def quantity_simplify(expr):
if expr.is_Atom:
return expr
if not expr.is_Mul:
return expr.func(*map(quantity_simplify, expr.args))
if expr.has(Prefix):
coeff, args = expr.as_coeff_mul(Prefix)
args = list(args)
for arg in args:
if isinstance(arg, Pow):
coeff = coeff * (arg.base.scale_factor ** arg.exp)
else:
coeff = coeff * arg.scale_factor
expr = coeff
coeff, args = expr.as_coeff_mul(Quantity)
args_pow = [arg.as_base_exp() for arg in args]
quantity_pow, other_pow = sift(args_pow, lambda x: isinstance(x[0], Quantity), binary=True)
quantity_pow_by_dim = sift(quantity_pow, lambda x: x[0].dimension)
# Just pick the first quantity:
ref_quantities = [i[0][0] for i in quantity_pow_by_dim.values()]
new_quantities = [
Mul.fromiter(
(quantity*i.scale_factor/quantity.scale_factor)**p for i, p in v)
if len(v) > 1 else v[0][0]**v[0][1]
for quantity, (k, v) in zip(ref_quantities, quantity_pow_by_dim.items())]
return coeff*Mul.fromiter(other_pow)*Mul.fromiter(new_quantities)
def quantity_simplify(expr):
if expr.is_Atom:
return expr
if not expr.is_Mul:
return expr.func(*map(quantity_simplify, expr.args))
if expr.has(Prefix):
coeff, args = expr.as_coeff_mul(Prefix)
args = list(args)
for arg in args:
if isinstance(arg, Pow):
coeff = coeff * (arg.base.scale_factor**arg.exp)
else:
coeff = coeff * arg.scale_factor
expr = coeff
coeff, args = expr.as_coeff_mul(Quantity)
args_pow = [arg.as_base_exp() for arg in args]
quantity_pow, other_pow = sift(args_pow,
lambda x: isinstance(x[0], Quantity),
binary=True)
quantity_pow_by_dim = sift(quantity_pow, lambda x: x[0].dimension)
# Just pick the first quantity:
ref_quantities = [i[0][0] for i in quantity_pow_by_dim.values()]
new_quantities = [
Mul.fromiter((quantity * i.scale_factor / quantity.scale_factor)**p
for i, p in v) if len(v) > 1 else v[0][0]**v[0][1]
for quantity, (k,
v) in zip(ref_quantities, quantity_pow_by_dim.items())
]
return coeff * Mul.fromiter(other_pow) * Mul.fromiter(new_quantities)
def _group_Add_terms(self, add):
numbers, non_num = sift(add.args,
lambda arg: arg.is_number,
binary=True)
numsum = sum(numbers)
terms_with_func, other = sift(non_num,
lambda arg: arg.has(self.func),
binary=True)
return numsum, terms_with_func, other
def _separate_imaginary_from_complex(cls, complexes):
from sympy.utilities.iterables import sift
def is_imag(c):
'''
return True if all roots are imaginary (ax**2 + b)
return False if no roots are imaginary
return None if 2 roots are imaginary (ax**N'''
u, f, k = c
deg = f.degree()
if f.length() == 2:
if deg == 2:
return True # both imag
elif _ispow2(deg):
if f.LC() * f.TC() < 0:
return None # 2 are imag
return False # none are imag
# separate according to the function
sifted = sift(complexes, lambda c: c[1])
del complexes
imag = []
complexes = []
for f in sifted:
isift = sift(sifted[f], lambda c: is_imag(c))
imag.extend(isift.pop(True, []))
complexes.extend(isift.pop(False, []))
mixed = isift.pop(None, [])
assert not isift
if not mixed:
continue
while True:
# the non-imaginary ones will be on one side or the other
# of the y-axis
i = 0
while i < len(mixed):
u, f, k = mixed[i]
if u.ax * u.bx > 0:
complexes.append(mixed.pop(i))
else:
i += 1
if len(mixed) == 2:
imag.extend(mixed)
break
# refine
for i, (u, f, k) in enumerate(mixed):
u = u._inner_refine()
mixed[i] = u, f, k
return imag, complexes
def _separate_imaginary_from_complex(cls, complexes):
from sympy.utilities.iterables import sift
def is_imag(c):
'''
return True if all roots are imaginary (ax**2 + b)
return False if no roots are imaginary
return None if 2 roots are imaginary (ax**N'''
u, f, k = c
deg = f.degree()
if f.length() == 2:
if deg == 2:
return True # both imag
elif _ispow2(deg):
if f.LC()*f.TC() < 0:
return None # 2 are imag
return False # none are imag
# separate according to the function
sifted = sift(complexes, lambda c: c[1])
del complexes
imag = []
complexes = []
for f in sifted:
isift = sift(sifted[f], lambda c: is_imag(c))
imag.extend(isift.pop(True, []))
complexes.extend(isift.pop(False, []))
mixed = isift.pop(None, [])
assert not isift
if not mixed:
continue
while True:
# the non-imaginary ones will be on one side or the other
# of the y-axis
i = 0
while i < len(mixed):
u, f, k = mixed[i]
if u.ax*u.bx > 0:
complexes.append(mixed.pop(i))
else:
i += 1
if len(mixed) == 2:
imag.extend(mixed)
break
# refine
for i, (u, f, k) in enumerate(mixed):
u = u._inner_refine()
mixed[i] = u, f, k
return imag, complexes
def _eval_expand_power_base(self, deep=True, **hints):
"""(a*b)**n -> a**n * b**n"""
force = hints.get('force', False)
b, ewas = self.args
if deep:
e = self.exp.expand(deep=deep, **hints)
else:
e = self.exp
if b.is_Mul:
bargs = b.args
if force or e.is_integer:
nonneg = bargs
other = []
elif ewas.is_Rational or len(
bargs) == 2 and bargs[0] is S.NegativeOne:
# the Rational exponent was already expanded automatically
# if there is a negative Number * foo, foo must be unknown
# or else it, too, would have automatically expanded;
# sqrt(-Number*foo) -> sqrt(Number)*sqrt(-foo); then
# sqrt(-foo) -> unchanged if foo is not positive else
# -> I*sqrt(foo)
# So...if we have a 2 arg Mul and the first is a Number
# that number is -1 and there is nothing more than can
# be done without the force=True hint
nonneg = []
else:
# this is just like what is happening automatically, except
# that now we are doing it for an arbitrary exponent for which
# no automatic expansion is done
def pred(x):
if x.is_polar is None:
return x.is_nonnegative
return x.is_polar
sifted = sift(b.args, pred)
nonneg = sifted.get(True, [])
other = sifted.get(None, [])
neg = sifted.get(False, [])
# make sure the Number gets pulled out
if neg and neg[0].is_Number and neg[0] is not S.NegativeOne:
nonneg.append(-neg[0])
neg[0] = S.NegativeOne
# leave behind a negative sign
oddneg = len(neg) % 2
if oddneg:
other.append(S.NegativeOne)
# negate all negatives and append to nonneg
nonneg += [-n for n in neg]
if nonneg: # then there's a new expression to return
other = [Pow(Mul(*other), e)]
if deep:
return Mul(*([Pow(b.expand(deep=deep, **hints), e)\
for b in nonneg] + other))
else:
return Mul(*([Pow(b, e) for b in nonneg] + other))
return Pow(b, e)
def _ncsplit(expr):
# this is not the same as args_cnc because here
# we don't assume expr is a Mul -- hence deal with args --
# and always return a set.
cpart, ncpart = sift(expr.args,
lambda arg: arg.is_commutative is True, binary=True)
return set(cpart), ncpart
def merge_explicit(vecadd):
""" Merge Vector arguments
Example
========
>>> from sympy.vector import CoordSys3D
>>> v1 = VectorSymbol("v1")
>>> v2 = VectorSymbol("v2")
>>> C = CoordSys3D("C")
>>> vn1 = 2 * C.i + 3 * C.j + 4 * C.k
>>> vn2 = x * C.i + y * C.j + x * y * C.k
>>> expr = v1 + v2 + vn1 + vn2
>>> pprint(expr)
(2*C.i + 3*C.j + 4*C.k) + (x*C.i + y*C.j + x*y*C.k) + v1 + v2
>>> pprint(merge_explicit(expr))
((2 + x)*C.i + (3 + y)*C.j + (4 + x*y)*C.k) + v1 + v2
"""
# there has to be some bug deep inside the core, I absolutely need this
# function to correctly perform addition
def recreate_args(args):
return [a.func(*a.args) for a in args]
# Adapted from sympy.matrices.expressions.matadd.py
groups = sift(vecadd.args, lambda arg: isinstance(arg, (Vector)))
if len(groups[True]) > 1:
return VecAdd(*(recreate_args(groups[False]) +
[reduce(add, recreate_args(groups[True]))]))
# return VecAdd(*(groups[False] + [reduce(add, groups[True])]))
else:
return vecadd
def factor_minmax(args):
is_other = lambda arg: isinstance(arg, other)
other_args, remaining_args = sift(args, is_other, binary=True)
if not other_args:
return args
# Min(Max(x, y, z), Max(x, y, u, v)) -> {x,y}, ({z}, {u,v})
arg_sets = [set(arg.args) for arg in other_args]
common = set.intersection(*arg_sets)
if not common:
return args
new_other_args = list(common)
arg_sets_diff = [arg_set - common for arg_set in arg_sets]
# If any set is empty after removing common then all can be
# discarded e.g. Min(Max(a, b, c), Max(a, b)) -> Max(a, b)
if all(arg_sets_diff):
other_args_diff = [
other(*s, evaluate=False) for s in arg_sets_diff
]
new_other_args.append(cls(*other_args_diff, evaluate=False))
other_args_factored = other(*new_other_args, evaluate=False)
return remaining_args + [other_args_factored]
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 quantity_simplify(expr):
"""Return an equivalent expression in which prefixes are replaced
with numerical values and all units of a given dimension are the
unified in a canonical manner.
