def test_ilcm():
assert ilcm(0, 0) == 0
assert ilcm(1, 0) == 0
assert ilcm(0, 1) == 0
assert ilcm(1, 1) == 1
assert ilcm(2, 1) == 2
assert ilcm(8, 2) == 8
assert ilcm(8, 6) == 24
assert ilcm(8, 7) == 56
raises(ValueError, lambda: ilcm(8.1, 7))
raises(ValueError, lambda: ilcm(8, 7.1))
def test_ilcm():
assert ilcm(0, 0) == 0
assert ilcm(1, 0) == 0
assert ilcm(0, 1) == 0
assert ilcm(1, 1) == 1
assert ilcm(2, 1) == 2
assert ilcm(8, 2) == 8
assert ilcm(8, 6) == 24
assert ilcm(8, 7) == 56
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)
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
if isinstance(symbols, (list, tuple)):
symbols_passed = symbols[:]
elif isinstance(symbols, set):
symbols_passed = list(symbols)
i = 0
for s in symbols:
if s.is_Symbol:
s_new = s
elif s.is_Function:
symbol_swapped = True
s_new = Symbol('F%d' % i, dummy=True)
elif s.is_Derivative:
symbol_swapped = True
s_new = Symbol('D%d' % i, dummy=True)
else:
raise TypeError('not a Symbol or a Function')
symbols_new.append(s_new)
i += 1
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)):
f = sympify(f)
# 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
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)
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_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)
if flags.get('simplified', True):
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:
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
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 _intersect(self, other):
from sympy.functions.elementary.integers import ceiling, floor
from sympy.functions.elementary.complexes import sign
if other is S.Naturals:
return self._intersect(Interval(1, S.Infinity))
if other is S.Integers:
return self
if other.is_Interval:
if not all(i.is_number for i in other.args[:2]):
return
# In case of null Range, return an EmptySet.
if self.size == 0:
return S.EmptySet
# trim down to self's size, and represent
# as a Range with step 1.
start = ceiling(max(other.inf, self.inf))
if start not in other:
start += 1
end = floor(min(other.sup, self.sup))
if end not in other:
end -= 1
return self.intersect(Range(start, end + 1))
if isinstance(other, Range):
from sympy.solvers.diophantine import diop_linear
from sympy.core.numbers import ilcm
# non-overlap quick exits
if not other:
return S.EmptySet
if not self:
return S.EmptySet
if other.sup < self.inf:
return S.EmptySet
if other.inf > self.sup:
return S.EmptySet
# work with finite end at the start
r1 = self
if r1.start.is_infinite:
r1 = r1.reversed
r2 = other
if r2.start.is_infinite:
r2 = r2.reversed
# this equation represents the values of the Range;
# it's a linear equation
eq = lambda r, i: r.start + i*r.step
# we want to know when the two equations might
# have integer solutions so we use the diophantine
# solver
a, b = diop_linear(eq(r1, Dummy()) - eq(r2, Dummy()))
# check for no solution
no_solution = a is None and b is None
if no_solution:
return S.EmptySet
# there is a solution
# -------------------
# find the coincident point, c
a0 = a.as_coeff_Add()[0]
c = eq(r1, a0)
# find the first point, if possible, in each range
# since c may not be that point
def _first_finite_point(r1, c):
if c == r1.start:
return c
# st is the signed step we need to take to
# get from c to r1.start
st = sign(r1.start - c)*step
# use Range to calculate the first point:
# we want to get as close as possible to
# r1.start; the Range will not be null since
# it will at least contain c
s1 = Range(c, r1.start + st, st)[-1]
if s1 == r1.start:
pass
else:
# if we didn't hit r1.start then, if the
# sign of st didn't match the sign of r1.step
# we are off by one and s1 is not in r1
if sign(r1.step) != sign(st):
s1 -= st
if s1 not in r1:
return
return s1
# calculate the step size of the new Range
step = abs(ilcm(r1.step, r2.step))
s1 = _first_finite_point(r1, c)
if s1 is None:
return S.EmptySet
s2 = _first_finite_point(r2, c)
if s2 is None:
return S.EmptySet
# replace the corresponding start or stop in
# the original Ranges with these points; the
# result must have at least one point since
# we know that s1 and s2 are in the Ranges
def _updated_range(r, first):
st = sign(r.step)*step
if r.start.is_finite:
rv = Range(first, r.stop, st)
else:
rv = Range(r.start, first + st, st)
return rv
r1 = _updated_range(self, s1)
r2 = _updated_range(other, s2)
# work with them both in the increasing direction
if sign(r1.step) < 0:
r1 = r1.reversed
if sign(r2.step) < 0:
r2 = r2.reversed
# return clipped Range with positive step; it
# can't be empty at this point
start = max(r1.start, r2.start)
stop = min(r1.stop, r2.stop)
return Range(start, stop, step)
else:
return
def poly_lcm(f, g, *symbols):
"""Computes least common multiple of two polynomials.
Given two univariate polynomials, the LCM is computed via
f*g = gcd(f, g)*lcm(f, g) formula. In multivariate case, we
compute the unique generator of the intersection of the two
ideals, generated by f and g. This is done by computing a
Groebner basis, with respect to any lexicographic ordering,
of t*f and (1 - t)*g, where t is an unrelated symbol and
filtering out solution that does not contain t.
