def split_stochastic(self):
'''
Split the expression into a stochastic and non-stochastic part.
Splits the expression into a tuple of one `Expression` objects f (the
non-stochastic part) and a dictionary mapping stochastic variables
to `Expression` objects. For example, an expression of the form
``f + g * xi_1 + h * xi_2`` would be returned as:
``(f, {'xi_1': g, 'xi_2': h})``
Note that the `Expression` objects for the stochastic parts do not
include the stochastic variable itself.
Returns
-------
(f, d) : (`Expression`, dict)
A tuple of an `Expression` object and a dictionary, the first
expression being the non-stochastic part of the equation and
the dictionary mapping stochastic variables (``xi`` or starting
with ``xi_``) to `Expression` objects. If no stochastic variable
is present in the code string, a tuple ``(self, None)`` will be
returned with the unchanged `Expression` object.
'''
stochastic_variables = []
for identifier in self.identifiers:
if identifier == 'xi' or identifier.startswith('xi_'):
stochastic_variables.append(identifier)
# No stochastic variable
if not len(stochastic_variables):
return (self, None)
s_expr = self.sympy_expr.expand()
stochastic_symbols = [
sympy.Symbol(variable, real=True)
for variable in stochastic_variables
]
f = sympy.Wild('f', exclude=stochastic_symbols) # non-stochastic part
match_objects = [
sympy.Wild('w_' + variable, exclude=stochastic_symbols)
for variable in stochastic_variables
]
match_expression = f
for symbol, match_object in zip(stochastic_symbols, match_objects):
match_expression += match_object * symbol
matches = s_expr.match(match_expression)
if matches is None:
raise ValueError(
('Expression "%s" cannot be separated into stochastic '
'and non-stochastic term') % self.code)
f_expr = Expression(sympy_to_str(matches[f]))
stochastic_expressions = dict(
(variable, Expression(sympy_to_str(matches[match_object])))
for (variable,
match_object) in zip(stochastic_variables, match_objects))
return (f_expr, stochastic_expressions)
def get_conditionally_linear_system(eqs):
'''
Convert equations into a linear system using sympy.
Parameters
----------
eqs : `Equations`
The model equations.
Returns
-------
coefficients : dict of (sympy expression, sympy expression) tuples
For every variable x, a tuple (M, B) containing the coefficients M and
B (as sympy expressions) for M * x + B
Raises
------
ValueError
If one of the equations cannot be converted into a M * x + B form.
Examples
--------
>>> from brian2 import Equations
>>> eqs = Equations("""
... dv/dt = (-v + w**2) / tau : 1
... dw/dt = -w / tau : 1
... """)
>>> system = get_conditionally_linear_system(eqs)
>>> print(system['v'])
(-1/tau, w**2.0/tau)
>>> print(system['w'])
(-1/tau, 0)
'''
diff_eqs = eqs.substituted_expressions
coefficients = {}
for name, expr in diff_eqs:
var = sp.Symbol(name, real=True)
# Coefficients
wildcard = sp.Wild('_A', exclude=[var])
#Additive constant
constant_wildcard = sp.Wild('_B', exclude=[var])
pattern = wildcard * var + constant_wildcard
# Factor out the variable
s_expr = sp.collect(expr.sympy_expr.expand(), var)
matches = s_expr.match(pattern)
if matches is None:
raise ValueError(
('The expression "%s", defining the variable %s, '
'could not be separated into linear components') %
(s_expr, name))
coefficients[name] = (matches[wildcard].simplify(),
matches[constant_wildcard].simplify())
return coefficients
