def _rebuild(expr):
if not isinstance(expr, Basic):
return expr
if not expr.args:
return expr
if iterable(expr):
new_args = [_rebuild(arg) for arg in expr]
return expr.func(*new_args)
if expr in subs:
return subs[expr]
orig_expr = expr
if expr in opt_subs:
expr = opt_subs[expr]
# If enabled, parse Muls and Adds arguments by order to ensure
# replacement order independent from hashes
if order != 'none':
if isinstance(expr, (Mul, MatMul)):
c, nc = expr.args_cnc()
if c == [1]:
args = nc
else:
args = list(ordered(c)) + nc
elif isinstance(expr, (Add, MatAdd)):
args = list(ordered(expr.args))
else:
args = expr.args
else:
args = expr.args
new_args = list(map(_rebuild, args))
if new_args != args:
new_expr = expr.func(*new_args)
else:
new_expr = expr
if orig_expr in to_eliminate:
try:
sym = next(symbols)
except StopIteration:
raise ValueError("Symbols iterator ran out of symbols.")
if isinstance(orig_expr, MatrixExpr):
sym = MatrixSymbol(sym.name, orig_expr.rows,
orig_expr.cols)
subs[orig_expr] = sym
replacements.append((sym, new_expr))
return sym
else:
return new_expr
def _linear_neq_order1_type3(match_):
r"""
System of n first-order nonconstant-coefficient linear homogeneous differential equations
.. math::
X' = A(t) X
where $X$ is the vector of $n$ dependent variables, $t$ is the dependent variable, $X'$
is the first order differential of $X$ with respect to $t$ and $A(t)$ is a $n \times n$
coefficient matrix.
Let us define $B$ as antiderivative of coefficient matrix $A$:
.. math::
B(t) = \int A(t) dt
If the system of ODEs defined above is such that its antiderivative $B(t)$ commutes with
$A(t)$ itself, then, the solution of the above system is given as:
.. math::
X = \exp(B(t)) C
where $C$ is the vector of constants.
"""
# Some parts of code is repeated, this needs to be taken care of
# The constant vector obtained here can be done so in the match
# function itself.
eq = match_['eq']
func = match_['func']
fc = match_['func_coeff']
n = len(eq)
t = list(list(eq[0].atoms(Derivative))[0].atoms(Symbol))[0]
constants = numbered_symbols(prefix='C', cls=Symbol, start=1)
# This needs to be modified in future so that fc is only of type Matrix
M = -fc if type(fc) is Matrix else Matrix(n, n, lambda i,j:-fc[i,func[j],0])
Cvect = Matrix(list(next(constants) for _ in range(n)))
# The code in if block will be removed when it is made sure
# that the code works without the statements in if block.
if "commutative_antiderivative" not in match_:
B, is_commuting = _is_commutative_anti_derivative(M, t)
# This course is subject to change
if not is_commuting:
return None
else:
B = match_['commutative_antiderivative']
sol_vector = B.exp() * Cvect
# The expand_mul is added to handle the solutions so that
# the exponential terms are collected properly.
sol_vector = [collect(expand_mul(s), ordered(s.atoms(exp)), exact=True) for s in sol_vector]
sol_dict = [Eq(func[i], sol_vector[i]) for i in range(n)]
return sol_dict
def _match_common_args(Func, funcs):
if order != 'none':
funcs = list(ordered(funcs))
else:
funcs = sorted(funcs, key=lambda x: len(x.args))
func_args = [set(e.args) for e in funcs]
for i in xrange(len(func_args)):
for j in xrange(i + 1, len(func_args)):
com_args = func_args[i].intersection(func_args[j])
if len(com_args) > 1:
com_func = Func(*com_args)
# for all sets, replace the common symbols by the function
# over them, to allow recursive matches
diff_i = func_args[i].difference(com_args)
func_args[i] = diff_i | set([com_func])
if diff_i:
opt_subs[funcs[i]] = Func(Func(*diff_i), com_func,
evaluate=False)
diff_j = func_args[j].difference(com_args)
func_args[j] = diff_j | set([com_func])
opt_subs[funcs[j]] = Func(Func(*diff_j), com_func,
evaluate=False)
for k in xrange(j + 1, len(func_args)):
if not com_args.difference(func_args[k]):
diff_k = func_args[k].difference(com_args)
func_args[k] = diff_k | set([com_func])
opt_subs[funcs[k]] = Func(Func(*diff_k), com_func,
evaluate=False)
def _eval_simplify(self, ratio, measure, rational, inverse):
args = [
a._eval_simplify(ratio, measure, rational, inverse)
for a in self.args
]
for i, (expr, cond) in enumerate(args):
# try to simplify conditions and the expression for
# equalities that are part of the condition, e.g.
# Piecewise((n, And(Eq(n,0), Eq(n + m, 0))), (1, True))
# -> Piecewise((0, And(Eq(n, 0), Eq(m, 0))), (1, True))
if isinstance(cond, And):
eqs, other = sift(cond.args,
lambda i: isinstance(i, Equality),
binary=True)
elif isinstance(cond, Equality):
eqs, other = [cond], []
else:
eqs = other = []
if eqs:
eqs = list(ordered(eqs))
for j, e in enumerate(eqs):
# these blessed lhs objects behave like Symbols
# and the rhs are simple replacements for the "symbols"
if isinstance(e.lhs, (Symbol, UndefinedFunction)) and \
isinstance(e.rhs,
(Rational, NumberSymbol,
Symbol, UndefinedFunction)):
expr = expr.subs(*e.args)
eqs[j + 1:] = [ei.subs(*e.args) for ei in eqs[j + 1:]]
other = [ei.subs(*e.args) for ei in other]
cond = And(*(eqs + other))
args[i] = args[i].func(expr, cond)
return self.func(*args)
def _match_common_args(Func, funcs):
if order != 'none':
funcs = list(ordered(funcs))
else:
funcs = sorted(funcs, key=lambda x: len(x.args))
func_args = [set(e.args) for e in funcs]
for i in xrange(len(func_args)):
for j in xrange(i + 1, len(func_args)):
com_args = func_args[i].intersection(func_args[j])
if len(com_args) > 1:
com_func = Func(*com_args)
# for all sets, replace the common symbols by the function
# over them, to allow recursive matches
diff_i = func_args[i].difference(com_args)
func_args[i] = diff_i | set([com_func])
if diff_i:
opt_subs[funcs[i]] = Func(Func(*diff_i), com_func,
evaluate=False)
diff_j = func_args[j].difference(com_args)
func_args[j] = diff_j | set([com_func])
opt_subs[funcs[j]] = Func(Func(*diff_j), com_func,
evaluate=False)
for k in xrange(j + 1, len(func_args)):
if not com_args.difference(func_args[k]):
diff_k = func_args[k].difference(com_args)
func_args[k] = diff_k | set([com_func])
opt_subs[funcs[k]] = Func(Func(*diff_k), com_func,
evaluate=False)
def _eval_simplify(self, ratio, measure, rational, inverse):
args = [a._eval_simplify(ratio, measure, rational, inverse)
for a in self.args]
for i, (expr, cond) in enumerate(args):
# try to simplify conditions and the expression for
# equalities that are part of the condition, e.g.
# Piecewise((n, And(Eq(n,0), Eq(n + m, 0))), (1, True))
# -> Piecewise((0, And(Eq(n, 0), Eq(m, 0))), (1, True))
if isinstance(cond, And):
eqs, other = sift(cond.args,
lambda i: isinstance(i, Equality), binary=True)
elif isinstance(cond, Equality):
eqs, other = [cond], []
else:
eqs = other = []
if eqs:
eqs = list(ordered(eqs))
for j, e in enumerate(eqs):
# these blessed lhs objects behave like Symbols
# and the rhs are simple replacements for the "symbols"
if isinstance(e.lhs, (Symbol, UndefinedFunction)) and \
isinstance(e.rhs,
(Rational, NumberSymbol,
Symbol, UndefinedFunction)):
expr = expr.subs(*e.args)
eqs[j + 1:] = [ei.subs(*e.args) for ei in eqs[j + 1:]]
other = [ei.subs(*e.args) for ei in other]
cond = And(*(eqs + other))
args[i] = args[i].func(expr, cond)
return self.func(*args)
def _rebuild(expr):
if expr.is_Atom:
return expr
if iterable(expr):
new_args = [_rebuild(arg) for arg in expr]
return expr.func(*new_args)
if expr in subs:
return subs[expr]
orig_expr = expr
if expr in opt_subs:
expr = opt_subs[expr]
# If enabled, parse Muls and Adds arguments by order to ensure
# replacement order independent from hashes
if order != 'none':
if expr.is_Mul:
c, nc = expr.args_cnc()
args = list(ordered(c)) + nc
elif expr.is_Add:
args = list(ordered(expr.args))
else:
args = expr.args
else:
args = expr.args
new_args = list(map(_rebuild, args))
if new_args != args:
new_expr = expr.func(*new_args)
else:
new_expr = expr
if orig_expr in to_eliminate:
try:
sym = next(symbols)
except StopIteration:
raise ValueError("Symbols iterator ran out of symbols.")
subs[orig_expr] = sym
replacements.append((sym, new_expr))
return sym
else:
return new_expr
def _rebuild(expr):
if not expr.args:
return expr
if iterable(expr):
new_args = [_rebuild(arg) for arg in expr]
return expr.func(*new_args)
if expr in subs:
return subs[expr]
orig_expr = expr
if expr in opt_subs:
expr = opt_subs[expr]
# If enabled, parse Muls and Adds arguments by order to ensure
# replacement order independent from hashes
if order != 'none':
if expr.is_Mul:
c, nc = expr.args_cnc()
args = list(ordered(c)) + nc
elif expr.is_Add:
args = list(ordered(expr.args))
else:
args = expr.args
else:
args = expr.args
new_args = list(map(_rebuild, args))
if new_args != args:
new_expr = expr.func(*new_args)
else:
new_expr = expr
if orig_expr in to_eliminate:
try:
sym = next(symbols)
except StopIteration:
raise ValueError("Symbols iterator ran out of symbols.")
subs[orig_expr] = sym
replacements.append((sym, new_expr))
return sym
else:
return new_expr
def identities(expr, **hints):
""" Apply to expr the identities specified in **hints.
In the following, w is scalar, wa/wb are vectors.
Hints: set a flag to True to apply the identity. By default all
flags are set to False.
prod_div = True
nabla & (w * wa) = (Grad(w) & wa) + (w * (nabla & wa))
prod_curl = True
nabla ^ (w * wa) = (Grad(w) ^ wa) + (w * (nabla ^ wa))
div_of_cross = True
nabla & (wa ^ wb) = ((nabla ^ wa) & wb) - ((nabla ^ wb) & wa)
curl_of_cross = True
nabla ^ (wa ^ wb) = ((nabla & wb) * wa) + Advection(wb, wa) - ((nabla & wa) * wb) - Advection(wa, wb)
grad_of_dot = True
Grad(wa & wb) = Advection(wa, wb) + Advection(wb, wa) + (wa ^ (nabla ^ wb)) + (wb ^ (nabla ^ wa))
curl_of_grad = True
nabla ^ Grad(w) = VectorZero()
div_of_curl = True
nabla & (nabla ^ wa) = S.Zero
curl_of_curl = True
nabla ^ (nabla ^ wa) = Grad(nabla & wa) - Laplace(wa)
"""
bac_cab = hints.get('bac_cab', False)
if bac_cab:
expr = bac_cab_backward(expr)
for hint, val in hints.items():
if hint in _id.keys() and (val == True):
pattern, subs = _id[hint]
found = list(ordered(list(expr.find(pattern))))
print("found list", found)
for i, f in enumerate(found):
# proceed only if there is a match
# Say we are in the previous iteration, i. Then we substitute
# the previous match into found[i+1]. Now we are into iteration
# i+1, but because we substituted a previous result into it, it
# might not be "a pattern" anymore. Need to check it!
m = f.match(pattern)
print("\tf", f, m)
if m:
fsubs = subs.xreplace(m)
# update found list
for j in range(i + 1, len(found)):
found[j] = found[j].subs(f, fsubs)
expr = expr.subs(f, fsubs)
return expr
def find_double_cross(expr):
""" Given a vector expression, return a list of elements satisfying
the pattern A x (B x C), where A, B, C are vectors and `x` is the
cross product. The list is ordered in such a way that nested matches
comes first (similarly to scanning the expression tree from bottom
to top).
