Example-08: Octupole (element)

# Comparison of octupole element with MADX-PTC and other features

from pathlib import Path
from os import system

import torch
from model.library.octupole import Octupole

# Tracking (paraxial)

ptc = Path('ptc')
obs = Path('track.obs0001.p0001')

exact = False
align = False

mo = + 50.0
dp = 0.005
length = 0.2
state = torch.tensor([0.01, -0.005, -0.005, 0.001], dtype=torch.float64)
qx, px, qy, py = state.tolist()

dx = align*torch.tensor(0.05, dtype=torch.float64)
dy = align*torch.tensor(-0.02, dtype=torch.float64)
dz = align*torch.tensor(0.05, dtype=torch.float64)

wx = align*torch.tensor(0.005, dtype=torch.float64)
wy = align*torch.tensor(-0.005, dtype=torch.float64)
wz = align*torch.tensor(0.1, dtype=torch.float64)

error = {'dx': dx, 'dy': dy, 'dz': dz, 'wx': wx, 'wy': wy, 'wz': wz}

code = f"""
mag: octupole, l={length},k3={mo};
map:line=(mag) ;
beam,energy=1.0E+6,particle=electron ;
set,format="20.20f","-20s" ;
use,period=map ;
select,flag=error,pattern="mag" ;
ealign,dx={dx.item()},dy={dy.item()},ds={dz.item()},dphi={wx.item()},dtheta={wy.item()},dpsi={wz.item()} ;
ptc_create_universe,sector_nmul_max=10,sector_nmul=10 ;
ptc_create_layout,model=1,method=6,nst=1000,exact={str(exact).lower()} ;
ptc_setswitch,fringe=false,time=true,totalpath=true,exact_mis=true ;
ptc_align ;
ptc_start,x={qx},px={px},y={qy},py={py},pt={dp},t=0.0 ;
ptc_track,icase=5,deltap=0.,turns=1,file=track,maxaper={{1.,1.,1.,1.,1.,1.}} ;
ptc_track_end ;
ptc_end ;
"""

with ptc.open('w') as stream:
    stream.write(code)

system(f'madx < {str(ptc)} > /dev/null')

with obs.open('r') as stream:
    for line in stream:
        continue
    _, _, qx, px, qy, py, *_ = line.split()

ref = torch.tensor([float(x) for x in (qx, px, qy, py)], dtype=torch.float64)

O = Octupole('O', length=length, mo=mo, dp=dp, exact=exact, order=5, ns=10)
res = O(state, alignment=align, data={**O.data(), **error})

print(ref.tolist())
print(res.tolist())
print((ref - res).tolist())

ptc.unlink()
obs.unlink()

[0.009004942132987458, -0.005000291994759593, -0.004801204788639501, 0.0009979801079392878]
[0.009004942132987486, -0.005000291994759635, -0.004801204788639504, 0.0009979801079392843]
[-2.7755575615628914e-17, 4.2500725161431774e-17, 3.469446951953614e-18, 3.469446951953614e-18]

# Tracking (exact)

ptc = Path('ptc')
obs = Path('track.obs0001.p0001')

exact = True
align = False

mo = + 50.0
dp = 0.005
length = 0.2
state = torch.tensor([0.01, -0.005, -0.005, 0.001], dtype=torch.float64)
qx, px, qy, py = state.tolist()

dx = align*torch.tensor(0.05, dtype=torch.float64)
dy = align*torch.tensor(-0.02, dtype=torch.float64)
dz = align*torch.tensor(0.05, dtype=torch.float64)

wx = align*torch.tensor(0.005, dtype=torch.float64)
wy = align*torch.tensor(-0.005, dtype=torch.float64)
wz = align*torch.tensor(0.1, dtype=torch.float64)

error = {'dx': dx, 'dy': dy, 'dz': dz, 'wx': wx, 'wy': wy, 'wz': wz}

code = f"""
mag: octupole, l={length},k3={mo};
map:line=(mag) ;
beam,energy=1.0E+6,particle=electron ;
set,format="20.20f","-20s" ;
use,period=map ;
select,flag=error,pattern="mag" ;
ealign,dx={dx.item()},dy={dy.item()},ds={dz.item()},dphi={wx.item()},dtheta={wy.item()},dpsi={wz.item()} ;
ptc_create_universe,sector_nmul_max=10,sector_nmul=10 ;
ptc_create_layout,model=1,method=6,nst=1000,exact={str(exact).lower()} ;
ptc_setswitch,fringe=false,time=true,totalpath=true,exact_mis=true ;
ptc_align ;
ptc_start,x={qx},px={px},y={qy},py={py},pt={dp},t=0.0 ;
ptc_track,icase=5,deltap=0.,turns=1,file=track,maxaper={{1.,1.,1.,1.,1.,1.}} ;
ptc_track_end ;
ptc_end ;
"""

with ptc.open('w') as stream:
    stream.write(code)

system(f'madx < {str(ptc)} > /dev/null')

with obs.open('r') as stream:
    for line in stream:
        continue
    _, _, qx, px, qy, py, *_ = line.split()

ref = torch.tensor([float(x) for x in (qx, px, qy, py)], dtype=torch.float64)