def trig_substitution_rule(integral):
integrand, symbol = integral
A = sympy.Wild('a', exclude=[0, symbol])
B = sympy.Wild('b', exclude=[0, symbol])
theta = sympy.Dummy("theta")
target_pattern = A + B*symbol**2
matches = integrand.find(target_pattern)
for expr in matches:
match = expr.match(target_pattern)
a = match.get(A, ZERO)
b = match.get(B, ZERO)
a_positive = ((a.is_number and a > 0) or a.is_positive)
b_positive = ((b.is_number and b > 0) or b.is_positive)
a_negative = ((a.is_number and a < 0) or a.is_negative)
b_negative = ((b.is_number and b < 0) or b.is_negative)
x_func = None
if a_positive and b_positive:
# a**2 + b*x**2. Assume sec(theta) > 0, -pi/2 < theta < pi/2
x_func = (sympy.sqrt(a)/sympy.sqrt(b)) * sympy.tan(theta)
# Do not restrict the domain: tan(theta) takes on any real
# value on the interval -pi/2 < theta < pi/2 so x takes on
# any value
restriction = True
elif a_positive and b_negative:
# a**2 - b*x**2. Assume cos(theta) > 0, -pi/2 < theta < pi/2
constant = sympy.sqrt(a)/sympy.sqrt(-b)
x_func = constant * sympy.sin(theta)
restriction = sympy.And(symbol > -constant, symbol < constant)
elif a_negative and b_positive:
# b*x**2 - a**2. Assume sin(theta) > 0, 0 < theta < pi
constant = sympy.sqrt(-a)/sympy.sqrt(b)
x_func = constant * sympy.sec(theta)
restriction = sympy.And(symbol > -constant, symbol < constant)
if x_func:
# Manually simplify sqrt(trig(theta)**2) to trig(theta)
# Valid due to assumed domain restriction
substitutions = {}
for f in [sympy.sin, sympy.cos, sympy.tan,
sympy.sec, sympy.csc, sympy.cot]:
substitutions[sympy.sqrt(f(theta)**2)] = f(theta)
substitutions[sympy.sqrt(f(theta)**(-2))] = 1/f(theta)
replaced = integrand.subs(symbol, x_func).trigsimp()
replaced = replaced.subs(substitutions)
if not replaced.has(symbol):
replaced *= manual_diff(x_func, theta)
replaced = replaced.trigsimp()
secants = replaced.find(1/sympy.cos(theta))
if secants:
replaced = replaced.xreplace({
1/sympy.cos(theta): sympy.sec(theta)
})
substep = integral_steps(replaced, theta)
if not contains_dont_know(substep):
return TrigSubstitutionRule(
theta, x_func, replaced, substep, restriction,
integrand, symbol)
def coeff(expr, term):
expr = sp.collect(expr, term)
symbols = list(term.atoms(sp.Symbol))
w = sp.Wild("coeff", exclude=symbols)
m = expr.match(w * term + sp.Wild("rest"))
if m:
return m[w]
def holstein(Hdef):
S = sympy.Symbol('S', real=True)
print 'holstein'
#print Hdef.atoms(sympy.Symbol)
Hdef = Hdef.expand()
#Hdef=Hdef.as_poly(S)
p = sympy.Wild('p', exclude='S')
q = sympy.Wild('q', exclude='S')
r = sympy.Wild('r', exclude='S')
l = sympy.Wild('l', exclude='S')
#Hlin=Hdef.coeffs[0]*S**2+Hdef.coeffs[1]*S
S2coeff = coeff(Hdef, S**2)
Scoeff = coeff(Hdef, S)
Hlin = None
#Hlin=coeff(Hdef,S**2)*S**2+coeff(Hdef,S)*S
#print 'S2Coeff', S2coeff
#print 'Scoeff',Scoeff
if Scoeff != None and S2coeff != None:
Hlin = coeff(Hdef, S**2) * S**2 + coeff(Hdef, S) * S
elif Scoeff == None and S2coeff != None:
Hlin = coeff(Hdef, S**2) * S**2
elif Scoeff != None and S2coeff == None:
#print 'S'
Hlin = coeff(Hdef, S) * S
return Hlin
def make_wilds(symbol):
a = sympy.Wild('a', exclude=[symbol])
b = sympy.Wild('b', exclude=[symbol])
m = sympy.Wild('m', exclude=[symbol], properties=[lambda n: isinstance(n, sympy.Integer)])
n = sympy.Wild('n', exclude=[symbol], properties=[lambda n: isinstance(n, sympy.Integer)])
return a, b, m, n
