def test_renorm_left_energy(self):
mpiprint(
0, '\n' + '=' * 50 +
'\nTesting left renormalization effect on Energy\n' + '-' * 50)
from cyclomps.tools.mps_tools import create_mps_list, make_mps_list_left, load_mps_list, calc_mps_norm, move_gauge
from cyclomps.mpo.tasep import return_mpo
from cyclomps.algs.dmrg1 import renormalize_left
# Create an MPS
d = 2
N = 4
mbd = 10
nState = 4
mps = create_mps_list([d] * N, mbd, nState)
mps = make_mps_list_left(mps)
# Load the MPO
mpo = return_mpo(N, (0.5, 0.5, 1.))
# Create a function to compute energy
def calc_energy(mpsL, mpo):
mps = load_mps_list(mpsL)
E = zeros(nState, dtype=mps[0][0].dtype)
norm = zeros(nState, dtype=mps[0][0].dtype)
for state in range(nState):
psi = einsum('apb,bqc,crd,dse->pqrs', mps[state][0],
mps[state][1], mps[state][2], mps[state][3])
op = einsum('iPpj,jQqk,kRrl,lSsm->PpQqRrSs', mpo[0][0],
mpo[0][1], mpo[0][2], mpo[0][3])
psi_conj = einsum('apb,bqc,crd,dse->pqrs', conj(mps[state][0]),
conj(mps[state][1]), conj(mps[state][2]),
conj(mps[state][3]))
E[state] = einsum('pqrs,PpQqRrSs,PQRS', psi, op, psi_conj)
norm[state] = calc_mps_norm(mpsL, state=state)
return E, norm
# Calc initial Energy
E1, norm1 = calc_energy(mps, mpo)
# Move gauge using renormalization
site = N - 1
(mps0, ) = retrieve_tensors(site - 1, mpsList=mps)
(mps1, ) = retrieve_tensors(site, mpsList=mps)
mps0, mps1, EE, EEs, wgt = renormalize_left(mps0,
mps1,
state_avg=False)
save_tensors(site - 1, mpsList=mps, mps=mps0)
save_tensors(site, mpsList=mps, mps=mps1)
# Calc Energy Again
E2, norm2 = calc_energy(mps, mpo)
# Check Energies
self.assertTrue(summ(abss(E1[0] - E2[0])) < 1e-10)
self.assertTrue(summ(abss(E1[1] - E2[1])) < 1e-10)
self.assertTrue(summ(abss(E1[2] - E2[2])) < 1e-10)
self.assertTrue(summ(abss(E1[3] - E2[3])) < 1e-10)
# Check for correct canonical form
for state in range(nState):
eye_check = einsum('apb,cpb->ac', mps1[state], conj(mps1[state]))
eye_actual = eye(eye_check.shape[0])
self.assertTrue(summ(abss(eye_check - eye_actual)) < 1e-10)
mpiprint(0, 'Passed\n' + '=' * 50)
def test_tasep_mpo(self):
mpiprint(
0, '\n' + '=' * 50 +
'\nTesting Initial Environment Using TASEP MPO\n' + '-' * 50)
# Create an MPS
from cyclomps.tools.mps_tools import create_mps_list, make_mps_list_right, load_mps_list, calc_mps_norm
d = 2
N = 4
mbd = 10
nState = 4
mpsList = create_mps_list([d] * N, mbd, nState)
# Load the MPO
from cyclomps.mpo.tasep import return_mpo
mpo = return_mpo(N, (0.5, 0.5, 1.))
