def _rootof_data(poly, indices):
"""Construct ``RootOf`` data from a polynomial and indices. """
(_, factors) = poly.factor_list()
reals = _rootof_get_reals(factors)
real_count = sum([k for _, _, k in reals])
if indices is None:
reals = _rootof_reals_sorted(reals)
for index in xrange(0, real_count):
yield _rootof_reals_index(reals, index)
else:
if any(index < real_count for index in indices):
reals = _rootof_reals_sorted(reals)
for index in indices:
if index < real_count:
yield _rootof_reals_index(reals, index)
if any(index >= real_count for index in indices):
complexes = _rootof_get_complexes(factors)
complexes = _rootof_complexes_sorted(complexes)
for index in indices:
if index >= real_count:
yield _rootof_complexes_index(complexes,
index - real_count)
def _rootof_data(poly, indices):
"""Construct ``RootOf`` data from a polynomial and indices. """
(_, factors) = poly.factor_list()
reals = _rootof_get_reals(factors)
real_count = sum([ k for _, _, k in reals ])
if indices is None:
reals = _rootof_reals_sorted(reals)
for index in xrange(0, real_count):
yield _rootof_reals_index(reals, index)
else:
if any(index < real_count for index in indices):
reals = _rootof_reals_sorted(reals)
for index in indices:
if index < real_count:
yield _rootof_reals_index(reals, index)
if any(index >= real_count for index in indices):
complexes = _rootof_get_complexes(factors)
complexes = _rootof_complexes_sorted(complexes)
for index in indices:
if index >= real_count:
yield _rootof_complexes_index(complexes, index-real_count)
def update(G, CP, h):
"""update G using the set of critical pairs CP and h = (expv,pi)
see [BW] page 230
"""
hexpv, hp = f[h]
# print 'DB10',hp
# filter new pairs (h,g), g in G
C = G.copy()
D = set()
while C:
# select a pair (h,g) by popping an element from C
g = C.pop()
gexpv = f[g][0]
LCMhg = lcm_expv(hexpv, gexpv)
def lcm_divides(p):
expv = lcm_expv(hexpv, f[p][0])
# LCM(LM(h), LM(p)) divides LCM(LM(h),LM(g))
return monomial_div(LCMhg, expv)
# HT(h) and HT(g) disjoint: hexpv + gexpv == LCMhg
if monomial_mul(hexpv, gexpv) == LCMhg or (
not any(lcm_divides(f) for f in C) and not any(lcm_divides(pr[1]) for pr in D)
):
D.add((h, g))
E = set()
while D:
# select h,g from D
h, g = D.pop()
gexpv = f[g][0]
LCMhg = lcm_expv(hexpv, gexpv)
if not monomial_mul(hexpv, gexpv) == LCMhg:
E.add((h, g))
# filter old pairs
B_new = set()
while CP:
# select g1,g2 from CP
g1, g2 = CP.pop()
g1expv = f[g1][0]
g2expv = f[g2][0]
LCM12 = lcm_expv(g1expv, g2expv)
# if HT(h) does not divide lcm(HT(g1),HT(g2))
if not monomial_div(LCM12, hexpv) or lcm_expv(g1expv, hexpv) == LCM12 or lcm_expv(g2expv, hexpv) == LCM12:
B_new.add((g1, g2))
B_new |= E
# filter polynomials
G_new = set()
while G:
g = G.pop()
if not monomial_div(f[g][0], hexpv):
G_new.add(g)
G_new.add(h)
return G_new, B_new
def is_univariate(f):
"""Returns True if 'f' is univariate in its last variable. """
for monom in f.monoms():
if any(m > 0 for m in monom[:-1]):
return False
return True
def solve_undetermined_coeffs(equ, coeffs, sym, **flags):
"""Solve equation of a type p(x; a_1, ..., a_k) == q(x) where both
p, q are univariate polynomials and f depends on k parameters.
The result of this functions is a dictionary with symbolic
values of those parameters with respect to coefficiens in q.
This functions accepts both Equations class instances and ordinary
SymPy expressions. Specification of parameters and variable is
obligatory for efficiency and simplicity reason.
>>> from sympy import *
>>> a, b, c, x = symbols('a', 'b', 'c', 'x')
>>> solve_undetermined_coeffs(Eq(2*a*x + a+b, x), [a, b], x)
{a: 1/2, b: -1/2}
>>> solve_undetermined_coeffs(Eq(a*c*x + a+b, x), [a, b], x)
{a: 1/c, b: -1/c}
"""
if isinstance(equ, Equality):
# got equation, so move all the
# terms to the left hand side
equ = equ.lhs - equ.rhs
system = collect(equ.expand(), sym, evaluate=False).values()
if not any([equ.has(sym) for equ in system]):
# consecutive powers in the input expressions have
# been successfully collected, so solve remaining
# system using Gaussian ellimination algorithm
return solve(system, *coeffs, **flags)
else:
return None # no solutions
def solve_undetermined_coeffs(equ, coeffs, sym, **flags):
"""Solve equation of a type p(x; a_1, ..., a_k) == q(x) where both
p, q are univariate polynomials and f depends on k parameters.
The result of this functions is a dictionary with symbolic
values of those parameters with respect to coefficiens in q.
This functions accepts both Equations class instances and ordinary
SymPy expressions. Specification of parameters and variable is
obligatory for efficiency and simplicity reason.
>>> from sympy import *
>>> a, b, c, x = symbols('a', 'b', 'c', 'x')
>>> solve_undetermined_coeffs(Eq(2*a*x + a+b, x), [a, b], x)
{a: 1/2, b: -1/2}
>>> solve_undetermined_coeffs(Eq(a*c*x + a+b, x), [a, b], x)
{a: 1/c, b: -1/c}
"""
if isinstance(equ, Equality):
# got equation, so move all the
# terms to the left hand side
equ = equ.lhs - equ.rhs
system = collect(equ.expand(), sym, evaluate=False).values()
if not any([ equ.has(sym) for equ in system ]):
# consecutive powers in the input expressions have
# been successfully collected, so solve remaining
# system using Gaussian ellimination algorithm
return solve(system, *coeffs, **flags)
else:
return None # no solutions
def is_univariate(f):
"""Returns True if 'f' is univariate in its last variable. """
for monom in f.monoms():
if any(m > 0 for m in monom[:-1]):
return False
return True
def gf_Qbasis(Q, p, K):
"""
Compute a basis of the kernel of ``Q``.
**Examples**
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.galoistools import gf_Qmatrix, gf_Qbasis
>>> gf_Qbasis(gf_Qmatrix([1, 0, 0, 0, 1], 5, ZZ), 5, ZZ)
[[1, 0, 0, 0], [0, 0, 1, 0]]
>>> gf_Qbasis(gf_Qmatrix([3, 2, 4], 5, ZZ), 5, ZZ)
[[1, 0]]
"""
Q, n = [list(q) for q in Q], len(Q)
for k in xrange(0, n):
Q[k][k] = (Q[k][k] - K.one) % p
for k in xrange(0, n):
for i in xrange(k, n):
if Q[k][i]:
break
else:
continue
inv = K.invert(Q[k][i], p)
for j in xrange(0, n):
Q[j][i] = (Q[j][i] * inv) % p
for j in xrange(0, n):
t = Q[j][k]
Q[j][k] = Q[j][i]
Q[j][i] = t
for i in xrange(0, n):
if i != k:
q = Q[k][i]
for j in xrange(0, n):
Q[j][i] = (Q[j][i] - Q[j][k] * q) % p
for i in xrange(0, n):
for j in xrange(0, n):
if i == j:
Q[i][j] = (K.one - Q[i][j]) % p
else:
Q[i][j] = (-Q[i][j]) % p
basis = []
for q in Q:
if any(q):
basis.append(q)
return basis
def gf_Qbasis(Q, p, K):
"""
Compute a basis of the kernel of ``Q``.
**Examples**
>>> from sympy.polys.domains import ZZ
>>> from sympy.polys.galoistools import gf_Qmatrix, gf_Qbasis
>>> gf_Qbasis(gf_Qmatrix([1, 0, 0, 0, 1], 5, ZZ), 5, ZZ)
[[1, 0, 0, 0], [0, 0, 1, 0]]
>>> gf_Qbasis(gf_Qmatrix([3, 2, 4], 5, ZZ), 5, ZZ)
[[1, 0]]
"""
Q, n = [ list(q) for q in Q ], len(Q)
for k in xrange(0, n):
Q[k][k] = (Q[k][k] - K.one) % p
for k in xrange(0, n):
for i in xrange(k, n):
if Q[k][i]:
break
else:
continue
inv = K.invert(Q[k][i], p)
for j in xrange(0, n):
Q[j][i] = (Q[j][i]*inv) % p
for j in xrange(0, n):
t = Q[j][k]
Q[j][k] = Q[j][i]
Q[j][i] = t
for i in xrange(0, n):
if i != k:
q = Q[k][i]
for j in xrange(0, n):
Q[j][i] = (Q[j][i] - Q[j][k]*q) % p
for i in xrange(0, n):
for j in xrange(0, n):
if i == j:
Q[i][j] = (K.one - Q[i][j]) % p
else:
Q[i][j] = (-Q[i][j]) % p
basis = []
for q in Q:
if any(q):
basis.append(q)
return basis
def denester(nested):
"""
Denests a list of expressions that contain nested square roots.
This method should not be called directly - use 'denest' instead.
This algorithm is based on <http://www.almaden.ibm.com/cs/people/fagin/symb85.pdf>.
It is assumed that all of the elements of 'nested' share the same
bottom-level radicand. (This is stated in the paper, on page 177, in
the paragraph immediately preceding the algorithm.)
When evaluating all of the arguments in parallel, the bottom-level
radicand only needs to be denested once. This means that calling
denester with x arguments results in a recursive invocation with x+1
arguments; hence denester has polynomial complexity.
However, if the arguments were evaluated separately, each call would
result in two recursive invocations, and the algorithm would have
exponential complexity.
This is discussed in the paper in the middle paragraph of page 179.
"""
if all((n ** 2).is_Number for n in nested): # If none of the arguments are nested
for f in subsets(len(nested)): # Test subset 'f' of nested
p = prod(nested[i] ** 2 for i in range(len(f)) if f[i]).expand()
if 1 in f and f.count(1) > 1 and f[-1]:
p = -p
if sqrt(p).is_Number:
return sqrt(p), f # If we got a perfect square, return its square root.
return nested[-1], [0] * len(nested) # Otherwise, return the radicand from the previous invocation.
else:
a, b, r, R = Wild("a"), Wild("b"), Wild("r"), None
values = [expr.match(sqrt(a + b * sqrt(r))) for expr in nested]
for v in values:
if r in v: # Since if b=0, r is not defined
if R is not None:
assert R == v[r] # All the 'r's should be the same.
else:
R = v[r]
d, f = denester([sqrt((v[a] ** 2).expand() - (R * v[b] ** 2).expand()) for v in values] + [sqrt(R)])
if not any([f[i] for i in range(len(nested))]): # If f[i]=0 for all i < len(nested)
v = values[-1]
return sqrt(v[a] + v[b] * d), f
else:
v = prod(nested[i] ** 2 for i in range(len(nested)) if f[i]).expand().match(a + b * sqrt(r))
if 1 in f and f.index(1) < len(nested) - 1 and f[len(nested) - 1]:
v[a] = -1 * v[a]
v[b] = -1 * v[b]
if not f[len(nested)]: # Solution denests with square roots
return (
(
sqrt((v[a] + d).expand() / 2) + sign(v[b]) * sqrt((v[b] ** 2 * R / (2 * (v[a] + d))).expand())
).expand(),
f,
)
else: # Solution requires a fourth root
FR, s = (R.expand() ** Rational(1, 4)), sqrt((v[b] * R).expand() + d)
return (s / (sqrt(2) * FR) + v[a] * FR / (sqrt(2) * s)).expand(), f
def _eval_nseries(self, x, n):
"""
This function does compute series for multivariate functions,
but the expansion is always in terms of *one* variable.
Examples:
>>> from sympy import atan2, O
>>> from sympy.abc import x, y
>>> atan2(x, y).series(x, n=2)
atan2(0, y) + x/y + O(x**2)
>>> atan2(x, y).series(y, n=2)
atan2(x, 0) - y/x + O(y**2)
"""
if self.func.nargs is None:
raise NotImplementedError('series for user-defined \
functions are not supported.')
args = self.args
args0 = [t.limit(x, 0) for t in args]
if any([t is S.NaN or t.is_bounded is False for t in args0]):
raise PoleError("Cannot expand %s around 0" % (args))
if (self.func.nargs == 1 and args0[0]) or self.func.nargs > 1:
e = self
e1 = e.expand()
if e == e1:
#for example when e = sin(x+1) or e = sin(cos(x))
#let's try the general algorithm
term = e.subs(x, S.Zero)
if term.is_bounded is False or term is S.NaN:
raise PoleError("Cannot expand %s around 0" % (self))
series = term
fact = S.One
for i in range(n-1):
i += 1
fact *= Rational(i)
e = e.diff(x)
subs = e.subs(x, S.Zero)
if subs is S.NaN:
# try to evaluate a limit if we have to
subs = e.limit(x, S.Zero)
if subs.is_bounded is False:
raise PoleError("Cannot expand %s around 0" % (self))
term = subs*(x**i)/fact
term = term.expand()
series += term
return series + C.Order(x**n, x)
return e1.nseries(x, n=n)
arg = self.args[0]
l = []
g = None
for i in xrange(n+2):
g = self.taylor_term(i, arg, g)
g = g.nseries(x, n=n)
l.append(g)
return Add(*l) + C.Order(x**n, x)
def _eval_nseries(self, x, n):
"""
This function does compute series for multivariate functions,
but the expansion is always in terms of *one* variable.
Examples:
>>> from sympy import atan2, O
>>> from sympy.abc import x, y
>>> atan2(x, y).series(x, n=2)
atan2(0, y) + x/y + O(x**2)
>>> atan2(x, y).series(y, n=2)
atan2(x, 0) - y/x + O(y**2)
"""
if self.func.nargs is None:
raise NotImplementedError('series for user-defined \
functions are not supported.')
args = self.args
args0 = [t.limit(x, 0) for t in args]
if any([t is S.NaN or t.is_bounded is False for t in args0]):
raise PoleError("Cannot expand %s around 0" % (args))
if (self.func.nargs == 1 and args0[0]) or self.func.nargs > 1:
e = self
e1 = e.expand()
if e == e1:
#for example when e = sin(x+1) or e = sin(cos(x))
#let's try the general algorithm
term = e.subs(x, S.Zero)
if term.is_bounded is False or term is S.NaN:
raise PoleError("Cannot expand %s around 0" % (self))
series = term
fact = S.One
for i in range(n - 1):
i += 1
fact *= Rational(i)
e = e.diff(x)
subs = e.subs(x, S.Zero)
if subs is S.NaN:
# try to evaluate a limit if we have to
subs = e.limit(x, S.Zero)
if subs.is_bounded is False:
raise PoleError("Cannot expand %s around 0" % (self))
term = subs * (x**i) / fact
term = term.expand()
series += term
return series + C.Order(x**n, x)
return e1.nseries(x, n=n)
arg = self.args[0]
l = []
g = None
for i in xrange(n + 2):
g = self.taylor_term(i, arg, g)
g = g.nseries(x, n=n)
l.append(g)
return Add(*l) + C.Order(x**n, x)
def dmp_zz_wang_lead_coeffs(f, T, cs, E, H, A, u, K):
"""Wang/EEZ: Compute correct leading coefficients. """
C, J, v = [], [0]*len(E), u-1
for h in H:
c = dmp_one(v, K)
d = dup_LC(h, K)*cs
for i in reversed(xrange(len(E))):
k, e, (t, _) = 0, E[i], T[i]
while not (d % e):
d, k = d//e, k+1
if k != 0:
c, J[i] = dmp_mul(c, dmp_pow(t, k, v, K), v, K), 1
C.append(c)
if any([ not j for j in J ]):
raise ExtraneousFactors # pragma: no cover
CC, HH = [], []
for c, h in zip(C, H):
d = dmp_eval_tail(c, A, v, K)
lc = dup_LC(h, K)
if K.is_one(cs):
cc = lc//d
else:
g = K.gcd(lc, d)
d, cc = d//g, lc//g
h, cs = dup_mul_ground(h, d, K), cs//d
c = dmp_mul_ground(c, cc, v, K)
CC.append(c)
HH.append(h)
if K.is_one(cs):
return f, HH, CC
CCC, HHH = [], []
for c, h in zip(CC, HH):
CCC.append(dmp_mul_ground(c, cs, v, K))
HHH.append(dmp_mul_ground(h, cs, 0, K))
f = dmp_mul_ground(f, cs**(len(H)-1), u, K)
return f, HHH, CCC
def dmp_zz_wang_lead_coeffs(f, T, cs, E, H, A, u, K):
"""Wang/EEZ: Compute correct leading coefficients. """
C, J, v = [], [0] * len(E), u - 1
for h in H:
c = dmp_one(v, K)
d = dup_LC(h, K) * cs
for i in reversed(xrange(len(E))):
k, e, (t, _) = 0, E[i], T[i]
while not (d % e):
d, k = d // e, k + 1
if k != 0:
c, J[i] = dmp_mul(c, dmp_pow(t, k, v, K), v, K), 1
C.append(c)
if any([not j for j in J]):
raise ExtraneousFactors # pragma: no cover
CC, HH = [], []
for c, h in zip(C, H):
d = dmp_eval_tail(c, A, v, K)
lc = dup_LC(h, K)
if K.is_one(cs):
cc = lc // d
else:
g = K.gcd(lc, d)
d, cc = d // g, lc // g
h, cs = dup_mul_ground(h, d, K), cs // d
c = dmp_mul_ground(c, cc, v, K)
CC.append(c)
HH.append(h)
if K.is_one(cs):
return f, HH, CC
CCC, HHH = [], []
for c, h in zip(CC, HH):
CCC.append(dmp_mul_ground(c, cs, v, K))
HHH.append(dmp_mul_ground(h, cs, 0, K))
f = dmp_mul_ground(f, cs**(len(H) - 1), u, K)
return f, HHH, CCC
def is_zero(self):
"""Since Integral doesn't autosimplify it it useful to see if
it would simplify to zero or not in a trivial manner, i.e. when
the function is 0 or two limits of a definite integral are the same.
