def d_target_d_params_finite(f_obs, xray_structure, eps=1.e-8):
result = flex.double()
scatterers = xray_structure.scatterers()
xray_structure_eps = xray_structure.deep_copy_scatterers()
scatterers_eps = xray_structure_eps.scatterers()
for i_scatterer in xrange(len(scatterers)):
if (not scatterers[i_scatterer].flags.use_u_aniso()):
np = 7
else:
np = 12
dx = []
for ix in xrange(np):
ts = []
for signed_eps in [eps, -eps]:
si_eps = scatterer_as_list(scatterers[i_scatterer])
si_eps[ix] += signed_eps
scatterers_eps[i_scatterer] = scatterer_from_list(si_eps)
sf = structure_factors(xray_structure=xray_structure_eps,
miller_set=f_obs)
sum_target_f = 0
for obs, f in zip(f_obs.data(), sf.fs()):
target = least_squares(obs=obs, calc=f)
sum_target_f += target.f()
ts.append(sum_target_f)
result.append((ts[0] - ts[1]) / (2 * eps))
scatterers_eps[i_scatterer] = scatterers[i_scatterer]
return result
def exercise(args):
verbose = "--verbose" in args
if (not verbose):
out = StringIO()
else:
out = sys.stdout
for i_trial in range(10):
for n_sites in range(2,5+1):
ops = []
for i in range(3):
ops.append(matrix.sqr(flex.random_double(size=9, factor=2)-1))
sites = []
for i in range(n_sites):
sites.append(matrix.col(flex.random_double(size=3, factor=4)-2))
hkl = matrix.row(flex.random_double(size=3, factor=4)-2)
sf = exp_i_hx(sites=sites, ops=ops, hkl=hkl)
for obs_factor in [1, 1.1]:
obs = abs(sf.f()) * obs_factor
grads_fin = d_exp_i_hx_d_sites_finite(
sites=sites, ops=ops, obs=obs, hkl=hkl)
print("grads_fin:", list(grads_fin), file=out)
tf = least_squares(obs=obs, calc=sf.f())
grads_ana = sf.d_target_d_sites(target=tf)
print("grads_ana:", list(grads_ana), file=out)
compare_derivatives(grads_ana, grads_fin)
curvs_fin = d2_exp_i_hx_d_sites_finite(
sites=sites, ops=ops, obs=obs, hkl=hkl)
print("curvs_fin:", list(curvs_fin), file=out)
curvs_ana = sf.d2_target_d_sites(target=tf)
print("curvs_ana:", list(curvs_ana), file=out)
compare_derivatives(curvs_ana, curvs_fin)
print(file=out)
print("OK")
def d_target_d_params_finite(f_obs, xray_structure, eps=1.e-8):
result = flex.double()
scatterers = xray_structure.scatterers()
xray_structure_eps = xray_structure.deep_copy_scatterers()
scatterers_eps = xray_structure_eps.scatterers()
for i_scatterer in xrange(len(scatterers)):
if (not scatterers[i_scatterer].flags.use_u_aniso()):
np = 7
else:
np = 12
dx = []
for ix in xrange(np):
ts = []
for signed_eps in [eps, -eps]:
si_eps = scatterer_as_list(scatterers[i_scatterer])
si_eps[ix] += signed_eps
scatterers_eps[i_scatterer] = scatterer_from_list(si_eps)
sf = structure_factors(
xray_structure=xray_structure_eps, miller_set=f_obs)
sum_target_f = 0
for obs,f in zip(f_obs.data(), sf.fs()):
target = least_squares(obs=obs, calc=f)
sum_target_f += target.f()
ts.append(sum_target_f)
result.append((ts[0]-ts[1])/(2*eps))
scatterers_eps[i_scatterer] = scatterers[i_scatterer]
return result
def exercise(args):
verbose = "--verbose" in args
if (not verbose):
out = StringIO()
else:
out = sys.stdout
for i_trial in xrange(10):
for n_sites in xrange(2,5+1):
ops = []
for i in xrange(3):
ops.append(matrix.sqr(flex.random_double(size=9, factor=2)-1))
sites = []
for i in xrange(n_sites):
sites.append(matrix.col(flex.random_double(size=3, factor=4)-2))
hkl = matrix.row(flex.random_double(size=3, factor=4)-2)