def can_be_applied(self, dim_exprs, variable_context, node_range,
orig_edges, dim_index, total_dims):
# Pattern does not support unions of expressions
if len(dim_exprs) > 1: return False
dexpr = dim_exprs[0]
# Pattern does not support ranges
if not isinstance(dexpr, sympy.Basic): return False
# Create wildcards
val = sympy.Wild('val')
mod = sympy.Wild('mod', exclude=variable_context[-1])
# Try to match an affine expression
matches = dexpr.match(val % mod)
if matches is None or len(matches) != 2:
return False
self.subexpr = matches[val]
self.modulo = matches[mod]
self.subpattern = None
for pattern_class in SeparableMemletPattern.s_smpatterns:
smpattern = pattern_class()
if smpattern.can_be_applied([self.subexpr], variable_context,
node_range, orig_edges, dim_index,
total_dims):
self.subpattern = smpattern
return self.subpattern is not None
def test_match_for_assignment_collection():
x, y = pystencils.fields('x, y: float32[3d]')
a, b, c, d = sp.symbols('a, b, c, d')
assignments = pystencils.AssignmentCollection({
a: sp.floor(1),
b: 2,
c: a + c,
y.center(): sp.ceiling(x.center()) + sp.floor(x.center())
})
w1 = sp.Wild('w1')
w2 = sp.Wild('w2')
w3 = sp.Wild('w3')
wild_ceiling = sp.ceiling(w1)
wild_addition = w1 + w2
assert assignments.match(pystencils.Assignment(w3, wild_ceiling + w2))[w1] == x.center()
assert assignments.match(pystencils.Assignment(w3, wild_ceiling + w2)) == {
w3: y.center(),
w2: sp.floor(x.center()),
w1: x.center()
}
assert assignments.find(wild_ceiling) == {sp.ceiling(x.center())}
assert len([a for a in assignments.find(wild_addition) if isinstance(a, sp.Add)]) == 2
def tick(self, tick):
Tm = tick.blackboard.get('Tm') # the current matrix equation
unknowns = tick.blackboard.get('unknowns')
R = tick.blackboard.get('Robot')
u = tick.blackboard.get('curr_unk')
if u.readytosolve:
if(self.BHdebug):
print "I'm trying to solve: ", u.symbol
print " Using: ",
print u.eqntosolve
if u.solvemethod == "algebra":
Aw = sp.Wild("Aw")
Bw = sp.Wild("Bw")
d = u.eqntosolve.RHS.match(Aw*u.symbol+Bw)
A = d[Aw]
B = d[Bw]
u.solutions.append( (u.eqntosolve.LHS-B)/A ) # one soluntion
u.nsolutions = 1 # or 1
u.set_solved(R,unknowns) # flag that this is solved
tick.blackboard.set('curr_unk', u)
tick.blackboard.set('unknowns', unknowns)
return b3.SUCCESS
def Max_Stress(stress,graph,dir=[],num_list=[],Number_Average=5):
a = syp.Wild('a')
b = syp.Wild('b')
c = syp.Wild('c')
x = syp.symbols('x')
y = syp.symbols('y')
ccc = syp.collect(stress,[x,y],evaluate=False)
try:
a1 = float(ccc[x])
except:
a1 = 0
try:
b1 = float(ccc[y])
except:
b1 = 0
try:
c1 = float(ccc[S(1)])
except:
c1 = 0
xdir = [float(dir[i][0]) for i in range(0,len(dir))]
ydir = [float(dir[i][1]) for i in range(0,len(dir))]
max_stress = 0
min_stress = 0
if num_list == []:
for v in graph:
for e in v.getConnections():
thick_ij = v.getWeight(e)
if thick_ij != 0:
xx = np.linspace(xdir[v.id],xdir[e.id],Number)
yy = np.linspace(ydir[v.id],ydir[e.id],Number)
zz = a1*xx + b1*yy + c1
a = max(zz)
if a > max_stress:
max_stress = a
b = min(zz)
if b < min_stress:
min_stress = b
if num_list:
for i in range(len(num_list)):
m = num_list[i][0]
n = num_list[i][1]
xx = np.linspace(xdir[m],xdir[n],Number_Average)
yy = np.linspace(ydir[m],ydir[n],Number_Average)
zz = a1*xx + b1*yy + c1
a = max(zz)
if a > max_stress:
max_stress = a
b = min(zz)
if b < min_stress:
min_stress = b
return [max_stress,min_stress]