O = Octupole('O', length=length, mo=mo, dp=dp, exact=exact, order=5, ns=10)
res = O(state, alignment=align, data={**O.data(), **error})

print(ref.tolist())
print(res.tolist())
print((ref - res).tolist())

ptc.unlink()
obs.unlink()

[0.00900492932570409, -0.005000291993307186, -0.004801202229721349, 0.0009979801112567327]
[0.009004929325704207, -0.0050002919933071705, -0.004801202229721335, 0.00099798011125672]
[-1.1622647289044608e-16, -1.5612511283791264e-17, -1.3877787807814457e-17, 1.2576745200831851e-17]

# Tracking (exact, alignment)

ptc = Path('ptc')
obs = Path('track.obs0001.p0001')

exact = True
align = True

mo = + 50.0
dp = 0.005
length = 0.2
state = torch.tensor([0.01, -0.005, -0.005, 0.001], dtype=torch.float64)
qx, px, qy, py = state.tolist()

dx = align*torch.tensor(0.05, dtype=torch.float64)
dy = align*torch.tensor(-0.02, dtype=torch.float64)
dz = align*torch.tensor(0.05, dtype=torch.float64)

wx = align*torch.tensor(0.005, dtype=torch.float64)
wy = align*torch.tensor(-0.005, dtype=torch.float64)
wz = align*torch.tensor(0.1, dtype=torch.float64)

error = {'dx': dx, 'dy': dy, 'dz': dz, 'wx': wx, 'wy': wy, 'wz': wz}

code = f"""
mag: octupole, l={length},k3={mo};
map:line=(mag) ;
beam,energy=1.0E+6,particle=electron ;
set,format="20.20f","-20s" ;
use,period=map ;
select,flag=error,pattern="mag" ;
ealign,dx={dx.item()},dy={dy.item()},ds={dz.item()},dphi={wx.item()},dtheta={wy.item()},dpsi={wz.item()} ;
ptc_create_universe,sector_nmul_max=10,sector_nmul=10 ;
ptc_create_layout,model=1,method=6,nst=1000,exact={str(exact).lower()} ;
ptc_setswitch,fringe=false,time=true,totalpath=true,exact_mis=true ;
ptc_align ;
ptc_start,x={qx},px={px},y={qy},py={py},pt={dp},t=0.0 ;
ptc_track,icase=5,deltap=0.,turns=1,file=track,maxaper={{1.,1.,1.,1.,1.,1.}} ;
ptc_track_end ;
ptc_end ;
"""

with ptc.open('w') as stream:
    stream.write(code)

system(f'madx < {str(ptc)} > /dev/null')

with obs.open('r') as stream:
    for line in stream:
        continue
    _, _, qx, px, qy, py, *_ = line.split()

ref = torch.tensor([float(x) for x in (qx, px, qy, py)], dtype=torch.float64)

O = Octupole('O', length=length, mo=mo, dp=dp, exact=exact, order=5, ns=10)
res = O(state, alignment=align, data={**O.data(), **error})

print(ref.tolist())
print(res.tolist())
print((ref - res).tolist())

ptc.unlink()
obs.unlink()

[0.009005660276793346, -0.004983923494470567, -0.004794436076349048, 0.0011299481578555504]
[0.009005660276792638, -0.004983923494470562, -0.004794436076348805, 0.0011299481578555233]
[7.077671781985373e-16, -5.204170427930421e-18, -2.42861286636753e-16, 2.710505431213761e-17]

# Deviation/error variables

mo = 50.0
dp = 0.005
length = 0.2
state = torch.tensor([0.01, -0.005, -0.005, 0.001], dtype=torch.float64)

dx = align*torch.tensor(0.05, dtype=torch.float64)
dy = align*torch.tensor(-0.02, dtype=torch.float64)
dz = align*torch.tensor(0.05, dtype=torch.float64)

wx = align*torch.tensor(0.005, dtype=torch.float64)
wy = align*torch.tensor(-0.005, dtype=torch.float64)
wz = align*torch.tensor(0.1, dtype=torch.float64)

error = {'dx': dx, 'dy': dy, 'dz': dz, 'wx': wx, 'wy': wy, 'wz': wz}

O = Octupole('O', length, mo, dp)

# Each element has two variant of a call method
# In the first case only state is passed, it is transformed using parameters specified on initializaton

print(O(state))
print()