Examples
========
>>> from sympy.physics.units.util import quantity_simplify
>>> from sympy.physics.units.prefixes import kilo
>>> from sympy.physics.units import foot, inch
>>> quantity_simplify(kilo*foot*inch)
250*foot**2/3
>>> quantity_simplify(foot - 6*inch)
foot/2
"""
if expr.is_Atom or not expr.has(Prefix, Quantity):
return expr
# replace all prefixes with numerical values
p = expr.atoms(Prefix)
expr = expr.xreplace({p: p.scale_factor for p in p})
# replace all quantities of given dimension with a canonical
# quantity, chosen from those in the expression
d = sift(expr.atoms(Quantity), lambda i: i.dimension)
for k in d:
if len(d[k]) == 1:
continue
v = list(ordered(d[k]))
ref = v[0]/v[0].scale_factor
expr = expr.xreplace({vi: ref*vi.scale_factor for vi in v[1:]})
return expr
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 MatMul_elements(matrix_predicate, scalar_predicate, expr, assumptions):
d = sift(expr.args, lambda x: isinstance(x, MatrixExpr))
factors, matrices = d[False], d[True]
return fuzzy_and([
test_closed_group(Basic(*factors), assumptions, scalar_predicate),
test_closed_group(Basic(*matrices), assumptions, matrix_predicate)
])
def quantity_simplify(expr):
"""Return an equivalent expression in which prefixes are replaced
with numerical values and all units of a given dimension are the
unified in a canonical manner.
Examples
========
>>> from sympy.physics.units.util import quantity_simplify
>>> from sympy.physics.units.prefixes import kilo
>>> from sympy.physics.units import foot, inch
>>> quantity_simplify(kilo*foot*inch)
250*foot**2/3
>>> quantity_simplify(foot - 6*inch)
foot/2
"""
if expr.is_Atom or not expr.has(Prefix, Quantity):
return expr
# replace all prefixes with numerical values
p = expr.atoms(Prefix)
expr = expr.xreplace({p: p.scale_factor for p in p})
# replace all quantities of given dimension with a canonical
# quantity, chosen from those in the expression
d = sift(expr.atoms(Quantity), lambda i: i.dimension)
for k in d:
if len(d[k]) == 1:
continue
v = list(ordered(d[k]))
ref = v[0] / v[0].scale_factor
expr = expr.xreplace({vi: ref * vi.scale_factor for vi in v[1:]})
return expr
def _ncsplit(expr):
# this is not the same as args_cnc because here
# we don't assume expr is a Mul -- hence deal with args --
# and always return a set.
cpart, ncpart = sift(expr.args,
lambda arg: arg.is_commutative is True, binary=True)
return set(cpart), ncpart
def replace_in_Add(self, e):
""" passed as second argument to Basic.replace(...) """
numsum, terms_with_func, other_non_num_terms = self._group_Add_terms(e)
if numsum == 0:
return e
substituted, untouched = [], []
for with_func in terms_with_func:
if with_func.is_Mul:
func, coeff = sift(with_func.args,
lambda arg: arg.func == self.func,
binary=True)
if len(func) == 1 and len(coeff) == 1:
func, coeff = func[0], coeff[0]
else:
coeff = None
elif with_func.func == self.func:
func, coeff = with_func, S.One
else:
coeff = None
if coeff is not None and coeff.is_number and sign(
coeff) == -sign(numsum):
if self.opportunistic:
do_substitute = abs(coeff + numsum) < abs(numsum)
else:
do_substitute = coeff + numsum == 0
if do_substitute: # advantageous substitution
numsum += coeff
substituted.append(coeff * self.func_m_1(*func.args))
continue
untouched.append(with_func)
return e.func(numsum, *substituted, *untouched, *other_non_num_terms)
def _eval_expand_power_base(self, deep=True, **hints):
"""(a*b)**n -> a**n * b**n"""
force = hints.get("force", False)
b, ewas = self.args
if deep:
e = self.exp.expand(deep=deep, **hints)
else:
e = self.exp
if b.is_Mul:
bargs = b.args
if force or e.is_integer:
nonneg = bargs
other = []
elif ewas.is_Rational or len(bargs) == 2 and bargs[0] is S.NegativeOne:
# the Rational exponent was already expanded automatically
# if there is a negative Number * foo, foo must be unknown
# or else it, too, would have automatically expanded;
# sqrt(-Number*foo) -> sqrt(Number)*sqrt(-foo); then
# sqrt(-foo) -> unchanged if foo is not positive else
# -> I*sqrt(foo)
# So...if we have a 2 arg Mul and the first is a Number
# that number is -1 and there is nothing more than can
# be done without the force=True hint
nonneg = []
else:
# this is just like what is happening automatically, except
# that now we are doing it for an arbitrary exponent for which
# no automatic expansion is done
def pred(x):
if x.is_polar is None:
return x.is_nonnegative
return x.is_polar
sifted = sift(b.args, pred)
nonneg = sifted.get(True, [])
other = sifted.get(None, [])
neg = sifted.get(False, [])
# make sure the Number gets pulled out
if neg and neg[0].is_Number and neg[0] is not S.NegativeOne:
nonneg.append(-neg[0])
neg[0] = S.NegativeOne
# leave behind a negative sign
oddneg = len(neg) % 2
if oddneg:
other.append(S.NegativeOne)
# negate all negatives and append to nonneg
nonneg += [-n for n in neg]
if nonneg: # then there's a new expression to return
other = [Pow(Mul(*other), e)]
if deep:
return Mul(*([Pow(b.expand(deep=deep, **hints), e) for b in nonneg] + other))
else:
return Mul(*([Pow(b, e) for b in nonneg] + other))
return Pow(b, e)
def __new__(cls, sym, condition, base_set):
# nonlinsolve uses ConditionSet to return an unsolved system
# of equations (see _return_conditionset in solveset) so until
# that is changed we do minimal checking of the args
unsolved = isinstance(sym, (Tuple, tuple))
if unsolved:
sym = Tuple(*sym)
condition = FiniteSet(*condition)
else:
condition = as_Boolean(condition)
if isinstance(base_set, set):
base_set = FiniteSet(*base_set)
elif not isinstance(base_set, Set):
raise TypeError('expecting set for base_set')
if condition == S.false:
return S.EmptySet
if condition == S.true:
return base_set
if isinstance(base_set, EmptySet):
return base_set
if not unsolved:
if isinstance(base_set, FiniteSet):
sifted = sift(base_set,
lambda _: fuzzy_bool(condition.subs(sym, _)))
if sifted[None]:
return Union(
FiniteSet(*sifted[True]),
Basic.__new__(cls, sym, condition,
FiniteSet(*sifted[None])))
else:
return FiniteSet(*sifted[True])
if isinstance(base_set, cls):
s, c, base_set = base_set.args
if sym == s:
condition = And(condition, c)
elif sym not in c.free_symbols:
condition = And(condition, c.xreplace({s: sym}))
elif s not in condition.free_symbols:
condition = And(condition.xreplace({sym: s}), c)
sym = s
else:
# user will have to use cls.sym to get symbol
dum = Symbol('lambda')
if dum in condition.free_symbols or \
dum in c.free_symbols:
dum = Dummy(str(dum))
condition = And(condition.xreplace({sym: dum}),
c.xreplace({s: dum}))
sym = dum
if sym in base_set.free_symbols or \
not isinstance(sym, Symbol):
s = Symbol('lambda')
if s in base_set.free_symbols:
s = Dummy('lambda')
condition = condition.xreplace({sym: s})
sym = s
return Basic.__new__(cls, sym, condition, base_set)
def conglomerate(expr):
""" Conglomerate together identical args x + x -> 2x """
groups = sift(expr.args, key)
counts = dict((k, sum(map(count, args))) for k, args in groups.items())
newargs = [combine(cnt, mat) for mat, cnt in counts.items()]
if set(newargs) != set(expr.args):
return new(type(expr), *newargs)
else:
return expr
def conglomerate(expr):
""" Conglomerate together identical args x + x -> 2x """
groups = sift(expr.args, key)
counts = dict((k, sum(map(count, args))) for k, args in groups.items())
newargs = [combine(cnt, mat) for mat, cnt in counts.items()]
if set(newargs) != set(expr.args):
return new(type(expr), *newargs)
else:
return expr
def __new__(cls, variable, condition, base_set=S.UniversalSet):
# nonlinsolve uses ConditionSet to return an unsolved system
# of equations (see _return_conditionset in solveset) so until
# that is changed we do minimal checking of the args
if isinstance(variable, (Tuple, tuple)): # unsolved eqns syntax
variable = Tuple(*variable)
condition = FiniteSet(*condition)
return Basic.__new__(cls, variable, condition, base_set)
condition = as_Boolean(condition)
if isinstance(base_set, set):
base_set = FiniteSet(*base_set)
elif not base_set.is_set:
raise TypeError('expecting set for base_set')
if condition is S.false:
return S.EmptySet
if condition is S.true:
return base_set
if isinstance(base_set, EmptySet):
return base_set
know = None
if isinstance(base_set, FiniteSet):
sifted = sift(
base_set, lambda _: fuzzy_bool(
condition.subs(variable, _)))
if sifted[None]:
know = FiniteSet(*sifted[True])
base_set = FiniteSet(*sifted[None])
else:
return FiniteSet(*sifted[True])
if isinstance(base_set, cls):
s, c, base_set = base_set.args
if variable == s:
condition = And(condition, c)
elif variable not in c.free_symbols:
condition = And(condition, c.xreplace({s: variable}))
elif s not in condition.free_symbols:
condition = And(condition.xreplace({variable: s}), c)
variable = s
else:
# user will have to use cls.variable to get symbol
dum = Symbol('lambda')
if dum in condition.free_symbols or \
dum in c.free_symbols:
dum = Dummy(str(dum))
condition = And(
condition.xreplace({variable: dum}),
c.xreplace({s: dum}))
variable = dum
from sympy.tensor.indexed import Slice, IndexedBase
assert isinstance(variable, (Symbol, Slice, IndexedBase))
# s = Dummy('lambda')
# if s not in condition.xreplace({variable: s}).free_symbols:
# raise ValueError('non-symbol dummy not recognized in condition')
if condition.is_BooleanFalse:
return S.EmptySet
rv = Basic.__new__(cls, variable, condition, base_set)
return rv if know is None else Union(know, rv)
def quantity_simplify(expr):
"""Return an equivalent expression in which prefixes are replaced
with numerical values and products of related quantities are
unified in a canonical manner.