This is a very naive and quick test, not intended to check for special
patterns like Integral(sin(m*x)*cos(n*x), (x, 0, 2*pi)) == 0.
"""
if self.function.is_zero or any(len(xab) == 3 and xab[1] == xab[2] for xab in self.limits):
return True
if not self.free_symbols and self.function.is_number:
# the integrand is a number and the limits are numerical
return False
def _parallel_dict_from_expr(exprs, opt):
"""Transform expressions into a multinomial form. """
if opt.expand is not False:
exprs = [expr.expand() for expr in exprs]
if any(expr.is_commutative is False for expr in exprs):
raise PolynomialError('non-commutative expressions are not supported')
if opt.gens:
reps, gens = _parallel_dict_from_expr_if_gens(exprs, opt)
else:
reps, gens = _parallel_dict_from_expr_no_gens(exprs, opt)
return reps, opt.clone({'gens': gens})
def _find(self, tests, obj, name, module, source_lines, globs, seen):
"""
Find tests for the given object and any contained objects, and
add them to `tests`.
"""
if self._verbose:
print 'Finding tests in %s' % name
# If we've already processed this object, then ignore it.
if id(obj) in seen:
return
seen[id(obj)] = 1
# Make sure we don't run doctests for classes outside of sympy, such
# as in numpy or scipy.
if inspect.isclass(obj):
if obj.__module__.split('.')[0] != 'sympy':
return
# Find a test for this object, and add it to the list of tests.
test = self._get_test(obj, name, module, globs, source_lines)
if test is not None:
tests.append(test)
# Look for tests in a module's contained objects.
if inspect.ismodule(obj) and self._recurse:
for rawname, val in obj.__dict__.items():
# Recurse to functions & classes.
if inspect.isfunction(val) or inspect.isclass(val):
in_module = self._from_module(module, val)
if not in_module:
# double check in case this function is decorated
# and just appears to come from a different module.
pat = r'\s*(def|class)\s+%s\s*\(' % rawname
PAT = pre.compile(pat)
in_module = any(
PAT.match(line) for line in source_lines)
if in_module:
try:
valname = '%s.%s' % (name, rawname)
self._find(tests, val, valname, module,
source_lines, globs, seen)
except ValueError, msg:
if "invalid option" in msg.args[0]:
# +SKIP raises ValueError in Python 2.4
pass
else:
raise
except:
pass
def preprocess(cls, gens):
if isinstance(gens, Basic):
gens = (gens,)
elif len(gens) == 1 and hasattr(gens[0], '__iter__'):
gens = gens[0]
if gens == (None,):
gens = ()
elif len(set(gens)) != len(gens):
raise GeneratorsError("duplicated generators: %s" % str(gens))
elif any(gen.is_commutative is False for gen in gens):
raise GeneratorsError("non-commutative generators: %s" % str(gens))
return tuple(gens)
def preprocess(cls, gens):
if isinstance(gens, Basic):
gens = (gens,)
elif len(gens) == 1 and hasattr(gens[0], '__iter__'):
gens = gens[0]
if gens == (None,):
gens = ()
elif len(set(gens)) != len(gens):
raise GeneratorsError("duplicated generators: %s" % str(gens))
elif any(gen.is_commutative is False for gen in gens):
raise GeneratorsError("non-commutative generators: %s" % str(gens))
return tuple(gens)
def _parallel_dict_from_expr(exprs, opt):
"""Transform expressions into a multinomial form. """
if opt.expand is not False:
exprs = [ expr.expand() for expr in exprs ]
if any(expr.is_commutative is False for expr in exprs):
raise PolynomialError('non-commutative expressions are not supported')
if opt.gens:
reps, gens = _parallel_dict_from_expr_if_gens(exprs, opt)
else:
reps, gens = _parallel_dict_from_expr_no_gens(exprs, opt)
return reps, opt.clone({'gens': gens})
def is_zero(self):
"""Since Integral doesn't autosimplify it it useful to see if
it would simplify to zero or not in a trivial manner, i.e. when
the function is 0 or two limits of a definite integral are the same.
This is a very naive and quick test, not intended to check for special
patterns like Integral(sin(m*x)*cos(n*x), (x, 0, 2*pi)) == 0.
"""
if (self.function.is_zero or
any(len(xab) == 3 and xab[1] == xab[2] for xab in self.limits)):
return True
if not self.free_symbols and self.function.is_number:
# the integrand is a number and the limits are numerical
return False
def __new__(cls, *args, **options):
# Handle calls like Function('f')
if cls is Function:
return UndefinedFunction(*args)
args = map(sympify, args)
evaluate = options.pop('evaluate', True)
if evaluate:
evaluated = cls.eval(*args)
if evaluated is not None:
return evaluated
result = super(Application, cls).__new__(cls, *args, **options)
if evaluate and any([cls._should_evalf(a) for a in args]):
return result.evalf()
return result
def __new__(cls, *args, **options):
# Handle calls like Function('f')
if cls is Function:
return UndefinedFunction(*args)
args = map(sympify, args)
evaluate = options.pop('evaluate', True)
if evaluate:
evaluated = cls.eval(*args)
if evaluated is not None:
return evaluated
result = super(Application, cls).__new__(cls, *args, **options)
if evaluate and any([cls._should_evalf(a) for a in args]):
return result.evalf()
return result
def _find(self, tests, obj, name, module, source_lines, globs, seen):
"""
Find tests for the given object and any contained objects, and
add them to `tests`.
"""
if self._verbose:
print 'Finding tests in %s' % name
# If we've already processed this object, then ignore it.
if id(obj) in seen:
return
seen[id(obj)] = 1
# Make sure we don't run doctests for classes outside of sympy, such
# as in numpy or scipy.
if inspect.isclass(obj):
if obj.__module__.split('.')[0] != 'sympy':
return
# Find a test for this object, and add it to the list of tests.
test = self._get_test(obj, name, module, globs, source_lines)
if test is not None:
tests.append(test)
# Look for tests in a module's contained objects.
if inspect.ismodule(obj) and self._recurse:
for rawname, val in obj.__dict__.items():
# Recurse to functions & classes.
if inspect.isfunction(val) or inspect.isclass(val):
in_module = self._from_module(module, val)
if not in_module:
# double check in case this function is decorated
# and just appears to come from a different module.
pat = r'\s*(def|class)\s+%s\s*\(' % rawname
PAT = pre.compile(pat)
in_module = any(PAT.match(line) for line in source_lines)
if in_module:
try:
valname = '%s.%s' % (name, rawname)
self._find(tests, val, valname, module, source_lines, globs, seen)
except ValueError, msg:
if "invalid option" in msg.args[0]:
# +SKIP raises ValueError in Python 2.4
pass
else:
raise
except:
pass
def gf_Qbasis(Q, p, K):
"""Compute a basis of the kernel of `Q`. """
Q, n = [ list(q) for q in Q ], len(Q)
for k in xrange(0, n):
Q[k][k] = (Q[k][k] - K.one) % p
for k in xrange(0, n):
for i in xrange(k, n):
if Q[k][i]:
break
else:
continue
inv = K.invert(Q[k][i], p)
for j in xrange(0, n):
Q[j][i] = (Q[j][i]*inv) % p
for j in xrange(0, n):
t = Q[j][k]
Q[j][k] = Q[j][i]
Q[j][i] = t
for i in xrange(0, n):
if i != k:
q = Q[k][i]
for j in xrange(0, n):
Q[j][i] = (Q[j][i] - Q[j][k]*q) % p
for i in xrange(0, n):
for j in xrange(0, n):
if i == j:
Q[i][j] = (K.one - Q[i][j]) % p
else:
Q[i][j] = (-Q[i][j]) % p
basis = []
for q in Q:
if any(q):
basis.append(q)
return basis
def dup_zz_cyclotomic_factor(f, K):
"""Efficiently factor polynomials `x**n - 1` and `x**n + 1` in `Z[x]`.
Given a univariate polynomial `f` in `Z[x]` returns a list of factors
of `f`, provided that `f` is in the form `x**n - 1` or `x**n + 1` for
`n >= 1`. Otherwise returns None.
Factorization is performed using using cyclotomic decomposition of `f`,
which makes this method much faster that any other direct factorization
approach (e.g. Zassenhaus's).
References
==========
.. [Weisstein09] Eric W. Weisstein, Cyclotomic Polynomial, From MathWorld - A
Wolfram Web Resource, http://mathworld.wolfram.com/CyclotomicPolynomial.html
"""
lc_f, tc_f = dup_LC(f, K), dup_TC(f, K)
if dup_degree(f) <= 0:
return None
if lc_f != 1 or tc_f not in [-1, 1]:
return None
if any([bool(cf) for cf in f[1:-1]]):
return None
n = dup_degree(f)
F = _dup_cyclotomic_decompose(n, K)
if not K.is_one(tc_f):
return F
else:
H = []
for h in _dup_cyclotomic_decompose(2 * n, K):
if h not in F:
H.append(h)
return H
def dup_zz_cyclotomic_factor(f, K):
"""Efficiently factor polynomials `x**n - 1` and `x**n + 1` in `Z[x]`.
Given a univariate polynomial `f` in `Z[x]` returns a list of factors
of `f`, provided that `f` is in the form `x**n - 1` or `x**n + 1` for
`n >= 1`. Otherwise returns None.
Factorization is performed using using cyclotomic decomposition of `f`,
which makes this method much faster that any other direct factorization
approach (e.g. Zassenhaus's).
References
==========
.. [Weisstein09] Eric W. Weisstein, Cyclotomic Polynomial, From MathWorld - A
Wolfram Web Resource, http://mathworld.wolfram.com/CyclotomicPolynomial.html
"""
lc_f, tc_f = dup_LC(f, K), dup_TC(f, K)
if dup_degree(f) <= 0:
return None
if lc_f != 1 or tc_f not in [-1, 1]:
return None
if any([ bool(cf) for cf in f[1:-1] ]):
return None
n = dup_degree(f)
F = _dup_cyclotomic_decompose(n, K)
if not K.is_one(tc_f):
return F
else:
H = []
for h in _dup_cyclotomic_decompose(2*n, K):
if h not in F:
H.append(h)
return H
def test(*args, **kwargs):
"""
Run all tests containing any of the given strings in their path.
If sort=False, run them in random order (not default).
Warning: Tests in *very* deeply nested directories are not found.
Examples:
>> import sympy
Run all tests:
>> sympy.test()
Run one file:
>> sympy.test("sympy/core/tests/test_basic.py")
Run all tests in sympy/functions/ and some particular file:
>> sympy.test("sympy/core/tests/test_basic.py", "sympy/functions")
Run all tests in sympy/core and sympy/utilities:
>> sympy.test("core", "util")
"""
from glob import glob
verbose = kwargs.get("verbose", False)
tb = kwargs.get("tb", "short")
kw = kwargs.get("kw", "")
post_mortem = kwargs.get("pdb", False)
colors = kwargs.get("colors", True)
sort = kwargs.get("sort", True)
r = PyTestReporter(verbose, tb, colors)
t = SymPyTests(r, kw, post_mortem)
if len(args) == 0:
t.add_paths(["sympy"])
else:
mypaths = []
for p in t.get_paths(dir='sympy'):
mypaths.extend(glob(p))
mypaths = set(mypaths)
t.add_paths([p for p in mypaths if any(a in p for a in args)])
return t.test(sort=sort)
def __new__(cls, *args, **options):
args = map(sympify, args)
# these lines should be refactored
for opt in ["nargs", "dummy", "comparable", "noncommutative", "commutative"]:
if opt in options:
del options[opt]
# up to here.
if not options.pop('evaluate', True):
return super(Application, cls).__new__(cls, *args, **options)
evaluated = cls.eval(*args)
if evaluated is not None:
return evaluated
# Just undefined functions have nargs == None
if not cls.nargs and hasattr(cls, 'undefined_Function'):
r = super(Application, cls).__new__(cls, *args, **options)
r.nargs = len(args)
return r
r = super(Application, cls).__new__(cls, *args, **options)
if any([cls._should_evalf(a) for a in args]):
return r.evalf()
return r
def __new__(cls, *args, **options):
args = map(sympify, args)
# these lines should be refactored
for opt in [
"nargs", "dummy", "comparable", "noncommutative", "commutative"
]:
if opt in options:
del options[opt]
# up to here.
if not options.pop('evaluate', True):
return super(Application, cls).__new__(cls, *args, **options)
evaluated = cls.eval(*args)
if evaluated is not None:
return evaluated
# Just undefined functions have nargs == None
if not cls.nargs and hasattr(cls, 'undefined_Function'):
r = super(Application, cls).__new__(cls, *args, **options)
r.nargs = len(args)
return r
r = super(Application, cls).__new__(cls, *args, **options)
if any([cls._should_evalf(a) for a in args]):
return r.evalf()
return r
def _construct_composite(coeffs, opt):
"""Handle composite domains, e.g.: ZZ[X], QQ[X], ZZ(X), QQ(X). """
numers, denoms = [], []
for coeff in coeffs:
numer, denom = coeff.as_numer_denom()
numers.append(numer)
denoms.append(denom)
polys, gens = parallel_dict_from_basic(numers + denoms) # XXX: sorting
if any(gen.is_number for gen in gens):
return None # generators are number-like so lets better use EX
n = len(gens)
k = len(polys)//2
numers = polys[:k]
denoms = polys[k:]
if opt.field:
fractions = True
else:
fractions, zeros = False, (0,)*n
for denom in denoms:
if len(denom) > 1 or zeros not in denom:
fractions = True
break
result = []
if not fractions:
coeffs = set([])
for numer, denom in zip(numers, denoms):
denom = denom[zeros]
for monom, coeff in numer.iteritems():
coeff /= denom
coeffs.add(coeff)
numer[monom] = coeff
rationals, reals = False, False
for coeff in coeffs:
if coeff.is_Rational:
if not coeff.is_Integer:
rationals = True
elif coeff.is_Real:
reals = True
break
if reals:
ground = RR
elif rationals:
ground = QQ
else:
ground = ZZ
domain = ground.poly_ring(*gens)
for numer in numers:
for monom, coeff in numer.iteritems():
numer[monom] = ground.from_sympy(coeff)
result.append(domain(numer))
else:
domain = ZZ.frac_field(*gens)
for numer, denom in zip(numers, denoms):
for monom, coeff in numer.iteritems():
numer[monom] = ZZ.from_sympy(coeff)
for monom, coeff in denom.iteritems():
denom[monom] = ZZ.from_sympy(coeff)
result.append(domain((numer, denom)))
return domain, result
def doctest(*paths, **kwargs):
"""
Runs doctests in all *py files in the sympy directory which match
any of the given strings in `paths` or all tests if paths=[].
