with respect to a deviation variable\n",
    "\n",
    "ms = torch.tensor(0.0, dtype=torch.float64)\n",
    "\n",
    "def wrapper(state, ms):\n",
    "    return S(state, data={**S.data(), **{'ms': ms}})\n",
    "\n",
    "print(torch.func.jacrev(wrapper, 1)(state, ms))\n",
    "print()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "6c3a722b-5737-4aa3-8cfc-e1106eeb49e9",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tensor([ 0.0023, -0.0052, -0.0039,  0.0006], 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",
    "ms = 10.0\n",
    "dp = 0.0\n",
    "length = 1.5\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",
    "S = Sextupole('S', length, ms, dp, exact=False, ns=10, output=True, matrix=True)\n",
    "\n",
    "# Final state is still returned\n",
    "\n",
    "print(S(state))\n",
    "\n",
    "# Data is added to special attributes (state and tangent matrix)\n",
    "\n",
    "print(S.container_output.shape)\n",
    "print(S.container_matrix.shape)\n",
    "\n",
    "# Number of integration steps can be changed\n",
    "\n",
    "S.ns = 100\n",
    "\n",
    "S(state)\n",
    "print(S.container_output.shape)\n",
    "print(S.container_matrix.shape)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "58abd0bc-7a66-494a-b8a8-ee1d690a4f29",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.0022880403040043407, -0.005176627884407675, -0.0038882248469424585, 0.0005834231060909752]\n",
      "[0.0022871610802911667, -0.005176794799148338, -0.0038891562083014203, 0.0005832353574511327]\n",
      "[8.792237131739246e-07, 1.6691474066278522e-07, 9.313613589618727e-07, 1.877486398425181e-07]\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",
    "ms = 10.0\n",
    "dp = 0.0\n",
    "length = 1.5\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",
    "S = Sextupole('S', length, ms, dp, order=0, exact=True)\n",
    "\n",
    "# For sextupole integration is always performed\n",
    "# In exact case, kinematic term error is added\n",
    "\n",
    "S.ns = 10\n",
    "ref = S(state)\n",
    "\n",
    "S.ns = 100\n",
    "res = S(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 = S._data\n",
    "print(maps)\n",
    "print(weights)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "9c53baf6-e0ad-478d-a252-232a20b11b9a",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tensor([0.2107, 0.1703], dtype=torch.float64)\n",
      "tensor([-0.2279, -0.2107], dtype=torch.float64)\n",
      "tensor([[0., 0.],\n",
      "        [0., 0.]], dtype=torch.float64)\n"
     ]
    }
   ],
   "source": [
    "# Derivatives of twiss parameters (chromaticity)\n",
    "\n",
    "# pip install git+https://github.com/i-a-morozov/twiss.git@main\n",
    "# pip install git+https://github.com/i-a-morozov/ndmap.git@main\n",
    "\n",
    "from twiss import twiss\n",
    "\n",
    "from ndmap.pfp import parametric_fixed_point\n",
    "from ndmap.evaluate import evaluate\n",
    "\n",
    "# Define elements\n",
    "\n",
    "QF = Quadrupole('QF', 0.5, +0.21)\n",
    "QD = Quadrupole('QD', 0.5, -0.19)\n",
    "SF = Sextupole('SF', 0.25)\n",
    "SD = Sextupole('SD', 0.25)\n",
    "DA = Drift('DR', 0.25)\n",
    "DB = Drift('DR', 4.00)\n",
    "\n",
    "# Define one-turn transformation\n",
    "\n",
    "def fodo(state, dp, ms):\n",
    "    dp, *_ = dp\n",