# Create a function to compute energy
def calc_energy(mps, mpo, load_mps=True):
if load_mps:
mps = load_mps_list(mps)
E = zeros(nState, dtype=mps[0][0].dtype)
for state in range(nState):
psi = einsum('apb,bqc,crd,dse->pqrs', mps[state][0],
mps[state][1], mps[state][2], mps[state][3])
op = einsum('iPpj,jQqk,kRrl,lSsm->PpQqRrSs', mpo[0][0],
mpo[0][1], mpo[0][2], mpo[0][3])
psi_conj = einsum('apb,bqc,crd,dse->pqrs', conj(mps[state][0]),
conj(mps[state][1]), conj(mps[state][2]),
conj(mps[state][3]))
E[state] = einsum('pqrs,PpQqRrSs,PQRS', psi, op, psi_conj)
norm = einsum('pqrs,pqrs->', psi, psi_conj)
E[state] /= norm
return E
# Compute energy
E1 = calc_energy(mpsList, mpo)
mpiprint(0, 'Initial Energy =\n{}'.format(E1))
# Right Canonicalize
mpsList = make_mps_list_right(mpsList)
# Compute energy again
E2 = calc_energy(mpsList, mpo)
mpiprint(0, 'Energy after canonicalization =\n{}'.format(E2))
# Calculate the environment
from cyclomps.tools.env_tools import calc_env, env_load_ten, load_env_list
envList = calc_env(mpsList, mpo)
# Calculate the energy using the environment
env = load_env_list(envList)
mps = load_mps_list(mpsList)
E3 = zeros(nState, dtype=mps[0][0].dtype)
for state in range(nState):
E3[state] = einsum('Ala,apb,lPpr,APB,Brb->', env[0][0], mps[0][0],
mpo[0][0], conj(mps[0][0]), env[0][1])
E3[state] /= calc_mps_norm(mpsList, state=state)
mpiprint(0, 'Energy from environment =\n{}'.format(E3))
# Check Results
self.assertTrue(summ(abss(E1[0] - E2[0])) < 1e-10)
self.assertTrue(summ(abss(E1[0] - E3[0])) < 1e-10)
mpiprint(0, 'Passed\n' + '=' * 50)
def test_move_gauge_left2(self):
mpiprint(
0, '\n' + '=' * 50 +
'\nTesting moving gauge left effect on Energy\n' + '-' * 50)
from cyclomps.mpo.tasep import return_mpo
import copy
N = 4
d = 2
mbd = 10
nState = 2
mpsL = create_mps_list([d] * N, mbd, nState)
mpo = return_mpo(N, (0.5, 0.5, 1.))
# Calculate initial Energy (very slow way)
mps = load_mps_list(mpsL)
E = zeros(nState, dtype=mps[0][0].dtype)
for state in range(nState):
psi = einsum('apb,bqc,crd,dse->pqrs', mps[state][0], mps[state][1],
mps[state][2], mps[state][3])
op = einsum('iPpj,jQqk,kRrl,lSsm->PpQqRrSs', mpo[0][0], mpo[0][1],
mpo[0][2], mpo[0][3])
psi_conj = einsum('apb,bqc,crd,dse->pqrs', conj(mps[state][0]),
conj(mps[state][1]), conj(mps[state][2]),
conj(mps[state][3]))
E[state] = einsum('pqrs,PpQqRrSs,PQRS', psi, op, psi_conj)
#E[state] /= calc_mps_norm(mpsL,state)
# Move guage one site left
for state in range(nState):
site = 3
ten1 = mps_load_ten(mpsL, state, site - 1)
ten2 = mps_load_ten(mpsL, state, site)
ten1, ten2, _, _, _ = move_gauge_left_tens(copy.deepcopy(ten1),
copy.deepcopy(ten2))
mps_save_ten(ten1, mpsL, state, site - 1)
mps_save_ten(ten2, mpsL, state, site)
#for state in range(nState):
# mpsL = move_gauge_left(mpsL,state,3,return_ent=False,return_wgt=False)
# Calculate Final Energy
mps = load_mps_list(mpsL)
E2 = zeros(nState, dtype=mps[0][0].dtype)
for state in range(nState):
psi = einsum('apb,bqc,crd,dse->pqrs', mps[state][0], mps[state][1],
mps[state][2], mps[state][3])
op = einsum('iPpj,jQqk,kRrl,lSsm->PpQqRrSs', mpo[0][0], mpo[0][1],
mpo[0][2], mpo[0][3])
psi_conj = einsum('apb,bqc,crd,dse->pqrs', conj(mps[state][0]),
conj(mps[state][1]), conj(mps[state][2]),
conj(mps[state][3]))
E2[state] = einsum('pqrs,PpQqRrSs,PQRS', psi, op, psi_conj)
#E2[state] /= calc_mps_norm(mpsL,state)
# Print
for state in range(nState):
self.assertTrue(abs(E[state] - E2[state]) < 1e-10)
mpiprint(0, 'Passed\n' + '=' * 50)
def test_move_gauge_right2(self):
mpiprint(
0, '\n' + '=' * 50 +
'\nTesting moving gauge right effect on Energy\n' + '-' * 50)
N = 4
d = 2
mbd = 10
nState = 2
mpsL = create_mps_list([d] * N, mbd, nState)
from cyclomps.mpo.tasep import return_mpo
mpo = return_mpo(N, (0.5, 0.5, 1.))
