def calculate(f: sp.Add, var: Tuple[sp.Symbol, sp.Symbol], eps: float = 0.001, debug: bool = False)\
-> List[Tuple[float, float]]:
x, y = var
first_diff = sp.Matrix([sp.diff(f, x), sp.diff(f, y)])
a = sp.Symbol('a')
point = sp.Matrix([0.0, 0.0])
prev_point = sp.Matrix([0.0, 0.0])
result: List[Tuple[float, float]] = [(point[0], point[1])]
iteration = 0
while iteration < 1 or sqrt((point[0] - prev_point[0])**2 + (point[1] - prev_point[1])**2) > eps:
if debug:
print()
prev_point = point
next_point = point - a * first_diff.subs({x: point[0], y: point[1]})
step = sp.solve(sp.diff(f.subs({x: next_point[0], y: next_point[1]}), a))[0]
point = next_point.subs({a: step})
result.append((point[0], point[1]))
iteration += 1
if debug:
print(f"Method of quick going down found solution of: {(point[0], point[1])}")
return result
def newtons_method(
function: sympy.Add, x0, error, max_iteration
): # function is a sympy expresion. x0 is the initial point.
x1 = x0
x2 = x0
iteration = 0
while iteration < max_iteration:
iteration += 1
x2 = x1 - (float(function.subs(x, x1)) /
float(sympy.diff(function, x).subs(x, x1)))
x1 = x2
if abs(float(function.subs(x, x2))) < error:
break
print("Newtons method: ", x2, "The answer was found after ", iteration,
"iterations")
return x2
def atomic_ordering_energy(self, dbe):
"""
Return the atomic ordering contribution in symbolic form.
Description follows Servant and Ansara, Calphad, 2001.
"""
phase = dbe.phases[self.phase_name]
ordered_phase_name = phase.model_hints.get("ordered_phase", None)
disordered_phase_name = phase.model_hints.get("disordered_phase", None)
if phase.name != ordered_phase_name:
return S.Zero
disordered_model = self.__class__(dbe, self.components, disordered_phase_name)
constituents = [
sorted(set(c).intersection(self.components)) for c in dbe.phases[ordered_phase_name].constituents
]
# Fix variable names
variable_rename_dict = {}
for atom in disordered_model.energy.atoms(v.SiteFraction):
# Replace disordered phase site fractions with mole fractions of
# ordered phase site fractions.
# Special case: Pure vacancy sublattices
all_species_in_sublattice = dbe.phases[disordered_phase_name].constituents[atom.sublattice_index]
if atom.species == "VA" and len(all_species_in_sublattice) == 1:
# Assume: Pure vacancy sublattices are always last
vacancy_subl_index = len(dbe.phases[ordered_phase_name].constituents) - 1
variable_rename_dict[atom] = v.SiteFraction(ordered_phase_name, vacancy_subl_index, atom.species)
else:
# All other cases: replace site fraction with mole fraction
variable_rename_dict[atom] = self.mole_fraction(
atom.species, ordered_phase_name, constituents, dbe.phases[ordered_phase_name].sublattices
)
# Save all of the ordered energy contributions
# This step is why this routine must be called _last_ in build_phase
ordered_energy = Add(*list(self.models.values()))
self.models.clear()
# Copy the disordered energy contributions into the correct bins
for name, value in disordered_model.models.items():
self.models[name] = value.xreplace(variable_rename_dict)
# All magnetic parameters will be defined in the disordered model
self.TC = self.curie_temperature = disordered_model.TC
self.TC = self.curie_temperature = self.TC.xreplace(variable_rename_dict)
molefraction_dict = {}
# Construct a dictionary that replaces every site fraction with its
# corresponding mole fraction in the disordered state
for sitefrac in ordered_energy.atoms(v.SiteFraction):
all_species_in_sublattice = dbe.phases[ordered_phase_name].constituents[sitefrac.sublattice_index]
if sitefrac.species == "VA" and len(all_species_in_sublattice) == 1:
# pure-vacancy sublattices should not be replaced
# this handles cases like AL,NI,VA:AL,NI,VA:VA and
# ensures the VA's don't get mixed up
continue
molefraction_dict[sitefrac] = self.mole_fraction(
sitefrac.species, ordered_phase_name, constituents, dbe.phases[ordered_phase_name].sublattices
)
return ordered_energy - ordered_energy.subs(molefraction_dict, simultaneous=True)
def _inverse_mellin_transform(F, s, x_, strip, as_meijerg=False):
""" A helper for the real inverse_mellin_transform function, this one here
assumes x to be real and positive. """
from sympy import (expand, expand_mul, hyperexpand, meijerg, And, Or, arg,
pi, re, factor, Heaviside, gamma, Add)
x = _dummy('t', 'inverse-mellin-transform', F, positive=True)