"""
A, B, C = [WVS(t) for t in ["A", "B", "C"]]
pattern = A ^ (B ^ C)
return list(ordered(list(expr.find(pattern))))
def _collect_coeff_from_product(expr, pattern):
""" Collect the coefficients of nested dot (or cross) products. For example:
(x * A) ^ ((y * B) ^ (z * C)) = (x * y * z) * A ^ (B ^ C)
"""
found = list(ordered(list(expr.find(pattern))))
for i, f in enumerate(found):
newf = _as_coeff_product(f)
for j in range(i + 1, len(found)):
found[j] = found[j].xreplace({f: newf})
expr = expr.subs({f: newf})
return expr
def test_ordered():
assert list(ordered((x, y), hash, default=False)) in [[x, y], [y, x]]
assert list(ordered((x, y), hash, default=False)) == list(ordered((y, x), hash, default=False))
assert list(ordered((x, y))) == [x, y]
seq, keys = [[[1, 2, 1], [0, 3, 1], [1, 1, 3], [2], [1]], (lambda x: len(x), lambda x: sum(x))]
assert list(ordered(seq, keys, default=False, warn=False)) == [[1], [2], [1, 2, 1], [0, 3, 1], [1, 1, 3]]
raises(ValueError, lambda: list(ordered(seq, keys, default=False, warn=True)))
def func(expr, pattern, prev=None):
# debug("\n", hints)
old = expr
found = list(ordered(list(expr.find(pattern))))
# print("func", expr)
# debug("found", found)
# print("found", found)
for f in found:
fexp = f.expand(**hints)
# debug("\t fexp", fexp)
if f != fexp:
expr = expr.xreplace({f: fexp})
# debug("expr", expr)
if old != expr:
expr = func(expr, pattern)
return expr
def test_ordered():
assert list(ordered((x, y), hash, default=False)) in [[x, y], [y, x]]
assert list(ordered((x, y), hash, default=False)) == \
list(ordered((y, x), hash, default=False))
assert list(ordered((x, y))) == [x, y]
seq, keys = [[[1, 2, 1], [0, 3, 1], [1, 1, 3], [2], [1]],
(lambda x: len(x), lambda x: sum(x))]
assert list(ordered(seq, keys, default=False, warn=False)) == \
[[1], [2], [1, 2, 1], [0, 3, 1], [1, 1, 3]]
raises(ValueError,
lambda: list(ordered(seq, keys, default=False, warn=True)))
def get_lambda(expr, **kwargs):
"""Create a lambda function to numerically evaluate expr by sorting
alphabetically the function arguments.
Parameters
----------
expr : Expr
The Sympy expression to convert.
**kwargs
Other keyword arguments to the function lambdify.
Returns
-------
s : list
The function signature: a list of the ordered function arguments.
f : lambda
The generated lambda function.ò
"""
signature = list(ordered(expr.free_symbols))
return signature, lambdify(signature, expr, **kwargs)
def _linear_neq_order1_type1(match_):
r"""
System of n first-order constant-coefficient linear homogeneous differential equations
.. math:: y'_k = a_{k1} y_1 + a_{k2} y_2 +...+ a_{kn} y_n; k = 1,2,...,n
or that can be written as `\vec{y'} = A . \vec{y}`
where `\vec{y}` is matrix of `y_k` for `k = 1,2,...n` and `A` is a `n \times n` matrix.
These equations are equivalent to a first order homogeneous linear
differential equation.
The system of ODEs described above has a unique solution, namely:
.. math ::
\vec{y} = \exp(A t) C
where $t$ is the independent variable and $C$ is a vector of n constants. These are constants
from the integration.
"""
eq = match_['eq']
func = match_['func']
fc = match_['func_coeff']
n = len(eq)
t = list(list(eq[0].atoms(Derivative))[0].atoms(Symbol))[0]
constants = numbered_symbols(prefix='C', cls=Symbol, start=1)
# This needs to be modified in future so that fc is only of type Matrix
M = -fc if type(fc) is Matrix else Matrix(n, n,
lambda i, j: -fc[i, func[j], 0])
P, J = matrix_exp_jordan_form(M, t)
P = simplify(P)
Cvect = Matrix(list(next(constants) for _ in range(n)))
sol_vector = P * (J * Cvect)
gens = sol_vector.atoms(exp)
sol_vector = [collect(s, ordered(gens), exact=True) for s in sol_vector]
sol_dict = [Eq(func[i], sol_vector[i]) for i in range(n)]
return sol_dict
def _find_and_replace(expr, pattern, rep, matches=None):
""" Given an expression, a search `pattern` and a replacement
pattern `rep`, search for `pattern` in the expression and substitute
it with `rep`. If matches is given, skip the pattern search and
substitute each match with the corresponding `rep` pattern.
"""
if not matches:
# list of matches ordered accordingly to the length of the match.
# The shorter the match expression, the lower it should be in the
# expression tree. Substitute from the bottom up!
found = list(ordered(list(expr.find(pattern))))
if len(found) > 0:
f = found[0]
expr = expr.xreplace({f: rep.xreplace(f.match(pattern))})
# repeat the procedure with the updated expression
expr = _find_and_replace(expr, pattern, rep)
return expr
if not isinstance(matches, (list, tuple)):
matches = [matches]
for match in matches:
expr = expr.xreplace({match: rep.xreplace(match.match(pattern))})
return expr
def get_lambda(expr, modules=["numpy", "scipy"], **kwargs):
""" Create a lambda function to numerically evaluate expr by sorting
alphabetically the function arguments.
Parameters
----------
expr : Expr
The Sympy expression to convert.
modules : str
The numerical module to use for evaluation. Default to
["numpy", "scipy"]. See help(lambdify) for other choices.
**kwargs
Other keyword arguments to the function lambdify.
Returns
-------
s : list
The function signature: a list of the ordered function arguments.
f : lambda
The generated lambda function.ò
"""
from sympy.utilities.iterables import ordered
signature = list(ordered(expr.free_symbols))
return signature, lambdify(signature, expr, modules=modules, **kwargs)
def _intervals(self, sym):
"""Return a list of unique tuples, (a, b, e, i), where a and b
are the lower and upper bounds in which the expression e of
argument i in self is defined and a < b (when involving
numbers) or a <= b when involving symbols.
If there are any relationals not involving sym, or any
relational cannot be solved for sym, NotImplementedError is
raised. The calling routine should have removed such
relationals before calling this routine.
The evaluated conditions will be returned as ranges.
Discontinuous ranges will be returned separately with
identical expressions. The first condition that evaluates to
True will be returned as the last tuple with a, b = -oo, oo.
"""
from sympy.solvers.inequalities import _solve_inequality
from sympy.logic.boolalg import to_cnf, distribute_or_over_and
assert isinstance(self, Piecewise)
def _solve_relational(r):
if sym not in r.free_symbols:
nonsymfail(r)
rv = _solve_inequality(r, sym)
if isinstance(rv, Relational):
free = rv.args[1].free_symbols
if rv.args[0] != sym or sym in free:
raise NotImplementedError(filldedent('''
Unable to solve relational
%s for %s.''' % (r, sym)))
if rv.rel_op == '==':
# this equality has been affirmed to have the form
# Eq(sym, rhs) where rhs is sym-free; it represents
# a zero-width interval which will be ignored
# whether it is an isolated condition or contained
# within an And or an Or
rv = S.false
elif rv.rel_op == '!=':
try:
rv = Or(sym < rv.rhs, sym > rv.rhs)
except TypeError:
# e.g. x != I ==> all real x satisfy
rv = S.true
elif rv == (S.NegativeInfinity < sym) & (sym < S.Infinity):
rv = S.true
return rv
def nonsymfail(cond):
raise NotImplementedError(filldedent('''
A condition not involving
%s appeared: %s''' % (sym, cond)))
# make self canonical wrt Relationals
reps = {
r: _solve_relational(r) for r in self.atoms(Relational)}
# process args individually so if any evaluate, their position
# in the original Piecewise will be known
args = [i.xreplace(reps) for i in self.args]
# precondition args
expr_cond = []
default = idefault = None
for i, (expr, cond) in enumerate(args):
if cond is S.false:
continue
elif cond is S.true:
default = expr
idefault = i
break
cond = to_cnf(cond)
if isinstance(cond, And):
cond = distribute_or_over_and(cond)
if isinstance(cond, Or):
expr_cond.extend(
[(i, expr, o) for o in cond.args
if not isinstance(o, Equality)])
elif cond is not S.false:
expr_cond.append((i, expr, cond))
# determine intervals represented by conditions
int_expr = []
for iarg, expr, cond in expr_cond:
if isinstance(cond, And):
lower = S.NegativeInfinity
upper = S.Infinity
exclude = []
for cond2 in cond.args:
if isinstance(cond2, Equality):
lower = upper # ignore
break
elif isinstance(cond2, Unequality):
l, r = cond2.args
if l == sym:
exclude.append(r)
elif r == sym:
exclude.append(l)
else:
nonsymfail(cond2)
continue
elif cond2.lts == sym:
upper = Min(cond2.gts, upper)
elif cond2.gts == sym:
lower = Max(cond2.lts, lower)
else:
nonsymfail(cond2) # should never get here
if exclude:
exclude = list(ordered(exclude))
newcond = []
for i, e in enumerate(exclude):
if e < lower == True or e > upper == True:
continue
if not newcond:
newcond.append((None, lower)) # add a primer
newcond.append((newcond[-1][1], e))
newcond.append((newcond[-1][1], upper))
newcond.pop(0) # remove the primer
expr_cond.extend([(iarg, expr, And(i[0] < sym, sym < i[1])) for i in newcond])
continue
elif isinstance(cond, Relational):
lower, upper = cond.lts, cond.gts # part 1: initialize with givens
if cond.lts == sym: # part 1a: expand the side ...
lower = S.NegativeInfinity # e.g. x <= 0 ---> -oo <= 0
elif cond.gts == sym: # part 1a: ... that can be expanded
upper = S.Infinity # e.g. x >= 0 ---> oo >= 0
else:
nonsymfail(cond)
else:
raise NotImplementedError(
'unrecognized condition: %s' % cond)
lower, upper = lower, Max(lower, upper)
if (lower >= upper) is not S.true:
int_expr.append((lower, upper, expr, iarg))
if default is not None:
int_expr.append(
(S.NegativeInfinity, S.Infinity, default, idefault))
return list(uniq(int_expr))
def _handle_irel(self, x, handler):
"""Return either None (if the conditions of self depend only on x) else
a Piecewise expression whose expressions (handled by the handler that
was passed) are paired with the governing x-independent relationals,
e.g. Piecewise((A, a(x) & b(y)), (B, c(x) | c(y)) ->
Piecewise(
(handler(Piecewise((A, a(x) & True), (B, c(x) | True)), b(y) & c(y)),
(handler(Piecewise((A, a(x) & True), (B, c(x) | False)), b(y)),
(handler(Piecewise((A, a(x) & False), (B, c(x) | True)), c(y)),
(handler(Piecewise((A, a(x) & False), (B, c(x) | False)), True))
"""
# identify governing relationals
rel = self.atoms(Relational)
irel = list(ordered([r for r in rel if x not in r.free_symbols
and r not in (S.true, S.false)]))
if irel:
args = {}
exprinorder = []
for truth in product((1, 0), repeat=len(irel)):
reps = dict(zip(irel, truth))
# only store the true conditions since the false are implied
# when they appear lower in the Piecewise args
if 1 not in truth:
cond = None # flag this one so it doesn't get combined
else:
andargs = Tuple(*[i for i in reps if reps[i]])
free = list(andargs.free_symbols)
if len(free) == 1:
from sympy.solvers.inequalities import (
reduce_inequalities, _solve_inequality)
try:
t = reduce_inequalities(andargs, free[0])
# ValueError when there are potentially
# nonvanishing imaginary parts
except (ValueError, NotImplementedError):
# at least isolate free symbol on left
t = And(*[_solve_inequality(
a, free[0], linear=True)
for a in andargs])
else:
t = And(*andargs)
if t is S.false:
continue # an impossible combination
cond = t
expr = handler(self.xreplace(reps))
if isinstance(expr, self.func) and len(expr.args) == 1:
expr, econd = expr.args[0]
cond = And(econd, True if cond is None else cond)
# the ec pairs are being collected since all possibilities
# are being enumerated, but don't put the last one in since
# its expr might match a previous expression and it
# must appear last in the args
if cond is not None:
args.setdefault(expr, []).append(cond)
# but since we only store the true conditions we must maintain
# the order so that the expression with the most true values
# comes first
exprinorder.append(expr)
# convert collected conditions as args of Or
for k in args:
args[k] = Or(*args[k])
# take them in the order obtained
args = [(e, args[e]) for e in uniq(exprinorder)]
# add in the last arg
args.append((expr, True))
# if any condition reduced to True, it needs to go last
# and there should only be one of them or else the exprs
# should agree
trues = [i for i in range(len(args)) if args[i][1] is S.true]
if not trues:
# make the last one True since all cases were enumerated
e, c = args[-1]
args[-1] = (e, S.true)
else:
assert len(set([e for e, c in [args[i] for i in trues]])) == 1
args.append(args.pop(trues.pop()))
while trues:
args.pop(trues.pop())
return Piecewise(*args)
def _collapse_arguments(cls, args, **assumptions):
"""Remove redundant args.