# Calculate initial Energy (very slow way)
mps = load_mps_list(mpsL)
E = zeros(nState, dtype=mps[0][0].dtype)
for state in range(nState):
psi = einsum('apb,bqc,crd,dse->pqrs', mps[state][0], mps[state][1],
mps[state][2], mps[state][3])
op = einsum('iPpj,jQqk,kRrl,lSsm->PpQqRrSs', mpo[0][0], mpo[0][1],
mpo[0][2], mpo[0][3])
psi_conj = einsum('apb,bqc,crd,dse->pqrs', conj(mps[state][0]),
conj(mps[state][1]), conj(mps[state][2]),
conj(mps[state][3]))
E[state] = einsum('pqrs,PpQqRrSs,PQRS', psi, op, psi_conj)
# Move guage one site left
for state in range(nState):
mpsL = move_gauge_right(mpsL,
state,
0,
return_ent=False,
return_wgt=False)
# Calculate Final Energy
mps = load_mps_list(mpsL)
E2 = zeros(nState, dtype=mps[0][0].dtype)
for state in range(nState):
psi = einsum('apb,bqc,crd,dse->pqrs', mps[state][0], mps[state][1],
mps[state][2], mps[state][3])
op = einsum('iPpj,jQqk,kRrl,lSsm->PpQqRrSs', mpo[0][0], mpo[0][1],
mpo[0][2], mpo[0][3])
psi_conj = einsum('apb,bqc,crd,dse->pqrs', conj(mps[state][0]),
conj(mps[state][1]), conj(mps[state][2]),
conj(mps[state][3]))
E2[state] = einsum('pqrs,PpQqRrSs,PQRS', psi, op, psi_conj)
# Print
for state in range(nState):
mpiprint(
0, 'State {} Energies: {},{},{}'.format(
state, E[state], E2[state],
summ(abss(E[state] - E2[state]))))
self.assertTrue(abs(E[state] - E2[state]) < 1e-10)
mpiprint(0, 'Passed\n' + '=' * 50)
def test_canonicalize_mps4(self):
mpiprint(
0, '\n' + '=' * 50 +
'\nTesting Canonicalization effect on Energy\n' + '-' * 50)
N = 4
d = 2
mbd = 10
nState = 5
mpsL = create_mps_list([d] * N, mbd, nState)
from cyclomps.mpo.tasep import return_mpo
mpo = return_mpo(N, (0.5, 0.5, 1.))
# Create function to calculate energy and norm
def calc_energy(mps, mpo, load_mps=True):
if load_mps:
mps = load_mps_list(mps)
E = zeros(nState, dtype=mps[0][0].dtype)
norm = zeros(nState, dtype=mps[0][0].dtype)
for state in range(nState):
psi = einsum('apb,bqc,crd,dse->pqrs', mps[state][0],
mps[state][1], mps[state][2], mps[state][3])
op = einsum('iPpj,jQqk,kRrl,lSsm->PpQqRrSs', mpo[0][0],
mpo[0][1], mpo[0][2], mpo[0][3])
psi_conj = einsum('apb,bqc,crd,dse->pqrs', conj(mps[state][0]),
conj(mps[state][1]), conj(mps[state][2]),
conj(mps[state][3]))
E[state] = einsum('pqrs,PpQqRrSs,PQRS', psi, op, psi_conj)
norm[state] = einsum('pqrs,pqrs->', psi, psi_conj)
return E, norm
# Calculate initial norm
E, norm = calc_energy(mpsL, mpo, load_mps=True)
# Make the MPS Right canonical
mpsL = make_mps_list_left(mpsL)
# Calculate Final norm
E2, norm2 = calc_energy(mpsL, mpo, load_mps=True)
# Print
for state in range(nState):
mpiprint(
0, 'Norm of State {}: {},{},{}'.format(
state, norm[state], norm2[state],
abs(norm[state] - norm2[state])))
self.assertTrue(abs(norm[state] - norm2[state]) < 1e-8)
mpiprint(
0, 'Energy of State {}: {},{},{}'.format(
state, E[state], E2[state], abs(E[state] - E2[state])))
self.assertTrue(abs(E[state] - E2[state]) < 1e-8)
mpiprint(0, 'Passed\n' + '=' * 50)
def test_dmrg2(self):
mpiprint(0,'\n'+'='*20+'\nTesting one-site dmrg\n'+\
'-Davidson\n-Fixed BD\n-not orthonormalized\n-calc left\n-Multiple BD\n'+'='*20)
# Use the TASEP MPO
from cyclomps.algs.dmrg1 import dmrg
from numpy import complex_
from cyclomps.mpo.tasep import return_mpo
mpo = return_mpo(10,(0.5,0.5,1.))