Note:
o paths can be entered in native system format or in unix,
forward-slash format.
o files that are on the blacklist can be tested by providing
their path; they are only excluded if no paths are given.
Examples:
>> froms sympy.utilities.runtests import doctest
Run all tests:
>> doctest()
Run one file:
>> doctest("sympy/core/basic.py")
>> doctest("polynomial.txt")
Run all tests in sympy/functions/ and some particular file:
>> doctest("/functions", "basic.py")
Run any file having polynomial in its name, doc/src/modules/polynomial.txt,
sympy\functions\special\polynomials.py, and sympy\polys\polynomial.py:
>> doctest("polynomial")
"""
normal = kwargs.get("normal", False)
verbose = kwargs.get("verbose", False)
blacklist = kwargs.get("blacklist", [])
blacklist.extend([
"sympy/thirdparty/pyglet", # segfaults
"doc/src/modules/mpmath", # needs to be fixed upstream
"sympy/mpmath", # needs to be fixed upstream
"doc/src/modules/plotting.txt", # generates live plots
"sympy/plotting", # generates live plots
"sympy/utilities/compilef.py", # needs tcc
"sympy/utilities/autowrap.py", # needs installed compiler
"sympy/galgebra/GA.py", # needs numpy
"sympy/galgebra/latex_ex.py", # needs numpy
"sympy/conftest.py", # needs py.test
"sympy/utilities/benchmarking.py", # needs py.test
])
blacklist = convert_to_native_paths(blacklist)
r = PyTestReporter(verbose)
t = SymPyDocTests(r, normal)
test_files = t.get_test_files('sympy')
not_blacklisted = [
f for f in test_files if not any(b in f for b in blacklist)
]
if len(paths) == 0:
t._testfiles.extend(not_blacklisted)
else:
# take only what was requested...but not blacklisted items
# and allow for partial match anywhere or fnmatch of name
paths = convert_to_native_paths(paths)
matched = []
for f in not_blacklisted:
basename = os.path.basename(f)
for p in paths:
if p in f or fnmatch(basename, p):
matched.append(f)
break
t._testfiles.extend(matched)
# run the tests and record the result for this *py portion of the tests
if t._testfiles:
failed = not t.test()
else:
failed = False
# test *txt files only if we are running python newer than 2.4
if sys.version_info[:2] > (2, 4):
# N.B.
# --------------------------------------------------------------------
# Here we test *.txt files at or below doc/src. Code from these must
# be self supporting in terms of imports since there is no importing
# of necessary modules by doctest.testfile. If you try to pass *.py
# files through this they might fail because they will lack the needed
# imports and smarter parsing that can be done with source code.
#
test_files = t.get_test_files('doc/src', '*.txt', init_only=False)
test_files.sort()
not_blacklisted = [
f for f in test_files if not any(b in f for b in blacklist)
]
if len(paths) == 0:
matched = not_blacklisted
else:
# Take only what was requested as long as it's not on the blacklist.
# Paths were already made native in *py tests so don't repeat here.
# There's no chance of having a *py file slip through since we
# only have *txt files in test_files.
matched = []
for f in not_blacklisted:
basename = os.path.basename(f)
for p in paths:
if p in f or fnmatch(basename, p):
matched.append(f)
break
setup_pprint()
first_report = True
for txt_file in matched:
if not os.path.isfile(txt_file):
continue
old_displayhook = sys.displayhook
try:
# out = pdoctest.testfile(txt_file, module_relative=False, encoding='utf-8',
# optionflags=pdoctest.ELLIPSIS | pdoctest.NORMALIZE_WHITESPACE)
out = sympytestfile(txt_file,
module_relative=False,
encoding='utf-8',
optionflags=pdoctest.ELLIPSIS
| pdoctest.NORMALIZE_WHITESPACE)
finally:
# make sure we return to the original displayhook in case some
# doctest has changed that
sys.displayhook = old_displayhook
txtfailed, tested = out
if tested:
failed = txtfailed or failed
if first_report:
first_report = False
msg = 'txt doctests start'
lhead = '=' * ((80 - len(msg)) // 2 - 1)
rhead = '=' * (79 - len(msg) - len(lhead) - 1)
print ' '.join([lhead, msg, rhead])
print
# use as the id, everything past the first 'sympy'
file_id = txt_file[txt_file.find('sympy') + len('sympy') + 1:]
print file_id, # get at least the name out so it is know who is being tested
wid = 80 - len(file_id) - 1 #update width
test_file = '[%s]' % (tested)
report = '[%s]' % (txtfailed or 'OK')
print ''.join([
test_file, ' ' * (wid - len(test_file) - len(report)),
report
])
# the doctests for *py will have printed this message already if there was
# a failure, so now only print it if there was intervening reporting by
# testing the *txt as evidenced by first_report no longer being True.
if not first_report and failed:
print
print("DO *NOT* COMMIT!")
return not failed
def tsolve(eq, sym):
"""
Solves a transcendental equation with respect to the given
symbol. Various equations containing mixed linear terms, powers,
and logarithms, can be solved.
Only a single solution is returned. This solution is generally
not unique. In some cases, a complex solution may be returned
even though a real solution exists.
>>> from sympy import tsolve, log
>>> from sympy.abc import x
>>> tsolve(3**(2*x+5)-4, x)
[(-5*log(3) + log(4))/(2*log(3))]
>>> tsolve(log(x) + 2*x, x)
[LambertW(2)/2]
"""
if patterns is None:
_generate_patterns()
eq = sympify(eq)
if isinstance(eq, Equality):
eq = eq.lhs - eq.rhs
sym = sympify(sym)
eq2 = eq.subs(sym, x)
# First see if the equation has a linear factor
# In that case, the other factor can contain x in any way (as long as it
# is finite), and we have a direct solution to which we add others that
# may be found for the remaining portion.
r = Wild("r")
m = eq2.match((a * x + b) * r)
if m and m[a]:
return [(-b / a).subs(m).subs(x, sym)] + solve(m[r], x)
for p, sol in patterns:
m = eq2.match(p)
if m:
return [sol.subs(m).subs(x, sym)]
# let's also try to inverse the equation
lhs = eq
rhs = S.Zero
while True:
indep, dep = lhs.as_independent(sym)
# dep + indep == rhs
if lhs.is_Add:
# this indicates we have done it all
if indep is S.Zero:
break
lhs = dep
rhs -= indep
# dep * indep == rhs
else:
# this indicates we have done it all
if indep is S.One:
break
lhs = dep
rhs /= indep
# -1
# f(x) = g -> x = f (g)
if lhs.is_Function and lhs.nargs == 1 and hasattr(lhs, "inverse"):
rhs = lhs.inverse()(rhs)
lhs = lhs.args[0]
sol = solve(lhs - rhs, sym)
return sol
elif lhs.is_Add:
# just a simple case - we do variable substitution for first function,
# and if it removes all functions - let's call solve.
# x -x -1
# UC: e + e = y -> t + t = y
t = Dummy("t")
terms = lhs.args
# find first term which is Function
for f1 in lhs.args:
if f1.is_Function:
break
else:
raise NotImplementedError(
"Unable to solve the equation" + "(tsolve: at least one Function expected at this point"
)
# perform the substitution
lhs_ = lhs.subs(f1, t)
# if no Functions left, we can proceed with usual solve
if not (lhs_.is_Function or any(term.is_Function for term in lhs_.args)):
cv_sols = solve(lhs_ - rhs, t)
for sol in cv_sols:
if sol.has(sym):
raise NotImplementedError("Unable to solve the equation")
cv_inv = solve(t - f1, sym)[0]
sols = list()
for sol in cv_sols:
sols.append(cv_inv.subs(t, sol))
return sols
raise NotImplementedError("Unable to solve the equation.")
def solve(f, *symbols, **flags):
"""Solves equations and systems of equations.
Currently supported are univariate polynomial, transcendental
equations, piecewise combinations thereof and systems of linear
and polynomial equations. Input is formed as a single expression
or an equation, or an iterable container in case of an equation
system. The type of output may vary and depends heavily on the
input. For more details refer to more problem specific functions.
By default all solutions are simplified to make the output more
readable. If this is not the expected behavior (e.g., because of
speed issues) set simplified=False in function arguments.
To solve equations and systems of equations like recurrence relations
or differential equations, use rsolve() or dsolve(), respectively.
>>> from sympy import I, solve
>>> from sympy.abc import x, y
Solve a polynomial equation:
>>> solve(x**4-1, x)
[1, -1, -I, I]
Solve a linear system:
>>> solve((x+5*y-2, -3*x+6*y-15), x, y)
{x: -3, y: 1}
"""
def sympit(w):
return map(sympify, iff(isinstance(w, (list, tuple, set)), w, [w]))
# make f and symbols into lists of sympified quantities
# keeping track of how f was passed since if it is a list
# a dictionary of results will be returned.
bare_f = not isinstance(f, (list, tuple, set))
f, symbols = (sympit(w) for w in [f, symbols])
if any(isinstance(fi, bool) or (fi.is_Relational and not fi.is_Equality) for fi in f):
return reduce_inequalities(f, assume=flags.get("assume"))
for i, fi in enumerate(f):
if fi.is_Equality:
f[i] = fi.lhs - fi.rhs
if not symbols:
# get symbols from equations or supply dummy symbols since
# solve(3,x) returns []...though it seems that it should raise some sort of error TODO
symbols = set([])
for fi in f:
symbols |= fi.atoms(Symbol) or set([Dummy("x")])
symbols = list(symbols)
if bare_f:
f = f[0]
if len(symbols) == 1:
if isinstance(symbols[0], (list, tuple, set)):
symbols = symbols[0]
result = list()
# Begin code handling for Function and Derivative instances
# Basic idea: store all the passed symbols in symbols_passed, check to see
# if any of them are Function or Derivative types, if so, use a dummy
# symbol in their place, and set symbol_swapped = True so that other parts
# of the code can be aware of the swap. Once all swapping is done, the
# continue on with regular solving as usual, and swap back at the end of
# the routine, so that whatever was passed in symbols is what is returned.
symbols_new = []
symbol_swapped = False
symbols_passed = list(symbols)
for i, s in enumerate(symbols):
if s.is_Symbol:
s_new = s
elif s.is_Function:
symbol_swapped = True
s_new = Dummy("F%d" % i)
elif s.is_Derivative:
symbol_swapped = True
s_new = Dummy("D%d" % i)
else:
raise TypeError("not a Symbol or a Function")
symbols_new.append(s_new)
if symbol_swapped:
swap_back_dict = dict(zip(symbols_new, symbols))
# End code for handling of Function and Derivative instances
if not isinstance(f, (tuple, list, set)):
# Create a swap dictionary for storing the passed symbols to be solved
# for, so that they may be swapped back.
if symbol_swapped:
swap_dict = zip(symbols, symbols_new)
f = f.subs(swap_dict)
symbols = symbols_new
# Any embedded piecewise functions need to be brought out to the
# top level so that the appropriate strategy gets selected.
f = piecewise_fold(f)
if len(symbols) != 1:
result = {}
for s in symbols:
result[s] = solve(f, s, **flags)
if flags.get("simplified", True):
for s, r in result.items():
result[s] = map(simplify, r)
return result
symbol = symbols[0]
strategy = guess_solve_strategy(f, symbol)
if strategy == GS_POLY:
poly = f.as_poly(symbol)
if poly is None:
raise NotImplementedError("Cannot solve equation " + str(f) + " for " + str(symbol))
# for cubics and quartics, if the flag wasn't set, DON'T do it
# by default since the results are quite long. Perhaps one could
# base this decision on a certain crtical length of the roots.
if poly.degree > 2:
flags["simplified"] = flags.get("simplified", False)
result = roots(poly, cubics=True, quartics=True).keys()
elif strategy == GS_RATIONAL:
P, Q = f.as_numer_denom()
# TODO: check for Q != 0
result = solve(P, symbol, **flags)
elif strategy == GS_POLY_CV_1:
args = list(f.args)
if isinstance(f, Add):
# we must search for a suitable change of variable
# collect exponents
exponents_denom = list()
for arg in args:
if isinstance(arg, Pow):
exponents_denom.append(arg.exp.q)
elif isinstance(arg, Mul):
for mul_arg in arg.args:
if isinstance(mul_arg, Pow):
exponents_denom.append(mul_arg.exp.q)
assert len(exponents_denom) > 0
if len(exponents_denom) == 1:
m = exponents_denom[0]
else:
# get the LCM of the denominators
m = reduce(ilcm, exponents_denom)
# x -> y**m.
# we assume positive for simplification purposes
t = Dummy("t", positive=True)
f_ = f.subs(symbol, t ** m)
if guess_solve_strategy(f_, t) != GS_POLY:
raise NotImplementedError("Could not convert to a polynomial equation: %s" % f_)
cv_sols = solve(f_, t)
for sol in cv_sols:
result.append(sol ** m)
elif isinstance(f, Mul):
for mul_arg in args:
result.extend(solve(mul_arg, symbol))
elif strategy == GS_POLY_CV_2:
m = 0
args = list(f.args)
if isinstance(f, Add):
for arg in args:
if isinstance(arg, Pow):
m = min(m, arg.exp)
elif isinstance(arg, Mul):
for mul_arg in arg.args:
if isinstance(mul_arg, Pow):
m = min(m, mul_arg.exp)
elif isinstance(f, Mul):
for mul_arg in args:
if isinstance(mul_arg, Pow):
m = min(m, mul_arg.exp)
f1 = simplify(f * symbol ** (-m))
result = solve(f1, symbol)
# TODO: we might have introduced unwanted solutions
# when multiplied by x**-m
elif strategy == GS_PIECEWISE:
result = set()
for expr, cond in f.args:
candidates = solve(expr, *symbols)
if isinstance(cond, bool) or cond.is_Number:
if not cond:
continue
# Only include solutions that do not match the condition
# of any of the other pieces.
for candidate in candidates:
matches_other_piece = False
for other_expr, other_cond in f.args:
if isinstance(other_cond, bool) or other_cond.is_Number:
continue
if bool(other_cond.subs(symbol, candidate)):
matches_other_piece = True
break
if not matches_other_piece:
result.add(candidate)
else:
for candidate in candidates:
if bool(cond.subs(symbol, candidate)):
result.add(candidate)
result = list(result)
elif strategy == GS_TRANSCENDENTAL:
# a, b = f.as_numer_denom()
# Let's throw away the denominator for now. When we have robust
# assumptions, it should be checked, that for the solution,
# b!=0.
result = tsolve(f, *symbols)
elif strategy == -1:
raise ValueError("Could not parse expression %s" % f)
else:
raise NotImplementedError("No algorithms are implemented to solve equation %s" % f)
# This symbol swap should not be necessary for the single symbol case: if you've
# solved for the symbol the it will not appear in the solution. Right now, however
# ode's are getting solutions for solve (even though they shouldn't be -- see the
# swap_back test in test_solvers).
if symbol_swapped:
result = [ri.subs(swap_back_dict) for ri in result]
if flags.get("simplified", True) and strategy != GS_RATIONAL:
return map(simplify, result)
else:
return result
else:
if not f:
return {}
else:
# Create a swap dictionary for storing the passed symbols to be
# solved for, so that they may be swapped back.
if symbol_swapped:
swap_dict = zip(symbols, symbols_new)
f = [fi.subs(swap_dict) for fi in f]
symbols = symbols_new
polys = []
for g in f:
poly = g.as_poly(*symbols)
if poly is not None:
polys.append(poly)
else:
raise NotImplementedError()
if all(p.is_linear for p in polys):
n, m = len(f), len(symbols)
matrix = zeros((n, m + 1))
for i, poly in enumerate(polys):
for monom, coeff in poly.terms():
try:
j = list(monom).index(1)
matrix[i, j] = coeff
except ValueError:
matrix[i, m] = -coeff
soln = solve_linear_system(matrix, *symbols, **flags)
else:
soln = solve_poly_system(polys)
# Use swap_dict to ensure we return the same type as what was
# passed
if symbol_swapped:
if isinstance(soln, dict):
res = {}
for k in soln.keys():
res.update({swap_back_dict[k]: soln[k]})
return res
else:
return soln
else:
return soln
def solve(f, *symbols, **flags):
"""Solves equations and systems of equations.