Examples
========
>>> from sympy import Min, Max
>>> from sympy.abc import a, b, c, d, e
Any arg in parent that appears in any
parent-like function in any of the flat args
of parent can be removed from that sub-arg:
>>> Min(a, Max(b, Min(a, c, d)))
Min(a, Max(b, Min(c, d)))
If the arg of parent appears in an opposite-than parent
function in any of the flat args of parent that function
can be replaced with the arg:
>>> Min(a, Max(b, Min(c, d, Max(a, e))))
Min(a, Max(b, Min(a, c, d)))
"""
from sympy.utilities.iterables import ordered
from sympy.simplify.simplify import walk
if not args:
return args
args = list(ordered(args))
if cls == Min:
other = Max
else:
other = Min
# find global comparable max of Max and min of Min if a new
# value is being introduced in these args at position 0 of
# the ordered args
if args[0].is_number:
sifted = mins, maxs = [], []
for i in args:
for v in walk(i, Min, Max):
if v.args[0].is_comparable:
sifted[isinstance(v, Max)].append(v)
small = Min.identity
for i in mins:
v = i.args[0]
if v.is_number and (v < small) == True:
small = v
big = Max.identity
for i in maxs:
v = i.args[0]
if v.is_number and (v > big) == True:
big = v
# at the point when this function is called from __new__,
# there may be more than one numeric arg present since
# local zeros have not been handled yet, so look through
# more than the first arg
if cls == Min:
for i in range(len(args)):
if not args[i].is_number:
break
if (args[i] < small) == True:
small = args[i]
elif cls == Max:
for i in range(len(args)):
if not args[i].is_number:
break
if (args[i] > big) == True:
big = args[i]
T = None
if cls == Min:
if small != Min.identity:
other = Max
T = small
elif big != Max.identity:
other = Min
T = big
if T is not None:
# remove numerical redundancy
for i in range(len(args)):
a = args[i]
if isinstance(a, other):
a0 = a.args[0]
if ((a0 > T) if other == Max else (a0 < T)) == True:
args[i] = cls.identity
# remove redundant symbolic args
def do(ai, a):
if not isinstance(ai, (Min, Max)):
return ai
cond = a in ai.args
if not cond:
return ai.func(*[do(i, a) for i in ai.args],
evaluate=False)
if isinstance(ai, cls):
return ai.func(*[do(i, a) for i in ai.args if i != a],
evaluate=False)
return a
for i, a in enumerate(args):
args[i + 1:] = [do(ai, a) for ai in args[i + 1:]]
# factor out common elements as for
# Min(Max(x, y), Max(x, z)) -> Max(x, Min(y, z))
# and vice versa when swapping Min/Max -- do this only for the
# easy case where all functions contain something in common;
# trying to find some optimal subset of args to modify takes
# too long
if len(args) > 1:
common = None
remove = []
sets = []
for i in range(len(args)):
a = args[i]
if not isinstance(a, other):
continue
s = set(a.args)
common = s if common is None else (common & s)
if not common:
break
sets.append(s)
remove.append(i)
if common:
sets = filter(None, [s - common for s in sets])
sets = [other(*s, evaluate=False) for s in sets]
for i in reversed(remove):
args.pop(i)
oargs = [cls(*sets)] if sets else []
oargs.extend(common)
args.append(other(*oargs, evaluate=False))
return args
def _handle_irel(self, x, handler):
"""Return either None (if the conditions of self depend only on x) else
a Piecewise expression whose expressions (handled by the handler that
was passed) are paired with the governing x-independent relationals,
e.g. Piecewise((A, a(x) & b(y)), (B, c(x) | c(y)) ->
Piecewise(
(handler(Piecewise((A, a(x) & True), (B, c(x) | True)), b(y) & c(y)),
(handler(Piecewise((A, a(x) & True), (B, c(x) | False)), b(y)),
(handler(Piecewise((A, a(x) & False), (B, c(x) | True)), c(y)),
(handler(Piecewise((A, a(x) & False), (B, c(x) | False)), True))
"""
# identify governing relationals
rel = self.atoms(Relational)
irel = list(
ordered([
r for r in rel
if x not in r.free_symbols and r not in (S.true, S.false)
]))
if irel:
args = {}
exprinorder = []
for truth in product((1, 0), repeat=len(irel)):
reps = dict(zip(irel, truth))
# only store the true conditions since the false are implied
# when they appear lower in the Piecewise args
if 1 not in truth:
cond = None # flag this one so it doesn't get combined
else:
andargs = Tuple(*[i for i in reps if reps[i]])
free = list(andargs.free_symbols)
if len(free) == 1:
from sympy.solvers.inequalities import (
reduce_inequalities, _solve_inequality)
try:
t = reduce_inequalities(andargs, free[0])
# ValueError when there are potentially
# nonvanishing imaginary parts
except (ValueError, NotImplementedError):
# at least isolate free symbol on left
t = And(*[
_solve_inequality(a, free[0], linear=True)
for a in andargs
])
else:
t = And(*andargs)
if t is S.false:
continue # an impossible combination
cond = t
expr = handler(self.xreplace(reps))
if isinstance(expr, self.func) and len(expr.args) == 1:
expr, econd = expr.args[0]
cond = And(econd, True if cond is None else cond)
# the ec pairs are being collected since all possibilities
# are being enumerated, but don't put the last one in since
# its expr might match a previous expression and it
# must appear last in the args
if cond is not None:
args.setdefault(expr, []).append(cond)
# but since we only store the true conditions we must maintain
# the order so that the expression with the most true values
# comes first
exprinorder.append(expr)
# convert collected conditions as args of Or
for k in args:
args[k] = Or(*args[k])
# take them in the order obtained
args = [(e, args[e]) for e in uniq(exprinorder)]
# add in the last arg
args.append((expr, True))
# if any condition reduced to True, it needs to go last
# and there should only be one of them or else the exprs
# should agree
trues = [i for i in range(len(args)) if args[i][1] is S.true]
if not trues:
# make the last one True since all cases were enumerated
e, c = args[-1]
args[-1] = (e, S.true)
else:
assert len(set([e for e, c in [args[i] for i in trues]])) == 1
args.append(args.pop(trues.pop()))
while trues:
args.pop(trues.pop())
return Piecewise(*args)
def piecewise_fold(expr):
"""
Takes an expression containing a piecewise function and returns the
expression in piecewise form. In addition, any ITE conditions are
rewritten in negation normal form and simplified.
Examples
========
>>> from sympy import Piecewise, piecewise_fold, sympify as S
>>> from sympy.abc import x
>>> p = Piecewise((x, x < 1), (1, S(1) <= x))
>>> piecewise_fold(x*p)
Piecewise((x**2, x < 1), (x, True))
See Also
========
Piecewise
"""
if not isinstance(expr, Basic) or not expr.has(Piecewise):
return expr
new_args = []
if isinstance(expr, (ExprCondPair, Piecewise)):
for e, c in expr.args:
if not isinstance(e, Piecewise):
e = piecewise_fold(e)
# we don't keep Piecewise in condition because
# it has to be checked to see that it's complete
# and we convert it to ITE at that time
assert not c.has(Piecewise) # pragma: no cover
if isinstance(c, ITE):
c = c.to_nnf()
c = simplify_logic(c, form='cnf')
if isinstance(e, Piecewise):
new_args.extend([(piecewise_fold(ei), And(ci, c))
for ei, ci in e.args])
else:
new_args.append((e, c))
else:
from sympy.utilities.iterables import cartes, sift, common_prefix
# Given
# P1 = Piecewise((e11, c1), (e12, c2), A)
# P2 = Piecewise((e21, c1), (e22, c2), B)
# ...
# the folding of f(P1, P2) is trivially
# Piecewise(
# (f(e11, e21), c1),
# (f(e12, e22), c2),
# (f(Piecewise(A), Piecewise(B)), True))
# Certain objects end up rewriting themselves as thus, so
# we do that grouping before the more generic folding.
# The following applies this idea when f = Add or f = Mul
# (and the expression is commutative).
if expr.is_Add or expr.is_Mul and expr.is_commutative:
p, args = sift(expr.args, lambda x: x.is_Piecewise, binary=True)
pc = sift(p, lambda x: tuple([c for e, c in x.args]))
for c in list(ordered(pc)):
if len(pc[c]) > 1:
pargs = [list(i.args) for i in pc[c]]
# the first one is the same; there may be more
com = common_prefix(*[[i.cond for i in j] for j in pargs])
n = len(com)
collected = []
for i in range(n):
collected.append(
(expr.func(*[ai[i].expr for ai in pargs]), com[i]))
remains = []
for a in pargs:
if n == len(a): # no more args
continue
if a[n].cond == True: # no longer Piecewise
remains.append(a[n].expr)
else: # restore the remaining Piecewise
remains.append(Piecewise(*a[n:], evaluate=False))
if remains:
collected.append((expr.func(*remains), True))
args.append(Piecewise(*collected, evaluate=False))
continue
args.extend(pc[c])
else:
args = expr.args
# fold
folded = list(map(piecewise_fold, args))
for ec in cartes(*[(i.args if isinstance(i, Piecewise) else [(i,
true)])
for i in folded]):
e, c = zip(*ec)
new_args.append((expr.func(*e), And(*c)))
return Piecewise(*new_args)
def __hash__(self): # Factors
keys = tuple(ordered(self.factors.keys()))
values = [self.factors[k] for k in keys]
return hash((keys, values))
def __repr__(self): # Factors
return "Factors({%s})" % ', '.join(
['%s: %s' % (k, v) for k, v in ordered(self.factors.items())])
def piecewise_fold(expr):
"""
Takes an expression containing a piecewise function and returns the
expression in piecewise form. In addition, any ITE conditions are
rewritten in negation normal form and simplified.
Examples
========
>>> from sympy import Piecewise, piecewise_fold, sympify as S
>>> from sympy.abc import x
>>> p = Piecewise((x, x < 1), (1, S(1) <= x))
>>> piecewise_fold(x*p)
Piecewise((x**2, x < 1), (x, True))
See Also
========
Piecewise
"""
if not isinstance(expr, Basic) or not expr.has(Piecewise):
return expr
new_args = []
if isinstance(expr, (ExprCondPair, Piecewise)):
for e, c in expr.args:
if not isinstance(e, Piecewise):
e = piecewise_fold(e)
# we don't keep Piecewise in condition because
# it has to be checked to see that it's complete
# and we convert it to ITE at that time
assert not c.has(Piecewise) # pragma: no cover
if isinstance(c, ITE):
c = c.to_nnf()
c = simplify_logic(c, form='cnf')
if isinstance(e, Piecewise):
new_args.extend([(piecewise_fold(ei), And(ci, c))
for ei, ci in e.args])
else:
new_args.append((e, c))
else:
from sympy.utilities.iterables import cartes, sift, common_prefix
# Given
# P1 = Piecewise((e11, c1), (e12, c2), A)
# P2 = Piecewise((e21, c1), (e22, c2), B)
# ...
# the folding of f(P1, P2) is trivially
# Piecewise(
# (f(e11, e21), c1),
# (f(e12, e22), c2),
# (f(Piecewise(A), Piecewise(B)), True))
# Certain objects end up rewriting themselves as thus, so
# we do that grouping before the more generic folding.
# The following applies this idea when f = Add or f = Mul
# (and the expression is commutative).
if expr.is_Add or expr.is_Mul and expr.is_commutative:
p, args = sift(expr.args, lambda x: x.is_Piecewise, binary=True)
pc = sift(p, lambda x: tuple([c for e,c in x.args]))
for c in list(ordered(pc)):
if len(pc[c]) > 1:
pargs = [list(i.args) for i in pc[c]]
# the first one is the same; there may be more
com = common_prefix(*[
[i.cond for i in j] for j in pargs])
n = len(com)
collected = []
for i in range(n):
collected.append((
expr.func(*[ai[i].expr for ai in pargs]),
com[i]))
remains = []
for a in pargs:
if n == len(a): # no more args
continue
if a[n].cond == True: # no longer Piecewise
remains.append(a[n].expr)
else: # restore the remaining Piecewise
remains.append(
Piecewise(*a[n:], evaluate=False))
if remains:
collected.append((expr.func(*remains), True))
args.append(Piecewise(*collected, evaluate=False))
continue
args.extend(pc[c])
else:
args = expr.args
# fold
folded = list(map(piecewise_fold, args))
for ec in cartes(*[
(i.args if isinstance(i, Piecewise) else
[(i, true)]) for i in folded]):
e, c = zip(*ec)
new_args.append((expr.func(*e), And(*c)))
return Piecewise(*new_args)
def _eval_simplify(self, ratio, measure, rational, inverse):
args = [a._eval_simplify(ratio, measure, rational, inverse)
for a in self.args]
for i, (expr, cond) in enumerate(args):