def heaviside_pattern(symbol):
m = sympy.Wild('m', exclude=[symbol])
b = sympy.Wild('b', exclude=[symbol])
g = sympy.Wild('g')
pattern = sympy.Heaviside(m * symbol + b) * g
return pattern, m, b, g
def tick(self, tick):
unknowns = tick.blackboard.get("unknowns")
R = tick.blackboard.get('Robot')
Tm = tick.blackboard.get('Tm')
solvedtag = False
thx = sp.Wild('thx')
thy = sp.Wild('thy')
sgn = sp.Wild('sgn')
#for unk in unknowns:
#if unk.n == 0 and unk.solved: #joint varible has order of 0
if len(Tm.auxeqns) > 0:
for e in Tm.auxeqns:
#d = unk.joint_eq.match(thx + sgn * thy)
print e
d = e.RHS.match(thx + sgn * thy)
unka = find_obj(d[thx], unknowns)
#print unka
unkb = find_obj(d[thy], unknowns)
#print unkb
if unka == None:
print "variable %s doesn't exist" % (d[thx])
elif unkb == None:
print "variable %s doesn't exist" % (d[thy])
else:
if unka.solved and (not unkb.solved):
sol = (e.LHS - unka.symbol) / d[sgn]
unkb.solutions.append(sol)
unkb.nsolutions = 1
unkb.set_solved(R, unknowns)
solvedtag = True
if self.BHdebug:
print "I'm solving %s from joint variable %s" % (
unkb.symbol, e.LHS)
print "solution: %s" % sol
elif unkb.solved and (not unka.solved):
sol = e.LHS - d[sgn] * unkb.symbol
unka.solutions.append(sol)
unka.nsolutions = 1
unka.set_solved(R, unknowns)
solvedtag = True
if self.BHdebug:
print "I'm solving %s from joint variable %s" % (
unka.symbol, e.LHS)
print "solution: %s" % sol
#tick.blackboard.set('test_id', test_id)
tick.blackboard.set('unknowns', unknowns)
tick.blackboard.set('Robot', R)
if solvedtag:
return b3.SUCCESS
return b3.FAILURE
def flip_variant(expr):
""" Change every momentum in expr from contravariant to
covariant or vice versa and returns the new expression.
"""
a = sy.Wild("a")
b = sy.Wild("b")
c = sy.Wild("c")
return expr.replace(Momentum(a, b, c), Momentum(a, b, 1 - c))
def tick(self, tick):
unknowns = tick.blackboard.get(
'unknowns') # the current list of unknowns
R = tick.blackboard.get('Robot')
one_unk = tick.blackboard.get('eqns_1u')
two_unk = tick.blackboard.get('eqns_2u')
u = tick.blackboard.get('curr_unk')
# identify unknowns in T where one equation can be solved by
# arcsin() or arccos()
Aw = sp.Wild("Aw")
Bw = sp.Wild("Bw")
Cw = sp.Wild('Cw')
Dw = sp.Wild('Dw')
found = False
if (not u.solved): # only if not already solved!
for e in one_unk: # only look at the eqns with one unknowns
print "Looking for unknown: ", u.symbol, " in equation: ",
print e
tmp = e.RHS - e.LHS
lhs = l_1 - l_1
if (tmp.has(sp.sin(u.symbol)) and tmp.has(sp.cos(u.symbol))):
es = tmp.expand()
# collect terms in sin(x) and cos(x)
es = es.collect(sp.sin(u.symbol))
es = es.collect(sp.cos(u.symbol))
d = {}
d[Aw] = es.coeff(sp.sin(u.symbol))
d[Bw] = es.coeff(sp.cos(u.symbol))
d[Cw] = es - d[Aw] * sp.sin(u.symbol) - d[Bw] * sp.cos(
u.symbol)
if (self.BHdebug):
print 'Sin AND Cos identifying: ', es
print 'Aw: ', d[Aw], ' Bw: ', d[Bw], ' Cw: ', d[Cw]
if not d[Cw].has(u.symbol):
u.readytosolve = True
u.eqntosolve = kc.kequation(lhs, es)
u.solvemethod = "sinANDcos"
found = True
break
tick.blackboard.set('curr_unk', u)
tick.blackboard.set('Robot', R)
tick.blackboard.set('unknowns',
unknowns) # the current list of unknowns
if found:
return b3.SUCCESS
else:
return b3.FAILURE
def test_wild_typed_symbol():
x = pystencils.fields('x: float32[3d]')
typed_symbol = pystencils.data_types.TypedSymbol('a', create_type('float64'))