sf = exp_i_hx(sites=sites, ops=ops, hkl=hkl)
for obs_factor in [1, 1.1]:
obs = abs(sf.f()) * obs_factor
grads_fin = d_exp_i_hx_d_sites_finite(
sites=sites, ops=ops, obs=obs, hkl=hkl)
print >> out, "grads_fin:", list(grads_fin)
tf = least_squares(obs=obs, calc=sf.f())
grads_ana = sf.d_target_d_sites(target=tf)
print >> out, "grads_ana:", list(grads_ana)
compare_derivatives(grads_ana, grads_fin)
curvs_fin = d2_exp_i_hx_d_sites_finite(
sites=sites, ops=ops, obs=obs, hkl=hkl)
print >> out, "curvs_fin:", list(curvs_fin)
curvs_ana = sf.d2_target_d_sites(target=tf)
print >> out, "curvs_ana:", list(curvs_ana)
compare_derivatives(curvs_ana, curvs_fin)
print >> out
print "OK"
def exercise(args):
verbose = "--verbose" in args
if (not verbose):
out = StringIO()
else:
out = sys.stdout
for n_alphas in xrange(2, 5):
for i_trial in xrange(5):
alphas = [2 * math.pi * random.random() for i in xrange(n_alphas)]
exp_sum = exp_i_alpha_sum(alphas=alphas)
obs = abs(exp_sum.f())
compare_analytical_and_finite(obs=obs, alphas=alphas, out=out)
compare_analytical_and_finite(obs=obs * (random.random() + 0.5),
alphas=alphas,
out=out)
for obs in [0, 0.1]:
for calc in [0j, 1.e-200j]:
target = least_squares(obs=obs, calc=calc)
assert target.f() == obs**2
assert target.da() == 0
assert target.db() == 0
if (obs == 0):
assert target.daa() == 2
assert target.dbb() == 2
assert target.dab() == 0
else:
assert target.daa() == -1.e160
assert target.dbb() == -1.e160
assert target.dab() == 1.e160
calc = 1 + 1j
while (calc != 0):
target = least_squares(obs=obs, calc=calc)
# exercise numerical stability without checking the results
target.f()
target.da()
target.db()
target.daa()
target.dbb()
target.dab()
calc /= 2
print "OK"
def d_target_d_alphas_finite(obs, alphas, eps=1.e-8):
result = []
for i_alpha in xrange(len(alphas)):
alphas_eps = list(alphas)
ts = []
for signed_eps in [eps, -eps]:
alphas_eps[i_alpha] = alphas[i_alpha] + signed_eps
exp_sum = exp_i_alpha_sum(alphas=alphas_eps)
target = least_squares(obs=obs, calc=exp_sum.f())
ts.append(target.f())
result.append((ts[0] - ts[1]) / (2 * eps))
return result
def d_target_d_alphas_finite(obs, alphas, eps=1.0e-8):
result = []
for i_alpha in xrange(len(alphas)):
alphas_eps = list(alphas)
ts = []
for signed_eps in [eps, -eps]:
alphas_eps[i_alpha] = alphas[i_alpha] + signed_eps
exp_sum = exp_i_alpha_sum(alphas=alphas_eps)
target = least_squares(obs=obs, calc=exp_sum.f())
ts.append(target.f())
result.append((ts[0] - ts[1]) / (2 * eps))
return result
def d2_target_d_alphas_finite(obs, alphas, eps=1.e-8):
result = []
for i_alpha in xrange(len(alphas)):
alphas_eps = list(alphas)
gs = []
for signed_eps in [eps, -eps]:
alphas_eps[i_alpha] = alphas[i_alpha] + signed_eps
exp_sum = exp_i_alpha_sum(alphas=alphas_eps)
target = least_squares(obs=obs, calc=exp_sum.f())
dalphas = exp_sum.d_target_d_alphas(target=target)
gs.append(dalphas)
result.append([(gp - gm) / (2 * eps) for gp, gm in zip(gs[0], gs[1])])
return result
def exercise(args):
verbose = "--verbose" in args
if not verbose:
out = StringIO()
else:
out = sys.stdout
for n_alphas in xrange(2, 5):
for i_trial in xrange(5):
alphas = [2 * math.pi * random.random() for i in xrange(n_alphas)]
exp_sum = exp_i_alpha_sum(alphas=alphas)
obs = abs(exp_sum.f())
compare_analytical_and_finite(obs=obs, alphas=alphas, out=out)