# try to simplify conditions and the expression for
# equalities that are part of the condition, e.g.
# Piecewise((n, And(Eq(n,0), Eq(n + m, 0))), (1, True))
# -> Piecewise((0, And(Eq(n, 0), Eq(m, 0))), (1, True))
if isinstance(cond, And):
eqs, other = sift(cond.args,
lambda i: isinstance(i, Equality), binary=True)
elif isinstance(cond, Equality):
eqs, other = [cond], []
else:
eqs = other = []
if eqs:
eqs = list(ordered(eqs))
for j, e in enumerate(eqs):
# these blessed lhs objects behave like Symbols
# and the rhs are simple replacements for the "symbols"
if isinstance(e.lhs, (Symbol, UndefinedFunction)) and \
isinstance(e.rhs,
(Rational, NumberSymbol,
Symbol, UndefinedFunction)):
expr = expr.subs(*e.args)
eqs[j + 1:] = [ei.subs(*e.args) for ei in eqs[j + 1:]]
other = [ei.subs(*e.args) for ei in other]
cond = And(*(eqs + other))
args[i] = args[i].func(expr, cond)
# See if expressions valid for a single point happens to evaluate
# to the same function as in the next piecewise segment, see:
# https://github.com/sympy/sympy/issues/8458
prevexpr = None
for i, (expr, cond) in reversed(list(enumerate(args))):
if prevexpr is not None:
_prevexpr = prevexpr
_expr = expr
if isinstance(cond, And):
eqs, other = sift(cond.args,
lambda i: isinstance(i, Equality), binary=True)
elif isinstance(cond, Equality):
eqs, other = [cond], []
else:
eqs = other = []
if eqs:
eqs = list(ordered(eqs))
for j, e in enumerate(eqs):
# these blessed lhs objects behave like Symbols
# and the rhs are simple replacements for the "symbols"
if isinstance(e.lhs, (Symbol, UndefinedFunction)) and \
isinstance(e.rhs,
(Rational, NumberSymbol,
Symbol, UndefinedFunction)):
_prevexpr = _prevexpr.subs(*e.args)
_expr = _expr.subs(*e.args)
if _prevexpr == _expr:
args[i] = args[i].func(args[i+1][0], cond)
else:
prevexpr = expr
else:
prevexpr = expr
return self.func(*args)
def tree_cse(exprs, symbols, opt_subs=None, order='canonical', ignore=()):
"""Perform raw CSE on expression tree, taking opt_subs into account.
Parameters
==========
exprs : list of sympy expressions
The expressions to reduce.
symbols : infinite iterator yielding unique Symbols
The symbols used to label the common subexpressions which are pulled
out.
opt_subs : dictionary of expression substitutions
The expressions to be substituted before any CSE action is performed.
order : string, 'none' or 'canonical'
The order by which Mul and Add arguments are processed. For large
expressions where speed is a concern, use the setting order='none'.
ignore : iterable of Symbols
Substitutions containing any Symbol from ``ignore`` will be ignored.
"""
from sympy.matrices.expressions import MatrixExpr, MatrixSymbol, MatMul, MatAdd
if opt_subs is None:
opt_subs = dict()
## Find repeated sub-expressions
to_eliminate = set()
seen_subexp = set()
def _find_repeated(expr):
if not isinstance(expr, Basic):
return
if expr.is_Atom or expr.is_Order:
return
if iterable(expr):
args = expr
else:
if expr in seen_subexp:
for ign in ignore:
if ign in expr.free_symbols:
break
else:
to_eliminate.add(expr)
return
seen_subexp.add(expr)
if expr in opt_subs:
expr = opt_subs[expr]
args = expr.args
list(map(_find_repeated, args))
for e in exprs:
if isinstance(e, Basic):
_find_repeated(e)
## Rebuild tree
replacements = []
subs = dict()
def _rebuild(expr):
if not isinstance(expr, Basic):
return expr
if not expr.args:
return expr
if iterable(expr):
new_args = [_rebuild(arg) for arg in expr]
return expr.func(*new_args)
if expr in subs:
return subs[expr]
orig_expr = expr
if expr in opt_subs:
expr = opt_subs[expr]
# If enabled, parse Muls and Adds arguments by order to ensure
# replacement order independent from hashes
if order != 'none':
if isinstance(expr, (Mul, MatMul)):
c, nc = expr.args_cnc()
if c == [1]:
args = nc
else:
args = list(ordered(c)) + nc
elif isinstance(expr, (Add, MatAdd)):
args = list(ordered(expr.args))
else:
args = expr.args
else:
args = expr.args
new_args = list(map(_rebuild, args))
if new_args != args:
new_expr = expr.func(*new_args)
else:
new_expr = expr
if orig_expr in to_eliminate:
try:
sym = next(symbols)
except StopIteration:
raise ValueError("Symbols iterator ran out of symbols.")
if isinstance(orig_expr, MatrixExpr):
sym = MatrixSymbol(sym.name, orig_expr.rows,
orig_expr.cols)
subs[orig_expr] = sym
replacements.append((sym, new_expr))
return sym
else:
return new_expr
reduced_exprs = []
for e in exprs:
if isinstance(e, Basic):
reduced_e = _rebuild(e)
else:
reduced_e = e
reduced_exprs.append(reduced_e)
# don't allow hollow nesting; restore expressions that were
# subject to "advantageous grouping"
# e.g if p = [b + 2*d + e + f, b + 2*d + f + g, a + c + d + f + g]
# and R, C = cse(p) then
# R = [(x0, d + f), (x1, b + d)]
# C = [e + x0 + x1, g + x0 + x1, a + c + d + f + g]
# but the args of C[-1] should not be `(a + c, d + f + g)`
subs_opt = list(ordered([(v, k) for k, v in opt_subs.items()]))
for i, e in enumerate(reduced_exprs):
reduced_exprs[i] = e.subs(subs_opt)
return replacements, reduced_exprs
def _match_common_args(Func, funcs):
if order != 'none':
funcs = list(ordered(funcs))
else:
funcs = sorted(funcs, key=lambda x: len(x.args))
if Func is Mul:
F = Pow
meth = 'as_powers_dict'
from sympy.core.add import _addsort as inplace_sorter
elif Func is Add:
F = Mul
meth = 'as_coefficients_dict'
from sympy.core.mul import _mulsort as inplace_sorter
else:
assert None # expected Mul or Add
# ----------------- helpers ---------------------------
def ufunc(*args):
# return a well formed unevaluated function from the args
# SHARES Func, inplace_sorter
args = list(args)
inplace_sorter(args)
return Func(*args, evaluate=False)
def as_dict(e):
# creates a dictionary of the expression using either
# as_coefficients_dict or as_powers_dict, depending on Func
# SHARES meth
d = getattr(e, meth, lambda: {a: S.One for a in e.args})()
for k in list(d.keys()):
try:
as_int(d[k])
except ValueError:
d[F(k, d.pop(k))] = S.One
return d
def from_dict(d):
# build expression from dict from
# as_coefficients_dict or as_powers_dict
# SHARES F
return ufunc(*[F(k, v) for k, v in d.items()])
def update(k):
# updates all of the info associated with k using
# the com_dict: func_dicts, func_args, opt_subs
# returns True if all values were updated, else None
# SHARES com_dict, com_func, func_dicts, func_args,
# opt_subs, funcs, verbose
for di in com_dict:
# don't allow a sign to change
if com_dict[di] > func_dicts[k][di]:
return
# remove it
if Func is Add:
take = min(func_dicts[k][i] for i in com_dict)
com_func_take = Mul(take, from_dict(com_dict), evaluate=False)
else:
take = igcd(*[func_dicts[k][i] for i in com_dict])
com_func_take = Pow(from_dict(com_dict), take, evaluate=False)
for di in com_dict:
func_dicts[k][di] -= take*com_dict[di]
# compute the remaining expression
rem = from_dict(func_dicts[k])
# reject hollow change, e.g extracting x + 1 from x + 3
if Func is Add and rem and rem.is_Integer and 1 in com_dict:
return
if verbose:
print('\nfunc %s (%s) \ncontains %s \nas %s \nleaving %s' %
(funcs[k], func_dicts[k], com_func, com_func_take, rem))
# recompute the dict since some keys may now
# have corresponding values of 0; one could
# keep track of which ones went to zero but
# this seems cleaner
func_dicts[k] = as_dict(rem)
# update associated info
func_dicts[k][com_func] = take
func_args[k] = set(func_dicts[k])
# keep the constant separate from the remaining
# part of the expression, e.g. 2*(a*b) rather than 2*a*b
opt_subs[funcs[k]] = ufunc(rem, com_func_take)
# everything was updated
return True
def get_copy(i):
return [func_dicts[i].copy(), func_args[i].copy(), funcs[i], i]
def restore(dafi):
i = dafi.pop()
func_dicts[i], func_args[i], funcs[i] = dafi
# ----------------- end helpers -----------------------
func_dicts = [as_dict(f) for f in funcs]
func_args = [set(d) for d in func_dicts]
while True:
hit = pairwise_most_common(func_args)
if not hit or len(hit[0][0]) <= 1:
break
changed = False
for com_args, ij in hit:
take = len(com_args)
ALL = list(ordered(com_args))
while take >= 2:
for com_args in subsets(ALL, take):
com_func = Func(*com_args)
com_dict = as_dict(com_func)
for i, j in ij:
dafi = None
if com_func != funcs[i]:
dafi = get_copy(i)
ch = update(i)
if not ch:
restore(dafi)
continue
if com_func != funcs[j]:
dafj = get_copy(j)
ch = update(j)
if not ch:
if dafi is not None:
restore(dafi)
restore(dafj)
continue
changed = True
if changed:
break
else:
take -= 1
continue
break
else:
continue
break
if not changed:
break
def tree_cse(exprs, symbols, opt_subs=None, order='canonical', ignore=()):
"""Perform raw CSE on expression tree, taking opt_subs into account.
Parameters
==========
exprs : list of sympy expressions
The expressions to reduce.
symbols : infinite iterator yielding unique Symbols
The symbols used to label the common subexpressions which are pulled
out.
opt_subs : dictionary of expression substitutions
The expressions to be substituted before any CSE action is performed.
order : string, 'none' or 'canonical'
The order by which Mul and Add arguments are processed. For large
expressions where speed is a concern, use the setting order='none'.
ignore : iterable of Symbols
Substitutions containing any Symbol from ``ignore`` will be ignored.
"""
from sympy.matrices.expressions import MatrixExpr, MatrixSymbol, MatMul, MatAdd
if opt_subs is None:
opt_subs = dict()
## Find repeated sub-expressions
to_eliminate = set()
seen_subexp = set()
def _find_repeated(expr):
if not isinstance(expr, Basic):
return
if expr.is_Atom or expr.is_Order:
return
if iterable(expr):
args = expr
else:
if expr in seen_subexp:
for ign in ignore:
if ign in expr.free_symbols:
break
else:
to_eliminate.add(expr)
return
seen_subexp.add(expr)
if expr in opt_subs:
expr = opt_subs[expr]
args = expr.args
list(map(_find_repeated, args))
for e in exprs:
if isinstance(e, Basic):
_find_repeated(e)
## Rebuild tree
replacements = []
subs = dict()
def _rebuild(expr):
if not isinstance(expr, Basic):
return expr
if not expr.args:
return expr
if iterable(expr):
new_args = [_rebuild(arg) for arg in expr]
return expr.func(*new_args)
if expr in subs:
return subs[expr]
orig_expr = expr
if expr in opt_subs:
expr = opt_subs[expr]
# If enabled, parse Muls and Adds arguments by order to ensure
# replacement order independent from hashes
if order != 'none':
if isinstance(expr, (Mul, MatMul)):
c, nc = expr.args_cnc()
if c == [1]:
args = nc
else:
args = list(ordered(c)) + nc
elif isinstance(expr, (Add, MatAdd)):
args = list(ordered(expr.args))
else:
args = expr.args
else:
args = expr.args
new_args = list(map(_rebuild, args))
if new_args != args:
new_expr = expr.func(*new_args)
else:
new_expr = expr
if orig_expr in to_eliminate:
try:
sym = next(symbols)
except StopIteration:
raise ValueError("Symbols iterator ran out of symbols.")
if isinstance(orig_expr, MatrixExpr):
sym = MatrixSymbol(sym.name, orig_expr.rows, orig_expr.cols)
subs[orig_expr] = sym
replacements.append((sym, new_expr))
return sym
else:
return new_expr
reduced_exprs = []
for e in exprs:
if isinstance(e, Basic):
reduced_e = _rebuild(e)
else:
reduced_e = e
reduced_exprs.append(reduced_e)
# don't allow hollow nesting; restore expressions that were
# subject to "advantageous grouping"
# e.g if p = [b + 2*d + e + f, b + 2*d + f + g, a + c + d + f + g]
# and R, C = cse(p) then
# R = [(x0, d + f), (x1, b + d)]
# C = [e + x0 + x1, g + x0 + x1, a + c + d + f + g]
# but the args of C[-1] should not be `(a + c, d + f + g)`
subs_opt = list(ordered([(v, k) for k, v in opt_subs.items()]))
for i, e in enumerate(reduced_exprs):
reduced_exprs[i] = e.subs(subs_opt)
return replacements, reduced_exprs
def cse(exprs, symbols=None, optimizations=None, postprocess=None):
""" Perform common subexpression elimination on an expression.