E1 = dmrg(mpo,
alg='davidson',
dtype=complex_,
tol=1e-5,
mbd=[10,20],
min_iter=3,
nStates=2,
fixed_bd=True,
left=False)
mpiprint(0,'Passed\n'+'='*20)
def test_dmrg1(self):
mpiprint(0,'\n'+'='*20+'\nTesting one-site dmrg\n'+\
'-Exact Diagonalization\n-not fixed\n-not orthonormalized\n-calc left\n'+'='*20)
# Use the TASEP MPO
from cyclomps.algs.dmrg1 import dmrg
from numpy import complex_
from cyclomps.mpo.tasep import return_mpo
mpo = return_mpo(10,(0.5,0.5,1.))
E0 = dmrg(mpo,
alg='exact',
dtype=complex_,
tol=1e-3,
max_iter=[4,2],
min_iter=[2,0],
mbd = [10,20],
mps_subdir='test_mps',
env_subdir='test_env',
nStates=1,
fixed_bd=False,
orthonormalize=False,
left=True,
end_gauge=5)
mpiprint(0,'Passed\n'+'='*20)
if __name__ == "__main__":
from cyclomps.mpo.tasep import return_mpo, return_tebd_ops
from cyclomps.algs.dmrg1 import dmrg
N = 30
a = 0.5
b = 0.5
D = 10
sVec = [
-1., -0.9, -0.8, -0.7, -0.6, -0.5, -0.4, -0.3, -0.2, -0.1, -0.05,
-0.01, 0., 0.01, 0.01, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0
]
for sind, s in enumerate(sVec):
# Get operators
ops = return_tebd_ops(N, (a, b, s))
ham = return_mpo(N, (a, b, s))
# Start by trying DMRG
E0, mps = dmrg(ham,
mbd=D,
max_iter=5,
nStates=1,
fixed_bd=True,
return_state=True)
# Try again with TEBD
E = tebd(ops,
H=ham,
D=[D, D, D, D, D, D, D, D, D],
step_size=[1., 0.9, 0.7, 0.6, 0.5, 0.4, 0.3, 0.2, 0.1],
n_step=[10, 10, 10, 10, 10, 10, 10, 10, 10],
conv_tol=1e-8)
print(s, E)
def test_move_gauge_left_(self):
mpiprint(
0, '\n' + '=' * 50 +
'\nTesting moving gauge left effect on Norm\n' + '-' * 50)
from cyclomps.mpo.tasep import return_mpo
N = 4
d = 2
mbd = 10
nState = 2
mpsL = create_mps_list([d] * N, mbd, nState)
mpo = return_mpo(N, (0.5, 0.5, 1.))