# Actually, we won't try integration at all. Instead we use the definition
# of the Meijer G function as a fairly general inverse mellin transform.
F = F.rewrite(gamma)
for g in [factor(F), expand_mul(F), expand(F)]:
if g.is_Add:
# do all terms separately
ress = [_inverse_mellin_transform(G, s, x, strip, as_meijerg,
noconds=False) \
for G in g.args]
conds = [p[1] for p in ress]
ress = [p[0] for p in ress]
res = Add(*ress)
if not as_meijerg:
res = factor(res, gens=res.atoms(Heaviside))
return res.subs(x, x_), And(*conds)
try:
a, b, C, e, fac = _rewrite_gamma(g, s, strip[0], strip[1])
except IntegralTransformError:
continue
G = meijerg(a, b, C / x**e)
if as_meijerg:
h = G
else:
h = hyperexpand(G)
if h.is_Piecewise and len(h.args) == 3:
# XXX we break modularity here!
h = Heaviside(x - abs(C))*h.args[0].args[0] \
+ Heaviside(abs(C) - x)*h.args[1].args[0]
# We must ensure that the intgral along the line we want converges,
# and return that value.
# See [L], 5.2
cond = [abs(arg(G.argument)) < G.delta * pi]
# Note: we allow ">=" here, this corresponds to convergence if we let
# limits go to oo symetrically. ">" corresponds to absolute convergence.
cond += [
And(Or(len(G.ap) != len(G.bq), 0 >= re(G.nu) + 1),
abs(arg(G.argument)) == G.delta * pi)
]
cond = Or(*cond)
if cond is False:
raise IntegralTransformError('Inverse Mellin', F,
'does not converge')
return (h * fac).subs(x, x_), cond
raise IntegralTransformError('Inverse Mellin', F, '')
def subs_Eqs(eq: Add, eqs: Sequence[Eq]) -> Add:
"""
TODO: Проверить какой класс является базовым для всех выражений sympy
@param eq: expression
@param eqs: Eq
@return: expression
Например: x = z+2*y
Это eq.subs({x: z+2*y})
"""
for eq in eqs:
left, right = eq.args[0], eq.args[1]
eq = eq.subs({left: right})
return eq
def _inverse_mellin_transform(F, s, x_, strip, as_meijerg=False):
""" A helper for the real inverse_mellin_transform function, this one here
assumes x to be real and positive. """
from sympy import (expand, expand_mul, hyperexpand, meijerg, And, Or,
arg, pi, re, factor, Heaviside, gamma, Add)
x = _dummy('t', 'inverse-mellin-transform', F, positive=True)
# Actually, we won't try integration at all. Instead we use the definition
# of the Meijer G function as a fairly general inverse mellin transform.
F = F.rewrite(gamma)
for g in [factor(F), expand_mul(F), expand(F)]:
if g.is_Add:
# do all terms separately
ress = [_inverse_mellin_transform(G, s, x, strip, as_meijerg,
noconds=False) \
for G in g.args]
conds = [p[1] for p in ress]
ress = [p[0] for p in ress]
res = Add(*ress)
if not as_meijerg:
res = factor(res, gens=res.atoms(Heaviside))
return res.subs(x, x_), And(*conds)
try:
a, b, C, e, fac = _rewrite_gamma(g, s, strip[0], strip[1])
except IntegralTransformError:
continue
G = meijerg(a, b, C/x**e)
if as_meijerg:
h = G
else:
h = hyperexpand(G)
if h.is_Piecewise and len(h.args) == 3:
# XXX we break modularity here!
h = Heaviside(x - abs(C))*h.args[0].args[0] \
+ Heaviside(abs(C) - x)*h.args[1].args[0]