compare_analytical_and_finite(obs=obs * (random.random() + 0.5), alphas=alphas, out=out)
for obs in [0, 0.1]:
for calc in [0j, 1.0e-200j]:
target = least_squares(obs=obs, calc=calc)
assert target.f() == obs ** 2
assert target.da() == 0
assert target.db() == 0
if obs == 0:
assert target.daa() == 2
assert target.dbb() == 2
assert target.dab() == 0
else:
assert target.daa() == -1.0e160
assert target.dbb() == -1.0e160
assert target.dab() == 1.0e160
calc = 1 + 1j
while calc != 0:
target = least_squares(obs=obs, calc=calc)
# exercise numerical stability without checking the results
target.f()
target.da()
target.db()
target.daa()
target.dbb()
target.dab()
calc /= 2
print "OK"
def d2_target_d_alphas_finite(obs, alphas, eps=1.0e-8):
result = []
for i_alpha in xrange(len(alphas)):
alphas_eps = list(alphas)
gs = []
for signed_eps in [eps, -eps]:
alphas_eps[i_alpha] = alphas[i_alpha] + signed_eps
exp_sum = exp_i_alpha_sum(alphas=alphas_eps)
target = least_squares(obs=obs, calc=exp_sum.f())
dalphas = exp_sum.d_target_d_alphas(target=target)
gs.append(dalphas)
result.append([(gp - gm) / (2 * eps) for gp, gm in zip(gs[0], gs[1])])
return result
def compare_analytical_and_finite(obs, params, out):
grads_fin = d_target_d_params_finite(obs=obs, params=params)
print("grads_fin:", pack_gradients(grads_fin), file=out)
exp_sum = g_exp_i_alpha_sum(params=params)
target = least_squares(obs=obs, calc=exp_sum.f())
grads_ana = exp_sum.d_target_d_params(target=target)
print("grads_ana:", pack_gradients(grads_ana), file=out)
assert approx_equal(pack_gradients(grads_ana), pack_gradients(grads_fin))
curvs_fin = d2_target_d_params_finite(obs=obs, params=params)
print("curvs_fin:", curvs_fin, file=out)
curvs_ana = list(exp_sum.d2_target_d_params(target=target))
print("curvs_ana:", curvs_ana, file=out)
assert approx_equal(curvs_ana, curvs_fin)
print(file=out)
def compare_analytical_and_finite(obs, alphas, out):
grads_fin = d_target_d_alphas_finite(obs=obs, alphas=alphas)
print >> out, "grads_fin:", grads_fin
exp_sum = exp_i_alpha_sum(alphas=alphas)
target = least_squares(obs=obs, calc=exp_sum.f())
grads_ana = exp_sum.d_target_d_alphas(target=target)
print >> out, "grads_ana:", grads_ana
assert approx_equal(grads_ana, grads_fin)
curvs_fin = d2_target_d_alphas_finite(obs=obs, alphas=alphas)
print >> out, "curvs_fin:", curvs_fin
curvs_ana = exp_sum.d2_target_d_alphas(target=target)
print >> out, "curvs_ana:", curvs_ana
assert approx_equal(curvs_ana, curvs_fin)
print >> out
def compare_analytical_and_finite(obs, params, out): grads_fin = d_target_d_params_finite(obs=obs, params=params) print >> out, "grads_fin:", pack_gradients(grads_fin) exp_sum = g_exp_i_alpha_sum(params=params) target = least_squares(obs=obs, calc=exp_sum.f()) grads_ana = exp_sum.d_target_d_params(target=target) print >> out, "grads_ana:", pack_gradients(grads_ana) assert approx_equal(pack_gradients(grads_ana), pack_gradients(grads_fin)) curvs_fin = d2_target_d_params_finite(obs=obs, params=params) print >> out, "curvs_fin:", curvs_fin curvs_ana = list(exp_sum.d2_target_d_params(target=target)) print >> out, "curvs_ana:", curvs_ana assert approx_equal(curvs_ana, curvs_fin) print >> out
def compare_analytical_and_finite(obs, alphas, out):
grads_fin = d_target_d_alphas_finite(obs=obs, alphas=alphas)
print >> out, "grads_fin:", grads_fin