Parameters
==========
exprs : list of sympy expressions, or a single sympy expression
The expressions to reduce.
symbols : infinite iterator yielding unique Symbols
The symbols used to label the common subexpressions which are pulled
out. The ``numbered_symbols`` generator is useful. The default is a
stream of symbols of the form "x0", "x1", etc. This must be an infinite
iterator.
optimizations : list of (callable, callable) pairs, optional
The (preprocessor, postprocessor) pairs. If not provided,
``sympy.simplify.cse.cse_optimizations`` is used.
postprocess : a function which accepts the two return values of cse and
returns the desired form of output from cse, e.g. if you want the
replacements reversed the function might be the following lambda:
lambda r, e: return reversed(r), e
Returns
=======
replacements : list of (Symbol, expression) pairs
All of the common subexpressions that were replaced. Subexpressions
earlier in this list might show up in subexpressions later in this list.
reduced_exprs : list of sympy expressions
The reduced expressions with all of the replacements above.
"""
from sympy.matrices import Matrix
if symbols is None:
symbols = numbered_symbols()
else:
# In case we get passed an iterable with an __iter__ method instead of
# an actual iterator.
symbols = iter(symbols)
tmp_symbols = numbered_symbols('_csetmp')
subexp_iv = dict()
muls = set()
adds = set()
if optimizations is None:
# Pull out the default here just in case there are some weird
# manipulations of the module-level list in some other thread.
optimizations = list(cse_optimizations)
# Handle the case if just one expression was passed.
if isinstance(exprs, Basic):
exprs = [exprs]
# Preprocess the expressions to give us better optimization opportunities.
prep_exprs = [preprocess_for_cse(e, optimizations) for e in exprs]
# Find all subexpressions.
def _parse(expr):
if expr.is_Atom:
# Exclude atoms, since there is no point in renaming them.
return expr
if iterable(expr):
return expr
subexpr = type(expr)(*map(_parse, expr.args))
if subexpr in subexp_iv:
return subexp_iv[subexpr]
if subexpr.is_Mul:
muls.add(subexpr)
elif subexpr.is_Add:
adds.add(subexpr)
ivar = next(tmp_symbols)
subexp_iv[subexpr] = ivar
return ivar
tmp_exprs = list()
for expr in prep_exprs:
if isinstance(expr, Basic):
tmp_exprs.append(_parse(expr))
else:
tmp_exprs.append(expr)
# process adds - any adds that weren't repeated might contain
# subpatterns that are repeated, e.g. x+y+z and x+y have x+y in common
adds = list(ordered(adds))
addargs = [set(a.args) for a in adds]
for i in xrange(len(addargs)):
for j in xrange(i + 1, len(addargs)):
com = addargs[i].intersection(addargs[j])
if len(com) > 1:
add_subexp = Add(*com)
diff_add_i = addargs[i].difference(com)
diff_add_j = addargs[j].difference(com)
if add_subexp in subexp_iv:
ivar = subexp_iv[add_subexp]
else:
ivar = next(tmp_symbols)
subexp_iv[add_subexp] = ivar
if diff_add_i:
newadd = Add(ivar,*diff_add_i)
subexp_iv[newadd] = subexp_iv.pop(adds[i])
adds[i] = newadd
#else add_i is itself subexp_iv[add_subexp] -> ivar
if diff_add_j:
newadd = Add(ivar,*diff_add_j)
subexp_iv[newadd] = subexp_iv.pop(adds[j])
adds[j] = newadd
#else add_j is itself subexp_iv[add_subexp] -> ivar
addargs[i] = diff_add_i
addargs[j] = diff_add_j
for k in xrange(j + 1, len(addargs)):
if com.issubset(addargs[k]):
diff_add_k = addargs[k].difference(com)
if diff_add_k:
newadd = Add(ivar,*diff_add_k)
subexp_iv[newadd] = subexp_iv.pop(adds[k])
adds[k] = newadd
#else add_k is itself subexp_iv[add_subexp] -> ivar
addargs[k] = diff_add_k
# process muls - any muls that weren't repeated might contain
# subpatterns that are repeated, e.g. x*y*z and x*y have x*y in common
# *assumes that there are no non-commutative parts*
muls = list(ordered(muls))
mulargs = [set(a.args) for a in muls]
for i in xrange(len(mulargs)):
for j in xrange(i + 1, len(mulargs)):
com = mulargs[i].intersection(mulargs[j])
if len(com) > 1:
mul_subexp = Mul(*com)
diff_mul_i = mulargs[i].difference(com)
diff_mul_j = mulargs[j].difference(com)
if mul_subexp in subexp_iv:
ivar = subexp_iv[mul_subexp]
else:
ivar = next(tmp_symbols)
subexp_iv[mul_subexp] = ivar
if diff_mul_i:
newmul = Mul(ivar,*diff_mul_i)
subexp_iv[newmul] = subexp_iv.pop(muls[i])
muls[i] = newmul
#else mul_i is itself subexp_iv[mul_subexp] -> ivar
if diff_mul_j:
newmul = Mul(ivar,*diff_mul_j)
subexp_iv[newmul] = subexp_iv.pop(muls[j])
muls[j] = newmul
#else mul_j is itself subexp_iv[mul_subexp] -> ivar
mulargs[i] = diff_mul_i
mulargs[j] = diff_mul_j
for k in xrange(j + 1, len(mulargs)):
if com.issubset(mulargs[k]):
diff_mul_k = mulargs[k].difference(com)
if diff_mul_k:
newmul = Mul(ivar,*diff_mul_k)
subexp_iv[newmul] = subexp_iv.pop(muls[k])
muls[k] = newmul
#else mul_k is itself subexp_iv[mul_subexp] -> ivar
mulargs[k] = diff_mul_k
# Find all of the repeated subexpressions.
ivar_se = {iv:se for se,iv in subexp_iv.iteritems()}
used_ivs = set()
repeated = set()
def _find_repeated_subexprs(subexpr):
if subexpr.is_Atom:
symbs = [subexpr]
else:
symbs = subexpr.args
for symb in symbs:
if symb in ivar_se:
if symb not in used_ivs:
_find_repeated_subexprs(ivar_se[symb])
used_ivs.add(symb)
else:
repeated.add(symb)
for expr in tmp_exprs:
_find_repeated_subexprs(expr)
# Substitute symbols for all of the repeated subexpressions.
# remove temporary replacements that weren't used more than once
tmpivs_ivs = dict()
ordered_iv_se = OrderedDict()
def _get_subexprs(args):
args = list(args)
for i,symb in enumerate(args):
if symb in ivar_se:
if symb in tmpivs_ivs:
args[i] = tmpivs_ivs[symb]
else:
subexpr = ivar_se[symb]
subexpr = type(subexpr)(*_get_subexprs(subexpr.args))
if symb in repeated:
ivar = next(symbols)
ordered_iv_se[ivar] = subexpr
tmpivs_ivs[symb] = ivar
args[i] = ivar
else:
args[i] = subexpr
return args
out_exprs = _get_subexprs(tmp_exprs)
# Postprocess the expressions to return the expressions to canonical form.
ordered_iv_se_notopt = ordered_iv_se
ordered_iv_se = OrderedDict()
for i, (ivar, subexpr) in enumerate(ordered_iv_se_notopt.items()):
subexpr = postprocess_for_cse(subexpr, optimizations)
ordered_iv_se[ivar] = subexpr
out_exprs = [postprocess_for_cse(e, optimizations) for e in out_exprs]
if isinstance(exprs, Matrix):
out_exprs = Matrix(exprs.rows, exprs.cols, out_exprs)
if postprocess is None:
return ordered_iv_se.items(), out_exprs
return postprocess(ordered_iv_se.items(), out_exprs)
def find_bac_cab(expr):
""" Given a vector expression, find the terms satisfying the pattern:
B * (A & C) - C * (A & B)
where & is the dot product. The list is ordered in such a way that
nested matches comes first (similarly to scanning the expression tree from
bottom to top).
"""
def _check_pairs(terms):
# c1 = Mul(*[a for a in terms[0].args if not hasattr(a, "is_Vector_Scalar")])
# c2 = Mul(*[a for a in terms[1].args if not hasattr(a, "is_Vector_Scalar")])
c1 = Mul(*[
a for a in terms[0].args if not isinstance(a, (Vector, VectorExpr))
])
c2 = Mul(*[
a for a in terms[1].args if not isinstance(a, (Vector, VectorExpr))
])
n1, _ = c1.as_coeff_mul()
n2, _ = c2.as_coeff_mul()
if n1 * n2 > 0:
# opposite sign
return False
if Abs(c1) != Abs(c2):
return False
# v1 = [a for a in terms[0].args if (hasattr(a, "is_Vector_Scalar") and a.is_Vector)][0]
# v2 = [a for a in terms[1].args if (hasattr(a, "is_Vector_Scalar") and a.is_Vector)][0]
v1 = [
a for a in terms[0].args
if (isinstance(a, (Vector, VectorExpr)) and a.is_Vector)
][0]
v2 = [
a for a in terms[1].args
if (isinstance(a, (Vector, VectorExpr)) and a.is_Vector)
][0]
if v1 == v2:
return False
dot1 = [a for a in terms[0].args if isinstance(a, VecDot)][0]
dot2 = [a for a in terms[1].args if isinstance(a, VecDot)][0]
if not ((v1 in dot2.args) and (v2 in dot1.args)):
return False
v = list(set(dot1.args).intersection(set(dot2.args)))[0]
if v == v1 or v == v2:
return False
return True
def _check(arg):
if (isinstance(arg, VecMul) and any([a.is_Vector for a in arg.args])
and any([isinstance(a, VecDot) for a in arg.args])):
return True
return False
found = set()
for arg in preorder_traversal(expr):
possible_args = list(filter(_check, arg.args))
# print("possible_args", possible_args)
# for p in possible_args:
# print("\t", p.func, p.is_Vector, p)
# # possible_args = list(arg.find(A * (B & C)))
# # possible_args = list(filter(lambda x: isinstance(x, VecMul), possible_args))
if len(possible_args) > 1:
combs = list(combinations(possible_args, 2))
# print("COMBINATIONS", combs)
for c in combs:
# print("\tC", c)
# print("\n\t\t".join(str(a.func) + ", " + str(a.is_Vector) + ", " + str(a) for a in c))
if _check_pairs(c):
# print("\t\t proceeding")
found.add(VecAdd(*c))
found = list(ordered(list(found)))
return found
def cse(exprs, symbols=None, optimizations=None, postprocess=None):
""" Perform common subexpression elimination on an expression.
Parameters
==========
exprs : list of sympy expressions, or a single sympy expression
The expressions to reduce.
symbols : infinite iterator yielding unique Symbols
The symbols used to label the common subexpressions which are pulled
out. The ``numbered_symbols`` generator is useful. The default is a
stream of symbols of the form "x0", "x1", etc. This must be an infinite
iterator.
optimizations : list of (callable, callable) pairs, optional
The (preprocessor, postprocessor) pairs. If not provided,
``sympy.simplify.cse.cse_optimizations`` is used.
postprocess : a function which accepts the two return values of cse and
returns the desired form of output from cse, e.g. if you want the
replacements reversed the function might be the following lambda:
lambda r, e: return reversed(r), e
Returns
=======
replacements : list of (Symbol, expression) pairs
All of the common subexpressions that were replaced. Subexpressions
earlier in this list might show up in subexpressions later in this list.
reduced_exprs : list of sympy expressions
The reduced expressions with all of the replacements above.