Examples
========
>>> from sympy.physics.units.util import quantity_simplify
>>> from sympy.physics.units.prefixes import kilo
>>> from sympy.physics.units import foot, inch
>>> quantity_simplify(kilo*foot*inch)
250*foot**2/3
"""
if expr.is_Atom:
return expr
if isinstance(expr, Prefix):
return expr.scale_factor
expr = expr.func(*map(quantity_simplify, expr.args))
if not expr.is_Mul or not expr.has(Quantity):
return expr
args_pow = [arg.as_base_exp() for arg in expr.args]
quantity_pow, other_pow = sift(args_pow,
lambda x: isinstance(x[0], Quantity),
binary=True)
coeff = Mul.fromiter([Pow(b, e, evaluate=False) for b, e in other_pow])
quantity_pow_by_dim = sift(quantity_pow, lambda x: x[0].dimension)
new_quantities = []
for _, bp in quantity_pow_by_dim.items():
if len(bp) == 1:
new_quantities.append(bp[0][0]**bp[0][1])
else:
# just let reference quantity be the first quantity,
# picked from an ordered list
bp = list(ordered(bp))
ref = bp[0][0] / bp[0][0].scale_factor
new_quantities.append(
Mul.fromiter(((ref * b.scale_factor)**p for b, p in bp)))
return coeff * Mul.fromiter(new_quantities)
def _filter_assumptions(kwargs):
"""Split the given dict into assumptions and non-assumptions.
Keys are taken as assumptions if they correspond to an
entry in ``_assume_defined``.
"""
assumptions, nonassumptions = map(
dict,
sift(kwargs.items(), lambda i: i[0] in _assume_defined, binary=True))
Symbol._sanitize(assumptions)
return assumptions, nonassumptions
def _handle_finite_sets(op, x, y, commutative):
# Handle finite sets:
fs_args, other = sift([x, y], lambda x: isinstance(x, FiniteSet), binary=True)
if len(fs_args) == 2:
return FiniteSet(*[op(i, j) for i in fs_args[0] for j in fs_args[1]])
elif len(fs_args) == 1:
sets = [_apply_operation(op, other[0], i, commutative) for i in fs_args[0]]
return Union(*sets)
else:
return None
def __new__(cls, sym, condition, base_set=S.UniversalSet):
# nonlinsolve uses ConditionSet to return an unsolved system
# of equations (see _return_conditionset in solveset) so until
# that is changed we do minimal checking of the args
if isinstance(sym, (Tuple, tuple)): # unsolved eqns syntax
sym = Tuple(*sym)
condition = FiniteSet(*condition)
return Basic.__new__(cls, sym, condition, base_set)
condition = as_Boolean(condition)
if isinstance(base_set, set):
base_set = FiniteSet(*base_set)
elif not isinstance(base_set, Set):
raise TypeError('expecting set for base_set')
if condition is S.false:
return S.EmptySet
if condition is S.true:
return base_set
if isinstance(base_set, EmptySet):
return base_set
know = None
if isinstance(base_set, FiniteSet):
sifted = sift(
base_set, lambda _: fuzzy_bool(
condition.subs(sym, _)))
if sifted[None]:
know = FiniteSet(*sifted[True])
base_set = FiniteSet(*sifted[None])
else:
return FiniteSet(*sifted[True])
if isinstance(base_set, cls):
s, c, base_set = base_set.args
if sym == s:
condition = And(condition, c)
elif sym not in c.free_symbols:
condition = And(condition, c.xreplace({s: sym}))
elif s not in condition.free_symbols:
condition = And(condition.xreplace({sym: s}), c)
sym = s
else:
# user will have to use cls.sym to get symbol
dum = Symbol('lambda')
if dum in condition.free_symbols or \
dum in c.free_symbols:
dum = Dummy(str(dum))
condition = And(
condition.xreplace({sym: dum}),
c.xreplace({s: dum}))
sym = dum
if not isinstance(sym, Symbol):
s = Dummy('lambda')
if s not in condition.xreplace({sym: s}).free_symbols:
raise ValueError(
'non-symbol dummy not recognized in condition')
rv = Basic.__new__(cls, sym, condition, base_set)
return rv if know is None else Union(know, rv)
def __new__(cls, sym, condition, base_set=S.UniversalSet):
# nonlinsolve uses ConditionSet to return an unsolved system
# of equations (see _return_conditionset in solveset) so until
# that is changed we do minimal checking of the args
if isinstance(sym, (Tuple, tuple)): # unsolved eqns syntax
sym = Tuple(*sym)
condition = FiniteSet(*condition)
return Basic.__new__(cls, sym, condition, base_set)
condition = as_Boolean(condition)
if isinstance(base_set, set):
base_set = FiniteSet(*base_set)
elif not isinstance(base_set, Set):
raise TypeError('expecting set for base_set')
if condition is S.false:
return S.EmptySet
if condition is S.true:
return base_set
if isinstance(base_set, EmptySet):
return base_set
know = None
if isinstance(base_set, FiniteSet):
sifted = sift(
base_set, lambda _: fuzzy_bool(
condition.subs(sym, _)))
if sifted[None]:
know = FiniteSet(*sifted[True])
base_set = FiniteSet(*sifted[None])
else:
return FiniteSet(*sifted[True])
if isinstance(base_set, cls):
s, c, base_set = base_set.args
if sym == s:
condition = And(condition, c)
elif sym not in c.free_symbols:
condition = And(condition, c.xreplace({s: sym}))
elif s not in condition.free_symbols:
condition = And(condition.xreplace({sym: s}), c)
sym = s
else:
# user will have to use cls.sym to get symbol
dum = Symbol('lambda')
if dum in condition.free_symbols or \
dum in c.free_symbols:
dum = Dummy(str(dum))
condition = And(
condition.xreplace({sym: dum}),
c.xreplace({s: dum}))
sym = dum
if not isinstance(sym, Symbol):
s = Dummy('lambda')
if s not in condition.xreplace({sym: s}).free_symbols:
raise ValueError(
'non-symbol dummy not recognized in condition')
rv = Basic.__new__(cls, sym, condition, base_set)
return rv if know is None else Union(know, rv)
def _expm1_value(e):
numbers, non_num = sift(e.args, lambda arg: arg.is_number, binary=True)
non_num_exp, non_num_other = sift(non_num, lambda arg: arg.has(exp),
binary=True)
numsum = sum(numbers)
new_exp_terms, done = [], False
for exp_term in non_num_exp:
if done:
new_exp_terms.append(exp_term)
else:
looking_at = exp_term + numsum
attempt = _try_expm1(looking_at)
if looking_at == attempt:
new_exp_terms.append(exp_term)
else:
done = True
new_exp_terms.append(attempt)
if not done:
new_exp_terms.append(numsum)
return e.func(*chain(new_exp_terms, non_num_other))
def _expm1_value(e):
numbers, non_num = sift(e.args, lambda arg: arg.is_number, binary=True)
non_num_exp, non_num_other = sift(non_num, lambda arg: arg.has(exp),
binary=True)
numsum = sum(numbers)
new_exp_terms, done = [], False
for exp_term in non_num_exp:
if done:
new_exp_terms.append(exp_term)
else:
looking_at = exp_term + numsum
attempt = _try_expm1(looking_at)
if looking_at == attempt:
new_exp_terms.append(exp_term)
else:
done = True
new_exp_terms.append(attempt)
if not done:
new_exp_terms.append(numsum)
return e.func(*chain(new_exp_terms, non_num_other))
def __call__(self, e):
numbers, non_num = sift(e.args, lambda arg: arg.is_number, binary=True)
non_num_func, non_num_other = sift(non_num, lambda arg: arg.has(self.func),
binary=True)
numsum = sum(numbers)
new_func_terms, done = [], False
for func_term in non_num_func:
if done:
new_func_terms.append(func_term)
else:
looking_at = func_term + numsum
attempt = self._try_func_m_1(looking_at)
if looking_at == attempt:
new_func_terms.append(func_term)
else:
done = True
new_func_terms.append(attempt)
if not done:
new_func_terms.append(numsum)
return e.func(*chain(new_func_terms, non_num_other))
def test_sift():
assert sift(list(range(5)), lambda _: _ % 2) == {1: [1, 3], 0: [0, 2, 4]}
assert sift([x, y], lambda _: _.has(x)) == {False: [y], True: [x]}
assert sift([S.One], lambda _: _.has(x)) == {False: [1]}
assert sift([0, 1, 2, 3], lambda x: x % 2, binary=True) == ([1, 3], [0, 2])
assert sift([0, 1, 2, 3], lambda x: x % 3 == 1, binary=True) == ([1], [0, 2, 3])
raises(ValueError, lambda: sift([0, 1, 2, 3], lambda x: x % 3, binary=True))
def bc_matadd(expr):
args = sift(expr.args, lambda M: isinstance(M, BlockMatrix))
blocks = args[True]
if not blocks:
return expr
nonblocks = args[False]
block = blocks[0]
for b in blocks[1:]:
block = block._blockadd(b)
if nonblocks:
return MatAdd(*nonblocks) + block
else:
return block
def _handle_finite_sets(op, x, y, commutative):
# Handle finite sets:
fs_args, other = sift([x, y],
lambda x: isinstance(x, FiniteSet),
binary=True)
if len(fs_args) == 2:
return FiniteSet(*[op(i, j) for i in fs_args[0] for j in fs_args[1]])
elif len(fs_args) == 1:
sets = [
_apply_operation(op, other[0], i, commutative) for i in fs_args[0]
]
return Union(*sets)
else:
return None
def _eval_factor(self, **hints):
if 1 == len(self.limits):
summand = self.function.factor(**hints)
if summand.is_Mul:
out = sift(summand.args, lambda w: w.is_commutative \
and not w.has(*self.variables))
return Mul(*out[True])*self.func(Mul(*out[False]), \
*self.limits)
else:
summand = self.func(self.function, self.limits[0:-1]).factor()
if not summand.has(self.variables[-1]):
return self.func(1, [self.limits[-1]]).doit() * summand
elif isinstance(summand, Mul):
return self.func(summand, self.limits[-1]).factor()
return self
def __new__(cls, sym, condition, base_set):
if condition == S.false:
return S.EmptySet
if condition == S.true:
return base_set
if isinstance(base_set, EmptySet):
return base_set
if isinstance(base_set, FiniteSet):
sifted = sift(base_set, lambda _: fuzzy_bool(condition.subs(sym, _)))
if sifted[None]:
return Union(FiniteSet(*sifted[True]),
Basic.__new__(cls, sym, condition, FiniteSet(*sifted[None])))
else:
return FiniteSet(*sifted[True])
return Basic.__new__(cls, sym, condition, base_set)
def _eval_factor(self, **hints):
if 1 == len(self.limits):
summand = self.function.factor(**hints)
if summand.is_Mul:
out = sift(summand.args, lambda w: w.is_commutative \
and not w.has(*self.variables))
return C.Mul(*out[True])*self.func(C.Mul(*out[False]), \
*self.limits)
else:
summand = self.func(self.function, self.limits[0:-1]).factor()
if not summand.has(self.variables[-1]):
return self.func(1, [self.limits[-1]]).doit()*summand
elif isinstance(summand, C.Mul):
return self.func(summand, self.limits[-1]).factor()
return self
def test_sift():
assert sift(list(range(5)), lambda _: _ % 2) == {1: [1, 3], 0: [0, 2, 4]}
assert sift([x, y], lambda _: _.has(x)) == {False: [y], True: [x]}
assert sift([S.One], lambda _: _.has(x)) == {False: [1]}
assert sift([0, 1, 2, 3], lambda x: x % 2, binary=True) == (
[1, 3], [0, 2])
assert sift([0, 1, 2, 3], lambda x: x % 3 == 1, binary=True) == (
[1], [0, 2, 3])
raises(ValueError, lambda:
sift([0, 1, 2, 3], lambda x: x % 3, binary=True))
def cse_separate(r, e):
"""Move expressions that are in the form (symbol, expr) out of the
expressions and sort them into the replacements using the reps_toposort.
Examples
========
>>> from sympy.simplify.cse_main import cse_separate
>>> from sympy.abc import x, y, z
>>> from sympy import cos, exp, cse, Eq
>>> eq = (x + 1 + exp((x + 1)/(y + 1)) + cos(y + 1))
>>> cse([eq, Eq(x, z + 1), z - 2], postprocess=cse_separate)
[[(x0, y + 1), (x, z + 1), (x1, x + 1)], [x1 + exp(x1/x0) + cos(x0), z - 2]]
"""
syms = set([k for k, v in r])
d = sift(e, lambda w: w.is_Equality and not bool(w.free_symbols & set(syms)))
r, e = [r + [w.args for w in d[True]], d[False]]
return [reps_toposort(r), e]
def cse_separate(r, e):
"""Move expressions that are in the form (symbol, expr) out of the
expressions and sort them into the replacements using the reps_toposort.
Examples
========
>>> from sympy.simplify.cse_main import cse_separate
>>> from sympy.abc import x, y, z
>>> from sympy import cos, exp, cse, Eq
>>> eq = (x + 1 + exp((x + 1)/(y + 1)) + cos(y + 1))
>>> cse([eq, Eq(x, z + 1), z - 2], postprocess=cse_separate)
[[(x0, y + 1), (x, z + 1), (x1, x + 1)], [x1 + exp(x1/x0) + cos(x0), z - 2]]
"""
syms = set([k for k, v in r])
d = sift(e,
lambda w: w.is_Equality and not bool(w.free_symbols & set(syms)))
r, e = [r + [w.args for w in d[True]], d[False]]
return [reps_toposort(r), e]
def __new__(cls, sym, condition, base_set):
if condition == S.false:
return S.EmptySet
if condition == S.true:
return base_set
if isinstance(base_set, EmptySet):
return base_set
if isinstance(base_set, FiniteSet):
sifted = sift(base_set,
lambda _: fuzzy_bool(condition.subs(sym, _)))
if sifted[None]:
return Union(
FiniteSet(*sifted[True]),
Basic.__new__(cls, sym, condition,
FiniteSet(*sifted[None])))
else:
return FiniteSet(*sifted[True])
return Basic.__new__(cls, sym, condition, base_set)
def as_coeff_add(self, *deps):
"""
Returns a tuple (coeff, args) where self is treated as an Add and coeff
is the Number term and args is a tuple of all other terms.
Examples
========
>>> from sympy.abc import x
>>> (7 + 3*x).as_coeff_add()
(7, (3*x,))
>>> (7*x).as_coeff_add()
(0, (7*x,))
"""
if deps:
l1, l2 = sift(self.args, lambda x: x.has_free(*deps), binary=True)
return self._new_rawargs(*l2), tuple(l1)
coeff, notrat = self.args[0].as_coeff_add()
if coeff is not S.Zero:
return coeff, notrat + self.args[1:]
return S.Zero, self.args
def merge_explicit_mul(vecmul):
""" Merge explicit Vector and Expr (Numbers, Symbols, ...) arguments
Example
========
>>> from sympy.vector import CoordSys3D
>>> v1 = VectorSymbol("v1")
>>> v2 = VectorSymbol("v2")
>>> C = CoordSys3D("C")
>>> vn1 = R.x * R.i + R.y * R.j + R.k * R.z
>>> expr = VecMul(2, v1.mag, vn1, x, evaluate=False)
>>> pprint(expr)
2*x*(R.x*R.i + R.y*R.j + R.z*R.k)*Magnitude(VectorSymbol(v1))
>>> pprint(merge_explicit(expr))
(2*R.x*x*R.i + 2*R.y*x*R.j + 2*R.z*x*R.k)*Magnitude(VectorSymbol(v1))
"""
groups = sift(vecmul.args, lambda arg: not isinstance(arg, VectorExpr))
print("\t", groups)
if len(groups[True]) > 1:
return VecMul(*(groups[False] + [reduce(mul, groups[True])]))
else:
return vecmul
def cse_separate(r, e):
"""Move expressions that are in the form (symbol, expr) out of the
expressions and sort them into the replacements using the reps_toposort.
Examples
========
>>> from sympy.simplify.cse_main import cse_separate
>>> from sympy.abc import x, y, z
>>> from sympy import cos, exp, cse, Eq, symbols
>>> x0, x1 = symbols('x:2')
>>> eq = (x + 1 + exp((x + 1)/(y + 1)) + cos(y + 1))
>>> cse([eq, Eq(x, z + 1), z - 2], postprocess=cse_separate) in [
... [[(x0, y + 1), (x, z + 1), (x1, x + 1)],
... [x1 + exp(x1/x0) + cos(x0), z - 2]],
... [[(x1, y + 1), (x, z + 1), (x0, x + 1)],
... [x0 + exp(x0/x1) + cos(x1), z - 2]]]
...