assert x.center().match(sp.Wild('w1'))
assert typed_symbol.match(sp.Wild('w1'))
wild_ceiling = sp.ceiling(sp.Wild('w1'))
assert sp.ceiling(x.center()).match(wild_ceiling)
assert sp.ceiling(typed_symbol).match(wild_ceiling)
def _check_map_conflicts(map, edge):
for itervar, (_, _, mapskip) in zip(map.params, map.range):
itersym = symbolic.pystr_to_symbolic(itervar)
a = sp.Wild('a', exclude=[itersym])
b = sp.Wild('b', exclude=[itersym])
if not _check_range_conflicts(edge.data.subset, a, itersym, b,
mapskip):
return False
# If matches all map params, good to go
return True
def generate_cross_section(N, arg, atom_list):
"""Generates the Cross-Section Formula for the one magnon case"""
S = sp.Symbol('S', commutative=True)
gam = sp.Symbol('gamma', commutative=True)
r = sp.Symbol('r0', commutative=True)
h = sp.Symbol('hbar', commutative=True)
k = sp.Symbol('k', commutative=True)
kp = sp.Symbol('kp', commutative=True)
g = sp.Symbol('g', commutative=True)
F = sp.Function('F')
kap = sp.Symbol('kappa', commutative=True)
kapx = sp.Symbol('kappax', commutative=True)
kapy = sp.Symbol('kappay', commutative=True)
w = sp.Symbol('w', commutative=True)
W = sp.Symbol('W', commutative=False)
t = sp.Symbol('t', commutative=True)
dif = sp.Symbol('diff', commutative=False)
A = sp.Wild('A', exclude=[0])
B = sp.Wild('B', exclude=[0])
C = sp.Wild('C', exclude=[0])
D = sp.Wild('D', exclude=[0])
front_constant = (gam * r)**2 / (2 * pi * h) * (kp / k) * N
front_func = (1. / 2.) * g * F(kap) * exp(-2 * W)
temp2 = []
temp3 = []
temp4 = []
# This is were the heart of the calculation comes in.
# First the exponentials are turned into delta functions:
# exp(I(wq*t - w*t)) ---> delta(wq-w)
# exp(I(wq*t - w*t)+I*(q-qp)*l) ---> delta(wq*t-w*t+q*l-qp*l) ---> delta(wq-w)*delta(q*l-qp*l) # NEEDS REVIEW
for i in range(len(arg)): # _
for j in range(N): # ^
arg[i][j] = (arg[i][j] * exp(-I * w * t)).expand() # |
arg[i][j] = sub_in(arg[i][j], exp(A * I * t + B * I * t),
sp.DiracDelta(A + B)) # |
arg[i][j] = sub_in(arg[i][j],
exp(I * t * A + I * t * B + I * C + I * D),
sp.DiracDelta(A * t + B * t + C + D)) # |
arg[i][j] = sub_in(arg[i][j], sp.DiracDelta(A * t + B * t + C + D),
sp.DiracDelta(A + B) *
sp.DiracDelta(C + D)) # |
temp2.append(exp(I * kap * atom_list[j]) * arg[i][j]) # |
temp3.append(sum(temp2)) # |
print "Converted to Delta Functions!" # |
for i in range(len(temp3)): # |
temp4.append((1 - kapx**2) * temp3[i]) # V
dif = front_constant * front_func**2 * (sum(temp4)) # _
print "Cross-section calculated!"
return dif
def write_conflicted_map_params(map, edge):
result = []
for itervar, (_, _, mapskip) in zip(map.params, map.range):
itersym = symbolic.pystr_to_symbolic(itervar)
a = sp.Wild('a', exclude=[itersym])
b = sp.Wild('b', exclude=[itersym])
if not _check_range_conflicts(edge.data.subset, a, itersym, b,
mapskip):
result.append(itervar)
return result
def _cases_from_branches(
edges: List[Edge[InterstateEdge]],
cblocks: Dict[Edge[InterstateEdge], GeneralBlock],
) -> Tuple[str, Dict[str, GeneralBlock]]:
"""
If the input list of edges correspond to a switch/case scope (with all
conditions being "x == y" for a unique symbolic x and integers y),
returns the switch/case scope parameters.
:param edges: List of inter-state edges.
:return: Tuple of (case variable C++ expression, mapping from case to
control flow block). If not a valid switch/case scope,
returns None.