"""
from sympy.matrices import Matrix
if symbols is None:
symbols = numbered_symbols()
else:
# In case we get passed an iterable with an __iter__ method instead of
# an actual iterator.
symbols = iter(symbols)
seen_subexp = set()
muls = set()
adds = set()
to_eliminate = set()
if optimizations is None:
# Pull out the default here just in case there are some weird
# manipulations of the module-level list in some other thread.
optimizations = list(cse_optimizations)
# Handle the case if just one expression was passed.
if isinstance(exprs, Basic):
exprs = [exprs]
# Preprocess the expressions to give us better optimization opportunities.
reduced_exprs = [preprocess_for_cse(e, optimizations) for e in exprs]
# Find all of the repeated subexpressions.
for expr in reduced_exprs:
if not isinstance(expr, Basic):
continue
pt = preorder_traversal(expr)
for subtree in pt:
inv = 1/subtree if subtree.is_Pow else None
if subtree.is_Atom or iterable(subtree) or inv and inv.is_Atom:
# Exclude atoms, since there is no point in renaming them.
continue
if subtree in seen_subexp:
if inv and _coeff_isneg(subtree.exp):
# save the form with positive exponent
subtree = inv
to_eliminate.add(subtree)
pt.skip()
continue
if inv and inv in seen_subexp:
if _coeff_isneg(subtree.exp):
# save the form with positive exponent
subtree = inv
to_eliminate.add(subtree)
pt.skip()
continue
elif subtree.is_Mul:
muls.add(subtree)
elif subtree.is_Add:
adds.add(subtree)
seen_subexp.add(subtree)
# process adds - any adds that weren't repeated might contain
# subpatterns that are repeated, e.g. x+y+z and x+y have x+y in common
adds = [set(a.args) for a in ordered(adds)]
for i in xrange(len(adds)):
for j in xrange(i + 1, len(adds)):
com = adds[i].intersection(adds[j])
if len(com) > 1:
to_eliminate.add(Add(*com))
# remove this set of symbols so it doesn't appear again
adds[i] = adds[i].difference(com)
adds[j] = adds[j].difference(com)
for k in xrange(j + 1, len(adds)):
if not com.difference(adds[k]):
adds[k] = adds[k].difference(com)
# process muls - any muls that weren't repeated might contain
# subpatterns that are repeated, e.g. x*y*z and x*y have x*y in common
# use SequenceMatcher on the nc part to find the longest common expression
# in common between the two nc parts
sm = difflib.SequenceMatcher()
muls = [a.args_cnc(cset=True) for a in ordered(muls)]
for i in xrange(len(muls)):
if muls[i][1]:
sm.set_seq1(muls[i][1])
for j in xrange(i + 1, len(muls)):
# the commutative part in common
ccom = muls[i][0].intersection(muls[j][0])
# the non-commutative part in common
if muls[i][1] and muls[j][1]:
# see if there is any chance of an nc match
ncom = set(muls[i][1]).intersection(set(muls[j][1]))
if len(ccom) + len(ncom) < 2:
continue
# now work harder to find the match
sm.set_seq2(muls[j][1])
i1, _, n = sm.find_longest_match(0, len(muls[i][1]),
0, len(muls[j][1]))
ncom = muls[i][1][i1:i1 + n]
else:
ncom = []
com = list(ccom) + ncom
if len(com) < 2:
continue
to_eliminate.add(Mul(*com))
# remove ccom from all if there was no ncom; to update the nc part
# would require finding the subexpr and then replacing it with a
# dummy to keep bounding nc symbols from being identified as a
# subexpr, e.g. removing B*C from A*B*C*D might allow A*D to be
# identified as a subexpr which would not be right.
if not ncom:
muls[i][0] = muls[i][0].difference(ccom)
for k in xrange(j, len(muls)):
if not ccom.difference(muls[k][0]):
muls[k][0] = muls[k][0].difference(ccom)
# make to_eliminate canonical; we will prefer non-Muls to Muls
# so select them first (non-Muls will have False for is_Mul and will
# be first in the ordering.
to_eliminate = list(ordered(to_eliminate, lambda _: _.is_Mul))
# Substitute symbols for all of the repeated subexpressions.
replacements = []
reduced_exprs = list(reduced_exprs)
hit = True
for i, subtree in enumerate(to_eliminate):
if hit:
sym = next(symbols)
hit = False
if subtree.is_Pow and subtree.exp.is_Rational:
update = lambda x: x.xreplace({subtree: sym, 1/subtree: 1/sym})
else:
update = lambda x: x.subs(subtree, sym)
# Make the substitution in all of the target expressions.
for j, expr in enumerate(reduced_exprs):
old = reduced_exprs[j]
reduced_exprs[j] = update(expr)
hit = hit or (old != reduced_exprs[j])
# Make the substitution in all of the subsequent substitutions.
for j in range(i + 1, len(to_eliminate)):
old = to_eliminate[j]
to_eliminate[j] = update(to_eliminate[j])
hit = hit or (old != to_eliminate[j])
if hit:
replacements.append((sym, subtree))
# Postprocess the expressions to return the expressions to canonical form.
for i, (sym, subtree) in enumerate(replacements):
subtree = postprocess_for_cse(subtree, optimizations)
replacements[i] = (sym, subtree)
reduced_exprs = [postprocess_for_cse(e, optimizations)
for e in reduced_exprs]
# remove replacements that weren't used more than once
_remove_singletons(replacements, reduced_exprs)
if isinstance(exprs, Matrix):
reduced_exprs = [Matrix(exprs.rows, exprs.cols, reduced_exprs)]
if postprocess is None:
return replacements, reduced_exprs
return postprocess(replacements, reduced_exprs)
def __hash__(self): # Factors
keys = tuple(ordered(self.factors.keys()))
values = [self.factors[k] for k in keys]
return hash((keys, values))
def _neq_linear_first_order_const_coeff_homogeneous(match_):
r"""
System of n first-order constant-coefficient linear homogeneous differential equations
.. math:: y'_k = a_{k1} y_1 + a_{k2} y_2 +...+ a_{kn} y_n; k = 1,2,...,n
or that can be written as `\vec{y'} = A . \vec{y}`
where `\vec{y}` is matrix of `y_k` for `k = 1,2,...n` and `A` is a `n \times n` matrix.
Since these equations are equivalent to a first order homogeneous linear
differential equation. So the general solution will contain `n` linearly
independent parts and solution will consist some type of exponential
functions. Assuming `y = \vec{v} e^{rt}` is a solution of the system where
`\vec{v}` is a vector of coefficients of `y_1,...,y_n`. Substituting `y` and
`y' = r v e^{r t}` into the equation `\vec{y'} = A . \vec{y}`, we get
.. math:: r \vec{v} e^{rt} = A \vec{v} e^{rt}
.. math:: r \vec{v} = A \vec{v}
where `r` comes out to be eigenvalue of `A` and vector `\vec{v}` is the eigenvector
of `A` corresponding to `r`. There are three possibilities of eigenvalues of `A`
- `n` distinct real eigenvalues
- complex conjugate eigenvalues
- eigenvalues with multiplicity `k`
1. When all eigenvalues `r_1,..,r_n` are distinct with `n` different eigenvectors
`v_1,...v_n` then the solution is given by
.. math:: \vec{y} = C_1 e^{r_1 t} \vec{v_1} + C_2 e^{r_2 t} \vec{v_2} +...+ C_n e^{r_n t} \vec{v_n}
where `C_1,C_2,...,C_n` are arbitrary constants.
2. When some eigenvalues are complex then in order to make the solution real,
we take a linear combination: if `r = a + bi` has an eigenvector
`\vec{v} = \vec{w_1} + i \vec{w_2}` then to obtain real-valued solutions to
the system, replace the complex-valued solutions `e^{rx} \vec{v}`
with real-valued solution `e^{ax} (\vec{w_1} \cos(bx) - \vec{w_2} \sin(bx))`
and for `r = a - bi` replace the solution `e^{-r x} \vec{v}` with
`e^{ax} (\vec{w_1} \sin(bx) + \vec{w_2} \cos(bx))`
3. If some eigenvalues are repeated. Then we get fewer than `n` linearly
independent eigenvectors, we miss some of the solutions and need to
construct the missing ones. We do this via generalized eigenvectors, vectors
which are not eigenvectors but are close enough that we can use to write
down the remaining solutions. For a eigenvalue `r` with eigenvector `\vec{w}`
we obtain `\vec{w_2},...,\vec{w_k}` using
.. math:: (A - r I) . \vec{w_2} = \vec{w}
.. math:: (A - r I) . \vec{w_3} = \vec{w_2}
.. math:: \vdots
.. math:: (A - r I) . \vec{w_k} = \vec{w_{k-1}}
Then the solutions to the system for the eigenspace are `e^{rt} [\vec{w}],
e^{rt} [t \vec{w} + \vec{w_2}], e^{rt} [\frac{t^2}{2} \vec{w} + t \vec{w_2} + \vec{w_3}],
...,e^{rt} [\frac{t^{k-1}}{(k-1)!} \vec{w} + \frac{t^{k-2}}{(k-2)!} \vec{w_2} +...+ t \vec{w_{k-1}}
+ \vec{w_k}]`
So, If `\vec{y_1},...,\vec{y_n}` are `n` solution of obtained from three
categories of `A`, then general solution to the system `\vec{y'} = A . \vec{y}`
.. math:: \vec{y} = C_1 \vec{y_1} + C_2 \vec{y_2} + \cdots + C_n \vec{y_n}
"""
eq = match_['eq']
func = match_['func']
fc = match_['func_coeff']
n = len(eq)
t = list(list(eq[0].atoms(Derivative))[0].atoms(Symbol))[0]
constants = numbered_symbols(prefix='C', cls=Symbol, start=1)
# This needs to be modified in future so that fc is only of type Matrix
M = -fc if type(fc) is Matrix else Matrix(n, n,
lambda i, j: -fc[i, func[j], 0])
P, J = matrix_exp_jordan_form(M, t)
P = simplify(P)
Cvect = Matrix(list(next(constants) for _ in range(n)))
sol_vector = P * (J * Cvect)
sol_vector = [
collect(s, ordered(J.atoms(exp)), exact=True) for s in sol_vector
]
sol_dict = [Eq(func[i], sol_vector[i]) for i in range(n)]
return sol_dict
def _collapse_arguments(cls, args, **assumptions):
"""Remove redundant args.
Examples
========
>>> from sympy import Min, Max
>>> from sympy.abc import a, b, c, d, e
Any arg in parent that appears in any
parent-like function in any of the flat args
of parent can be removed from that sub-arg:
>>> Min(a, Max(b, Min(a, c, d)))
Min(a, Max(b, Min(c, d)))
If the arg of parent appears in an opposite-than parent
function in any of the flat args of parent that function
can be replaced with the arg:
>>> Min(a, Max(b, Min(c, d, Max(a, e))))
Min(a, Max(b, Min(a, c, d)))
"""
from sympy.utilities.iterables import ordered
from sympy.simplify.simplify import walk
if not args:
return args
args = list(ordered(args))
if cls == Min:
other = Max
else:
other = Min
# find global comparable max of Max and min of Min if a new
# value is being introduced in these args at position 0 of
# the ordered args
if args[0].is_number:
sifted = mins, maxs = [], []
for i in args:
for v in walk(i, Min, Max):
if v.args[0].is_comparable:
sifted[isinstance(v, Max)].append(v)
small = Min.identity
for i in mins:
v = i.args[0]
if v.is_number and (v < small) == True:
small = v
big = Max.identity
for i in maxs:
v = i.args[0]
if v.is_number and (v > big) == True:
big = v
# at the point when this function is called from __new__,
# there may be more than one numeric arg present since
# local zeros have not been handled yet, so look through
# more than the first arg
if cls == Min:
for i in range(len(args)):
if not args[i].is_number:
break
if (args[i] < small) == True:
small = args[i]
elif cls == Max:
for i in range(len(args)):
if not args[i].is_number:
break
if (args[i] > big) == True:
big = args[i]
T = None
if cls == Min:
if small != Min.identity:
other = Max
T = small
elif big != Max.identity:
other = Min
T = big
if T is not None:
# remove numerical redundancy
for i in range(len(args)):
a = args[i]
if isinstance(a, other):
a0 = a.args[0]
if ((a0 > T) if other == Max else (a0 < T)) == True:
args[i] = cls.identity
# remove redundant symbolic args
def do(ai, a):
if not isinstance(ai, (Min, Max)):
return ai
cond = a in ai.args
if not cond:
return ai.func(*[do(i, a) for i in ai.args], evaluate=False)
if isinstance(ai, cls):
return ai.func(*[do(i, a) for i in ai.args if i != a],
evaluate=False)
return a
for i, a in enumerate(args):
args[i + 1:] = [do(ai, a) for ai in args[i + 1:]]
# factor out common elements as for
# Min(Max(x, y), Max(x, z)) -> Max(x, Min(y, z))
# and vice versa when swapping Min/Max -- do this only for the
# easy case where all functions contain something in common;
# trying to find some optimal subset of args to modify takes
# too long
if len(args) > 1:
common = None
remove = []
sets = []
for i in range(len(args)):
a = args[i]
if not isinstance(a, other):
continue
s = set(a.args)
common = s if common is None else (common & s)
if not common:
break
sets.append(s)
remove.append(i)
if common:
sets = filter(None, [s - common for s in sets])
sets = [other(*s, evaluate=False) for s in sets]
for i in reversed(remove):
args.pop(i)
oargs = [cls(*sets)] if sets else []
oargs.extend(common)
args.append(other(*oargs, evaluate=False))
return args
def __repr__(self): # Factors
return "Factors({%s})" % ", ".join(["%s: %s" % (k, v) for k, v in ordered(self.factors.items())])
def __repr__(self): # Factors
return "Factors({%s})" % ', '.join(
['%s: %s' % (k, v) for k, v in ordered(self.factors.items())])
def collect_const(expr, *vars, **kwargs):
""" This is the very same code of sympy.simplify.radsimp.py collect_const,
with modification: the original method used Mul._from_args
and Add._from_args, which do not call a post-processor, hence I obtained the
wrong result. Here, I use VecAdd, VecMul...