True
"""
d = sift(e, lambda w: w.is_Equality and w.lhs.is_Symbol)
r = r + [w.args for w in d[True]]
e = d[False]
return [reps_toposort(r), e]
def cse_separate(r, e):
"""Move expressions that are in the form (symbol, expr) out of the
expressions and sort them into the replacements using the reps_toposort.
Examples
========
>>> from sympy.simplify.cse_main import cse_separate
>>> from sympy.abc import x, y, z
>>> from sympy import cos, exp, cse, Eq, symbols
>>> x0, x1 = symbols('x:2')
>>> eq = (x + 1 + exp((x + 1)/(y + 1)) + cos(y + 1))
>>> cse([eq, Eq(x, z + 1), z - 2], postprocess=cse_separate) in [
... [[(x0, y + 1), (x, z + 1), (x1, x + 1)],
... [x1 + exp(x1/x0) + cos(x0), z - 2]],
... [[(x1, y + 1), (x, z + 1), (x0, x + 1)],
... [x0 + exp(x0/x1) + cos(x1), z - 2]]]
...
True
"""
d = sift(e, lambda w: w.is_Equality and w.lhs.is_Symbol)
r = r + [w.args for w in d[True]]
e = d[False]
return [reps_toposort(r), e]
def solve_univariate_inequality(expr,
gen,
relational=True,
domain=S.Reals,
continuous=False):
"""Solves a real univariate inequality.
Parameters
==========
expr : Relational
The target inequality
gen : Symbol
The variable for which the inequality is solved
relational : bool
A Relational type output is expected or not
domain : Set
The domain over which the equation is solved
continuous: bool
True if expr is known to be continuous over the given domain
(and so continuous_domain() doesn't need to be called on it)
Raises
======
NotImplementedError
The solution of the inequality cannot be determined due to limitation
in `solvify`.
Notes
=====
Currently, we cannot solve all the inequalities due to limitations in
`solvify`. Also, the solution returned for trigonometric inequalities
are restricted in its periodic interval.
See Also
========
solvify: solver returning solveset solutions with solve's output API
Examples
========
>>> from sympy.solvers.inequalities import solve_univariate_inequality
>>> from sympy import Symbol, sin, Interval, S
>>> x = Symbol('x')
>>> solve_univariate_inequality(x**2 >= 4, x)
((2 <= x) & (x < oo)) | ((x <= -2) & (-oo < x))
>>> solve_univariate_inequality(x**2 >= 4, x, relational=False)
Union(Interval(-oo, -2), Interval(2, oo))
>>> domain = Interval(0, S.Infinity)
>>> solve_univariate_inequality(x**2 >= 4, x, False, domain)
Interval(2, oo)
>>> solve_univariate_inequality(sin(x) > 0, x, relational=False)
Interval.open(0, pi)
"""
from sympy import im
from sympy.calculus.util import (continuous_domain, periodicity,
function_range)
from sympy.solvers.solvers import denoms
from sympy.solvers.solveset import solveset_real, solvify, solveset
from sympy.solvers.solvers import solve
# This keeps the function independent of the assumptions about `gen`.
# `solveset` makes sure this function is called only when the domain is
# real.
_gen = gen
_domain = domain
if gen.is_real is False:
rv = S.EmptySet
return rv if not relational else rv.as_relational(_gen)
elif gen.is_real is None:
gen = Dummy('gen', real=True)
try:
expr = expr.xreplace({_gen: gen})
except TypeError:
raise TypeError(
filldedent('''
When gen is real, the relational has a complex part
which leads to an invalid comparison like I < 0.
'''))
rv = None
if expr is S.true:
rv = domain
elif expr is S.false:
rv = S.EmptySet
else:
e = expr.lhs - expr.rhs
period = periodicity(e, gen)
if period is S.Zero:
e = expand_mul(e)
const = expr.func(e, 0)
if const is S.true:
rv = domain
elif const is S.false:
rv = S.EmptySet
elif period is not None:
frange = function_range(e, gen, domain)
rel = expr.rel_op
if rel == '<' or rel == '<=':
if expr.func(frange.sup, 0):
rv = domain
elif not expr.func(frange.inf, 0):
rv = S.EmptySet
elif rel == '>' or rel == '>=':
if expr.func(frange.inf, 0):
rv = domain
elif not expr.func(frange.sup, 0):
rv = S.EmptySet
inf, sup = domain.inf, domain.sup
if sup - inf is S.Infinity:
domain = Interval(0, period, False, True)
if rv is None:
n, d = e.as_numer_denom()
try:
if gen not in n.free_symbols and len(e.free_symbols) > 1:
raise ValueError
# this might raise ValueError on its own
# or it might give None...
solns = solvify(e, gen, domain)
if solns is None:
# in which case we raise ValueError
raise ValueError
except (ValueError, NotImplementedError):
# replace gen with generic x since it's
# univariate anyway
raise NotImplementedError(
filldedent('''
The inequality, %s, cannot be solved using
solve_univariate_inequality.
''' % expr.subs(gen, Symbol('x'))))
expanded_e = expand_mul(e)
def valid(x):
# this is used to see if gen=x satisfies the
# relational by substituting it into the
# expanded form and testing against 0, e.g.
# if expr = x*(x + 1) < 2 then e = x*(x + 1) - 2
# and expanded_e = x**2 + x - 2; the test is
# whether a given value of x satisfies
# x**2 + x - 2 < 0
#
# expanded_e, expr and gen used from enclosing scope
v = expanded_e.subs(gen, expand_mul(x))
try:
r = expr.func(v, 0)
except TypeError:
r = S.false
if r in (S.true, S.false):
return r
if v.is_real is False:
return S.false
else:
v = v.n(2)
if v.is_comparable:
return expr.func(v, 0)
# not comparable or couldn't be evaluated
raise NotImplementedError(
'relationship did not evaluate: %s' % r)
singularities = []
for d in denoms(expr, gen):
singularities.extend(solvify(d, gen, domain))
if not continuous:
domain = continuous_domain(expanded_e, gen, domain)
include_x = '=' in expr.rel_op and expr.rel_op != '!='
try:
discontinuities = set(domain.boundary -
FiniteSet(domain.inf, domain.sup))
# remove points that are not between inf and sup of domain
critical_points = FiniteSet(
*(solns + singularities +
list(discontinuities))).intersection(
Interval(domain.inf, domain.sup, domain.inf
not in domain, domain.sup not in domain))
if all(r.is_number for r in critical_points):
reals = _nsort(critical_points, separated=True)[0]
else:
from sympy.utilities.iterables import sift
sifted = sift(critical_points, lambda x: x.is_real)
if sifted[None]:
# there were some roots that weren't known
# to be real
raise NotImplementedError
try:
reals = sifted[True]
if len(reals) > 1:
reals = list(sorted(reals))
except TypeError:
raise NotImplementedError
except NotImplementedError:
raise NotImplementedError(
'sorting of these roots is not supported')
# If expr contains imaginary coefficients, only take real
# values of x for which the imaginary part is 0
make_real = S.Reals
if im(expanded_e) != S.Zero:
check = True
im_sol = FiniteSet()
try:
a = solveset(im(expanded_e), gen, domain)
if not isinstance(a, Interval):
for z in a:
if z not in singularities and valid(
z) and z.is_real:
im_sol += FiniteSet(z)
else:
start, end = a.inf, a.sup
for z in _nsort(critical_points + FiniteSet(end)):
valid_start = valid(start)
if start != end:
valid_z = valid(z)
pt = _pt(start, z)
if pt not in singularities and pt.is_real and valid(
pt):
if valid_start and valid_z:
im_sol += Interval(start, z)
elif valid_start:
im_sol += Interval.Ropen(start, z)
elif valid_z:
im_sol += Interval.Lopen(start, z)
else:
im_sol += Interval.open(start, z)
start = z
for s in singularities:
im_sol -= FiniteSet(s)
except (TypeError):
im_sol = S.Reals
check = False
if isinstance(im_sol, EmptySet):
raise ValueError(
filldedent('''
%s contains imaginary parts which cannot be
made 0 for any value of %s satisfying the
inequality, leading to relations like I < 0.