"""
cond = edges[0].data.condition_sympy()
if not isinstance(cond, sp.Basic):
return None
a = sp.Wild('a')
b = sp.Wild('b', properties=[lambda k: k.is_Integer])
m = cond.match(sp.Eq(a, b))
if m:
# Obtain original code for variable
call_or_compare = edges[0].data.condition.code[0].value
if isinstance(call_or_compare, ast.Call):
astvar = call_or_compare.args[0]
else: # Binary comparison
astvar = call_or_compare.left
else:
# Try integer == symbol
m = cond.match(sp.Eq(b, a))
if m:
call_or_compare = edges[0].data.condition.code[0].value
if isinstance(call_or_compare, ast.Call):
astvar = call_or_compare.args[1]
else: # Binary comparison
astvar = call_or_compare.right
else:
return None
# Get C++ expression from AST
switchvar = cppunparse.pyexpr2cpp(astvar)
# Check that all edges match criteria
result = {}
for e in edges:
ematch = e.data.condition_sympy().match(sp.Eq(m[a], b))
if not ematch:
ematch = e.data.condition_sympy().match(sp.Eq(b, m[a]))
if not ematch:
return None
# Create mapping to codeblocks
result[cpp.sym2cpp(ematch[b])] = cblocks[e]
return switchvar, result
def sympy_ceiling_fix(expr):
""" Fix for SymPy printing out reciprocal values when they should be
integral in "ceiling/floor" sympy functions.
"""
nexpr = expr
if not isinstance(expr, sympy.Basic):
return expr
# The properties avoid matching the silly case "ceiling(N/32)" as
# ceiling of 1/N and 1/32
a = sympy.Wild('a', properties=[lambda k: k.is_Symbol or k.is_Integer])
b = sympy.Wild('b', properties=[lambda k: k.is_Symbol or k.is_Integer])
c = sympy.Wild('c')
d = sympy.Wild('d')
int_ceil = sympy.Function('int_ceil')
int_floor = sympy.Function('int_floor')
processed = 1
while processed > 0:
processed = 0
for ceil in nexpr.find(sympy.ceiling):
# Simple ceiling
m = ceil.match(sympy.ceiling(a / b))
if m is not None:
nexpr = nexpr.subs(ceil, int_ceil(m[a], m[b]))
processed += 1
continue
# Ceiling of ceiling: "ceil(ceil(c/d) / b)"
m = ceil.match(sympy.ceiling(int_ceil(c, d) / b))
if m is not None:
nexpr = nexpr.subs(ceil, int_ceil(int_ceil(m[c], m[d]), m[b]))
processed += 1
continue
# Ceiling of ceiling: "ceil(a / ceil(c/d))"
m = ceil.match(sympy.ceiling(a / int_ceil(c, d)))
if m is not None:
nexpr = nexpr.subs(ceil, int_ceil(m[a], int_ceil(m[c], m[d])))
processed += 1
continue
# Match ceiling of multiplication with our custom integer functions
m = ceil.match(sympy.ceiling(a * int_floor(c, d)))
if m is not None:
nexpr = nexpr.subs(ceil, m[a] * int_floor(m[c], m[d]))
processed += 1
continue
m = ceil.match(sympy.ceiling(a * int_ceil(c, d)))
if m is not None:
nexpr = nexpr.subs(ceil, m[a] * int_ceil(m[c], m[d]))
processed += 1
continue
return nexpr
def arctan_rule(integral):
integrand, symbol = integral
base, exp = integrand.as_base_exp()
if sympy.simplify(exp + 1) == 0:
a = sympy.Wild('a', exclude=[symbol])
b = sympy.Wild('b', exclude=[symbol])
match = base.match(a + b * symbol**2)
if match:
a, b = match[a], match[b]
if ((isinstance(a, sympy.Number) and a < 0)
or (isinstance(b, sympy.Number) and b < 0)):
return
if (sympy.ask(
sympy.Q.negative(a) | sympy.Q.negative(b)
| sympy.Q.is_true(a <= 0) | sympy.Q.is_true(b <= 0))):
return
if a != 1 or b != 1:
b_condition = b >= 0
u_var = sympy.Dummy("u")
rewritten = (sympy.Integer(1) / a) * (base / a)**(-1)
u_func = sympy.sqrt(sympy.sympify(b) / a) * symbol
constant = 1 / sympy.sqrt(sympy.sympify(b) / a)
substituted = rewritten.subs(u_func, u_var)
if a == b:
substep = ArctanRule(integrand, symbol)
else:
subrule = ArctanRule(substituted, u_var)
if constant != 1:
b_condition = b > 0
subrule = ConstantTimesRule(constant, substituted,
subrule, substituted,
symbol)
substep = URule(u_var, u_func, constant, subrule,
integrand, symbol)
if a != 1:
other = (base / a)**(-1)
substep = ConstantTimesRule(