# We must ensure that the intgral along the line we want converges,
# and return that value.
# See [L], 5.2
cond = [abs(arg(G.argument)) < G.delta*pi]
# Note: we allow ">=" here, this corresponds to convergence if we let
# limits go to oo symetrically. ">" corresponds to absolute convergence.
cond += [And(Or(len(G.ap) != len(G.bq), 0 >= re(G.nu) + 1),
abs(arg(G.argument)) == G.delta*pi)]
cond = Or(*cond)
if cond is False:
raise IntegralTransformError('Inverse Mellin', F, 'does not converge')
return (h*fac).subs(x, x_), cond
raise IntegralTransformError('Inverse Mellin', F, '')
def atomic_ordering_energy(self, dbe):
"""
Return the atomic ordering contribution in symbolic form.
Description follows Servant and Ansara, Calphad, 2001.
"""
phase = dbe.phases[self.phase_name]
ordered_phase_name = phase.model_hints.get('ordered_phase', None)
disordered_phase_name = phase.model_hints.get('disordered_phase', None)
if phase.name != ordered_phase_name:
return S.Zero
disordered_model = self.__class__(dbe, sorted(self.components),
disordered_phase_name)
constituents = [sorted(set(c).intersection(self.components)) \
for c in dbe.phases[ordered_phase_name].constituents]
# Fix variable names
variable_rename_dict = {}
for atom in disordered_model.energy.atoms(v.SiteFraction):
# Replace disordered phase site fractions with mole fractions of
# ordered phase site fractions.
# Special case: Pure vacancy sublattices
all_species_in_sublattice = \
dbe.phases[disordered_phase_name].constituents[
atom.sublattice_index]
if atom.species == 'VA' and len(all_species_in_sublattice) == 1:
# Assume: Pure vacancy sublattices are always last
vacancy_subl_index = \
len(dbe.phases[ordered_phase_name].constituents)-1
variable_rename_dict[atom] = \
v.SiteFraction(
ordered_phase_name, vacancy_subl_index, atom.species)
else:
# All other cases: replace site fraction with mole fraction
variable_rename_dict[atom] = \
self.mole_fraction(
atom.species,
ordered_phase_name,
constituents,
dbe.phases[ordered_phase_name].sublattices
)
# Save all of the ordered energy contributions
# This step is why this routine must be called _last_ in build_phase
ordered_energy = Add(*list(self.models.values()))
self.models.clear()
# Copy the disordered energy contributions into the correct bins
for name, value in disordered_model.models.items():
self.models[name] = value.xreplace(variable_rename_dict)
# All magnetic parameters will be defined in the disordered model
self.TC = self.curie_temperature = disordered_model.TC
self.TC = self.curie_temperature = self.TC.xreplace(variable_rename_dict)
molefraction_dict = {}
# Construct a dictionary that replaces every site fraction with its
# corresponding mole fraction in the disordered state
for sitefrac in ordered_energy.atoms(v.SiteFraction):
all_species_in_sublattice = \
dbe.phases[ordered_phase_name].constituents[
sitefrac.sublattice_index]
if sitefrac.species == 'VA' and len(all_species_in_sublattice) == 1:
# pure-vacancy sublattices should not be replaced
# this handles cases like AL,NI,VA:AL,NI,VA:VA and
# ensures the VA's don't get mixed up
continue
molefraction_dict[sitefrac] = \
self.mole_fraction(sitefrac.species,
ordered_phase_name, constituents,
dbe.phases[ordered_phase_name].sublattices)
return ordered_energy - ordered_energy.subs(molefraction_dict,
simultaneous=True)