    "    msf, msd, *_ = ms\n",
    "    state = QF(state, data={**QF.data(), **{'dp': dp}})\n",
    "    state = DA(state, data={**DA.data(), **{'dp': dp}})\n",
    "    state = SF(state, data={**SF.data(), **{'dp': dp, 'ms': msf}})\n",
    "    state = DB(state, data={**DB.data(), **{'dp': dp}})\n",
    "    state = SD(state, data={**SD.data(), **{'dp': dp, 'ms': msd}})\n",
    "    state = DA(state, data={**DA.data(), **{'dp': dp}})\n",
    "    state = QD(state, data={**QD.data(), **{'dp': dp}})\n",
    "    state = QD(state, data={**QD.data(), **{'dp': dp}})\n",
    "    state = DA(state, data={**DA.data(), **{'dp': dp}})\n",
    "    state = SD(state, data={**SD.data(), **{'dp': dp, 'ms': msd}})\n",
    "    state = DB(state, data={**DB.data(), **{'dp': dp}})\n",
    "    state = SF(state, data={**SF.data(), **{'dp': dp, 'ms': msf}})\n",
    "    state = DA(state, data={**DA.data(), **{'dp': dp}})\n",
    "    state = QF(state, data={**QF.data(), **{'dp': dp}})\n",
    "    return state\n",
    "\n",
    "# Set deviation parameters\n",
    "\n",
    "msf = torch.tensor(0.0, dtype=torch.float64)\n",
    "msd = torch.tensor(0.0, dtype=torch.float64)\n",
    "ms = torch.stack([msf, msd])\n",
    "dp = torch.tensor([0.0], dtype=torch.float64)\n",
    "\n",
    "# Set fixed point\n",
    "\n",
    "fp = torch.tensor([0.0, 0.0, 0.0, 0.0], dtype=torch.float64)\n",
    "\n",
    "\n",
    "# Compute parametrix fixed point (first order in momentum deviation)\n",
    "# Note, all parameters must be vectors\n",
    "\n",
    "pfp, *_ = parametric_fixed_point((1, ), fp, [dp], fodo, ms)\n",
    "\n",
    "# Define transformation around fixed point\n",
    "\n",
    "def pfp_fodo(state, dp, ms):\n",
    "    return fodo(state + evaluate(pfp, [dp]), dp, ms) - evaluate(pfp, [dp])\n",
    "\n",
    "# Tune\n",
    "\n",
    "def tune(dp, ms):\n",
    "    matrix = torch.func.jacrev(pfp_fodo)(fp, dp, ms)\n",
    "    tune, *_ = twiss(matrix)\n",
    "    return tune\n",
    "\n",
    "# Chromaticity\n",
    "\n",
    "def chromaticity(ms):\n",
    "    return torch.func.jacrev(tune)(dp, ms)\n",
    "\n",
    "# Compute tunes\n",
    "\n",
    "tunes = tune(dp, ms)\n",
    "print(tunes)\n",
    "\n",
    "# Compute chromaticity\n",
    "\n",
    "chromaticities = chromaticity(ms)\n",
    "print(chromaticities.squeeze())\n",
    "\n",
    "# Compute derivative of chromaticities \n",
    "# The result is zero, since there is no dispersion to feed sextupoles down\n",
    "\n",
    "print(torch.func.jacrev(chromaticity)(ms).squeeze())"
   ]
  }
 ],
 "metadata": {
  "colab": {
   "collapsed_sections": [
    "myt0_gMIOq7b",
    "5d97819c"
   ],
   "name": "03_frequency.ipynb",
   "provenance": []
  },
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "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": "3.12.1"
  },
  "latex_envs": {
   "LaTeX_envs_menu_present": true,
   "autoclose": false,
   "autocomplete": true,
   "bibliofile": "biblio.bib",
   "cite_by": "apalike",
   "current_citInitial": 1,
   "eqLabelWithNumbers": true,
   "eqNumInitial": 1,
   "hotkeys": {
    "equation": "Ctrl-E",
    "itemize": "Ctrl-I"
   },
   "labels_anchors": false,
   "latex_user_defs": false,
   "report_style_numbering": false,
   "user_envs_cfg": false
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}