"""
if not expr.is_Add:
return expr
recurse = False
Numbers = kwargs.get('Numbers', True)
if not vars:
recurse = True
vars = set()
for a in expr.args:
for m in Mul.make_args(a):
if m.is_number:
vars.add(m)
else:
vars = sympify(vars)
if not Numbers:
vars = [v for v in vars if not v.is_Number]
vars = list(ordered(vars))
for v in vars:
terms = defaultdict(list)
Fv = Factors(v)
for m in Add.make_args(expr):
f = Factors(m)
q, r = f.div(Fv)
if r.is_one:
# only accept this as a true factor if
# it didn't change an exponent from an Integer
# to a non-Integer, e.g. 2/sqrt(2) -> sqrt(2)
# -- we aren't looking for this sort of change
fwas = f.factors.copy()
fnow = q.factors
if not any(k in fwas and fwas[k].is_Integer
and not fnow[k].is_Integer for k in fnow):
terms[v].append(q.as_expr())
continue
terms[S.One].append(m)
args = []
hit = False
uneval = False
for k in ordered(terms):
v = terms[k]
if k is S.One:
args.extend(v)
continue
if len(v) > 1:
v = Add(*v)
hit = True
if recurse and v != expr:
vars.append(v)
else:
v = v[0]
# be careful not to let uneval become True unless
# it must be because it's going to be more expensive
# to rebuild the expression as an unevaluated one
if Numbers and k.is_Number and v.is_Add:
# args.append(_keep_coeff(k, v, sign=True))
args.append(VecMul(*[k, v], evaluate=False))
uneval = True
else:
args.append(k * v)
if hit:
if uneval:
# expr = Add(*args)
expr = VecAdd(*args, evaluate=False)
else:
# expr = Add(*args)
expr = VecAdd(*args)
if not expr.is_Add:
break
return expr
def piecewise_simplify(expr, **kwargs):
expr = piecewise_simplify_arguments(expr, **kwargs)
args = list(expr.args)
_blessed = lambda e: getattr(e.lhs, '_diff_wrt', False) and (getattr(
e.rhs, '_diff_wrt', None) or isinstance(e.rhs,
(Rational, NumberSymbol)))
for i, (expr, cond) in enumerate(args):
# try to simplify conditions and the expression for
# equalities that are part of the condition, e.g.
# Piecewise((n, And(Eq(n,0), Eq(n + m, 0))), (1, True))
# -> Piecewise((0, And(Eq(n, 0), Eq(m, 0))), (1, True))
if isinstance(cond, And):
eqs, other = sift(cond.args,
lambda i: isinstance(i, Equality),
binary=True)
elif isinstance(cond, Equality):
eqs, other = [cond], []
else:
eqs = other = []
if eqs:
eqs = list(ordered(eqs))
for j, e in enumerate(eqs):
# these blessed lhs objects behave like Symbols
# and the rhs are simple replacements for the "symbols"
if _blessed(e):
expr = expr.subs(*e.args)
eqs[j + 1:] = [ei.subs(*e.args) for ei in eqs[j + 1:]]
other = [ei.subs(*e.args) for ei in other]
cond = And(*(eqs + other))
args[i] = args[i].func(expr, cond)
# See if expressions valid for an Equal expression happens to evaluate
# to the same function as in the next piecewise segment, see:
# https://github.com/sympy/sympy/issues/8458
prevexpr = None
for i, (expr, cond) in reversed(list(enumerate(args))):
if prevexpr is not None:
if isinstance(cond, And):
eqs, other = sift(cond.args,
lambda i: isinstance(i, Equality),
binary=True)
elif isinstance(cond, Equality):
eqs, other = [cond], []
else:
eqs = other = []
_prevexpr = prevexpr
_expr = expr
if eqs and not other:
eqs = list(ordered(eqs))
for e in eqs:
# these blessed lhs objects behave like Symbols
# and the rhs are simple replacements for the "symbols"
if _blessed(e):
_prevexpr = _prevexpr.subs(*e.args)
_expr = _expr.subs(*e.args)
# Did it evaluate to the same?
if _prevexpr == _expr:
# Set the expression for the Not equal section to the same
# as the next. These will be merged when creating the new
# Piecewise
args[i] = args[i].func(args[i + 1][0], cond)
else:
# Update the expression that we compare against
prevexpr = expr
else:
prevexpr = expr
return Piecewise(*args)
def _match_common_args(Func, funcs):
if order != 'none':
funcs = list(ordered(funcs))
else:
funcs = sorted(funcs, key=lambda x: len(x.args))
if Func is Mul:
F = Pow
meth = 'as_powers_dict'
from sympy.core.add import _addsort as inplace_sorter
elif Func is Add:
F = Mul
meth = 'as_coefficients_dict'
from sympy.core.mul import _mulsort as inplace_sorter
else:
assert None # expected Mul or Add
# ----------------- helpers ---------------------------
def ufunc(*args):
# return a well formed unevaluated function from the args
# SHARES Func, inplace_sorter
args = list(args)
inplace_sorter(args)
return Func(*args, evaluate=False)
def as_dict(e):
# creates a dictionary of the expression using either
# as_coefficients_dict or as_powers_dict, depending on Func
# SHARES meth
d = getattr(e, meth, lambda: {a: S.One for a in e.args})()
for k in list(d.keys()):
try:
as_int(d[k])
except ValueError:
d[F(k, d.pop(k))] = S.One
return d
def from_dict(d):
# build expression from dict from
# as_coefficients_dict or as_powers_dict
# SHARES F
return ufunc(*[F(k, v) for k, v in d.items()])
def update(k):
# updates all of the info associated with k using
# the com_dict: func_dicts, func_args, opt_subs
# returns True if all values were updated, else None
# SHARES com_dict, com_func, func_dicts, func_args,
# opt_subs, funcs, verbose
for di in com_dict:
# don't allow a sign to change
if com_dict[di] > func_dicts[k][di]:
return
# remove it
if Func is Add:
take = min(func_dicts[k][i] for i in com_dict)
_sum = from_dict(com_dict)
if take == 1:
com_func_take = _sum
else:
com_func_take = Mul(take, _sum, evaluate=False)
else:
take = igcd(*[func_dicts[k][i] for i in com_dict])
base = from_dict(com_dict)
if take == 1:
com_func_take = base
else:
com_func_take = Pow(base, take, evaluate=False)
for di in com_dict:
func_dicts[k][di] -= take * com_dict[di]
# compute the remaining expression
rem = from_dict(func_dicts[k])
# reject hollow change, e.g extracting x + 1 from x + 3
if Func is Add and rem and rem.is_Integer and 1 in com_dict:
return
if verbose:
print('\nfunc %s (%s) \ncontains %s \nas %s \nleaving %s' %
(funcs[k], func_dicts[k], com_func, com_func_take, rem))
# recompute the dict since some keys may now
# have corresponding values of 0; one could
# keep track of which ones went to zero but
# this seems cleaner
func_dicts[k] = as_dict(rem)
# update associated info
func_dicts[k][com_func] = take
func_args[k] = set(func_dicts[k])
# keep the constant separate from the remaining
# part of the expression, e.g. 2*(a*b) rather than 2*a*b
opt_subs[funcs[k]] = ufunc(rem, com_func_take)
# everything was updated
return True
def get_copy(i):
return [func_dicts[i].copy(), func_args[i].copy(), funcs[i], i]
def restore(dafi):
i = dafi.pop()
func_dicts[i], func_args[i], funcs[i] = dafi
# ----------------- end helpers -----------------------
func_dicts = [as_dict(f) for f in funcs]
func_args = [set(d) for d in func_dicts]
while True:
hit = pairwise_most_common(func_args)
if not hit or len(hit[0][0]) <= 1:
break
changed = False
for com_args, ij in hit:
take = len(com_args)
ALL = list(ordered(com_args))
while take >= 2:
for com_args in subsets(ALL, take):
com_func = Func(*com_args)
com_dict = as_dict(com_func)
for i, j in ij:
dafi = None
if com_func != funcs[i]:
dafi = get_copy(i)
ch = update(i)
if not ch:
restore(dafi)
continue
if com_func != funcs[j]:
dafj = get_copy(j)
ch = update(j)
if not ch:
if dafi is not None:
restore(dafi)
restore(dafj)
continue
changed = True
if changed:
break
else:
take -= 1
continue
break
else:
continue
break
if not changed:
break
def cse(exprs, symbols=None, optimizations=None, postprocess=None):
""" Perform common subexpression elimination on an expression.
Parameters
==========
exprs : list of sympy expressions, or a single sympy expression
The expressions to reduce.
symbols : infinite iterator yielding unique Symbols
The symbols used to label the common subexpressions which are pulled
out. The ``numbered_symbols`` generator is useful. The default is a
stream of symbols of the form "x0", "x1", etc. This must be an infinite
iterator.
optimizations : list of (callable, callable) pairs, optional
The (preprocessor, postprocessor) pairs. If not provided,
``sympy.simplify.cse.cse_optimizations`` is used.
postprocess : a function which accepts the two return values of cse and
returns the desired form of output from cse, e.g. if you want the
replacements reversed the function might be the following lambda:
lambda r, e: return reversed(r), e
Returns
=======
replacements : list of (Symbol, expression) pairs
All of the common subexpressions that were replaced. Subexpressions
earlier in this list might show up in subexpressions later in this list.
reduced_exprs : list of sympy expressions
The reduced expressions with all of the replacements above.