# Deviation errors can be also passed to call method
# These variables are added to corresponding parameters specified on initializaton
# For example, element lenght can changed

print(O(state, data={**O.data(), **{'dl': -O.length}}))
print()

# In the above O.data() creates default deviation dictionary (with zero values for each deviaton)
# {**O.data(), **{'dl': -O.length}} replaces the 'dl' key value

# Additionaly, alignment errors are passed as deivation variables
# They are used if alignment flag is raised

print(O(state, data={**O.data(), **error}, alignment=True))
print()

# The following elements can be made equivalent using deviation variables

OA = Octupole('OA', length, mo, dp)
OB = Octupole('OB', length - 0.1, mo, dp)

print(OA(state) - OB(state, data={**OB.data(), **{'dl': torch.tensor(+0.1, dtype=OB.dtype)}}))

# Note, while in some cases float values can be passed as values to deviation variables
# The correct behaviour in guaranteed only for tensors

tensor([ 0.0090, -0.0050, -0.0048,  0.0010], dtype=torch.float64)

tensor([ 0.0100, -0.0050, -0.0050,  0.0010], dtype=torch.float64)

tensor([ 0.0090, -0.0050, -0.0048,  0.0011], dtype=torch.float64)

tensor([0., 0., 0., 0.], dtype=torch.float64)

# Insertion element

# In this mode elements are treated as thin insertions (at the center)
# Using parameters specified on initialization, transport two matrices are computed
# These matrices are used to insert the element
# Input state is transformed from the element center to its entrance
# Next, transformation from the entrance frame to the exit frame is performed
# This transformation can contain errors
# The final step is to transform state from the exit frame back to the element center
# Without errors, this results in identity transformation for linear elements

mo = 50.0
dp = 0.005
length = 0.2
state = torch.tensor([0.01, -0.005, -0.005, 0.001], dtype=torch.float64)

dx = torch.tensor(0.05, dtype=torch.float64)
dy = torch.tensor(-0.02, dtype=torch.float64)
dz = torch.tensor(0.05, dtype=torch.float64)

wx = torch.tensor(0.005, dtype=torch.float64)
wy = torch.tensor(-0.005, dtype=torch.float64)
wz = torch.tensor(0.1, dtype=torch.float64)

error = {'dx': dx, 'dy': dy, 'dz': dz, 'wx': wx, 'wy': wy, 'wz': wz}

O = Octupole('O', length, mo, dp, exact=False, insertion=True)

# Since octupole is a nonlinear element, insertion is an identity transformation only for zero strenght

print(O(state) - state)
print(O(state, data={**O.data(), **{'mo': -mo}}) - state)

# Represents effect of an error (any nonzero value of strengh or a change in other parameter)

print(O(state, data={**O.data(), **{'dl': 0.1}}) - state)

# Exact tracking corresponds to inclusion of kinematic term as errors

O = Octupole('O', length, mo, dp, exact=True, insertion=True, ns=100, order=1)

print(O(state) - state)

tensor([ 0.0000e+00, -4.1667e-07,  0.0000e+00, -2.2917e-06],
       dtype=torch.float64)
tensor([0., 0., 0., 0.], dtype=torch.float64)
tensor([-4.9754e-04, -5.2588e-07,  9.9342e-05, -3.2270e-06],
       dtype=torch.float64)
tensor([-1.7351e-08, -4.1976e-07, -6.9314e-09, -2.2953e-06],
       dtype=torch.float64)

# Mapping over a set of initial conditions

# Call method can be used to map over a set of initial conditions
# Note, device can be set to cpu or gpu via base element classvariables

mo = 50.0
dp = 0.005
length = 0.2

dx = torch.tensor(0.05, dtype=torch.float64)
dy = torch.tensor(-0.02, dtype=torch.float64)
dz = torch.tensor(0.05, dtype=torch.float64)

wx = torch.tensor(0.005, dtype=torch.float64)
wy = torch.tensor(-0.005, dtype=torch.float64)
wz = torch.tensor(0.1, dtype=torch.float64)

error = {'dx': dx, 'dy': dy, 'dz': dz, 'wx': wx, 'wy': wy, 'wz': wz}

O = Octupole('O', length, mo, dp, exact=True)

state = 1.0E-3*torch.randn((512, 4), dtype=O.dtype, device=O.device)

print(torch.vmap(O)(state).shape)

# To map over deviations parameters a wrapper function (or a lambda expression) can be used

def wrapper(state, dp):
    return O(state, data={**O.data(), **{'dp': dp}})

dp = 1.0E-3*torch.randn(512, dtype=O.dtype, device=O.device)

print(torch.vmap(wrapper)(state, dp).shape)

torch.Size([512, 4])
torch.Size([512, 4])