''' % (expr.subs(gen, _gen), _gen)))
make_real = make_real.intersect(im_sol)
empty = sol_sets = [S.EmptySet]
start = domain.inf
if valid(start) and start.is_finite:
sol_sets.append(FiniteSet(start))
for x in reals:
end = x
if valid(_pt(start, end)):
sol_sets.append(Interval(start, end, True, True))
if x in singularities:
singularities.remove(x)
else:
if x in discontinuities:
discontinuities.remove(x)
_valid = valid(x)
else: # it's a solution
_valid = include_x
if _valid:
sol_sets.append(FiniteSet(x))
start = end
end = domain.sup
if valid(end) and end.is_finite:
sol_sets.append(FiniteSet(end))
if valid(_pt(start, end)):
sol_sets.append(Interval.open(start, end))
if im(expanded_e) != S.Zero and check:
rv = (make_real).intersect(_domain)
else:
rv = Intersection((Union(*sol_sets)), make_real,
_domain).subs(gen, _gen)
return rv if not relational else rv.as_relational(_gen)
def split_real_imag(expr):
real_imag = lambda t: (
'real' if is_extended_real(t, assumptions) else 'imag'
if is_extended_real(I * t, assumptions) else None)
return sift(Add.make_args(expr), real_imag)
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 _eval_expand_power_base(self, **hints):
"""(a*b)**n -> a**n * b**n"""
force = hints.get('force', False)
b = self.base
e = self.exp
if not b.is_Mul:
return self
cargs, nc = b.args_cnc(split_1=False)
# expand each term - this is top-level-only
# expansion but we have to watch out for things
# that don't have an _eval_expand method
if nc:
nc = [i._eval_expand_power_base(**hints)
if hasattr(i, '_eval_expand_power_base') else i
for i in nc]
if e.is_Integer:
if e.is_positive:
rv = Mul(*nc*e)
else:
rv = 1/Mul(*nc*-e)
if cargs:
rv *= Mul(*cargs)**e
return rv
if not cargs:
return self.func(Mul(*nc), e, evaluate=False)
nc = [Mul(*nc)]
# sift the commutative bases
def pred(x):
if x is S.ImaginaryUnit:
return S.ImaginaryUnit
polar = x.is_polar
if polar:
return True
if polar is None:
return fuzzy_bool(x.is_nonnegative)
sifted = sift(cargs, pred)
nonneg = sifted[True]
other = sifted[None]
neg = sifted[False]
imag = sifted[S.ImaginaryUnit]
if imag:
I = S.ImaginaryUnit
i = len(imag) % 4
if i == 0:
pass
elif i == 1:
other.append(I)
elif i == 2:
if neg:
nonn = -neg.pop()
if nonn is not S.One:
nonneg.append(nonn)
else:
neg.append(S.NegativeOne)
else:
if neg:
nonn = -neg.pop()
if nonn is not S.One:
nonneg.append(nonn)
else:
neg.append(S.NegativeOne)
other.append(I)
del imag
# bring out the bases that can be separated from the base
if force or e.is_integer:
# treat all commutatives the same and put nc in other
cargs = nonneg + neg + other
other = nc
else:
# this is just like what is happening automatically, except
# that now we are doing it for an arbitrary exponent for which
# no automatic expansion is done
assert not e.is_Integer
# handle negatives by making them all positive and putting
# the residual -1 in other
if len(neg) > 1:
o = S.One
if not other and neg[0].is_Number:
o *= neg.pop(0)
if len(neg) % 2:
o = -o
for n in neg:
nonneg.append(-n)
if o is not S.One:
other.append(o)
elif neg and other:
if neg[0].is_Number and neg[0] is not S.NegativeOne:
other.append(S.NegativeOne)
nonneg.append(-neg[0])
else:
other.extend(neg)
else:
other.extend(neg)
del neg
cargs = nonneg
other += nc
rv = S.One
if cargs:
rv *= Mul(*[self.func(b, e, evaluate=False) for b in cargs])
if other:
rv *= self.func(Mul(*other), e, evaluate=False)
return rv
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: x.args[0].cond)
for c in 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 rule_gamma(expr, level=0):
""" Simplify products of gamma functions further. """
if expr.is_Atom:
return expr
def gamma_rat(x):
# helper to simplify ratios of gammas
was = x.count(gamma)
xx = x.replace(gamma, lambda n: _rf(1, (n - 1).expand()
).replace(_rf, lambda a, b: gamma(a + b)/gamma(a)))
if xx.count(gamma) < was:
x = xx
return x
def gamma_factor(x):
# return True if there is a gamma factor in shallow args
if isinstance(x, gamma):
return True
if x.is_Add or x.is_Mul:
return any(gamma_factor(xi) for xi in x.args)
if x.is_Pow and (x.exp.is_integer or x.base.is_positive):
return gamma_factor(x.base)
return False
# recursion step
if level == 0:
expr = expr.func(*[rule_gamma(x, level + 1) for x in expr.args])
level += 1
if not expr.is_Mul:
return expr
# non-commutative step
if level == 1:
args, nc = expr.args_cnc()
if not args:
return expr
if nc:
return rule_gamma(Mul._from_args(args), level + 1)*Mul._from_args(nc)
level += 1
# pure gamma handling, not factor absorption
if level == 2:
T, F = sift(expr.args, gamma_factor, binary=True)
gamma_ind = Mul(*F)
d = Mul(*T)
nd, dd = d.as_numer_denom()
for ipass in range(2):
args = list(ordered(Mul.make_args(nd)))
for i, ni in enumerate(args):
if ni.is_Add:
ni, dd = Add(*[
rule_gamma(gamma_rat(a/dd), level + 1) for a in ni.args]
).as_numer_denom()
args[i] = ni
if not dd.has(gamma):
break
nd = Mul(*args)
if ipass == 0 and not gamma_factor(nd):
break
nd, dd = dd, nd # now process in reversed order
expr = gamma_ind*nd/dd
if not (expr.is_Mul and (gamma_factor(dd) or gamma_factor(nd))):
return expr
level += 1
# iteration until constant
if level == 3:
while True:
was = expr
expr = rule_gamma(expr, 4)
if expr == was:
return expr
numer_gammas = []
denom_gammas = []
numer_others = []
denom_others = []
def explicate(p):
if p is S.One:
return None, []
b, e = p.as_base_exp()
if e.is_Integer:
if isinstance(b, gamma):
return True, [b.args[0]]*e
else:
return False, [b]*e
else:
return False, [p]
newargs = list(ordered(expr.args))
while newargs:
n, d = newargs.pop().as_numer_denom()
isg, l = explicate(n)
if isg:
numer_gammas.extend(l)
elif isg is False:
numer_others.extend(l)
isg, l = explicate(d)
if isg:
denom_gammas.extend(l)
elif isg is False:
denom_others.extend(l)
# =========== level 2 work: pure gamma manipulation =========
if not as_comb:
# Try to reduce the number of gamma factors by applying the
# reflection formula gamma(x)*gamma(1-x) = pi/sin(pi*x)
for gammas, numer, denom in [(
numer_gammas, numer_others, denom_others),
(denom_gammas, denom_others, numer_others)]:
new = []
while gammas:
g1 = gammas.pop()
if g1.is_integer:
new.append(g1)
continue
for i, g2 in enumerate(gammas):
n = g1 + g2 - 1
if not n.is_Integer:
continue
numer.append(S.Pi)
denom.append(sin(S.Pi*g1))
gammas.pop(i)
if n > 0:
for k in range(n):
numer.append(1 - g1 + k)
elif n < 0:
for k in range(-n):
denom.append(-g1 - k)
break
else:
new.append(g1)
# /!\ updating IN PLACE
gammas[:] = new
# Try to reduce the number of gammas by using the duplication
# theorem to cancel an upper and lower: gamma(2*s)/gamma(s) =
# 2**(2*s + 1)/(4*sqrt(pi))*gamma(s + 1/2). Although this could
# be done with higher argument ratios like gamma(3*x)/gamma(x),
# this would not reduce the number of gammas as in this case.
for ng, dg, no, do in [(numer_gammas, denom_gammas, numer_others,
denom_others),
(denom_gammas, numer_gammas, denom_others,
numer_others)]:
while True:
for x in ng:
for y in dg:
n = x - 2*y
if n.is_Integer:
break
else:
continue
break
else:
break
ng.remove(x)
dg.remove(y)
if n > 0:
for k in range(n):
no.append(2*y + k)
elif n < 0:
for k in range(-n):
do.append(2*y - 1 - k)
ng.append(y + S(1)/2)
no.append(2**(2*y - 1))
do.append(sqrt(S.Pi))
# Try to reduce the number of gamma factors by applying the
# multiplication theorem (used when n gammas with args differing
# by 1/n mod 1 are encountered).
#
# run of 2 with args differing by 1/2
#
# >>> gammasimp(gamma(x)*gamma(x+S.Half))
# 2*sqrt(2)*2**(-2*x - 1/2)*sqrt(pi)*gamma(2*x)
#
# run of 3 args differing by 1/3 (mod 1)
#
# >>> gammasimp(gamma(x)*gamma(x+S(1)/3)*gamma(x+S(2)/3))
# 6*3**(-3*x - 1/2)*pi*gamma(3*x)
# >>> gammasimp(gamma(x)*gamma(x+S(1)/3)*gamma(x+S(5)/3))
# 2*3**(-3*x - 1/2)*pi*(3*x + 2)*gamma(3*x)
#
def _run(coeffs):
# find runs in coeffs such that the difference in terms (mod 1)
# of t1, t2, ..., tn is 1/n
u = list(uniq(coeffs))
for i in range(len(u)):
dj = ([((u[j] - u[i]) % 1, j) for j in range(i + 1, len(u))])
for one, j in dj:
if one.p == 1 and one.q != 1:
n = one.q
got = [i]
get = list(range(1, n))
for d, j in dj:
m = n*d
if m.is_Integer and m in get:
get.remove(m)
got.append(j)
if not get:
break
else:
continue
for i, j in enumerate(got):
c = u[j]
coeffs.remove(c)
got[i] = c
return one.q, got[0], got[1:]
def _mult_thm(gammas, numer, denom):
# pull off and analyze the leading coefficient from each gamma arg
# looking for runs in those Rationals
# expr -> coeff + resid -> rats[resid] = coeff
rats = {}
for g in gammas:
c, resid = g.as_coeff_Add()
rats.setdefault(resid, []).append(c)
# look for runs in Rationals for each resid
keys = sorted(rats, key=default_sort_key)
for resid in keys:
coeffs = list(sorted(rats[resid]))
new = []
while True:
run = _run(coeffs)
if run is None:
break
# process the sequence that was found:
# 1) convert all the gamma functions to have the right
# argument (could be off by an integer)
# 2) append the factors corresponding to the theorem
# 3) append the new gamma function
n, ui, other = run
# (1)
for u in other:
con = resid + u - 1
for k in range(int(u - ui)):
numer.append(con - k)
con = n*(resid + ui) # for (2) and (3)
# (2)
numer.append((2*S.Pi)**(S(n - 1)/2)*
n**(S(1)/2 - con))
# (3)
new.append(con)
# restore resid to coeffs
rats[resid] = [resid + c for c in coeffs] + new
# rebuild the gamma arguments
g = []
for resid in keys:
g += rats[resid]
# /!\ updating IN PLACE
gammas[:] = g
for l, numer, denom in [(numer_gammas, numer_others, denom_others),
(denom_gammas, denom_others, numer_others)]:
_mult_thm(l, numer, denom)
# =========== level >= 2 work: factor absorption =========
if level >= 2:
# Try to absorb factors into the gammas: x*gamma(x) -> gamma(x + 1)
# and gamma(x)/(x - 1) -> gamma(x - 1)
# This code (in particular repeated calls to find_fuzzy) can be very
# slow.
def find_fuzzy(l, x):
if not l:
return
S1, T1 = compute_ST(x)
for y in l:
S2, T2 = inv[y]
if T1 != T2 or (not S1.intersection(S2) and
(S1 != set() or S2 != set())):
continue
# XXX we want some simplification (e.g. cancel or
# simplify) but no matter what it's slow.
a = len(cancel(x/y).free_symbols)
b = len(x.free_symbols)
c = len(y.free_symbols)
# TODO is there a better heuristic?
if a == 0 and (b > 0 or c > 0):
return y
# We thus try to avoid expensive calls by building the following
# "invariants": For every factor or gamma function argument
# - the set of free symbols S
# - the set of functional components T
# We will only try to absorb if T1==T2 and (S1 intersect S2 != emptyset
# or S1 == S2 == emptyset)
inv = {}
def compute_ST(expr):
if expr in inv:
return inv[expr]
return (expr.free_symbols, expr.atoms(Function).union(
set(e.exp for e in expr.atoms(Pow))))
def update_ST(expr):
inv[expr] = compute_ST(expr)
for expr in numer_gammas + denom_gammas + numer_others + denom_others:
update_ST(expr)
for gammas, numer, denom in [(
numer_gammas, numer_others, denom_others),
(denom_gammas, denom_others, numer_others)]:
new = []
while gammas:
g = gammas.pop()
cont = True
while cont:
cont = False
y = find_fuzzy(numer, g)
if y is not None:
numer.remove(y)
if y != g:
numer.append(y/g)
update_ST(y/g)
g += 1
cont = True
y = find_fuzzy(denom, g - 1)
if y is not None:
denom.remove(y)
if y != g - 1:
numer.append((g - 1)/y)
update_ST((g - 1)/y)
g -= 1
cont = True
new.append(g)
# /!\ updating IN PLACE
gammas[:] = new
# =========== rebuild expr ==================================
return Mul(*[gamma(g) for g in numer_gammas]) \
/ Mul(*[gamma(g) for g in denom_gammas]) \
* Mul(*numer_others) / Mul(*denom_others)
def solve_univariate_inequality(expr, gen, relational=True, domain=S.Reals, continuous=False):
"""Solves a real univariate inequality.
Parameters
==========
expr : Relational
The target inequality
gen : Symbol
The variable for which the inequality is solved
relational : bool
A Relational type output is expected or not
domain : Set
The domain over which the equation is solved
continuous: bool
True if expr is known to be continuous over the given domain
(and so continuous_domain() doesn't need to be called on it)
Raises
======
NotImplementedError
The solution of the inequality cannot be determined due to limitation
in `solvify`.
Notes
=====
Currently, we cannot solve all the inequalities due to limitations in
`solvify`. Also, the solution returned for trigonometric inequalities
are restricted in its periodic interval.
See Also
========
solvify: solver returning solveset solutions with solve's output API
Examples
========
>>> from sympy.solvers.inequalities import solve_univariate_inequality
>>> from sympy import Symbol, sin, Interval, S
>>> x = Symbol('x')
>>> solve_univariate_inequality(x**2 >= 4, x)
((2 <= x) & (x < oo)) | ((x <= -2) & (-oo < x))
>>> solve_univariate_inequality(x**2 >= 4, x, relational=False)
Union(Interval(-oo, -2), Interval(2, oo))
>>> domain = Interval(0, S.Infinity)
>>> solve_univariate_inequality(x**2 >= 4, x, False, domain)
Interval(2, oo)
>>> solve_univariate_inequality(sin(x) > 0, x, relational=False)
Interval.open(0, pi)
"""
from sympy import im
from sympy.calculus.util import (continuous_domain, periodicity,
function_range)
from sympy.solvers.solvers import denoms
from sympy.solvers.solveset import solveset_real, solvify, solveset
from sympy.solvers.solvers import solve
# This keeps the function independent of the assumptions about `gen`.
# `solveset` makes sure this function is called only when the domain is
# real.
_gen = gen
_domain = domain
if gen.is_real is False:
rv = S.EmptySet
return rv if not relational else rv.as_relational(_gen)
elif gen.is_real is None:
gen = Dummy('gen', real=True)
try:
expr = expr.xreplace({_gen: gen})
except TypeError:
raise TypeError(filldedent('''
When gen is real, the relational has a complex part
which leads to an invalid comparison like I < 0.
'''))
rv = None
if expr is S.true:
rv = domain
elif expr is S.false:
rv = S.EmptySet
else:
e = expr.lhs - expr.rhs
period = periodicity(e, gen)
if period is not None:
frange = function_range(e, gen, domain)
rel = expr.rel_op
if rel == '<' or rel == '<=':
if expr.func(frange.sup, 0):
rv = domain
elif not expr.func(frange.inf, 0):
rv = S.EmptySet
elif rel == '>' or rel == '>=':
if expr.func(frange.inf, 0):
rv = domain
elif not expr.func(frange.sup, 0):
rv = S.EmptySet
inf, sup = domain.inf, domain.sup
if sup - inf is S.Infinity:
domain = Interval(0, period, False, True)
if rv is None:
n, d = e.as_numer_denom()
try:
if gen not in n.free_symbols and len(e.free_symbols) > 1:
raise ValueError
# this might raise ValueError on its own
# or it might give None...
solns = solvify(e, gen, domain)
if solns is None:
# in which case we raise ValueError
raise ValueError
except (ValueError, NotImplementedError):
raise NotImplementedError(filldedent('''
The inequality cannot be solved using solve_univariate_inequality.
'''))
expanded_e = expand_mul(e)
def valid(x):
# this is used to see if gen=x satisfies the
# relational by substituting it into the
# expanded form and testing against 0, e.g.
# if expr = x*(x + 1) < 2 then e = x*(x + 1) - 2
# and expanded_e = x**2 + x - 2; the test is
# whether a given value of x satisfies
# x**2 + x - 2 < 0
#
# expanded_e, expr and gen used from enclosing scope
v = expanded_e.subs(gen, x)
try:
r = expr.func(v, 0)
except TypeError:
r = S.false
if r in (S.true, S.false):
return r
if v.is_real is False:
return S.false
else:
v = v.n(2)
if v.is_comparable:
return expr.func(v, 0)
# not comparable or couldn't be evaluated
raise NotImplementedError(
'relationship did not evaluate: %s' % r)
singularities = []
for d in denoms(expr, gen):
singularities.extend(solvify(d, gen, domain))
if not continuous:
domain = continuous_domain(e, gen, domain)
include_x = '=' in expr.rel_op and expr.rel_op != '!='
try:
discontinuities = set(domain.boundary -
FiniteSet(domain.inf, domain.sup))
# remove points that are not between inf and sup of domain
critical_points = FiniteSet(*(solns + singularities + list(
discontinuities))).intersection(
Interval(domain.inf, domain.sup,
domain.inf not in domain, domain.sup not in domain))
if all(r.is_number for r in critical_points):
reals = _nsort(critical_points, separated=True)[0]
else:
from sympy.utilities.iterables import sift
sifted = sift(critical_points, lambda x: x.is_real)
if sifted[None]:
# there were some roots that weren't known
# to be real
raise NotImplementedError
try:
reals = sifted[True]
if len(reals) > 1:
reals = list(sorted(reals))
except TypeError:
raise NotImplementedError
except NotImplementedError:
raise NotImplementedError('sorting of these roots is not supported')
#If expr contains imaginary coefficients
#Only real values of x for which the imaginary part is 0 are taken
make_real = S.Reals
if im(expanded_e) != S.Zero:
check = True
im_sol = FiniteSet()
try:
a = solveset(im(expanded_e), gen, domain)
if not isinstance(a, Interval):
for z in a:
if z not in singularities and valid(z) and z.is_real:
im_sol += FiniteSet(z)
else:
start, end = a.inf, a.sup
for z in _nsort(critical_points + FiniteSet(end)):
valid_start = valid(start)
if start != end:
valid_z = valid(z)
pt = _pt(start, z)
if pt not in singularities and pt.is_real and valid(pt):
if valid_start and valid_z:
im_sol += Interval(start, z)
elif valid_start:
im_sol += Interval.Ropen(start, z)
elif valid_z:
im_sol += Interval.Lopen(start, z)
else:
im_sol += Interval.open(start, z)
start = z
for s in singularities:
im_sol -= FiniteSet(s)
except (TypeError):
im_sol = S.Reals
check = False
if isinstance(im_sol, EmptySet):
raise ValueError(filldedent('''
%s contains imaginary parts which cannot be made 0 for any value of %s
satisfying the inequality, leading to relations like I < 0.