exp_sum = exp_i_alpha_sum(alphas=alphas)
target = least_squares(obs=obs, calc=exp_sum.f())
grads_ana = exp_sum.d_target_d_alphas(target=target)
print >> out, "grads_ana:", grads_ana
assert approx_equal(grads_ana, grads_fin)
curvs_fin = d2_target_d_alphas_finite(obs=obs, alphas=alphas)
print >> out, "curvs_fin:", curvs_fin
curvs_ana = exp_sum.d2_target_d_alphas(target=target)
print >> out, "curvs_ana:", curvs_ana
assert approx_equal(curvs_ana, curvs_fin)
print >> out
def compare_analytical_and_finite(obs, hkl, d_star_sq, params, out):
grads_fin = d_target_d_params_finite(
obs=obs, hkl=hkl, d_star_sq=d_star_sq, params=params)
print >> out, "grads_fin:", pack_gradients(grads_fin)
sf = structure_factor(hkl=hkl, d_star_sq=d_star_sq, params=params)
target = least_squares(obs=obs, calc=sf.f())
grads_ana = sf.d_target_d_params(target=target)
print >> out, "grads_ana:", pack_gradients(grads_ana)
assert approx_equal(pack_gradients(grads_ana), pack_gradients(grads_fin))
curvs_fin = d2_target_d_params_finite(
obs=obs, hkl=hkl, d_star_sq=d_star_sq, params=params)
print >> out, "curvs_fin:", curvs_fin
curvs_ana = sf.d2_target_d_params(target=target)
print >> out, "curvs_ana:", curvs_ana
assert approx_equal(curvs_ana, curvs_fin, 1.e-5)
print >> out
def d2_exp_i_hx_d_sites_finite(sites, ops, hkl, obs, eps=1.e-8):
result = flex.double()
sites_eps = list(sites)
for js,site in enumerate(sites):
site_eps = list(site)
for jp in range(3):
vs = []
for signed_eps in [eps, -eps]:
site_eps[jp] = site[jp] + signed_eps
sites_eps[js] = matrix.col(site_eps)
sf = exp_i_hx(sites=sites_eps, ops=ops, hkl=hkl)
tf = least_squares(obs=obs, calc=sf.f())
vs.append(sf.d_target_d_sites(target=tf))
result.extend(((vs[0]-vs[1])/(2*eps)).as_1d())
sites_eps[js] = sites[js]
return result
def d2_exp_i_hx_d_sites_finite(sites, ops, hkl, obs, eps=1.e-8):
result = flex.double()
sites_eps = list(sites)
for js,site in enumerate(sites):
site_eps = list(site)
for jp in xrange(3):
vs = []
for signed_eps in [eps, -eps]:
site_eps[jp] = site[jp] + signed_eps
sites_eps[js] = matrix.col(site_eps)
sf = exp_i_hx(sites=sites_eps, ops=ops, hkl=hkl)
tf = least_squares(obs=obs, calc=sf.f())
vs.append(sf.d_target_d_sites(target=tf))
result.extend(((vs[0]-vs[1])/(2*eps)).as_1d())
sites_eps[js] = sites[js]
return result
def d2_target_d_params_finite(obs, hkl, d_star_sq, params, eps=1.e-8):
result = []
params_eps = copy.deepcopy(params)
for i_param in xrange(len(params)):
for ix in xrange(7):
gs = []
for signed_eps in [eps, -eps]:
pi_eps = params[i_param].as_list()
pi_eps[ix] += signed_eps
params_eps[i_param] = parameters(pi_eps[:3], *pi_eps[3:])
sf = structure_factor(hkl=hkl, d_star_sq=d_star_sq, params=params_eps)
target = least_squares(obs=obs, calc=sf.f())
dp = sf.d_target_d_params(target=target)
gs.append(pack_gradients(dp))
result.append([(gp-gm)/(2*eps) for gp,gm in zip(gs[0],gs[1])])
params_eps[i_param] = params[i_param]
return result
def d2_target_d_params_finite(obs, params, eps=1.e-8):
result = []
params_eps = copy.deepcopy(params)
for i_param in xrange(len(params)):
for ix in xrange(4):
gs = []
for signed_eps in [eps, -eps]:
pi_eps = params[i_param].as_list()
pi_eps[ix] += signed_eps
params_eps[i_param] = parameters(*pi_eps)
exp_sum = g_exp_i_alpha_sum(params=params_eps)
target = least_squares(obs=obs, calc=exp_sum.f())