sympy.Integer(1) / a, other, substep, integrand,
symbol)
return PiecewiseRule(
[(substep, sympy.And(a > 0, b_condition))], integrand,
symbol)
return ArctanRule(integrand, symbol)
def canonise_log(equation):
expanded_log = sympy.expand_log(equation, force=True)
terms = expanded_log.as_ordered_terms()
a, b = sympy.Wild('a'), sympy.Wild('b')
total_interior = 1
for term in terms:
if sympy.ask(sympy.Q.complex(term)):
term_interior *= -1
else:
term_interior = term.match(sympy.log(a) / b)[a]
if term.could_extract_minus_sign():
total_interior /= term_interior
else:
total_interior *= term_interior
if isinstance(total_interior, sympy.Add):
invert = False
elif isinstance(total_interior, sympy.Mul):
match = total_interior.together().match(x / b) # for some reason, (x/3).match(a/b) gives {a: 1/3, b: 1/x} so we have to use a workaround
if match is not None:
invert = False
else:
match = total_interior.together().match(a / b)
degree_numerator = 0 if isinstance(match[a], sympy.Rational) else match[a].as_poly().degree()
degree_denominator = 0 if isinstance(match[b], sympy.Rational) else match[b].as_poly().degree()
if degree_numerator < degree_denominator:
invert = True
else:
invert = False
elif isinstance(total_interior, sympy.Pow):
index = total_interior.as_base_exp()[1]
if index < 0:
invert = True
else:
invert = False
else: # for debugging - wtf kind of c-c-c-class is it???
print(total_interior, type(total_interior))
if invert:
return -sympy.log((1 / total_interior).together(), evaluate=False) / terms[0].as_coeff_Mul()[0].q
else:
return sympy.log(total_interior.together(), evaluate=False) / terms[0].as_coeff_Mul()[0].q
def _set_axis_and_point(self, cond_eq: sympy.Equality, coords: cs.CoordinateSystem):
coord_vars = coords.axises()
self.axis, self.axis_point = None, None
U = sympy.Function("u")
u_entrances = list(cond_eq.find(U))
if len(u_entrances) != 1:
raise ValueError(
"Cannot contain different args in U for derivative and u function"
)
a1 = sympy.Wild("a1", properties=[lambda x: x.is_constant])
a2 = sympy.Wild("a2", properties=[lambda x: x.is_constant])
u_pow = sympy.Wild("u_pow", properties=[lambda x: x.is_constant])
axis = sympy.Wild("axis", properties=[lambda x: x.is_symbol])
U_pattern = sympy.WildFunction("U_pattern", nargs=len(coords.axises()))
res = cond_eq.lhs.match(
a1 * U_pattern ** u_pow + a2 * sympy.Derivative(U_pattern, axis)
)
if res is None:
raise ValueError(f"Cannot parse {cond_eq}")
self.a1 = res[a1]
self.a2 = res[a2]
u_with_args = res[U_pattern]
if self.a1 != 0 and self.a2 == 0:
self.kind = ConditionKind.First.name
elif self.a1 == 0 and self.a2 != 0:
self.kind = ConditionKind.Second.name
elif self.a1 != 0 and self.a2 != 0:
self.kind = ConditionKind.Third.name
else:
raise ValueError(
f"Left side of boundary condition: {cond_eq} does"
"not contain u function"
)
for i, arg in enumerate(u_with_args.args):
if arg.is_constant():
self.axis = coord_vars[i]
self.axis_point = sympy.fcode(arg.evalf())
self._axis_point = arg
break
if self.axis is None:
raise ValueError(
"Bondary condition cannot be parsed"
"Probably you did not specified U arguments or"
"there is no constant in arguments"
)
def quadratic_denom_rule(integral):
integrand, symbol = integral
a = sympy.Wild('a', exclude=[symbol])
b = sympy.Wild('b', exclude=[symbol])
c = sympy.Wild('c', exclude=[symbol])
match = integrand.match(a / (b * symbol ** 2 + c))
if not match:
return
a, b, c = match[a], match[b], match[c]
return PiecewiseRule([(ArctanRule(a, b, c, integrand, symbol), sympy.Gt(c / b, 0)),
(ArccothRule(a, b, c, integrand, symbol), sympy.And(sympy.Gt(symbol ** 2, -c / b), sympy.Lt(c / b, 0))),
(ArctanhRule(a, b, c, integrand, symbol), sympy.And(sympy.Lt(symbol ** 2, -c / b), sympy.Lt(c / b, 0))),
], integrand, symbol)
def pinv(A_expr):