Currently supported are univariate polynomial and transcendental
equations and systems of linear and polynomial equations. Input
is formed as a single expression or an equation, or an iterable
container in case of an equation system. The type of output may
vary and depends heavily on the input. For more details refer to
more problem specific functions.
By default all solutions are simplified to make the output more
readable. If this is not the expected behavior, eg. because of
speed issues, set simplified=False in function arguments.
To solve equations and systems of equations of other kind, eg.
recurrence relations of differential equations use rsolve() or
dsolve() functions respectively.
>>> from sympy import *
>>> x,y = symbols('xy')
Solve a polynomial equation:
>>> solve(x**4-1, x)
[1, -1, -I, I]
Solve a linear system:
>>> solve((x+5*y-2, -3*x+6*y-15), x, y)
{x: -3, y: 1}
"""
if not symbols:
raise ValueError('no symbols were given')
if len(symbols) == 1:
if isinstance(symbols[0], (list, tuple, set)):
symbols = symbols[0]
symbols = map(sympify, symbols)
if any(not s.is_Symbol for s in symbols):
raise TypeError('not a Symbol')
if not isinstance(f, (tuple, list, set)):
f = sympify(f)
if isinstance(f, Equality):
f = f.lhs - f.rhs
if len(symbols) != 1:
raise NotImplementedError('multivariate equation')
symbol = symbols[0]
strategy = guess_solve_strategy(f, symbol)
if strategy == GS_POLY:
poly = f.as_poly( symbol )
assert poly is not None
result = roots(poly, cubics=True, quartics=True).keys()
elif strategy == GS_RATIONAL:
P, Q = f.as_numer_denom()
#TODO: check for Q != 0
return solve(P, symbol, **flags)
elif strategy == GS_POLY_CV_1:
args = list(f.args)
if isinstance(f, Add):
# we must search for a suitable change of variable
# collect exponents
exponents_denom = list()
for arg in args:
if isinstance(arg, Pow):
exponents_denom.append(arg.exp.q)
elif isinstance(arg, Mul):
for mul_arg in arg.args:
if isinstance(mul_arg, Pow):
exponents_denom.append(mul_arg.exp.q)
assert len(exponents_denom) > 0
if len(exponents_denom) == 1:
m = exponents_denom[0]
else:
# get the GCD of the denominators
m = ilcm(*exponents_denom)
# x -> y**m.
# we assume positive for simplification purposes
t = Symbol('t', positive=True, dummy=True)
f_ = f.subs(symbol, t**m)
if guess_solve_strategy(f_, t) != GS_POLY:
raise TypeError("Could not convert to a polynomial equation: %s" % f_)
cv_sols = solve(f_, t)
result = list()
for sol in cv_sols:
result.append(sol**m)
elif isinstance(f, Mul):
result = []
for mul_arg in args:
result.extend(solve(mul_arg, symbol))
elif strategy == GS_POLY_CV_2:
m = 0
args = list(f.args)
if isinstance(f, Add):
for arg in args:
if isinstance(arg, Pow):
m = min(m, arg.exp)
elif isinstance(arg, Mul):
for mul_arg in arg.args:
if isinstance(mul_arg, Pow):
m = min(m, mul_arg.exp)
elif isinstance(f, Mul):
for mul_arg in args:
if isinstance(mul_arg, Pow):
m = min(m, mul_arg.exp)
f1 = simplify(f*symbol**(-m))
result = solve(f1, symbol)
# TODO: we might have introduced unwanted solutions
# when multiplied by x**-m
elif strategy == GS_TRASCENDENTAL:
#a, b = f.as_numer_denom()
# Let's throw away the denominator for now. When we have robust
# assumptions, it should be checked, that for the solution,
# b!=0.
result = tsolve(f, *symbols)
elif strategy == -1:
raise Exception('Could not parse expression %s' % f)
else:
raise NotImplementedError("No algorithms where implemented to solve equation %s" % f)
if flags.get('simplified', True):
return map(simplify, result)
else:
return result
else:
if not f:
return {}
else:
polys = []
for g in f:
g = sympify(g)
if isinstance(g, Equality):
g = g.lhs - g.rhs
poly = g.as_poly(*symbols)
if poly is not None:
polys.append(poly)
else:
raise NotImplementedError()
if all(p.is_linear for p in polys):
n, m = len(f), len(symbols)
matrix = zeros((n, m + 1))
for i, poly in enumerate(polys):
for coeff, monom in poly.iter_terms():
try:
j = list(monom).index(1)
matrix[i, j] = coeff
except ValueError:
matrix[i, m] = -coeff
return solve_linear_system(matrix, *symbols, **flags)
else:
return solve_poly_system(polys)
def doctest(*paths, **kwargs):
"""
Runs doctests in all *py files in the sympy directory which match
any of the given strings in `paths` or all tests if paths=[].
Note:
o paths can be entered in native system format or in unix,
forward-slash format.
o files that are on the blacklist can be tested by providing
their path; they are only excluded if no paths are given.
Examples:
>> import sympy
Run all tests:
>> sympy.doctest()
Run one file:
>> sympy.doctest("sympy/core/basic.py")
>> sympy.doctest("polynomial.txt")
Run all tests in sympy/functions/ and some particular file:
>> sympy.doctest("/functions", "basic.py")
Run any file having polynomial in its name, doc/src/modules/polynomial.txt,
sympy\functions\special\polynomials.py, and sympy\polys\polynomial.py:
>> sympy.doctest("polynomial")
"""
normal = kwargs.get("normal", False)
verbose = kwargs.get("verbose", False)
blacklist = kwargs.get("blacklist", [])
blacklist.extend([
"sympy/thirdparty/pyglet", # segfaults
"doc/src/modules/mpmath", # needs to be fixed upstream
"sympy/mpmath", # needs to be fixed upstream
"doc/src/modules/plotting.txt", # generates live plots
"sympy/plotting", # generates live plots
"sympy/utilities/compilef.py", # needs tcc
"sympy/utilities/autowrap.py", # needs installed compiler
"sympy/galgebra/GA.py", # needs numpy
"sympy/galgebra/latex_ex.py", # needs numpy
"sympy/conftest.py", # needs py.test
"sympy/utilities/benchmarking.py", # needs py.test
])
blacklist = convert_to_native_paths(blacklist)
# Disable warnings for external modules
import sympy.external
sympy.external.importtools.WARN_OLD_VERSION = False
sympy.external.importtools.WARN_NOT_INSTALLED = False
r = PyTestReporter(verbose)
t = SymPyDocTests(r, normal)
test_files = t.get_test_files('sympy')
not_blacklisted = [f for f in test_files
if not any(b in f for b in blacklist)]
if len(paths) == 0:
t._testfiles.extend(not_blacklisted)
else:
# take only what was requested...but not blacklisted items
# and allow for partial match anywhere or fnmatch of name
paths = convert_to_native_paths(paths)
matched = []
for f in not_blacklisted:
basename = os.path.basename(f)
for p in paths:
if p in f or fnmatch(basename, p):
matched.append(f)
break
t._testfiles.extend(matched)
# run the tests and record the result for this *py portion of the tests
if t._testfiles:
failed = not t.test()
else:
failed = False
# test *txt files only if we are running python newer than 2.4
if sys.version_info[:2] > (2,4):
# N.B.
# --------------------------------------------------------------------
# Here we test *.txt files at or below doc/src. Code from these must
# be self supporting in terms of imports since there is no importing
# of necessary modules by doctest.testfile. If you try to pass *.py
# files through this they might fail because they will lack the needed
# imports and smarter parsing that can be done with source code.
#
test_files = t.get_test_files('doc/src', '*.txt', init_only=False)
test_files.sort()
not_blacklisted = [f for f in test_files
if not any(b in f for b in blacklist)]
if len(paths) == 0:
matched = not_blacklisted
else:
# Take only what was requested as long as it's not on the blacklist.
# Paths were already made native in *py tests so don't repeat here.
# There's no chance of having a *py file slip through since we
# only have *txt files in test_files.
matched = []
for f in not_blacklisted:
basename = os.path.basename(f)
for p in paths:
if p in f or fnmatch(basename, p):
matched.append(f)
break
setup_pprint()
first_report = True
for txt_file in matched:
if not os.path.isfile(txt_file):
continue
old_displayhook = sys.displayhook
try:
# out = pdoctest.testfile(txt_file, module_relative=False, encoding='utf-8',
# optionflags=pdoctest.ELLIPSIS | pdoctest.NORMALIZE_WHITESPACE)
out = sympytestfile(txt_file, module_relative=False, encoding='utf-8',
optionflags=pdoctest.ELLIPSIS | pdoctest.NORMALIZE_WHITESPACE)
finally:
# make sure we return to the original displayhook in case some
# doctest has changed that
sys.displayhook = old_displayhook
txtfailed, tested = out
if tested:
failed = txtfailed or failed
if first_report:
first_report = False
msg = 'txt doctests start'
lhead = '='*((80 - len(msg))//2 - 1)
rhead = '='*(79 - len(msg) - len(lhead) - 1)
print ' '.join([lhead, msg, rhead])
print
# use as the id, everything past the first 'sympy'
file_id = txt_file[txt_file.find('sympy') + len('sympy') + 1:]
print file_id, # get at least the name out so it is know who is being tested
wid = 80 - len(file_id) - 1 #update width
test_file = '[%s]' % (tested)
report = '[%s]' % (txtfailed or 'OK')
print ''.join([test_file,' '*(wid-len(test_file)-len(report)), report])
# the doctests for *py will have printed this message already if there was
# a failure, so now only print it if there was intervening reporting by
# testing the *txt as evidenced by first_report no longer being True.
if not first_report and failed:
print
print("DO *NOT* COMMIT!")
return not failed
def update(G, B, ih):
# update G using the set of critical pairs B and h
# [BW] page 230
h = f[ih]
mh = sdp_LM(h, u)
# filter new pairs (h, g), g in G
C = G.copy()
D = set()
while C:
# select a pair (h, g) by popping an element from C
ig = C.pop()
g = f[ig]
mg = sdp_LM(g, u)
LCMhg = monomial_lcm(mh, mg)
def lcm_divides(ip):
# LCM(LM(h), LM(p)) divides LCM(LM(h), LM(g))
m = monomial_lcm(mh, sdp_LM(f[ip], u))
return monomial_div(LCMhg, m)
# HT(h) and HT(g) disjoint: mh*mg == LCMhg
if monomial_mul(mh, mg) == LCMhg or (
not any(lcm_divides(ipx) for ipx in C) and not any(lcm_divides(pr[1]) for pr in D)
):
D.add((ih, ig))
E = set()
while D:
# select h, g from D (h the same as above)
ih, ig = D.pop()
mg = sdp_LM(f[ig], u)
LCMhg = monomial_lcm(mh, mg)
if not monomial_mul(mh, mg) == LCMhg:
E.add((ih, ig))
# filter old pairs
B_new = set()
while B:
# select g1, g2 from B (-> CP)
ig1, ig2 = B.pop()
mg1 = sdp_LM(f[ig1], u)
mg2 = sdp_LM(f[ig2], u)
LCM12 = monomial_lcm(mg1, mg2)
# if HT(h) does not divide lcm(HT(g1), HT(g2))
if not monomial_div(LCM12, mh) or monomial_lcm(mg1, mh) == LCM12 or monomial_lcm(mg2, mh) == LCM12:
B_new.add((ig1, ig2))
B_new |= E
# filter polynomials
G_new = set()
while G:
ig = G.pop()
mg = sdp_LM(f[ig], u)
if not monomial_div(mg, mh):
G_new.add(ig)
G_new.add(ih)
return G_new, B_new
def _eval_nseries(self, x, n, logx):
"""
This function does compute series for multivariate functions,
but the expansion is always in terms of *one* variable.
Examples:
>>> from sympy import atan2, O
>>> from sympy.abc import x, y
>>> atan2(x, y).series(x, n=2)
atan2(0, y) + x/y + O(x**2)
>>> atan2(x, y).series(y, n=2)
atan2(x, 0) - y/x + O(y**2)
This function also computes asymptotic expansions, if necessary
and possible:
>>> from sympy import loggamma
>>> loggamma(1/x)._eval_nseries(x,0,None)
log(x)/2 - log(x)/x - 1/x + O(1)
"""
if self.func.nargs is None:
raise NotImplementedError('series for user-defined \
functions are not supported.')
args = self.args
args0 = [t.limit(x, 0) for t in args]
if any([t.is_bounded == False for t in args0]):
from sympy import Dummy, oo, zoo, nan
a = [t.compute_leading_term(x, logx=logx) for t in args]
a0 = [t.limit(x, 0) for t in a]
if any([t.has(oo, -oo, zoo, nan) for t in a0]):
return self._eval_aseries(n, args0, x,
logx)._eval_nseries(x, n, logx)
# Careful: the argument goes to oo, but only logarithmically so. We
# are supposed to do a power series expansion "around the
# logarithmic term". e.g.
# f(1+x+log(x))
# -> f(1+logx) + x*f'(1+logx) + O(x**2)
# where 'logx' is given in the argument
a = [t._eval_nseries(x, n, logx) for t in args]
z = [r - r0 for (r, r0) in zip(a, a0)]
p = [Dummy() for t in z]
q = []
v = None
w = None
for ai, zi, pi in zip(a0, z, p):
if zi.has(x):
if v is not None: raise NotImplementedError
q.append(ai + pi)
v = pi
w = zi
else:
q.append(ai)
e1 = self.func(*q)
if v is None:
return e1
s = e1._eval_nseries(v, n, logx)
o = s.getO()
s = s.removeO()
s = s.subs(v, zi).expand() + C.Order(o.expr.subs(v, zi), x)
return s
if (self.func.nargs == 1 and args0[0]) or self.func.nargs > 1:
e = self
e1 = e.expand()
if e == e1:
#for example when e = sin(x+1) or e = sin(cos(x))
#let's try the general algorithm
term = e.subs(x, S.Zero)
if term.is_bounded is False or term is S.NaN:
raise PoleError("Cannot expand %s around 0" % (self))
series = term
fact = S.One
for i in range(n - 1):
i += 1
fact *= Rational(i)
e = e.diff(x)
subs = e.subs(x, S.Zero)
if subs is S.NaN:
# try to evaluate a limit if we have to
subs = e.limit(x, S.Zero)
if subs.is_bounded is False:
raise PoleError("Cannot expand %s around 0" % (self))
term = subs * (x**i) / fact
term = term.expand()
series += term
return series + C.Order(x**n, x)
return e1.nseries(x, n=n, logx=logx)
arg = self.args[0]
l = []
g = None
for i in xrange(n + 2):
g = self.taylor_term(i, arg, g)
g = g.nseries(x, n=n, logx=logx)
l.append(g)
return Add(*l) + C.Order(x**n, x)
def solve(f, *symbols, **flags):
"""
Algebraically solves equations and systems of equations.
Currently supported are:
- univariate polynomial,
- transcendental
- piecewise combinations of the above
- systems of linear and polynomial equations
- sytems containing relational expressions.