dp = exp_sum.d_target_d_params(target=target)
gs.append(pack_gradients(dp))
result.append([(gp-gm)/(2*eps) for gp,gm in zip(gs[0],gs[1])])
params_eps[i_param] = params[i_param]
return result
def d_target_d_params_finite(obs, params, eps=1.e-8):
result = []
params_eps = copy.deepcopy(params)
for i_param in xrange(len(params)):
dx = []
for ix in xrange(4):
ts = []
for signed_eps in [eps, -eps]:
pi_eps = params[i_param].as_list()
pi_eps[ix] += signed_eps
params_eps[i_param] = parameters(*pi_eps)
exp_sum = g_exp_i_alpha_sum(params=params_eps)
target = least_squares(obs=obs, calc=exp_sum.f())
ts.append(target.f())
dx.append((ts[0]-ts[1])/(2*eps))
result.append(gradients(*dx))
params_eps[i_param] = params[i_param]
return result
def d_target_d_params_finite(obs, hkl, d_star_sq, params, eps=1.e-8):
result = []
params_eps = copy.deepcopy(params)
for i_param in xrange(len(params)):
dx = []
for ix in xrange(7):
ts = []
for signed_eps in [eps, -eps]:
pi_eps = params[i_param].as_list()
pi_eps[ix] += signed_eps
params_eps[i_param] = parameters(pi_eps[:3], *pi_eps[3:])
sf = structure_factor(hkl=hkl, d_star_sq=d_star_sq, params=params_eps)
target = least_squares(obs=obs, calc=sf.f())
ts.append(target.f())
dx.append((ts[0]-ts[1])/(2*eps))
result.append(gradients(dx[:3], *dx[3:]))
params_eps[i_param] = params[i_param]
return result
def d2_target_d_params_finite(obs, params, eps=1.e-8):
result = []
params_eps = copy.deepcopy(params)
for i_param in range(len(params)):
for ix in range(4):
gs = []
for signed_eps in [eps, -eps]:
pi_eps = params[i_param].as_list()
pi_eps[ix] += signed_eps
params_eps[i_param] = parameters(*pi_eps)
exp_sum = g_exp_i_alpha_sum(params=params_eps)
target = least_squares(obs=obs, calc=exp_sum.f())
dp = exp_sum.d_target_d_params(target=target)
gs.append(pack_gradients(dp))
result.append([(gp - gm) / (2 * eps)
for gp, gm in zip(gs[0], gs[1])])
params_eps[i_param] = params[i_param]
return result
def d_target_d_params_finite(obs, params, eps=1.e-8):
result = []
params_eps = copy.deepcopy(params)
for i_param in range(len(params)):
dx = []
for ix in range(4):
ts = []
for signed_eps in [eps, -eps]:
pi_eps = params[i_param].as_list()
pi_eps[ix] += signed_eps
params_eps[i_param] = parameters(*pi_eps)
exp_sum = g_exp_i_alpha_sum(params=params_eps)
target = least_squares(obs=obs, calc=exp_sum.f())
ts.append(target.f())
dx.append((ts[0] - ts[1]) / (2 * eps))
result.append(gradients(*dx))
params_eps[i_param] = params[i_param]
return result
def compare_analytical_and_finite(obs, hkl, d_star_sq, params, out):
grads_fin = d_target_d_params_finite(obs=obs,
hkl=hkl,
d_star_sq=d_star_sq,
params=params)
print >> out, "grads_fin:", pack_gradients(grads_fin)
sf = structure_factor(hkl=hkl, d_star_sq=d_star_sq, params=params)
target = least_squares(obs=obs, calc=sf.f())
grads_ana = sf.d_target_d_params(target=target)
print >> out, "grads_ana:", pack_gradients(grads_ana)
assert approx_equal(pack_gradients(grads_ana), pack_gradients(grads_fin))
curvs_fin = d2_target_d_params_finite(obs=obs,
hkl=hkl,
d_star_sq=d_star_sq,
params=params)
print >> out, "curvs_fin:", curvs_fin
curvs_ana = sf.d2_target_d_params(target=target)
print >> out, "curvs_ana:", curvs_ana
assert approx_equal(curvs_ana, curvs_fin, 1.e-5)
print >> out
def d2_target_d_params_finite(obs, hkl, d_star_sq, params, eps=1.e-8):