a_wild_expr = sympy.Wild("a_wild")
assert A_expr == A_expr.replace(sympy.conjugate(a_wild_expr),a_wild_expr)
A_pinv_expr = A_expr.pinv()
A_pinv_expr = A_pinv_expr.replace(sympy.conjugate(a_wild_expr),a_wild_expr)
return A_pinv_expr
def can_be_applied(self, dim_exprs, variable_context, node_range,
orig_edges, dim_index, total_dims):
# Pattern does not support unions of expressions. TODO: Support
if len(dim_exprs) > 1: return False
dexpr = dim_exprs[0]
# Create a wildcard that excludes current map's parameters
cst = sympy.Wild('cst', exclude=variable_context[-1])
# Range case
if isinstance(dexpr, tuple) and len(dexpr) == 3:
# Try to match a constant expression for the range
for rngelem in dexpr:
if dtypes.isconstant(rngelem):
continue
matches = rngelem.match(cst)
if matches is None or len(matches) != 1:
return False
if not matches[cst].is_constant():
return False
else: # Single element case
# Try to match a constant expression
if not dtypes.isconstant(dexpr):
matches = dexpr.match(cst)
if matches is None or len(matches) != 1:
return False
if not matches[cst].is_constant():
return False
return True
def test_floor_ceil_float_no_optimization():
x, y = pystencils.fields('x,y: float32[2d]')
a, b, c = sp.symbols('a, b, c')
int_symbol = sp.Symbol('int_symbol', integer=True)
typed_symbol = pystencils.TypedSymbol('typed_symbol',
create_type('float32'))
assignments = pystencils.AssignmentCollection({
a:
sp.floor(1),
b:
sp.ceiling(typed_symbol),
c:
sp.floor(int_symbol),
y.center():
sp.ceiling(x.center()) + sp.floor(x.center())
})
assert not typed_symbol.is_integer
print(sp.simplify(sp.ceiling(typed_symbol)))
print(assignments)
wild_floor = sp.floor(sp.Wild('w1'))
assert not sp.floor(int_symbol).match(wild_floor)
assert sp.floor(a).match(wild_floor)
assert assignments.find(wild_floor)
def trig_sindouble_rule(integral):
integrand, symbol = integral
a = sympy.Wild('a', exclude=[sympy.sin(2*symbol)])
match = integrand.match(sympy.sin(2*symbol)*a)
if match:
sin_double = 2*sympy.sin(symbol)*sympy.cos(symbol)/sympy.sin(2*symbol)
return integral_steps(integrand * sin_double, symbol)
def _modify_cond(self, condition, var, step):
condition = pystr_to_symbolic(condition.as_string)
itersym = pystr_to_symbolic(var)
# Find condition by matching expressions
end: Optional[sp.Expr] = None
a = sp.Wild('a')
op = ''
match = condition.match(itersym < a)
if match:
op = '<'
end = match[a] - self.count * step
if end is None:
match = condition.match(itersym <= a)
if match:
op = '<='
end = match[a] - self.count * step
if end is None:
match = condition.match(itersym > a)
if match:
op = '>'
end = match[a] - self.count * step
if end is None:
match = condition.match(itersym >= a)
if match:
op = '>='
end = match[a] - self.count * step
if len(op) == 0:
raise ValueError('Cannot match loop condition for peeling')
res = str(itersym) + op + str(end)
return res
def expr_to_code(expr, symbols_replace, function_replace, display=None):
import re
if display is not None:
print("+ expression to code {}".format(display), end="\r")
# sympy function to replace into STL C++ functions
math_to_stl = [(f, sp.Function("std::" + str(f), nargs=1))
for f in (sp.sin, sp.cos, sp.exp)]
math_to_stl.append((sp.sqrt, sp.Function("std::sqrt", nargs=1)))
# first step: replace symbols
tmp = expr.subs(symbols_replace)
# second step: use STL functions
for old, new in math_to_stl:
tmp = tmp.subs(old, new)
# next step: use user-defined functions
for old, new in function_replace:
tmp = tmp.replace(old, new)
# last step: convert all division by 2 by multiplication by 0.5
a = sp.Wild('a')
tmp = tmp.replace(a / 2, 0.5 * a, exact=True).evalf()
# and return a string (where remove every `1.0*` pattern where there is not a number before)
return re.sub(r"([^0-9])1\.0\*", r"\1", str(tmp))