Input is formed as:
f
- a single Expr or Poly that must be zero,
- an Equality
- a Relational expression or boolean
- iterable of one or more of the above
symbols (Symbol, Function or Derivative) specified as
- none given (all free symbols will be used)
- single symbol
- denested list of symbols
e.g. solve(f, x, y)
- ordered iterable of symbols
e.g. solve(f, [x, y])
flags
- ``simplified``, when False, will not simplify solutions
(default=True except for polynomials of
order 3 or greater)
The output varies according to the input and can be seen by example:
>>> from sympy import solve, Poly, Eq, Function, exp
>>> from sympy.abc import x, y, z, a, b
o boolean or univariate Relational
>>> solve(x < 3)
And(im(x) == 0, re(x) < 3)
o single expression and single symbol that is in the expression
>>> solve(x - y, x)
[y]
>>> solve(x - 3, x)
[3]
>>> solve(Eq(x, 3), x)
[3]
>>> solve(Poly(x - 3), x)
[3]
>>> solve(x**2 - y**2, x)
[y, -y]
>>> solve(x**4 - 1, x)
[1, -1, -I, I]
o single expression with no symbol that is in the expression
>>> solve(3, x)
[]
>>> solve(x - 3, y)
[]
o when no symbol is given then all free symbols will be used
and sorted with default_sort_key and the result will be the
same as above as if those symbols had been supplied
>>> solve(x - 3)
[3]
>>> solve(x**2 - y**2)
[y, -y]
o when a Function or Derivative is given as a symbol, it is isolated
algebraically and an implicit solution may be obtained
>>> f = Function('f')
>>> solve(f(x) - x, f(x))
[x]
>>> solve(f(x).diff(x) - f(x) - x, f(x).diff(x))
[x + f(x)]
o single expression and more than 1 symbol
when there is a linear solution
>>> solve(x - y**2, x, y)
{x: y**2}
>>> solve(x**2 - y, x, y)
{y: x**2}
when undetermined coefficients are identified
that are linear
>>> solve((a + b)*x - b + 2, a, b)
{a: -2, b: 2}
that are nonlinear
>>> solve((a + b)*x - b**2 + 2, a, b)
[(-2**(1/2), 2**(1/2)), (2**(1/2), -2**(1/2))]
if there is no linear solution then the first successful
attempt for a nonlinear solution will be returned
>>> solve(x**2 - y**2, x, y)
[y, -y]
>>> solve(x**2 - y**2/exp(x), x, y)
[x*exp(x/2), -x*exp(x/2)]
o iterable of one or more of the above
involving relationals or bools
>>> solve([x < 3, x - 2])
And(im(x) == 0, re(x) == 2)
>>> solve([x > 3, x - 2])
False
when the system is linear
with a solution
>>> solve([x - 3], x)
{x: 3}
>>> solve((x + 5*y - 2, -3*x + 6*y - 15), x, y)
{x: -3, y: 1}
>>> solve((x + 5*y - 2, -3*x + 6*y - 15), x, y, z)
{x: -3, y: 1}
>>> solve((x + 5*y - 2, -3*x + 6*y - z), z, x, y)
{x: -5*y + 2, z: 21*y - 6}
without a solution
>>> solve([x + 3, x - 3])
when the system is not linear
>>> solve([x**2 + y -2, y**2 - 4], x, y)
[(-2, -2), (0, 2), (0, 2), (2, -2)]
Warning: there is a possibility of obtaining ambiguous results
if no symbols are given for a nonlinear system of equations or
are given as a set since the symbols are not presently reported
with the solution. A warning will be issued in this situation.
>>> solve([x - 2, x**2 + y])
<BLANKLINE>
For nonlinear systems of equations, symbols should be
given as a list so as to avoid ambiguity in the results.
solve sorted the symbols as [x, y]
[(2, -4)]
>>> solve([x - 2, x**2 + f(x)], set([f(x), x]))
<BLANKLINE>
For nonlinear systems of equations, symbols should be
given as a list so as to avoid ambiguity in the results.
solve sorted the symbols as [x, f(x)]
[(2, -4)]
See also:
rsolve() for solving recurrence relationships
dsolve() for solving differential equations
"""
# make f and symbols into lists of sympified quantities
# keeping track of how f was passed since if it is a list
# a dictionary of results will be returned.
###########################################################################
def sympified_list(w):
return map(sympify, iff(iterable(w), w, [w]))
bare_f = not iterable(f)
ordered_symbols = (symbols and
symbols[0] and
(isinstance(symbols[0], Symbol) or
ordered_iter(symbols[0], include=GeneratorType)
)
)
f, symbols = (sympified_list(w) for w in [f, symbols])
# preprocess equation(s)
###########################################################################
for i, fi in enumerate(f):
if isinstance(fi, Equality):
f[i] = fi.lhs - fi.rhs
elif isinstance(fi, Poly):
f[i] = fi.as_expr()
elif isinstance(fi, bool) or fi.is_Relational:
return reduce_inequalities(f, assume=flags.get('assume'))
# Any embedded piecewise functions need to be brought out to the
# top level so that the appropriate strategy gets selected.
f[i] = piecewise_fold(f[i])
# preprocess symbol(s)
###########################################################################
if not symbols:
# get symbols from equations or supply dummy symbols so solve(3) behaves
# like solve(3, x).
symbols = set([])
for fi in f:
symbols |= fi.free_symbols or set([Dummy()])
ordered_symbols = False
elif len(symbols) == 1 and iterable(symbols[0]):
symbols = symbols[0]
if not ordered_symbols:
# we do this to make the results returned canonical in case f
# contains a system of nonlinear equations; all other cases should
# be unambiguous
symbols = sorted(symbols, key=lambda i: i.sort_key())
# we can solve for Function and Derivative instances by replacing them
# with Dummy symbols
symbols_new = []
symbol_swapped = False
symbols_passed = list(symbols)
for i, s in enumerate(symbols):
if s.is_Symbol:
s_new = s
elif s.is_Function:
symbol_swapped = True
s_new = Dummy('F%d' % i)
elif s.is_Derivative:
symbol_swapped = True
s_new = Dummy('D%d' % i)
else:
msg = 'expected Symbol, Function or Derivative but got %s'
raise TypeError(msg % type(s))
symbols_new.append(s_new)
if symbol_swapped:
swap_back_dict = dict(zip(symbols_new, symbols))
swap_dict = zip(symbols, symbols_new)
f = [fi.subs(swap_dict) for fi in f]
symbols = symbols_new
#
# try to get a solution
###########################################################################
if bare_f:
# pass f the way it was passed to solve; if it wasn't a list then
# a list of solutions will be returned, otherwise a dictionary is
# going to be returned
f = f[0]
solution = _solve(f, *symbols, **flags)
#
# postprocessing
###########################################################################
# Restore original Functions and Derivatives if a dictionary is returned.
# This is not necessary for
# - the single equation, single unknown case
# since the symbol will have been removed from the solution;
# - the nonlinear poly_system since that only support zero-dimensional
# systems and those results come back as a list
if symbol_swapped and type(solution) is dict:
solution = dict([(swap_back_dict[k], v.subs(swap_back_dict))
for k, v in solution.iteritems()])
# warn if ambiguous results are being obtained
# XXX agree on how to make this unambiguous
# see issue 2405 for logic in how Polys chooses ordering and
# for discussion of what to return see http://groups.google.com/group/sympy
# Apr 18, 2011 posting 'using results from solve'
elif (not ordered_symbols and len(symbols) > 1 and solution and
ordered_iter(solution) and ordered_iter(solution[0]) and
any(len(set(s)) > 1 for s in solution)):
msg = ('\n\tFor nonlinear systems of equations, symbols should be' +
'\n\tgiven as a list so as to avoid ambiguity in the results.' +
'\n\tsolve sorted the symbols as %s')
print msg % str(bool(symbol_swapped) and list(zip(*swap_dict)[0]) or symbols)
#
# done
###########################################################################
return solution
def powsimp(expr, deep=False, combine='all'):
"""
== Usage ==
Reduces expression by combining powers with similar bases and exponents.
== Notes ==
If deep is True then powsimp() will also simplify arguments of
functions. By default deep is set to False.
You can make powsimp() only combine bases or only combine exponents by
changing combine='base' or combine='exp'. By default, combine='all',
which does both. combine='base' will only combine::
a a a 2x x
x * y => (x*y) as well as things like 2 => 4
and combine='exp' will only combine
::
a b (a + b)
x * x => x
combine='exp' will strictly only combine exponents in the way that used
to be automatic. Also use deep=True if you need the old behavior.
== Examples ==
>>> from sympy import *
>>> x,n = map(Symbol, 'xn')
>>> e = x**n * (x*n)**(-n) * n
>>> powsimp(e)
n**(1 - n)
>>> powsimp(log(e))
log(n*x**n*(n*x)**(-n))
>>> powsimp(log(e), deep=True)
log(n**(1 - n))
>>> powsimp(e, combine='exp')
n*x**n*(n*x)**(-n)
>>> powsimp(e, combine='base')
n*(1/n)**n
>>> y, z = symbols('yz')
>>> a = x**y*x**z*n**z*n**y
>>> powsimp(a, combine='exp')
n**(y + z)*x**(y + z)
>>> powsimp(a, combine='base')
(n*x)**y*(n*x)**z
>>> powsimp(a, combine='all')
(n*x)**(y + z)
"""
if combine not in ['all', 'exp', 'base']:
raise ValueError, "combine must be one of ('all', 'exp', 'base')."
if combine in ('all', 'base'):
expr = separate(expr, deep)
y = Symbol('y', dummy=True)
if expr.is_Pow:
if deep:
return powsimp(y*powsimp(expr.base, deep, combine)**powsimp(\
expr.exp, deep, combine), deep, combine)/y
else:
return powsimp(y * expr, deep,
combine) / y # Trick it into being a Mul
elif expr.is_Function:
if expr.func == exp and deep:
# Exp should really be like Pow
return powsimp(y * exp(powsimp(expr.args[0], deep, combine)), deep,
combine) / y
elif expr.func == exp and not deep:
return powsimp(y * expr, deep, combine) / y
elif deep:
return expr.func(*[powsimp(t, deep, combine) for t in expr.args])
else:
return expr
elif expr.is_Add:
return C.Add(*[powsimp(t, deep, combine) for t in expr.args])
elif expr.is_Mul:
if combine in ('exp', 'all'):
# Collect base/exp data, while maintaining order in the
# non-commutative parts of the product
if combine is 'all' and deep and any(
(t.is_Add for t in expr.args)):
# Once we get to 'base', there is no more 'exp', so we need to
# distribute here.
return powsimp(expand_mul(expr, deep=False), deep, combine)
c_powers = {}
nc_part = []
newexpr = sympify(1)
for term in expr.args:
if term.is_Add and deep:
newexpr *= powsimp(term, deep, combine)
else:
if term.is_commutative:
b, e = term.as_base_exp()
c_powers[b] = c_powers.get(b, 0) + e
else:
nc_part.append(term)
newexpr = Mul(newexpr,
Mul(*(Pow(b, e) for b, e in c_powers.items())))
if combine is 'exp':
return Mul(newexpr, Mul(*nc_part))
else:
# combine is 'all', get stuff ready for 'base'
if deep:
newexpr = expand_mul(newexpr, deep=False)
if newexpr.is_Add:
return powsimp(Mul(*nc_part), deep, combine='base') * Add(
*(powsimp(i, deep, combine='base')
for i in newexpr.args))
else:
return powsimp(Mul(*nc_part), deep, combine='base')*\
powsimp(newexpr, deep, combine='base')
else:
# combine is 'base'
if deep:
expr = expand_mul(expr, deep=False)
if expr.is_Add:
return Add(*(powsimp(i, deep, combine) for i in expr.args))
else:
# Build c_powers and nc_part. These must both be lists not
# dicts because exp's are not combined.
c_powers = []
nc_part = []
for term in expr.args:
if term.is_commutative:
c_powers.append(list(term.as_base_exp()))
else:
nc_part.append(term)
# Pull out numerical coefficients from exponent
# e.g., 2**(2*x) => 4**x
for i in xrange(len(c_powers)):
b, e = c_powers[i]
exp_c, exp_t = e.as_coeff_terms()
if not (exp_c is S.One) and exp_t:
c_powers[i] = [C.Pow(b, exp_c), C.Mul(*exp_t)]
# Combine bases whenever they have the same exponent which is
# not numeric
c_exp = {}
for b, e in c_powers:
if e in c_exp:
c_exp[e].append(b)
else:
c_exp[e] = [b]
# Merge back in the results of the above to form a new product
for e in c_exp:
bases = c_exp[e]
if deep:
simpe = powsimp(e, deep, combine)
c_exp = {}
for b, ex in c_powers:
if ex in c_exp:
c_exp[ex].append(b)
else:
c_exp[ex] = [b]
del c_exp[e]
c_exp[simpe] = bases
else:
simpe = e
if len(bases) > 1:
for b in bases:
c_powers.remove([b, e])
new_base = Mul(*bases)
in_c_powers = False
for i in xrange(len(c_powers)):
if c_powers[i][0] == new_base:
if combine == 'all':
c_powers[i][1] += simpe
else:
c_powers.append([new_base, simpe])
in_c_powers = True
if not in_c_powers:
c_powers.append([new_base, simpe])
c_part = [C.Pow(b, e) for b, e in c_powers]
return C.Mul(*(c_part + nc_part))
else:
return expr
def update(G, CP, h):
"""update G using the set of critical pairs CP and h = (expv,pi)
see [BW] page 230
"""
hexpv, hp = f[h]
#print 'DB10',hp
# filter new pairs (h,g), g in G
C = G.copy()
D = set()
while C:
# select a pair (h,g) by popping an element from C
g = C.pop()
gexpv = f[g][0]
LCMhg = lcm_expv(hexpv, gexpv)
def lcm_divides(p):
expv = lcm_expv(hexpv, f[p][0])
# LCM(LM(h), LM(p)) divides LCM(LM(h),LM(g))
return monomial_div(LCMhg, expv)
# HT(h) and HT(g) disjoint: hexpv + gexpv == LCMhg
if monomial_mul(hexpv,gexpv) == LCMhg or (\
not any( lcm_divides(f) for f in C ) and \
not any( lcm_divides(pr[1]) for pr in D )):
D.add((h, g))
E = set()
while D:
# select h,g from D
h, g = D.pop()
gexpv = f[g][0]
LCMhg = lcm_expv(hexpv, gexpv)
if not monomial_mul(hexpv, gexpv) == LCMhg:
E.add((h, g))
# filter old pairs
B_new = set()
while CP:
# select g1,g2 from CP
g1, g2 = CP.pop()
g1expv = f[g1][0]
g2expv = f[g2][0]
LCM12 = lcm_expv(g1expv, g2expv)
# if HT(h) does not divide lcm(HT(g1),HT(g2))
if not monomial_div(LCM12, hexpv) or \
lcm_expv(g1expv,hexpv) == LCM12 or \
lcm_expv(g2expv,hexpv) == LCM12:
B_new.add((g1, g2))
B_new |= E
# filter polynomials
G_new = set()
while G:
g = G.pop()
if not monomial_div(f[g][0], hexpv):
G_new.add(g)
G_new.add(h)
return G_new, B_new
def solve_linear_system(system, *symbols, **flags):
"""Solve system of N linear equations with M variables, which means
both Cramer and over defined systems are supported. The possible
number of solutions is zero, one or infinite. Respectively this
procedure will return None or dictionary with solutions. In the
case of over defined system all arbitrary parameters are skipped.
This may cause situation in with empty dictionary is returned.
In this case it means all symbols can be assigned arbitrary values.
Input to this functions is a Nx(M+1) matrix, which means it has
to be in augmented form. If you are unhappy with such setting
use 'solve' method instead, where you can input equations
explicitly. And don't worry about the matrix, this function
is persistent and will make a local copy of it.
The algorithm used here is fraction free Gaussian elimination,
which results, after elimination, in upper-triangular matrix.
Then solutions are found using back-substitution. This approach
is more efficient and compact than the Gauss-Jordan method.