# Calculate initial norm
mps = load_mps_list(mpsL)
norm = zeros(nState, dtype=mpsL[0][0]['dtype'])
E = zeros(nState, dtype=mpsL[0][0]['dtype'])
for state in range(nState):
norm[state] = calc_mps_norm(mpsL, state=state)
psi = einsum('apb,bqc,crd,dse->pqrs', mps[state][0], mps[state][1],
mps[state][2], mps[state][3])
op = einsum('iPpj,jQqk,kRrl,lSsm->PpQqRrSs', mpo[0][0], mpo[0][1],
mpo[0][2], mpo[0][3])
psi_conj = einsum('apb,bqc,crd,dse->pqrs', conj(mps[state][0]),
conj(mps[state][1]), conj(mps[state][2]),
conj(mps[state][3]))
E[state] = einsum('pqrs,PpQqRrSs,PQRS', psi, op, psi_conj)
# Move guage one site left
for state in range(nState):
mpsL = move_gauge_left(mpsL,
state,
3,
return_ent=False,
return_wgt=False)
# Calculate Final norm
mps2 = load_mps_list(mpsL)
norm2 = zeros(nState, dtype=mpsL[0][0]['dtype'])
E_ = zeros(nState, dtype=mpsL[0][0]['dtype'])
E2 = zeros(nState, dtype=mpsL[0][0]['dtype'])
for state in range(nState):
norm2[state] = calc_mps_norm(mpsL, state=state)
psi = einsum('apb,bqc,crd,dse->pqrs', mps[state][0], mps[state][1],
mps[state][2], mps[state][3])
op = einsum('iPpj,jQqk,kRrl,lSsm->PpQqRrSs', mpo[0][0], mpo[0][1],
mpo[0][2], mpo[0][3])
psi_conj = einsum('apb,bqc,crd,dse->pqrs', conj(mps[state][0]),
conj(mps[state][1]), conj(mps[state][2]),
conj(mps[state][3]))
E_[state] = einsum('pqrs,PpQqRrSs,PQRS', psi, op, psi_conj)
psi = einsum('apb,bqc,crd,dse->pqrs', mps2[state][0],
mps2[state][1], mps2[state][2], mps2[state][3])
op = einsum('iPpj,jQqk,kRrl,lSsm->PpQqRrSs', mpo[0][0], mpo[0][1],
mpo[0][2], mpo[0][3])
psi_conj = einsum('apb,bqc,crd,dse->pqrs', conj(mps2[state][0]),
conj(mps2[state][1]), conj(mps2[state][2]),
conj(mps2[state][3]))
E2[state] = einsum('pqrs,PpQqRrSs,PQRS', psi, op, psi_conj)
# Print
for state in range(nState):
mpiprint(
0, 'Norm of State {}: {},{}'.format(state, norm[state],
norm2[state]))
mpiprint(
0, 'Energies: {}, {}, {}'.format(E[state], E_[state],
E2[state]))
self.assertTrue(abs(norm[state] - norm2[state]) < 1e-8)
self.assertTrue(abs(E[state] - E2[state]) < 1e-8)
mpiprint(0, 'Passed\n' + '=' * 50)
def test_renorm_avg_left_energy2(self):
mpiprint(
0, '\n' + '=' * 50 +
'\nTesting state avg left renormalization effect on Energy\n' +
'-' * 50)
from cyclomps.tools.mps_tools import create_mps_list, make_mps_list_left, load_mps_list, calc_mps_norm, move_gauge
from cyclomps.mpo.tasep import return_mpo
from cyclomps.algs.dmrg1 import renormalize_left
# Create an MPS
d = 2
N = 4
mbd = 10
nState = 4
mpsL = create_mps_list([d] * N, mbd, nState)
mpsL = make_mps_list_left(mpsL)
# Load the MPO
mpo = return_mpo(N, (0.5, 0.5, 1.))