"""
from sympy.matrices import Matrix
if symbols is None:
symbols = numbered_symbols()
else:
# In case we get passed an iterable with an __iter__ method instead of
# an actual iterator.
symbols = iter(symbols)
seen_subexp = set()
muls = set()
adds = set()
to_eliminate = set()
if optimizations is None:
# Pull out the default here just in case there are some weird
# manipulations of the module-level list in some other thread.
optimizations = list(cse_optimizations)
# Handle the case if just one expression was passed.
if isinstance(exprs, Basic):
exprs = [exprs]
# Preprocess the expressions to give us better optimization opportunities.
reduced_exprs = [preprocess_for_cse(e, optimizations) for e in exprs]
# Find all of the repeated subexpressions.
for expr in reduced_exprs:
if not isinstance(expr, Basic):
continue
pt = preorder_traversal(expr)
for subtree in pt:
inv = 1 / subtree if subtree.is_Pow else None
if subtree.is_Atom or iterable(subtree) or inv and inv.is_Atom:
# Exclude atoms, since there is no point in renaming them.
continue
if subtree in seen_subexp:
if inv and _coeff_isneg(subtree.exp):
# save the form with positive exponent
subtree = inv
to_eliminate.add(subtree)
pt.skip()
continue
if inv and inv in seen_subexp:
if _coeff_isneg(subtree.exp):
# save the form with positive exponent
subtree = inv
to_eliminate.add(subtree)
pt.skip()
continue
elif subtree.is_Mul:
muls.add(subtree)
elif subtree.is_Add:
adds.add(subtree)
seen_subexp.add(subtree)
# process adds - any adds that weren't repeated might contain
# subpatterns that are repeated, e.g. x+y+z and x+y have x+y in common
adds = [set(a.args) for a in ordered(adds)]
for i in xrange(len(adds)):
for j in xrange(i + 1, len(adds)):
com = adds[i].intersection(adds[j])
if len(com) > 1:
to_eliminate.add(Add(*com))
# remove this set of symbols so it doesn't appear again
adds[i] = adds[i].difference(com)
adds[j] = adds[j].difference(com)
for k in xrange(j + 1, len(adds)):
if not com.difference(adds[k]):
adds[k] = adds[k].difference(com)
# process muls - any muls that weren't repeated might contain
# subpatterns that are repeated, e.g. x*y*z and x*y have x*y in common
# use SequenceMatcher on the nc part to find the longest common expression
# in common between the two nc parts
sm = difflib.SequenceMatcher()
muls = [a.args_cnc(cset=True) for a in ordered(muls)]
for i in xrange(len(muls)):
if muls[i][1]:
sm.set_seq1(muls[i][1])
for j in xrange(i + 1, len(muls)):
# the commutative part in common
ccom = muls[i][0].intersection(muls[j][0])
# the non-commutative part in common
if muls[i][1] and muls[j][1]:
# see if there is any chance of an nc match
ncom = set(muls[i][1]).intersection(set(muls[j][1]))
if len(ccom) + len(ncom) < 2:
continue
# now work harder to find the match
sm.set_seq2(muls[j][1])
i1, _, n = sm.find_longest_match(0, len(muls[i][1]), 0,
len(muls[j][1]))
ncom = muls[i][1][i1:i1 + n]
else:
ncom = []
com = list(ccom) + ncom
if len(com) < 2:
continue
to_eliminate.add(Mul(*com))
# remove ccom from all if there was no ncom; to update the nc part
# would require finding the subexpr and then replacing it with a
# dummy to keep bounding nc symbols from being identified as a
# subexpr, e.g. removing B*C from A*B*C*D might allow A*D to be
# identified as a subexpr which would not be right.
if not ncom:
muls[i][0] = muls[i][0].difference(ccom)
for k in xrange(j, len(muls)):
if not ccom.difference(muls[k][0]):
muls[k][0] = muls[k][0].difference(ccom)
# make to_eliminate canonical; we will prefer non-Muls to Muls
# so select them first (non-Muls will have False for is_Mul and will
# be first in the ordering.
to_eliminate = list(ordered(to_eliminate, lambda _: _.is_Mul))
# Substitute symbols for all of the repeated subexpressions.
replacements = []
reduced_exprs = list(reduced_exprs)
hit = True
for i, subtree in enumerate(to_eliminate):
if hit:
sym = next(symbols)
hit = False
if subtree.is_Pow and subtree.exp.is_Rational:
update = lambda x: x.xreplace({subtree: sym, 1 / subtree: 1 / sym})
else:
update = lambda x: x.subs(subtree, sym)
# Make the substitution in all of the target expressions.
for j, expr in enumerate(reduced_exprs):
old = reduced_exprs[j]
reduced_exprs[j] = update(expr)
hit = hit or (old != reduced_exprs[j])
# Make the substitution in all of the subsequent substitutions.
for j in range(i + 1, len(to_eliminate)):
old = to_eliminate[j]
to_eliminate[j] = update(to_eliminate[j])
hit = hit or (old != to_eliminate[j])
if hit:
replacements.append((sym, subtree))
# Postprocess the expressions to return the expressions to canonical form.
for i, (sym, subtree) in enumerate(replacements):
subtree = postprocess_for_cse(subtree, optimizations)
replacements[i] = (sym, subtree)
reduced_exprs = [
postprocess_for_cse(e, optimizations) for e in reduced_exprs
]
# remove replacements that weren't used more than once
_remove_singletons(replacements, reduced_exprs)
if isinstance(exprs, Matrix):
reduced_exprs = [Matrix(exprs.rows, exprs.cols, reduced_exprs)]
if postprocess is None:
return replacements, reduced_exprs
return postprocess(replacements, reduced_exprs)
def linodesolve(A, t, b=None, B=None, type="auto", doit=False):
r"""
System of n equations linear first-order differential equations
Explanation
===========
This solver solves the system of ODEs of the follwing form:
.. math::
X'(t) = A(t) X(t) + b(t)
Here, $A(t)$ is the coefficient matrix, $X(t)$ is the vector of n independent variables,
$b(t)$ is the non-homogeneous term and $X'(t)$ is the derivative of $X(t)$
Depending on the properties of $A(t)$ and $b(t)$, this solver evaluates the solution
differently.
When $A(t)$ is constant coefficient matrix and $b(t)$ is zero vector i.e. system is homogeneous,
the solution is:
.. math::
X(t) = \exp(A t) C
Here, $C$ is a vector of constants and $A$ is the constant coefficient matrix.
When $A(t)$ is constant coefficient matrix and $b(t)$ is non-zero i.e. system is non-homogeneous,
the solution is:
.. math::
X(t) = e^{A t} ( \int e^{- A t} b \,dt + C)
When $A(t)$ is coefficient matrix such that its commutative with its antiderivative $B(t)$ and
$b(t)$ is a zero vector i.e. system is homogeneous, the solution is:
.. math::
X(t) = \exp(B(t)) C
When $A(t)$ is commutative with its antiderivative $B(t)$ and $b(t)$ is non-zero i.e. system is
non-homogeneous, the solution is:
.. math::
X(t) = e^{B(t)} ( \int e^{-B(t)} b(t) \,dt + C)
The final solution is the general solution for all the four equations since a constant coefficient
matrix is always commutative with its antidervative.
Parameters
==========
A : Matrix
Coefficient matrix of the system of linear first order ODEs.
t : Symbol
Independent variable in the system of ODEs.
b : Matrix or None
Non-homogeneous term in the system of ODEs. If None is passed,
a homogeneous system of ODEs is assumed.
B : Matrix or None
Antiderivative of the coefficient matrix. If the antiderivative
is not passed and the solution requires the term, then the solver
would compute it internally.
type : String
Type of the system of ODEs passed. Depending on the type, the
solution is evaluated. The type values allowed and the corresponding
system it solves are: "type1" for constant coefficient homogeneous
"type2" for constant coefficient non-homogeneous, "type3" for non-constant
coefficient homogeneous and "type4" for non-constant coefficient non-homogeneous.
The default value is "auto" which will let the solver decide the correct type of
the system passed.
doit : Boolean
Evaluate the solution if True, default value is False
Examples
========
To solve the system of ODEs using this function directly, several things must be
done in the right order. Wrong inputs to the function will lead to incorrect results.
>>> from sympy import symbols, Function, Eq
>>> from sympy.solvers.ode.systems import canonical_odes, linear_ode_to_matrix, linodesolve, linodesolve_type
>>> from sympy.solvers.ode.subscheck import checkodesol
>>> f, g = symbols("f, g", cls=Function)
>>> x, a = symbols("x, a")
>>> funcs = [f(x), g(x)]
>>> eqs = [Eq(f(x).diff(x) - f(x), a*g(x) + 1), Eq(g(x).diff(x) + g(x), a*f(x))]
Here, it is important to note that before we derive the coefficient matrix, it is
important to get the system of ODEs into the desired form. For that we will use
:obj:`sympy.solvers.ode.systems.canonical_odes()`.
>>> eqs = canonical_odes(eqs, funcs, x)
>>> eqs
[[Eq(Derivative(f(x), x), a*g(x) + f(x) + 1), Eq(Derivative(g(x), x), a*f(x) - g(x))]]
Now, we will use :obj:`sympy.solvers.ode.systems.linear_ode_to_matrix()` to get the coefficient matrix and the
non-homogeneous term if it is there.
>>> eqs = eqs[0]
>>> (A1, A0), b = linear_ode_to_matrix(eqs, funcs, x, 1)
>>> A = A0
We have the coefficient matrices and the non-homogeneous term ready. Now, we can use
:obj:`sympy.solvers.ode.systems.linodesolve_type()` to get the information for the system of ODEs
to finally pass it to the solver.
>>> system_info = linodesolve_type(A, x, b=b)
>>> sol_vector = linodesolve(A, x, b=b, B=system_info['antiderivative'], type=system_info['type'])
Now, we can prove if the solution is correct or not by using :obj:`sympy.solvers.ode.checkodesol()`
>>> sol = [Eq(f, s) for f, s in zip(funcs, sol_vector)]
>>> checkodesol(eqs, sol)
(True, [0, 0])
We can also use the doit method to evaluate the solutions passed by the function.
>>> sol_vector_evaluated = linodesolve(A, x, b=b, type="type2", doit=True)
Now, we will look at a system of ODEs which is non-constant.
>>> eqs = [Eq(f(x).diff(x), f(x) + x*g(x)), Eq(g(x).diff(x), -x*f(x) + g(x))]
The system defined above is already in the desired form, so we don't have to convert it.
>>> (A1, A0), b = linear_ode_to_matrix(eqs, funcs, x, 1)
>>> A = A0
A user can also pass the commutative antiderivative required for type3 and type4 system of ODEs.
Passing an incorrect one will lead to incorrect results. If the coefficient matrix is not commutative
with its antiderivative, then :obj:`sympy.solvers.ode.systems.linodesolve_type()` raises a NotImplementedError.
If it does have a commutative antiderivative, then the function just returns the information about the system.
>>> system_info = linodesolve_type(A, x, b=b)
Now, we can pass the antiderivative as an argument to get the solution. If the system information is not
passed, then the solver will compute the required arguments internally.
>>> sol_vector = linodesolve(A, x, b=b)
Once again, we can verify the solution obtained.
>>> sol = [Eq(f, s) for f, s in zip(funcs, sol_vector)]
>>> checkodesol(eqs, sol)
(True, [0, 0])
Returns
=======
List
Raises
======
ValueError
This error is raised when the coefficient matrix, non-homogeneous term
or the antiderivative, if passed, aren't a matrix or
don't have correct dimensions
NonSquareMatrixError
When the coefficient matrix or its antiderivative, if passed isn't a square
matrix
NotImplementedError
If the coefficient matrix doesn't have a commutative antiderivative
See Also
========
linear_ode_to_matrix: Coefficient matrix computation function
canonical_odes: System of ODEs representation change
linodesolve_type: Getting information about systems of ODEs to pass in this solver
"""
if not isinstance(A, MatrixBase):
raise ValueError(
filldedent('''\
The coefficients of the system of ODEs should be of type Matrix
'''))
if not A.is_square:
raise NonSquareMatrixError(
filldedent('''\
The coefficient matrix must be a square
'''))
if b is not None:
if not isinstance(b, MatrixBase):
raise ValueError(
filldedent('''\
The non-homogeneous terms of the system of ODEs should be of type Matrix
'''))
if A.rows != b.rows:
raise ValueError(
filldedent('''\
The system of ODEs should have the same number of non-homogeneous terms and the number of
equations
'''))
if B is not None:
if not isinstance(B, MatrixBase):
raise ValueError(
filldedent('''\
The antiderivative of coefficients of the system of ODEs should be of type Matrix
'''))
if not B.is_square:
raise NonSquareMatrixError(
filldedent('''\
The antiderivative of the coefficient matrix must be a square
'''))
if A.rows != B.rows:
raise ValueError(
filldedent('''\
The coefficient matrix and its antiderivative should have same dimensions
'''))
if not any(type == "type{}".format(i)
for i in range(1, 5)) and not type == "auto":
raise ValueError(
filldedent('''\
The input type should be a valid one
'''))
n = A.rows
# constants = numbered_symbols(prefix='C', cls=Dummy, start=const_idx+1)
Cvect = Matrix(list(Dummy() for _ in range(n)))
if (type == "type2" or type == "type4") and b is None:
b = zeros(n, 1)
if type == "auto":
system_info = linodesolve_type(A, t, b=b)
type = system_info["type"]
B = system_info["antiderivative"]
if type == "type1" or type == "type2":
P, J = matrix_exp_jordan_form(A, t)
P = simplify(P)
if type == "type1":
sol_vector = P * (J * Cvect)
else:
sol_vector = P * J * (
(J.inv() * P.inv() * b).applyfunc(lambda x: Integral(x, t)) +
Cvect)
else:
if B is None:
B, _ = _is_commutative_anti_derivative(A, t)
if type == "type3":
sol_vector = B.exp() * Cvect
else:
sol_vector = B.exp() * ((
(-B).exp() * b).applyfunc(lambda x: Integral(x, t)) + Cvect)
gens = sol_vector.atoms(exp)
if type != "type1":
sol_vector = [expand_mul(s) for s in sol_vector]
sol_vector = [collect(s, ordered(gens), exact=True) for s in sol_vector]
if doit:
sol_vector = [s.doit() for s in sol_vector]
return sol_vector