''' % (expr.subs(gen, _gen), _gen)))
make_real = make_real.intersect(im_sol)
empty = sol_sets = [S.EmptySet]
start = domain.inf
if valid(start) and start.is_finite:
sol_sets.append(FiniteSet(start))
for x in reals:
end = x
if valid(_pt(start, end)):
sol_sets.append(Interval(start, end, True, True))
if x in singularities:
singularities.remove(x)
else:
if x in discontinuities:
discontinuities.remove(x)
_valid = valid(x)
else: # it's a solution
_valid = include_x
if _valid:
sol_sets.append(FiniteSet(x))
start = end
end = domain.sup
if valid(end) and end.is_finite:
sol_sets.append(FiniteSet(end))
if valid(_pt(start, end)):
sol_sets.append(Interval.open(start, end))
if im(expanded_e) != S.Zero and check:
rv = (make_real).intersect(_domain)
else:
rv = Intersection(
(Union(*sol_sets)), make_real, _domain).subs(gen, _gen)
return rv if not relational else rv.as_relational(_gen)
def _sort_args(self, args):
lists, atoms = sift(args,
lambda a: isinstance(a, (tuple, list)), binary=True)
return atoms, lists
def test_sift():
assert sift(range(5), lambda _: _ % 2) == {1: [1, 3], 0: [0, 2, 4]}
assert sift([x, y], lambda _: _.has(x)) == {False: [y], True: [x]}
assert sift([S.One], lambda _: _.has(x)) == {False: [1]}
def MatMul_elements(matrix_predicate, scalar_predicate, expr, assumptions):
d = sift(expr.args, lambda x: isinstance(x, MatrixExpr))
factors, matrices = d[False], d[True]
return fuzzy_and([
test_closed_group(Basic(*factors), assumptions, scalar_predicate),
test_closed_group(Basic(*matrices), assumptions, matrix_predicate)])
def _separate_sq(p):
"""
helper function for ``_minimal_polynomial_sq``
It selects a rational ``g`` such that the polynomial ``p``
consists of a sum of terms whose surds squared have gcd equal to ``g``
and a sum of terms with surds squared prime with ``g``;
then it takes the field norm to eliminate ``sqrt(g)``
See simplify.simplify.split_surds and polytools.sqf_norm.
Examples
========
>>> from sympy import sqrt
>>> from sympy.abc import x
>>> from sympy.polys.numberfields import _separate_sq
>>> p= -x + sqrt(2) + sqrt(3) + sqrt(7)
>>> p = _separate_sq(p); p
-x**2 + 2*sqrt(3)*x + 2*sqrt(7)*x - 2*sqrt(21) - 8
>>> p = _separate_sq(p); p
-x**4 + 4*sqrt(7)*x**3 - 32*x**2 + 8*sqrt(7)*x + 20
>>> p = _separate_sq(p); p
-x**8 + 48*x**6 - 536*x**4 + 1728*x**2 - 400
"""
from sympy.simplify.simplify import _split_gcd, _mexpand
from sympy.utilities.iterables import sift
def is_sqrt(expr):
return expr.is_Pow and expr.exp is S.Half
# p = c1*sqrt(q1) + ... + cn*sqrt(qn) -> a = [(c1, q1), .., (cn, qn)]
a = []
for y in p.args:
if not y.is_Mul:
if is_sqrt(y):
a.append((S.One, y**2))
elif y.is_Atom:
a.append((y, S.One))
elif y.is_Pow and y.exp.is_integer:
a.append((y, S.One))
else:
raise NotImplementedError
continue
sifted = sift(y.args, is_sqrt)
a.append((Mul(*sifted[False]), Mul(*sifted[True])**2))
a.sort(key=lambda z: z[1])
if a[-1][1] is S.One:
# there are no surds
return p
surds = [z for y, z in a]
for i in range(len(surds)):
if surds[i] != 1:
break
g, b1, b2 = _split_gcd(*surds[i:])
a1 = []
a2 = []
for y, z in a:
if z in b1:
a1.append(y*z**S.Half)
else:
a2.append(y*z**S.Half)
p1 = Add(*a1)
p2 = Add(*a2)
p = _mexpand(p1**2) - _mexpand(p2**2)
return p
# short entries will be ignored. The ORDER MATTERS
# so don't re-order the lines for a given address.
# Other tidying up could be done but we won't do that here.
def short_entry(line):
if line.count('<') == 2:
if line.split('>', 1)[1].split('<')[0].strip():
return False
return True
if len(who[k]) == 1:
line = who[k][0]
if not line.strip():
continue # ignore blank lines
out.append(line)
else:
uniq = list(OrderedDict.fromkeys(who[k]))
short, long = sift(uniq, short_entry, binary=True)
out.extend(long)
out.extend(short)
if out != lines or not blankline:
# write lines
with codecs.open(os.path.realpath(os.path.join(
__file__, os.path.pardir, os.path.pardir, ".mailmap")),
"w", "utf-8") as fd:
fd.write('\n'.join(out))
fd.write('\n')
print()
if out != lines:
print(yellow('.mailmap lines were re-ordered.'))
else:
print(yellow('blank line added to end of .mailmap'))
def _matches_commutative(self, expr, repl_dict={}, old=False):
"""
Matches Add/Mul "pattern" to an expression "expr".
repl_dict ... a dictionary of (wild: expression) pairs, that get
returned with the results
This function is the main workhorse for Add/Mul.
For instance:
>>> from sympy import symbols, Wild, sin
>>> a = Wild("a")
>>> b = Wild("b")
>>> c = Wild("c")
>>> x, y, z = symbols("x y z")
>>> (a+sin(b)*c)._matches_commutative(x+sin(y)*z)
{a_: x, b_: y, c_: z}
In the example above, "a+sin(b)*c" is the pattern, and "x+sin(y)*z" is
the expression.
The repl_dict contains parts that were already matched. For example
here:
>>> (x+sin(b)*c)._matches_commutative(x+sin(y)*z, repl_dict={a: x})
{a_: x, b_: y, c_: z}
the only function of the repl_dict is to return it in the
result, e.g. if you omit it:
>>> (x+sin(b)*c)._matches_commutative(x+sin(y)*z)
{b_: y, c_: z}
the "a: x" is not returned in the result, but otherwise it is
equivalent.
"""
# make sure expr is Expr if pattern is Expr
from .expr import Add, Expr
from sympy import Mul
if isinstance(self, Expr) and not isinstance(expr, Expr):
return None
# handle simple patterns
if self == expr:
return repl_dict
d = self._matches_simple(expr, repl_dict)
if d is not None:
return d
# eliminate exact part from pattern: (2+a+w1+w2).matches(expr) -> (w1+w2).matches(expr-a-2)
from .function import WildFunction
from .symbol import Wild
wild_part, exact_part = sift(self.args, lambda p:
p.has(Wild, WildFunction) and not expr.has(p),
binary=True)
if not exact_part:
wild_part = list(ordered(wild_part))
else:
exact = self._new_rawargs(*exact_part)
free = expr.free_symbols
if free and (exact.free_symbols - free):
# there are symbols in the exact part that are not
# in the expr; but if there are no free symbols, let
# the matching continue
return None
newexpr = self._combine_inverse(expr, exact)
if not old and (expr.is_Add or expr.is_Mul):
if newexpr.count_ops() > expr.count_ops():
return None
newpattern = self._new_rawargs(*wild_part)
return newpattern.matches(newexpr, repl_dict)
# now to real work ;)
i = 0
saw = set()
while expr not in saw:
saw.add(expr)
expr_list = (self.identity,) + tuple(ordered(self.make_args(expr)))
for last_op in reversed(expr_list):
for w in reversed(wild_part):
d1 = w.matches(last_op, repl_dict)
if d1 is not None:
d2 = self.xreplace(d1).matches(expr, d1)
if d2 is not None:
return d2
if i == 0:
if self.is_Mul:
# make e**i look like Mul
if expr.is_Pow and expr.exp.is_Integer:
if expr.exp > 0:
expr = Mul(*[expr.base, expr.base**(expr.exp - 1)], evaluate=False)
else:
expr = Mul(*[1/expr.base, expr.base**(expr.exp + 1)], evaluate=False)
i += 1
continue
elif self.is_Add:
# make i*e look like Add
c, e = expr.as_coeff_Mul()
if abs(c) > 1:
if c > 0:
expr = Add(*[e, (c - 1)*e], evaluate=False)
else:
expr = Add(*[-e, (c + 1)*e], evaluate=False)
i += 1
continue
# try collection on non-Wild symbols
from sympy.simplify.radsimp import collect
was = expr
did = set()
for w in reversed(wild_part):
c, w = w.as_coeff_mul(Wild)
free = c.free_symbols - did
if free:
did.update(free)
expr = collect(expr, free)
if expr != was:
i += 0
continue
break # if we didn't continue, there is nothing more to do
return