>>> from sympy import Matrix, solve_linear_system
>>> from sympy.abc import x, y
Solve the following system:
x + 4 y == 2
-2 x + y == 14
>>> system = Matrix(( (1, 4, 2), (-2, 1, 14)))
>>> solve_linear_system(system, x, y)
{x: -6, y: 2}
"""
matrix = system[:, :]
syms = list(symbols)
i, m = 0, matrix.cols - 1 # don't count augmentation
while i < matrix.rows:
if i == m:
# an overdetermined system
if any(matrix[i:, m]):
return None # no solutions
else:
# remove trailing rows
matrix = matrix[:i, :]
break
if not matrix[i, i]:
# there is no pivot in current column
# so try to find one in other columns
for k in xrange(i + 1, m):
if matrix[i, k]:
break
else:
if matrix[i, m]:
return None # no solutions
else:
# zero row or was a linear combination of
# other rows so now we can safely skip it
matrix.row_del(i)
continue
# we want to change the order of colums so
# the order of variables must also change
syms[i], syms[k] = syms[k], syms[i]
matrix.col_swap(i, k)
pivot_inv = S.One / matrix[i, i]
# divide all elements in the current row by the pivot
matrix.row(i, lambda x, _: x * pivot_inv)
for k in xrange(i + 1, matrix.rows):
if matrix[k, i]:
coeff = matrix[k, i]
# subtract from the current row the row containing
# pivot and multiplied by extracted coefficient
matrix.row(k, lambda x, j: simplify(x - matrix[i, j] * coeff))
i += 1
# if there weren't any problems, augmented matrix is now
# in row-echelon form so we can check how many solutions
# there are and extract them using back substitution
simplified = flags.get('simplified', True)
if len(syms) == matrix.rows:
# this system is Cramer equivalent so there is
# exactly one solution to this system of equations
k, solutions = i - 1, {}
while k >= 0:
content = matrix[k, m]
# run back-substitution for variables
for j in xrange(k + 1, m):
content -= matrix[k, j] * solutions[syms[j]]
if simplified:
solutions[syms[k]] = simplify(content)
else:
solutions[syms[k]] = content
k -= 1
return solutions
elif len(syms) > matrix.rows:
# this system will have infinite number of solutions
# dependent on exactly len(syms) - i parameters
k, solutions = i - 1, {}
while k >= 0:
content = matrix[k, m]
# run back-substitution for variables
for j in xrange(k + 1, i):
content -= matrix[k, j] * solutions[syms[j]]
# run back-substitution for parameters
for j in xrange(i, m):
content -= matrix[k, j] * syms[j]
if simplified:
solutions[syms[k]] = simplify(content)
else:
solutions[syms[k]] = content
k -= 1
return solutions
else:
return None # no solutions
def tsolve(eq, sym):
"""
Solves a transcendental equation with respect to the given
symbol. Various equations containing mixed linear terms, powers,
and logarithms, can be solved.
Only a single solution is returned. This solution is generally
not unique. In some cases, a complex solution may be returned
even though a real solution exists.
>>> from sympy import tsolve, log
>>> from sympy.abc import x
>>> tsolve(3**(2*x+5)-4, x)
[(-5*log(3) + log(4))/(2*log(3))]
>>> tsolve(log(x) + 2*x, x)
[LambertW(2)/2]
"""
if patterns is None:
_generate_patterns()
eq = sympify(eq)
if isinstance(eq, Equality):
eq = eq.lhs - eq.rhs
sym = sympify(sym)
eq2 = eq.subs(sym, x)
# First see if the equation has a linear factor
# In that case, the other factor can contain x in any way (as long as it
# is finite), and we have a direct solution to which we add others that
# may be found for the remaining portion.
r = Wild('r')
m = eq2.match((a * x + b) * r)
if m and m[a]:
return [(-b / a).subs(m).subs(x, sym)] + solve(m[r], x)
for p, sol in patterns:
m = eq2.match(p)
if m:
return [sol.subs(m).subs(x, sym)]
# let's also try to inverse the equation
lhs = eq
rhs = S.Zero
while True:
indep, dep = lhs.as_independent(sym)
# dep + indep == rhs
if lhs.is_Add:
# this indicates we have done it all
if indep is S.Zero:
break
lhs = dep
rhs -= indep
# dep * indep == rhs
else:
# this indicates we have done it all
if indep is S.One:
break
lhs = dep
rhs /= indep
# -1
# f(x) = g -> x = f (g)
if lhs.is_Function and lhs.nargs == 1 and hasattr(lhs, 'inverse'):
rhs = lhs.inverse()(rhs)
lhs = lhs.args[0]
sol = solve(lhs - rhs, sym)
return sol
elif lhs.is_Add:
# just a simple case - we do variable substitution for first function,
# and if it removes all functions - let's call solve.
# x -x -1
# UC: e + e = y -> t + t = y
t = Dummy('t')
terms = lhs.args
# find first term which is Function
for f1 in lhs.args:
if f1.is_Function:
break
else:
raise NotImplementedError("Unable to solve the equation" + \
"(tsolve: at least one Function expected at this point")
# perform the substitution
lhs_ = lhs.subs(f1, t)
# if no Functions left, we can proceed with usual solve
if not (lhs_.is_Function or any(term.is_Function
for term in lhs_.args)):
cv_sols = solve(lhs_ - rhs, t)
for sol in cv_sols:
if sol.has(sym):
raise NotImplementedError("Unable to solve the equation")
cv_inv = solve(t - f1, sym)[0]
sols = list()
for sol in cv_sols:
sols.append(cv_inv.subs(t, sol))
return sols
raise NotImplementedError("Unable to solve the equation.")
def _find(self, tests, obj, name, module, source_lines, globs, seen):
"""
Find tests for the given object and any contained objects, and
add them to `tests`.
"""
if self._verbose:
print 'Finding tests in %s' % name
# If we've already processed this object, then ignore it.
if id(obj) in seen:
return
seen[id(obj)] = 1
# Find a test for this object, and add it to the list of tests.
test = self._get_test(obj, name, module, globs, source_lines)
if test is not None:
tests.append(test)
# Look for tests in a module's contained objects.
if inspect.ismodule(obj) and self._recurse:
for rawname, val in obj.__dict__.items():
# Recurse to functions & classes.
if inspect.isfunction(val) or inspect.isclass(val):
in_module = self._from_module(module, val)
if not in_module:
# double check in case this function is decorated
# and just appears to come from a different module.
pat = r'\s*(def|class)\s+%s\s*\(' % rawname
PAT = pre.compile(pat)
in_module = any(PAT.match(line) for line in source_lines)
if in_module:
try:
valname = '%s.%s' % (name, rawname)
self._find(tests, val, valname, module, source_lines, globs, seen)
except:
pass
# Look for tests in a module's __test__ dictionary.
if inspect.ismodule(obj) and self._recurse:
for valname, val in getattr(obj, '__test__', {}).items():
if not isinstance(valname, basestring):
raise ValueError("SymPyDocTestFinder.find: __test__ keys "
"must be strings: %r" %
(type(valname),))
if not (inspect.isfunction(val) or inspect.isclass(val) or
inspect.ismethod(val) or inspect.ismodule(val) or
isinstance(val, basestring)):
raise ValueError("SymPyDocTestFinder.find: __test__ values "
"must be strings, functions, methods, "
"classes, or modules: %r" %
(type(val),))
valname = '%s.__test__.%s' % (name, valname)
self._find(tests, val, valname, module, source_lines,
globs, seen)
# Look for tests in a class's contained objects.
if inspect.isclass(obj) and self._recurse:
for valname, val in obj.__dict__.items():
# Special handling for staticmethod/classmethod.
if isinstance(val, staticmethod):
val = getattr(obj, valname)
if isinstance(val, classmethod):
val = getattr(obj, valname).im_func
# Recurse to methods, properties, and nested classes.
if (inspect.isfunction(val) or
inspect.isclass(val) or
isinstance(val, property)):
in_module = self._from_module(module, val)
if not in_module:
# "double check" again
pat = r'\s*(def|class)\s+%s\s*\(' % valname
PAT = pre.compile(pat)
in_module = any(PAT.match(line) for line in source_lines)
if in_module:
valname = '%s.%s' % (name, valname)
self._find(tests, val, valname, module, source_lines,
globs, seen)
# Special handling for staticmethod/classmethod.
if isinstance(val, staticmethod):
val = getattr(obj, valname)
if isinstance(val, classmethod):
val = getattr(obj, valname).im_func
# Recurse to methods, properties, and nested classes.
if (inspect.isfunction(val) or
inspect.isclass(val) or
isinstance(val, property)):
in_module = self._from_module(module, val)
if not in_module:
# "double check" again
pat = r'\s*(def|class)\s+%s\s*\(' % valname
PAT = pre.compile(pat)
in_module = any(PAT.match(line) for line in source_lines)
if in_module:
valname = '%s.%s' % (name, valname)
self._find(tests, val, valname, module, source_lines,
globs, seen)
def _get_test(self, obj, name, module, globs, source_lines):
"""
Return a DocTest for the given object, if it defines a docstring;
otherwise, return None.
"""
# Extract the object's docstring. If it doesn't have one,
# then return None (no test for this object).
if isinstance(obj, basestring):
docstring = obj
else:
def denester(nested):
"""
Denests a list of expressions that contain nested square roots.
This method should not be called directly - use 'denest' instead.
This algorithm is based on <http://www.almaden.ibm.com/cs/people/fagin/symb85.pdf>.
It is assumed that all of the elements of 'nested' share the same
bottom-level radicand. (This is stated in the paper, on page 177, in
the paragraph immediately preceding the algorithm.)
When evaluating all of the arguments in parallel, the bottom-level
radicand only needs to be denested once. This means that calling
denester with x arguments results in a recursive invocation with x+1
arguments; hence denester has polynomial complexity.
However, if the arguments were evaluated separately, each call would
result in two recursive invocations, and the algorithm would have
exponential complexity.
This is discussed in the paper in the middle paragraph of page 179.
"""
if all((n**2).is_Number
for n in nested): #If none of the arguments are nested
for f in subsets(len(nested)): #Test subset 'f' of nested
p = prod(nested[i]**2 for i in range(len(f)) if f[i]).expand()
if 1 in f and f.count(1) > 1 and f[-1]: p = -p
if sqrt(p).is_Number:
return sqrt(
p), f #If we got a perfect square, return its square root.
return nested[-1], [0] * len(
nested
) #Otherwise, return the radicand from the previous invocation.
else:
a, b, r, R = Wild('a'), Wild('b'), Wild('r'), None
values = [expr.match(sqrt(a + b * sqrt(r))) for expr in nested]
for v in values:
if r in v: #Since if b=0, r is not defined
if R is not None:
assert R == v[r] #All the 'r's should be the same.
else:
R = v[r]
d, f = denester([
sqrt((v[a]**2).expand() - (R * v[b]**2).expand()) for v in values
] + [sqrt(R)])
if not any([f[i] for i in range(len(nested))
]): #If f[i]=0 for all i < len(nested)
v = values[-1]
return sqrt(v[a] + v[b] * d), f
else:
v = prod(nested[i]**2 for i in range(len(nested))
if f[i]).expand().match(a + b * sqrt(r))
if 1 in f and f.index(1) < len(nested) - 1 and f[len(nested) - 1]:
v[a] = -1 * v[a]
v[b] = -1 * v[b]
if not f[len(nested)]: #Solution denests with square roots
return (sqrt((v[a] + d).expand() / 2) + sign(v[b]) * sqrt(
(v[b]**2 * R / (2 * (v[a] + d))).expand())).expand(), f
else: #Solution requires a fourth root
FR, s = (R.expand()**Rational(
1, 4)), sqrt((v[b] * R).expand() + d)
return (s / (sqrt(2) * FR) + v[a] * FR /
(sqrt(2) * s)).expand(), f
def update(G, B, ih):
# update G using the set of critical pairs B and h
# [BW] page 230
h = f[ih]
mh = sdp_LM(h, u)
# filter new pairs (h, g), g in G
C = G.copy()
D = set()
while C:
# select a pair (h, g) by popping an element from C
ig = C.pop()
g = f[ig]
mg = sdp_LM(g, u)
LCMhg = monomial_lcm(mh, mg)
def lcm_divides(ip):
# LCM(LM(h), LM(p)) divides LCM(LM(h), LM(g))
m = monomial_lcm(mh, sdp_LM(f[ip], u))
return monomial_div(LCMhg, m)
# HT(h) and HT(g) disjoint: mh*mg == LCMhg
if monomial_mul(mh, mg) == LCMhg or (not any(
lcm_divides(ipx)
for ipx in C) and not any(lcm_divides(pr[1]) for pr in D)):
D.add((ih, ig))
E = set()
while D:
# select h, g from D (h the same as above)
ih, ig = D.pop()
mg = sdp_LM(f[ig], u)
LCMhg = monomial_lcm(mh, mg)
if not monomial_mul(mh, mg) == LCMhg:
E.add((ih, ig))
# filter old pairs
B_new = set()
while B:
# select g1, g2 from B (-> CP)
ig1, ig2 = B.pop()
mg1 = sdp_LM(f[ig1], u)
mg2 = sdp_LM(f[ig2], u)
LCM12 = monomial_lcm(mg1, mg2)
# if HT(h) does not divide lcm(HT(g1), HT(g2))
if not monomial_div(LCM12, mh) or \
monomial_lcm(mg1, mh) == LCM12 or \
monomial_lcm(mg2, mh) == LCM12:
B_new.add((ig1, ig2))
B_new |= E
# filter polynomials
G_new = set()
while G:
ig = G.pop()
mg = sdp_LM(f[ig], u)
if not monomial_div(mg, mh):
G_new.add(ig)
G_new.add(ih)
return G_new, B_new
def _eval_nseries(self, x, n, logx):
"""
This function does compute series for multivariate functions,
but the expansion is always in terms of *one* variable.
Examples:
>>> from sympy import atan2, O
>>> from sympy.abc import x, y
>>> atan2(x, y).series(x, n=2)
atan2(0, y) + x/y + O(x**2)
>>> atan2(x, y).series(y, n=2)
atan2(x, 0) - y/x + O(y**2)
This function also computes asymptotic expansions, if necessary
and possible:
>>> from sympy import loggamma
>>> loggamma(1/x)._eval_nseries(x,0,None)
log(x)/2 - log(x)/x - 1/x + O(1)
"""
if self.func.nargs is None:
raise NotImplementedError('series for user-defined \
functions are not supported.')
args = self.args
args0 = [t.limit(x, 0) for t in args]
if any([t.is_bounded == False for t in args0]):
from sympy import Dummy, oo, zoo, nan
a = [t.compute_leading_term(x, logx=logx) for t in args]
a0 = [t.limit(x, 0) for t in a]
if any ([t.has(oo, -oo, zoo, nan) for t in a0]):
return self._eval_aseries(n, args0, x, logx)._eval_nseries(x, n, logx)
# Careful: the argument goes to oo, but only logarithmically so. We
# are supposed to do a power series expansion "around the
# logarithmic term". e.g.
# f(1+x+log(x))
# -> f(1+logx) + x*f'(1+logx) + O(x**2)
# where 'logx' is given in the argument
a = [t._eval_nseries(x, n, logx) for t in args]
z = [r - r0 for (r, r0) in zip(a, a0)]
p = [Dummy() for t in z]
q = []
v = None
for ai, zi, pi in zip(a0, z, p):
if zi.has(x):
if v is not None: raise NotImplementedError
q.append(ai + pi)
v = pi
else:
q.append(ai)
e1 = self.func(*q)
if v is None:
return e1
s = e1._eval_nseries(v, n, logx)
o = s.getO()
s = s.removeO()
s = s.subs(v, zi).expand() + C.Order(o.expr.subs(v, zi), x)
return s
if (self.func.nargs == 1 and args0[0]) or self.func.nargs > 1:
e = self
e1 = e.expand()
if e == e1:
#for example when e = sin(x+1) or e = sin(cos(x))
#let's try the general algorithm
term = e.subs(x, S.Zero)
if term.is_bounded is False or term is S.NaN:
raise PoleError("Cannot expand %s around 0" % (self))
series = term
fact = S.One
for i in range(n-1):
i += 1
fact *= Rational(i)
e = e.diff(x)
subs = e.subs(x, S.Zero)
if subs is S.NaN:
# try to evaluate a limit if we have to
subs = e.limit(x, S.Zero)
if subs.is_bounded is False:
raise PoleError("Cannot expand %s around 0" % (self))
term = subs*(x**i)/fact
term = term.expand()
series += term
return series + C.Order(x**n, x)
return e1.nseries(x, n=n, logx=logx)
arg = self.args[0]
l = []
g = None
for i in xrange(n+2):
g = self.taylor_term(i, arg, g)
g = g.nseries(x, n=n, logx=logx)
l.append(g)
return Add(*l) + C.Order(x**n, x)
def doit(self, **hints):
if not hints.get('integrals', True):
return self
deep = hints.get('deep', True)
# check for the trivial case of equal upper and lower limits
if self.is_zero:
return S.Zero
# now compute and check the function
function = self.function
if deep:
function = function.doit(**hints)
if function.is_zero:
return S.Zero
# There is no trivial answer, so continue
undone_limits = []
ulj = set() # free symbols of any undone limits' upper and lower limits
for xab in self.limits:
# compute uli, the free symbols in the
# Upper and Lower limits of limit I
if len(xab) == 1:
uli = set(xab[:1])
elif len(xab) == 2:
uli = xab[1].free_symbols
elif len(xab) == 3:
uli = xab[1].free_symbols.union(xab[2].free_symbols)
# this integral can be done as long as there is no blocking
# limit that has been undone. An undone limit is blocking if
# it contains an integration variable that is in this limit's
# upper or lower free symbols or vice versa
if xab[0] in ulj or any(v[0] in uli for v in undone_limits):
undone_limits.append(xab)
ulj.update(uli)
continue
antideriv = self._eval_integral(function, xab[0])
if antideriv is None:
undone_limits.append(xab)
else:
if len(xab) == 1:
function = antideriv
else:
if len(xab) == 3:
x, a, b = xab
if len(xab) == 2:
x, b = xab
a = None
if deep:
if isinstance(a, Basic):
a = a.doit(**hints)
if isinstance(b, Basic):
b = b.doit(**hints)
if antideriv.is_Poly:
gens = list(antideriv.gens)
gens.remove(x)
antideriv = antideriv.as_basic()
function = antideriv._eval_interval(x, a, b)
function = Poly(function, *gens)
else:
function = antideriv._eval_interval(x, a, b)
if undone_limits:
return self.new(*([function] + undone_limits))
return function
def solve(f, *symbols, **flags):
"""
Algebraically solves equations and systems of equations.