# Differentiability

# Both call methods are differentiable
# Derivative with respect to state can be computed directly
# For deviation variables, wrapping is required

mo = 50.0
dp = 0.005
length = 0.2
state = torch.tensor([0.01, -0.005, -0.005, 0.001], dtype=torch.float64)

dx = torch.tensor(0.05, dtype=torch.float64)
dy = torch.tensor(-0.02, dtype=torch.float64)
dz = torch.tensor(0.05, dtype=torch.float64)

wx = torch.tensor(0.005, dtype=torch.float64)
wy = torch.tensor(-0.005, dtype=torch.float64)
wz = torch.tensor(0.1, dtype=torch.float64)

error = {'dx': dx, 'dy': dy, 'dz': dz, 'wx': wx, 'wy': wy, 'wz': wz}

O = Octupole('O', length, mo, dp, exact=False)

# Compute derivative with respect to state

print(torch.func.jacrev(O)(state))
print()

# Compute derivative with respect to a deviation variable

mo = torch.tensor(0.0, dtype=torch.float64)

def wrapper(state, mo):
    return O(state, data={**O.data(), **{'mo': mo}})

print(torch.func.jacrev(wrapper, 1)(state, mo))
print()

tensor([[ 9.9997e-01,  1.9900e-01, -4.6335e-05, -4.6105e-06],
        [-3.3141e-04,  9.9997e-01, -4.6567e-04, -4.6335e-05],
        [-4.6335e-05, -4.6105e-06,  1.0000e+00,  1.9901e-01],
        [-4.6567e-04, -4.6335e-05,  3.3141e-04,  1.0000e+00]],
       dtype=torch.float64)

tensor([-5.7528e-10, -5.7815e-09, -4.0127e-09, -4.0327e-08],
       dtype=torch.float64)

# Output at each step

# It is possible to collect output of state or tangent matrix at each integration step
# Number of integratin steps is controlled by ns parameter on initialization
# Alternatively, desired integration step length can be passed
# Number of integration steps is computed as ceil(length/ds)

mo = 50.0
dp = 0.005
length = 0.2
state = torch.tensor([0.01, -0.005, -0.005, 0.001], dtype=torch.float64)

dx = torch.tensor(0.05, dtype=torch.float64)
dy = torch.tensor(-0.02, dtype=torch.float64)
dz = torch.tensor(0.05, dtype=torch.float64)

wx = torch.tensor(0.005, dtype=torch.float64)
wy = torch.tensor(-0.005, dtype=torch.float64)
wz = torch.tensor(0.1, dtype=torch.float64)

error = {'dx': dx, 'dy': dy, 'dz': dz, 'wx': wx, 'wy': wy, 'wz': wz}

O = Octupole('O', length, mo, dp, exact=False, ns=10, output=True, matrix=True)

# Final state is still returned

print(O(state))

# Data is added to special attributes (state and tangent matrix)

print(O.container_output.shape)
print(O.container_matrix.shape)

# Number of integration steps can be changed

O.ns = 100

O(state)
print(O.container_output.shape)
print(O.container_matrix.shape)

tensor([ 0.0090, -0.0050, -0.0048,  0.0010], dtype=torch.float64)
torch.Size([10, 4])
torch.Size([10, 4, 4])
torch.Size([100, 4])
torch.Size([100, 4, 4])

# Integration order is set on initialization (default value is zero)
# This order is related to difference order as 2n + 2
# Thus, zero corresponds to second order difference method

mo = 50.0
dp = 0.005
length = 0.2
state = torch.tensor([0.01, -0.005, -0.005, 0.001], dtype=torch.float64)

dx = torch.tensor(0.05, dtype=torch.float64)
dy = torch.tensor(-0.02, dtype=torch.float64)
dz = torch.tensor(0.05, dtype=torch.float64)

wx = torch.tensor(0.005, dtype=torch.float64)
wy = torch.tensor(-0.005, dtype=torch.float64)
wz = torch.tensor(0.1, dtype=torch.float64)

error = {'dx': dx, 'dy': dy, 'dz': dz, 'wx': wx, 'wy': wy, 'wz': wz}

O = Octupole('O', length, mo, dp, order=0, exact=True)

# For octupole integration is always performed
# In exact case, kinematic term error is added

O.ns = 10
ref = O(state)

O.ns = 100
res = O(state)

print(ref.tolist())
print(res.tolist())
print((ref - res).tolist())
print()

# Integrator parameters are stored in data attribute (if integration is actually performed)

maps, weights = O._data
print(maps)
print(weights)

[0.00900492936806216, -0.005000291964120008, -0.004801202138358545, 0.0009979801465156844]
[0.00900492932612774, -0.005000291993015304, -0.004801202228807697, 0.0009979801116093128]
[4.193442121325219e-11, 2.889529587823958e-11, 9.044915164069245e-11, 3.4906371629978006e-11]

[0, 1, 2, 1, 0]
[0.5, 0.5, 1.0, 0.5, 0.5]