# Create a function to compute energy
def calc_energy(mpsL, mpo, load_mps=True):
if load_mps:
mps = load_mps_list(mpsL)
else:
mps = mpsL
E = zeros(nState, dtype=mps[0][0].dtype)
for state in range(nState):
psi = einsum('apb,bqc,crd,dse->pqrs', mps[state][0],
mps[state][1], mps[state][2], mps[state][3])
op = einsum('iPpj,jQqk,kRrl,lSsm->PpQqRrSs', mpo[0][0],
mpo[0][1], mpo[0][2], mpo[0][3])
psi_conj = einsum('apb,bqc,crd,dse->pqrs', conj(mps[state][0]),
conj(mps[state][1]), conj(mps[state][2]),
conj(mps[state][3]))
E[state] = einsum('pqrs,PpQqRrSs,PQRS', psi, op, psi_conj)
return E
# Calc initial Energy
E1 = calc_energy(mpsL, mpo)
mps = load_mps_list(mpsL)
E1_ = calc_energy(mps, mpo, load_mps=False)
E1__ = calc_energy(mps, mpo, load_mps=False)
# Move gauge using renormalization
site = N - 1
(mps0, ) = retrieve_tensors(site - 1, mpsList=mpsL)
(mps1, ) = retrieve_tensors(site, mpsList=mpsL)
mps0_ = copy.deepcopy(mps0)
mps1_ = copy.deepcopy(mps1)
mps0, mps1, EE, EEs, wgt = renormalize_left(mps0, mps1, state_avg=True)
save_tensors(site - 1, mpsList=mpsL, mps=mps0)
save_tensors(site, mpsList=mpsL, mps=mps1)
# Check to make sure contracted states and local energies are the same
for state in range(nState):
finalState = einsum('apb,bqc->apqc', conj(mps0[state]),
conj(mps1[state]))
initState = einsum('apb,bqc->apqc', conj(mps0_[state]),
conj(mps1_[state]))
mpoComb = einsum('lPpm,mQqn->lPpQqn', mpo[0][site - 1],
mpo[0][site])
initLocalE = einsum('apqc,lPpQqn,APQC->AlaCnc', initState, mpoComb,
conj(initState))
finalLocalE = einsum('apqc,lPpQqn,APQC->AlaCnc', finalState,
mpoComb, conj(finalState))
self.assertTrue(summ(abss(finalState - initState)) < 1e-10)
self.assertTrue(summ(abss(initLocalE - initLocalE)) < 1e-10)
# Check to make sure we get the same energies from resulting states
for state in range(nState):
mps[state][site - 1] = mps0[state]
mps[state][site] = mps1[state]
# Calc Energy Again
E2 = calc_energy(mps, mpo, load_mps=False)
# Check Energies
self.assertTrue(summ(abss(E1[0] - E2[0])) < 1e-10)
self.assertTrue(summ(abss(E1[1] - E2[1])) < 1e-10)
self.assertTrue(summ(abss(E1[2] - E2[2])) < 1e-10)
self.assertTrue(summ(abss(E1[3] - E2[3])) < 1e-10)
# Check for correct canonical form
for state in range(nState):
eye_check = einsum('apb,cpb->ac', mps1[state], conj(mps1[state]))
eye_actual = eye(eye_check.shape[0])
self.assertTrue(summ(abss(eye_check - eye_actual)) < 1e-10)
mpiprint(0, 'Passed\n' + '=' * 50)
beta = 0.5
s = 1.
# Maximum bond dimension
mbd = 100
# Specify number of processors
n = 28
# Calculation Settings
alg = "'davidson'"
tol = 1e-5
max_iter = 1
min_iter = 0
mps_dir = "'tasep_mps'"
env_dir = "'tasep_env'"
nStates = 1
fixed_bd = True
state_avg = False
orthonormalize = False
end_gauge = 0
left = False
# Set up mpo
mpo = return_mpo(N,(alpha,beta,s))
# Run DMRG
cProfile.run("E0 = dmrg(mpo,alg="+alg+",dtype=complex_,mbd = "+str(mbd)+",tol="+str(tol)+",max_iter="+str(max_iter)+",min_iter="+str(min_iter)+",mps_subdir="+mps_dir+",env_subdir="+env_dir+",nStates="+str(nStates)+",fixed_bd="+str(fixed_bd)+",state_avg="+str(state_avg)+",orthonormalize="+str(orthonormalize)+",end_gauge="+str(end_gauge)+",left="+str(left)+")",'n'+str(n)+'_dmrg_stats_'+str(mbd))
if RANK == 0:
p = pstats.Stats('n'+str(n)+'_dmrg_stats_'+str(mbd))
import unittest
from cyclomps.tools.utils import *
from cyclomps.algs.dmrg1 import dmrg
from numpy import complex_
from cyclomps.mpo.tasep import return_mpo
import cProfile
import pstats
mpo = return_mpo(10, (0.5, 0.5, 1.))
mbd = 10
n = 28
cProfile.run(
"E0 = dmrg(mpo,alg='davidson',dtype=complex_,mbd = " + str(mbd) +
",tol=1e-5,max_iter=2,min_iter=2,mps_subdir='tasep_mps',env_subdir='tasep_env',nStates=1,fixed_bd=True,state_avg=False,orthonormalize=False,end_gauge=0,left=False)",
'n' + str(n) + '_dmrg_stats_' + str(mbd))
if RANK == 0:
p = pstats.Stats('n' + str(n) + '_dmrg_stats_' + str(mbd))