Currently supported are:
- univariate polynomial,
- transcendental
- piecewise combinations of the above
- systems of linear and polynomial equations
- sytems containing relational expressions.
Input is formed as:
f
- a single Expr or Poly that must be zero,
- an Equality
- a Relational expression or boolean
- iterable of one or more of the above
symbols (Symbol, Function or Derivative) specified as
- none given (all free symbols will be used)
- single symbol
- denested list of symbols
e.g. solve(f, x, y)
- ordered iterable of symbols
e.g. solve(f, [x, y])
flags
- ``simplified``, when False, will not simplify solutions
(default=True except for polynomials of
order 3 or greater)
The output varies according to the input and can be seen by example:
>>> from sympy import solve, Poly, Eq, Function, exp
>>> from sympy.abc import x, y, z, a, b
o boolean or univariate Relational
>>> solve(x < 3)
And(im(x) == 0, re(x) < 3)
o single expression and single symbol that is in the expression
>>> solve(x - y, x)
[y]
>>> solve(x - 3, x)
[3]
>>> solve(Eq(x, 3), x)
[3]
>>> solve(Poly(x - 3), x)
[3]
>>> solve(x**2 - y**2, x)
[y, -y]
>>> solve(x**4 - 1, x)
[1, -1, -I, I]
o single expression with no symbol that is in the expression
>>> solve(3, x)
[]
>>> solve(x - 3, y)
[]
o when no symbol is given then all free symbols will be used
and sorted with default_sort_key and the result will be the
same as above as if those symbols had been supplied
>>> solve(x - 3)
[3]
>>> solve(x**2 - y**2)
[y, -y]
o when a Function or Derivative is given as a symbol, it is isolated
algebraically and an implicit solution may be obtained
>>> f = Function('f')
>>> solve(f(x) - x, f(x))
[x]
>>> solve(f(x).diff(x) - f(x) - x, f(x).diff(x))
[x + f(x)]
o single expression and more than 1 symbol
when there is a linear solution
>>> solve(x - y**2, x, y)
{x: y**2}
>>> solve(x**2 - y, x, y)
{y: x**2}
when undetermined coefficients are identified
that are linear
>>> solve((a + b)*x - b + 2, a, b)
{a: -2, b: 2}
that are nonlinear
>>> solve((a + b)*x - b**2 + 2, a, b)
[(-2**(1/2), 2**(1/2)), (2**(1/2), -2**(1/2))]
if there is no linear solution then the first successful
attempt for a nonlinear solution will be returned
>>> solve(x**2 - y**2, x, y)
[y, -y]
>>> solve(x**2 - y**2/exp(x), x, y)
[x*exp(x/2), -x*exp(x/2)]
o iterable of one or more of the above
involving relationals or bools
>>> solve([x < 3, x - 2])
And(im(x) == 0, re(x) == 2)
>>> solve([x > 3, x - 2])
False
when the system is linear
with a solution
>>> solve([x - 3], x)
{x: 3}
>>> solve((x + 5*y - 2, -3*x + 6*y - 15), x, y)
{x: -3, y: 1}
>>> solve((x + 5*y - 2, -3*x + 6*y - 15), x, y, z)
{x: -3, y: 1}
>>> solve((x + 5*y - 2, -3*x + 6*y - z), z, x, y)
{x: -5*y + 2, z: 21*y - 6}
without a solution
>>> solve([x + 3, x - 3])
when the system is not linear
>>> solve([x**2 + y -2, y**2 - 4], x, y)
[(-2, -2), (0, 2), (0, 2), (2, -2)]
Warning: there is a possibility of obtaining ambiguous results
if no symbols are given for a nonlinear system of equations or
are given as a set since the symbols are not presently reported
with the solution. A warning will be issued in this situation.
>>> solve([x - 2, x**2 + y])
<BLANKLINE>
For nonlinear systems of equations, symbols should be
given as a list so as to avoid ambiguity in the results.
solve sorted the symbols as [x, y]
[(2, -4)]
>>> solve([x - 2, x**2 + f(x)], set([f(x), x]))
<BLANKLINE>
For nonlinear systems of equations, symbols should be
given as a list so as to avoid ambiguity in the results.
solve sorted the symbols as [x, f(x)]
[(2, -4)]
See also:
rsolve() for solving recurrence relationships
dsolve() for solving differential equations
"""
# make f and symbols into lists of sympified quantities
# keeping track of how f was passed since if it is a list
# a dictionary of results will be returned.
###########################################################################
def sympified_list(w):
return map(sympify, iff(iterable(w), w, [w]))
bare_f = not iterable(f)
ordered_symbols = (symbols and symbols[0] and
(isinstance(symbols[0], Symbol)
or ordered_iter(symbols[0], include=GeneratorType)))
f, symbols = (sympified_list(w) for w in [f, symbols])
# preprocess equation(s)
###########################################################################
for i, fi in enumerate(f):
if isinstance(fi, Equality):
f[i] = fi.lhs - fi.rhs
elif isinstance(fi, Poly):
f[i] = fi.as_expr()
elif isinstance(fi, bool) or fi.is_Relational:
return reduce_inequalities(f, assume=flags.get('assume'))
# Any embedded piecewise functions need to be brought out to the
# top level so that the appropriate strategy gets selected.
f[i] = piecewise_fold(f[i])
# preprocess symbol(s)
###########################################################################
if not symbols:
# get symbols from equations or supply dummy symbols so solve(3) behaves
# like solve(3, x).
symbols = set([])
for fi in f:
symbols |= fi.free_symbols or set([Dummy()])
ordered_symbols = False
elif len(symbols) == 1 and iterable(symbols[0]):
symbols = symbols[0]
if not ordered_symbols:
# we do this to make the results returned canonical in case f
# contains a system of nonlinear equations; all other cases should
# be unambiguous
symbols = sorted(symbols, key=lambda i: i.sort_key())
# we can solve for Function and Derivative instances by replacing them
# with Dummy symbols
symbols_new = []
symbol_swapped = False
symbols_passed = list(symbols)
for i, s in enumerate(symbols):
if s.is_Symbol:
s_new = s
elif s.is_Function:
symbol_swapped = True
s_new = Dummy('F%d' % i)
elif s.is_Derivative:
symbol_swapped = True
s_new = Dummy('D%d' % i)
else:
msg = 'expected Symbol, Function or Derivative but got %s'
raise TypeError(msg % type(s))
symbols_new.append(s_new)
if symbol_swapped:
swap_back_dict = dict(zip(symbols_new, symbols))
swap_dict = zip(symbols, symbols_new)
f = [fi.subs(swap_dict) for fi in f]
symbols = symbols_new
#
# try to get a solution
###########################################################################
if bare_f:
# pass f the way it was passed to solve; if it wasn't a list then
# a list of solutions will be returned, otherwise a dictionary is
# going to be returned
f = f[0]
solution = _solve(f, *symbols, **flags)
#
# postprocessing
###########################################################################
# Restore original Functions and Derivatives if a dictionary is returned.
# This is not necessary for
# - the single equation, single unknown case
# since the symbol will have been removed from the solution;
# - the nonlinear poly_system since that only support zero-dimensional
# systems and those results come back as a list
if symbol_swapped and type(solution) is dict:
solution = dict([(swap_back_dict[k], v.subs(swap_back_dict))
for k, v in solution.iteritems()])
# warn if ambiguous results are being obtained
# XXX agree on how to make this unambiguous
# see issue 2405 for logic in how Polys chooses ordering and
# for discussion of what to return see http://groups.google.com/group/sympy
# Apr 18, 2011 posting 'using results from solve'
elif (not ordered_symbols and len(symbols) > 1 and solution
and ordered_iter(solution) and ordered_iter(solution[0])
and any(len(set(s)) > 1 for s in solution)):
msg = ('\n\tFor nonlinear systems of equations, symbols should be' +
'\n\tgiven as a list so as to avoid ambiguity in the results.' +
'\n\tsolve sorted the symbols as %s')
print msg % str(
bool(symbol_swapped) and list(zip(*swap_dict)[0]) or symbols)
#
# done
###########################################################################
return solution
def solve_linear_system(system, *symbols, **flags):
"""Solve system of N linear equations with M variables, which means
both Cramer and over defined systems are supported. The possible
number of solutions is zero, one or infinite. Respectively this
procedure will return None or dictionary with solutions. In the
case of over defined system all arbitrary parameters are skipped.
This may cause situation in with empty dictionary is returned.
In this case it means all symbols can be assigned arbitrary values.
Input to this functions is a Nx(M+1) matrix, which means it has
to be in augmented form. If you are unhappy with such setting
use 'solve' method instead, where you can input equations
explicitly. And don't worry about the matrix, this function
is persistent and will make a local copy of it.
The algorithm used here is fraction free Gaussian elimination,
which results, after elimination, in upper-triangular matrix.
Then solutions are found using back-substitution. This approach
is more efficient and compact than the Gauss-Jordan method.
>>> from sympy import Matrix, solve_linear_system
>>> from sympy.abc import x, y
Solve the following system:
x + 4 y == 2
-2 x + y == 14
>>> system = Matrix(( (1, 4, 2), (-2, 1, 14)))
>>> solve_linear_system(system, x, y)
{x: -6, y: 2}
"""
matrix = system[:, :]
syms = list(symbols)
i, m = 0, matrix.cols - 1 # don't count augmentation
while i < matrix.rows:
if i == m:
# an overdetermined system
if any(matrix[i:, m]):
return None # no solutions
else:
# remove trailing rows
matrix = matrix[:i, :]
break
if not matrix[i, i]:
# there is no pivot in current column
# so try to find one in other columns
for k in xrange(i + 1, m):
if matrix[i, k]:
break
else:
if matrix[i, m]:
return None # no solutions
else:
# zero row or was a linear combination of
# other rows so now we can safely skip it
matrix.row_del(i)
continue
# we want to change the order of colums so
# the order of variables must also change
syms[i], syms[k] = syms[k], syms[i]
matrix.col_swap(i, k)
pivot_inv = S.One / matrix[i, i]
# divide all elements in the current row by the pivot
matrix.row(i, lambda x, _: x * pivot_inv)
for k in xrange(i + 1, matrix.rows):
if matrix[k, i]:
coeff = matrix[k, i]
# subtract from the current row the row containing
# pivot and multiplied by extracted coefficient
matrix.row(k, lambda x, j: simplify(x - matrix[i, j] * coeff))
i += 1
# if there weren't any problems, augmented matrix is now
# in row-echelon form so we can check how many solutions
# there are and extract them using back substitution
simplified = flags.get("simplified", True)
if len(syms) == matrix.rows:
# this system is Cramer equivalent so there is
# exactly one solution to this system of equations
k, solutions = i - 1, {}
while k >= 0:
content = matrix[k, m]
# run back-substitution for variables
for j in xrange(k + 1, m):
content -= matrix[k, j] * solutions[syms[j]]
if simplified:
solutions[syms[k]] = simplify(content)
else:
solutions[syms[k]] = content
k -= 1
return solutions
elif len(syms) > matrix.rows:
# this system will have infinite number of solutions
# dependent on exactly len(syms) - i parameters
k, solutions = i - 1, {}
while k >= 0:
content = matrix[k, m]
# run back-substitution for variables
for j in xrange(k + 1, i):
content -= matrix[k, j] * solutions[syms[j]]
# run back-substitution for parameters
for j in xrange(i, m):
content -= matrix[k, j] * syms[j]
if simplified:
solutions[syms[k]] = simplify(content)
else:
solutions[syms[k]] = content
k -= 1
return solutions
else:
return None # no solutions
def doctest(*paths, **kwargs):
"""
Runs doctests in all *py files in the sympy directory which match
any of the given strings in `paths` or all tests if paths=[].
Note:
o paths can be entered in native system format or in unix,
forward-slash format.
o files that are on the blacklist can be tested by providing
their path; they are only excluded if no paths are given.
Examples:
>> import sympy
Run all tests:
>> sympy.doctest()
Run one file:
>> sympy.doctest("sympy/core/basic.py")
>> sympy.doctest("polynomial.txt")
Run all tests in sympy/functions/ and some particular file:
>> sympy.doctest("/functions", "basic.py")
Run any file having polynomial in its name, doc/src/modules/polynomial.txt,
sympy\functions\special\polynomials.py, and sympy\polys\polynomial.py:
>> sympy.doctest("polynomial")
"""
normal = kwargs.get("normal", False)
verbose = kwargs.get("verbose", False)
blacklist = kwargs.get("blacklist", [])
blacklist.extend([
"sympy/thirdparty/pyglet", # segfaults
"doc/src/modules/mpmath", # needs to be fixed upstream
"sympy/mpmath", # needs to be fixed upstream
"doc/src/modules/plotting.txt", # generates live plots
"sympy/plotting", # generates live plots
"sympy/utilities/compilef.py", # needs tcc
"sympy/galgebra/GA.py", # needs numpy
"sympy/galgebra/latex_ex.py", # needs numpy
"sympy/conftest.py", # needs py.test
"sympy/utilities/benchmarking.py", # needs py.test
])
blacklist = convert_to_native_paths(blacklist)
r = PyTestReporter(verbose)
t = SymPyDocTests(r, normal)
test_files = t.get_test_files('sympy')
not_blacklisted = [f for f in test_files
if not any(b in f for b in blacklist)]
if len(paths) == 0:
t._tests.extend(not_blacklisted)
else:
# take only what was requested...but not blacklisted items
# and allow for partial match anywhere or fnmatch of name
paths = convert_to_native_paths(paths)
matched = []
for f in not_blacklisted:
basename = os.path.basename(f)
for p in paths:
if p in f or fnmatch(basename, p):
matched.append(f)
break
t._tests.extend(matched)
# run the tests and record the result for this *py portion of the tests
if t._tests:
doc_tests_succeeded = t.test()
else:
doc_tests_succeeded = True
# test *txt files only if we are running python newer than 2.4
if sys.version_info[:2] > (2,4):
# N.B.
# --------------------------------------------------------------------
# Here we test *.txt files at or below doc/src. Code from these must
# be self supporting in terms of imports since there is no importing
# of necessary modules by doctest.testfile. If you try to pass *.py
# files through this they might fail because they will lack the needed
# imports and smarter parsing that can be done with source code.
#
test_files = t.get_test_files('doc/src', '*.txt', init_only=False)
test_files.sort()
not_blacklisted = [f for f in test_files
if not any(b in f for b in blacklist)]
if len(paths) == 0:
matched = not_blacklisted
else:
# Take only what was requested as long as it's not on the blacklist.
# Paths were already made native in *py tests so don't repeat here.
# There's no chance of having a *py file slip through since we
# only have *txt files in test_files.
matched = []
for f in not_blacklisted:
basename = os.path.basename(f)
for p in paths:
if p in f or fnmatch(basename, p):
matched.append(f)
break
setup_pprint()
for txt_file in matched:
if not os.path.isfile(txt_file):
continue
old_displayhook = sys.displayhook
try:
out = pdoctest.testfile(txt_file, module_relative=False,
optionflags=pdoctest.ELLIPSIS | \
pdoctest.NORMALIZE_WHITESPACE)
finally:
# make sure we return to the original displayhook in case some
# doctest has changed that
sys.displayhook = old_displayhook
print "Testing ", txt_file
print "Failed %s, tested %s" % out
if out[0] != 0:
doc_tests_succeeded = False
# the doctests for *py will have printed this message already if there was
# a failure, so now only print it if there was intervening reporting by
# testing the *txt.
if matched and not doc_tests_succeeded:
print("DO *NOT* COMMIT!")
return doc_tests_succeeded
for valname, val in obj.__dict__.items():
# Special handling for staticmethod/classmethod.
if isinstance(val, staticmethod):
val = getattr(obj, valname)
if isinstance(val, classmethod):
val = getattr(obj, valname).im_func
# Recurse to methods, properties, and nested classes.
if (inspect.isfunction(val) or inspect.isclass(val)
or isinstance(val, property)):
in_module = self._from_module(module, val)
if not in_module:
# "double check" again
pat = r'\s*(def|class)\s+%s\s*\(' % valname
PAT = pre.compile(pat)
in_module = any(
PAT.match(line) for line in source_lines)
if in_module:
valname = '%s.%s' % (name, valname)
self._find(tests, val, valname, module, source_lines,
globs, seen)
def _get_test(self, obj, name, module, globs, source_lines):
"""
Return a DocTest for the given object, if it defines a docstring;
otherwise, return None.