result = []
params_eps = copy.deepcopy(params)
for i_param in xrange(len(params)):
for ix in xrange(7):
gs = []
for signed_eps in [eps, -eps]:
pi_eps = params[i_param].as_list()
pi_eps[ix] += signed_eps
params_eps[i_param] = parameters(pi_eps[:3], *pi_eps[3:])
sf = structure_factor(hkl=hkl,
d_star_sq=d_star_sq,
params=params_eps)
target = least_squares(obs=obs, calc=sf.f())
dp = sf.d_target_d_params(target=target)
gs.append(pack_gradients(dp))
result.append([(gp - gm) / (2 * eps)
for gp, gm in zip(gs[0], gs[1])])
params_eps[i_param] = params[i_param]
return result
def d_target_d_params_finite(obs, hkl, d_star_sq, params, eps=1.e-8):
result = []
params_eps = copy.deepcopy(params)
for i_param in xrange(len(params)):
dx = []
for ix in xrange(7):
ts = []
for signed_eps in [eps, -eps]:
pi_eps = params[i_param].as_list()
pi_eps[ix] += signed_eps
params_eps[i_param] = parameters(pi_eps[:3], *pi_eps[3:])
sf = structure_factor(hkl=hkl,
d_star_sq=d_star_sq,
params=params_eps)
target = least_squares(obs=obs, calc=sf.f())
ts.append(target.f())
dx.append((ts[0] - ts[1]) / (2 * eps))
result.append(gradients(dx[:3], *dx[3:]))
params_eps[i_param] = params[i_param]
return result
def d_target_d_params_finite(d_order, f_obs, xray_structure, eps=1.e-8):
assert d_order in [1,2]
result = flex.double()
scatterers = xray_structure.scatterers()
site_symmetry_table = xray_structure.site_symmetry_table()
xray_structure_eps = xray_structure.deep_copy_scatterers()
scatterers_eps = xray_structure_eps.scatterers()
for i_scatterer,scatterer in enumerate(scatterers):
if (not site_symmetry_table.is_special_position(i_scatterer)):
site_symmetry_ops = None
if (not scatterer.flags.use_u_aniso()):
ips = range(7)
else:
ips = range(12)
else:
site_symmetry_ops = site_symmetry_table.get(i_scatterer)
site_constraints = site_symmetry_ops.site_constraints()
ips = list(site_constraints.independent_indices)
if (not scatterer.flags.use_u_aniso()):
ips.extend(range(3,7))
else:
adp_constraints = site_symmetry_ops.adp_constraints()
ips.extend([i+3 for i in adp_constraints.independent_indices])
ips.extend(range(9,12))
dx = []
for ip in ips:
vs = []
for signed_eps in [eps, -eps]:
si_eps = scatterer_as_list(scatterer)
si_eps[ip] += signed_eps
if (site_symmetry_ops is not None):
if (ip < 3):
all_params = site_constraints.all_params(
independent_params=site_constraints.independent_params(
all_params=si_eps[:3]))
si_eps = list(all_params) + si_eps[3:]
elif (scatterer.flags.use_u_aniso() and ip < 9):
all_params = adp_constraints.all_params(
independent_params=adp_constraints.independent_params(
all_params=si_eps[3:9]))
si_eps = si_eps[:3] + list(all_params) + si_eps[9:]
scatterers_eps[i_scatterer] = scatterer_from_list(si_eps)
scatterers_eps[i_scatterer].scattering_type = scatterer.scattering_type
xray_structure_eps.re_apply_symmetry(i_scatterer)
sf = structure_factors(
xray_structure=xray_structure_eps, miller_set=f_obs)
if (d_order == 1):
sum_target_f = 0
for obs,f in zip(f_obs.data(), sf.fs()):
target = least_squares(obs=obs, calc=f)
sum_target_f += target.f()
vs.append(sum_target_f)
else:
dp = sf.d_target_d_params(f_obs=f_obs, target_type=least_squares)
vs.append(dp)
diff = (vs[0]-vs[1])/(2*eps)
if (d_order == 1):
result.append(diff)
else:
result.extend(diff)
scatterers_eps[i_scatterer] = scatterer
return result