"""
# Extract the object's docstring. If it doesn't have one,
# then return None (no test for this object).
if isinstance(obj, basestring):
docstring = obj
else:
def solve(f, *symbols, **flags):
"""Solves equations and systems of equations.
Currently supported are univariate polynomial and transcendental
equations and systems of linear and polynomial equations. Input
is formed as a single expression or an equation, or an iterable
container in case of an equation system. The type of output may
vary and depends heavily on the input. For more details refer to
more problem specific functions.
By default all solutions are simplified to make the output more
readable. If this is not the expected behavior, eg. because of
speed issues, set simplified=False in function arguments.
To solve equations and systems of equations of other kind, eg.
recurrence relations of differential equations use rsolve() or
dsolve() functions respectively.
>>> from sympy import *
>>> x,y = symbols('xy')
Solve a polynomial equation:
>>> solve(x**4-1, x)
[1, -1, -I, I]
Solve a linear system:
>>> solve((x+5*y-2, -3*x+6*y-15), x, y)
{x: -3, y: 1}
"""
if not symbols:
raise ValueError('no symbols were given')
if len(symbols) == 1:
if isinstance(symbols[0], (list, tuple, set)):
symbols = symbols[0]
symbols = map(sympify, symbols)
if any(not s.is_Symbol for s in symbols):
raise TypeError('not a Symbol')
if not isinstance(f, (tuple, list, set)):
f = sympify(f)
if isinstance(f, Equality):
f = f.lhs - f.rhs
if len(symbols) != 1:
raise NotImplementedError('multivariate equation')
symbol = symbols[0]
strategy = guess_solve_strategy(f, symbol)
if strategy == GS_POLY:
poly = f.as_poly(symbol)
assert poly is not None
result = roots(poly, cubics=True, quartics=True).keys()
elif strategy == GS_RATIONAL:
P, Q = f.as_numer_denom()
#TODO: check for Q != 0
return solve(P, symbol, **flags)
elif strategy == GS_POLY_CV_1:
# we must search for a suitable change of variable
# collect exponents
exponents_denom = list()
args = list(f.args)
if isinstance(f, Add):
for arg in args:
if isinstance(arg, Pow):
exponents_denom.append(arg.exp.q)
elif isinstance(arg, Mul):
for mul_arg in arg.args:
if isinstance(mul_arg, Pow):
exponents_denom.append(mul_arg.exp.q)
elif isinstance(f, Mul):
for mul_arg in args:
if isinstance(mul_arg, Pow):
exponents_denom.append(mul_arg.exp.q)
assert len(exponents_denom) > 0
if len(exponents_denom) == 1:
m = exponents_denom[0]
else:
# get the GCD of the denominators
m = ilcm(*exponents_denom)
# x -> y**m.
# we assume positive for simplification purposes
t = Symbol('t', positive=True, dummy=True)
f_ = f.subs(symbol, t**m)
if guess_solve_strategy(f_, t) != GS_POLY:
raise TypeError(
"Could not convert to a polynomial equation: %s" % f_)
cv_sols = solve(f_, t)
result = list()
for sol in cv_sols:
result.append(sol**(S.One / m))
elif strategy == GS_POLY_CV_2:
m = 0
args = list(f.args)
if isinstance(f, Add):
for arg in args:
if isinstance(arg, Pow):
m = min(m, arg.exp)
elif isinstance(arg, Mul):
for mul_arg in arg.args:
if isinstance(mul_arg, Pow):
m = min(m, mul_arg.exp)
elif isinstance(f, Mul):
for mul_arg in args:
if isinstance(mul_arg, Pow):
m = min(m, mul_arg.exp)
f1 = simplify(f * symbol**(-m))
result = solve(f1, symbol)
# TODO: we might have introduced unwanted solutions
# when multiplied by x**-m
elif strategy == GS_TRASCENDENTAL:
#a, b = f.as_numer_denom()
# Let's throw away the denominator for now. When we have robust
# assumptions, it should be checked, that for the solution,
# b!=0.
result = tsolve(f, *symbols)
elif strategy == -1:
raise Exception('Could not parse expression %s' % f)
else:
raise NotImplementedError(
"No algorithms where implemented to solve equation %s" % f)
if flags.get('simplified', True):
return map(simplify, result)
else:
return result
else:
if not f:
return {}
else:
polys = []
for g in f:
g = sympify(g)
if isinstance(g, Equality):
g = g.lhs - g.rhs
poly = g.as_poly(*symbols)
if poly is not None:
polys.append(poly)
else:
raise NotImplementedError
if all(p.is_linear for p in polys):
n, m = len(f), len(symbols)
matrix = zeros((n, m + 1))
for i, poly in enumerate(polys):
for coeff, monom in poly.iter_terms():
try:
j = list(monom).index(1)
matrix[i, j] = coeff
except ValueError:
matrix[i, m] = -coeff
return solve_linear_system(matrix, *symbols, **flags)
else:
return solve_poly_system(polys)
def _construct_composite(coeffs, opt):
"""Handle composite domains, e.g.: ZZ[X], QQ[X], ZZ(X), QQ(X). """
numers, denoms = [], []
for coeff in coeffs:
numer, denom = coeff.as_numer_denom()
numers.append(numer)
denoms.append(denom)
polys, gens = parallel_dict_from_basic(numers + denoms) # XXX: sorting
if any(gen.is_number for gen in gens):
return None # generators are number-like so lets better use EX
n = len(gens)
k = len(polys) // 2
numers = polys[:k]
denoms = polys[k:]
if opt.field:
fractions = True
else:
fractions, zeros = False, (0,) * n
for denom in denoms:
if len(denom) > 1 or zeros not in denom:
fractions = True
break
result = []
if not fractions:
coeffs = set([])
for numer, denom in zip(numers, denoms):
denom = denom[zeros]
for monom, coeff in numer.iteritems():
coeff /= denom
coeffs.add(coeff)
numer[monom] = coeff
rationals, reals = False, False
for coeff in coeffs:
if coeff.is_Rational:
if not coeff.is_Integer:
rationals = True
elif coeff.is_Real:
reals = True
break
if reals:
ground = RR
elif rationals:
ground = QQ
else:
ground = ZZ
domain = ground.poly_ring(*gens)
for numer in numers:
for monom, coeff in numer.iteritems():
numer[monom] = ground.from_sympy(coeff)
result.append(domain(numer))
else:
domain = ZZ.frac_field(*gens)
for numer, denom in zip(numers, denoms):
for monom, coeff in numer.iteritems():
numer[monom] = ZZ.from_sympy(coeff)
for monom, coeff in denom.iteritems():
denom[monom] = ZZ.from_sympy(coeff)
result.append(domain((numer, denom)))
return domain, result
def solve(f, *symbols, **flags):
"""Solves equations and systems of equations.
Currently supported are univariate polynomial, transcendental
equations, piecewise combinations thereof and systems of linear
and polynomial equations. Input is formed as a single expression
or an equation, or an iterable container in case of an equation
system. The type of output may vary and depends heavily on the
input. For more details refer to more problem specific functions.
By default all solutions are simplified to make the output more
readable. If this is not the expected behavior (e.g., because of
speed issues) set simplified=False in function arguments.
To solve equations and systems of equations like recurrence relations
or differential equations, use rsolve() or dsolve(), respectively.
>>> from sympy import I, solve
>>> from sympy.abc import x, y
Solve a polynomial equation:
>>> solve(x**4-1, x)
[1, -1, -I, I]
Solve a linear system:
>>> solve((x+5*y-2, -3*x+6*y-15), x, y)
{x: -3, y: 1}
"""
def sympit(w):
return map(sympify, iff(isinstance(w,(list, tuple, set)), w, [w]))
# make f and symbols into lists of sympified quantities
# keeping track of how f was passed since if it is a list
# a dictionary of results will be returned.
bare_f = not isinstance(f, (list, tuple, set))
f, symbols = (sympit(w) for w in [f, symbols])
if any(isinstance(fi, bool) or (fi.is_Relational and not fi.is_Equality) for fi in f):
return reduce_inequalities(f, assume=flags.get('assume'))
for i, fi in enumerate(f):
if fi.is_Equality:
f[i] = fi.lhs - fi.rhs
if not symbols:
#get symbols from equations or supply dummy symbols since
#solve(3,x) returns []...though it seems that it should raise some sort of error TODO
symbols = set([])
for fi in f:
symbols |= fi.atoms(Symbol) or set([Dummy('x')])
symbols = list(symbols)
if bare_f:
f=f[0]
if len(symbols) == 1:
if isinstance(symbols[0], (list, tuple, set)):
symbols = symbols[0]
result = list()
# Begin code handling for Function and Derivative instances
# Basic idea: store all the passed symbols in symbols_passed, check to see
# if any of them are Function or Derivative types, if so, use a dummy
# symbol in their place, and set symbol_swapped = True so that other parts
# of the code can be aware of the swap. Once all swapping is done, the
# continue on with regular solving as usual, and swap back at the end of
# the routine, so that whatever was passed in symbols is what is returned.
symbols_new = []
symbol_swapped = False
symbols_passed = list(symbols)
for i, s in enumerate(symbols):
if s.is_Symbol:
s_new = s
elif s.is_Function:
symbol_swapped = True
s_new = Dummy('F%d' % i)
elif s.is_Derivative:
symbol_swapped = True
s_new = Dummy('D%d' % i)
else:
raise TypeError('not a Symbol or a Function')
symbols_new.append(s_new)
if symbol_swapped:
swap_back_dict = dict(zip(symbols_new, symbols))
# End code for handling of Function and Derivative instances
if not isinstance(f, (tuple, list, set)):
# Create a swap dictionary for storing the passed symbols to be solved
# for, so that they may be swapped back.
if symbol_swapped:
swap_dict = zip(symbols, symbols_new)
f = f.subs(swap_dict)
symbols = symbols_new
# Any embedded piecewise functions need to be brought out to the
# top level so that the appropriate strategy gets selected.
f = piecewise_fold(f)
if len(symbols) != 1:
result = {}
for s in symbols:
result[s] = solve(f, s, **flags)
if flags.get('simplified', True):
for s, r in result.items():
result[s] = map(simplify, r)
return result
symbol = symbols[0]
strategy = guess_solve_strategy(f, symbol)
if strategy == GS_POLY:
poly = f.as_poly( symbol )
if poly is None:
raise NotImplementedError("Cannot solve equation " + str(f) + " for "
+ str(symbol))
# for cubics and quartics, if the flag wasn't set, DON'T do it
# by default since the results are quite long. Perhaps one could
# base this decision on a certain crtical length of the roots.
if poly.degree > 2:
flags['simplified'] = flags.get('simplified', False)
result = roots(poly, cubics=True, quartics=True).keys()
elif strategy == GS_RATIONAL:
P, Q = f.as_numer_denom()
#TODO: check for Q != 0
result = solve(P, symbol, **flags)
elif strategy == GS_POLY_CV_1:
args = list(f.args)
if isinstance(f, Add):
# we must search for a suitable change of variable
# collect exponents
exponents_denom = list()
for arg in args:
if isinstance(arg, Pow):
exponents_denom.append(arg.exp.q)
elif isinstance(arg, Mul):
for mul_arg in arg.args:
if isinstance(mul_arg, Pow):
exponents_denom.append(mul_arg.exp.q)
assert len(exponents_denom) > 0
if len(exponents_denom) == 1:
m = exponents_denom[0]
else:
# get the LCM of the denominators
m = reduce(ilcm, exponents_denom)
# x -> y**m.
# we assume positive for simplification purposes
t = Dummy('t', positive=True)
f_ = f.subs(symbol, t**m)
if guess_solve_strategy(f_, t) != GS_POLY:
raise NotImplementedError("Could not convert to a polynomial equation: %s" % f_)
cv_sols = solve(f_, t)
for sol in cv_sols:
result.append(sol**m)
elif isinstance(f, Mul):
for mul_arg in args:
result.extend(solve(mul_arg, symbol))
elif strategy == GS_POLY_CV_2:
m = 0
args = list(f.args)
if isinstance(f, Add):
for arg in args:
if isinstance(arg, Pow):
m = min(m, arg.exp)
elif isinstance(arg, Mul):
for mul_arg in arg.args:
if isinstance(mul_arg, Pow):
m = min(m, mul_arg.exp)
elif isinstance(f, Mul):
for mul_arg in args:
if isinstance(mul_arg, Pow):
m = min(m, mul_arg.exp)
f1 = simplify(f*symbol**(-m))
result = solve(f1, symbol)
# TODO: we might have introduced unwanted solutions
# when multiplied by x**-m
elif strategy == GS_PIECEWISE:
result = set()
for expr, cond in f.args:
candidates = solve(expr, *symbols)
if isinstance(cond, bool) or cond.is_Number:
if not cond:
continue
# Only include solutions that do not match the condition
# of any of the other pieces.
for candidate in candidates:
matches_other_piece = False
for other_expr, other_cond in f.args:
if isinstance(other_cond, bool) \
or other_cond.is_Number:
continue
if bool(other_cond.subs(symbol, candidate)):
matches_other_piece = True
break
if not matches_other_piece:
result.add(candidate)
else:
for candidate in candidates:
if bool(cond.subs(symbol, candidate)):
result.add(candidate)
result = list(result)
elif strategy == GS_TRANSCENDENTAL:
#a, b = f.as_numer_denom()
# Let's throw away the denominator for now. When we have robust
# assumptions, it should be checked, that for the solution,
# b!=0.
result = tsolve(f, *symbols)
elif strategy == -1:
raise ValueError('Could not parse expression %s' % f)
else:
raise NotImplementedError("No algorithms are implemented to solve equation %s" % f)
# This symbol swap should not be necessary for the single symbol case: if you've
# solved for the symbol the it will not appear in the solution. Right now, however
# ode's are getting solutions for solve (even though they shouldn't be -- see the
# swap_back test in test_solvers).
if symbol_swapped:
result = [ri.subs(swap_back_dict) for ri in result]
if flags.get('simplified', True) and strategy != GS_RATIONAL:
return map(simplify, result)
else:
return result
else:
if not f:
return {}
else:
# Create a swap dictionary for storing the passed symbols to be
# solved for, so that they may be swapped back.
if symbol_swapped:
swap_dict = zip(symbols, symbols_new)
f = [fi.subs(swap_dict) for fi in f]
symbols = symbols_new
polys = []
for g in f:
poly = g.as_poly(*symbols)
if poly is not None:
polys.append(poly)
else:
raise NotImplementedError()
if all(p.is_linear for p in polys):
n, m = len(f), len(symbols)
matrix = zeros((n, m + 1))
for i, poly in enumerate(polys):
for monom, coeff in poly.terms():
try:
j = list(monom).index(1)
matrix[i, j] = coeff
except ValueError:
matrix[i, m] = -coeff
soln = solve_linear_system(matrix, *symbols, **flags)
else:
soln = solve_poly_system(polys)
# Use swap_dict to ensure we return the same type as what was
# passed
if symbol_swapped:
if isinstance(soln, dict):
res = {}
for k in soln.keys():
res.update({swap_back_dict[k]: soln[k]})
return res
else:
return soln
else:
return soln