For more information on the implemented algorithm refer to:
[1] D. Cox, J. Little, D. O'Shea, Ideals, Varieties and
Algorithms, Springer, Second Edition, 1997, pp. 187
"""
if not isinstance(f, Poly):
f = Poly(f, *symbols)
elif symbols:
raise SymbolsError("Redundant symbols were given")
f, g = f.unify_with(g)
symbols, flags = f.symbols, f.flags
if f.is_monomial and g.is_monomial:
monom = monomial_lcm(f.LM, g.LM)
fc, gc = f.LC, g.LC
if fc.is_Rational and gc.is_Rational:
coeff = Integer(ilcm(fc.p, gc.p))
else:
coeff = S.One
return Poly((coeff, monom), *symbols, **flags)
fc, f = f.as_primitive()
gc, g = g.as_primitive()
lcm = ilcm(int(fc), int(gc))
if f.is_multivariate:
t = Symbol('t', dummy=True)
lex = { 'order' : 'lex' }
f_monoms = [ (1,) + monom for monom in f.monoms ]
F = Poly((f.coeffs, f_monoms), t, *symbols, **lex)
g_monoms = [ (0,) + monom for monom in g.monoms ] + \
[ (1,) + monom for monom in g.monoms ]
g_coeffs = list(g.coeffs) + [ -coeff for coeff in g.coeffs ]
G = Poly(dict(zip(g_monoms, g_coeffs)), t, *symbols, **lex)
def independent(h):
return all(not monom[0] for monom in h.monoms)
H = [ h for h in poly_groebner((F, G)) if independent(h) ]
if lcm != 1:
h_coeffs = [ coeff*lcm for coeff in H[0].coeffs ]
else:
h_coeffs = H[0].coeffs
h_monoms = [ monom[1:] for monom in H[0].monoms ]
return Poly(dict(zip(h_monoms, h_coeffs)), *symbols, **flags)
else:
h = poly_div(f * g, poly_gcd(f, g))[0]
if lcm != 1:
return h.mul_term(lcm / h.LC)
else:
return h.as_monic()
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 test_ilcm():
assert ilcm(0, 0) == 0
assert ilcm(1, 0) == 0
assert ilcm(0, 1) == 0
assert ilcm(1, 1) == 1
assert ilcm(2, 1) == 2
assert ilcm(8, 2) == 8
assert ilcm(8, 6) == 24
assert ilcm(8, 7) == 56
assert ilcm(*[10, 20, 30]) == 60
raises(ValueError, lambda: ilcm(8.1, 7))
raises(ValueError, lambda: ilcm(8, 7.1))
def intersection_sets(a, b):
from sympy.solvers.diophantine import diop_linear
from sympy.core.numbers import ilcm
from sympy import sign
# non-overlap quick exits
if not b:
return S.EmptySet
if not a:
return S.EmptySet
if b.sup < a.inf:
return S.EmptySet
if b.inf > a.sup:
return S.EmptySet
# work with finite end at the start
r1 = a
if r1.start.is_infinite:
r1 = r1.reversed
r2 = b
if r2.start.is_infinite:
r2 = r2.reversed
# this equation represents the values of the Range;
# it's a linear equation
eq = lambda r, i: r.start + i*r.step
# we want to know when the two equations might
# have integer solutions so we use the diophantine
# solver
va, vb = diop_linear(eq(r1, Dummy()) - eq(r2, Dummy()))
# check for no solution
no_solution = va is None and vb is None
if no_solution:
return S.EmptySet
# there is a solution
# -------------------
# find the coincident point, c
a0 = va.as_coeff_Add()[0]
c = eq(r1, a0)
# find the first point, if possible, in each range
# since c may not be that point
def _first_finite_point(r1, c):
if c == r1.start:
return c
# st is the signed step we need to take to
# get from c to r1.start
st = sign(r1.start - c)*step
# use Range to calculate the first point:
# we want to get as close as possible to
# r1.start; the Range will not be null since
# it will at least contain c
s1 = Range(c, r1.start + st, st)[-1]
if s1 == r1.start:
pass
else:
# if we didn't hit r1.start then, if the
# sign of st didn't match the sign of r1.step
# we are off by one and s1 is not in r1
if sign(r1.step) != sign(st):
s1 -= st
if s1 not in r1:
return
return s1
# calculate the step size of the new Range
step = abs(ilcm(r1.step, r2.step))
s1 = _first_finite_point(r1, c)
if s1 is None:
return S.EmptySet
s2 = _first_finite_point(r2, c)
if s2 is None:
return S.EmptySet
# replace the corresponding start or stop in
# the original Ranges with these points; the
# result must have at least one point since
# we know that s1 and s2 are in the Ranges
def _updated_range(r, first):
st = sign(r.step)*step
if r.start.is_finite:
rv = Range(first, r.stop, st)
else:
rv = Range(r.start, first + st, st)
return rv
r1 = _updated_range(a, s1)
r2 = _updated_range(b, s2)
# work with them both in the increasing direction
if sign(r1.step) < 0:
r1 = r1.reversed
if sign(r2.step) < 0:
r2 = r2.reversed
# return clipped Range with positive step; it
# can't be empty at this point
start = max(r1.start, r2.start)
stop = min(r1.stop, r2.stop)
return Range(start, stop, step)
def poly_lcm(f, g, *symbols):
"""Computes least common multiple of two polynomials.
Given two univariate polynomials, the LCM is computed via
f*g = gcd(f, g)*lcm(f, g) formula. In multivariate case, we
compute the unique generator of the intersection of the two
ideals, generated by f and g. This is done by computing a
Groebner basis, with respect to any lexicographic ordering,
of t*f and (1 - t)*g, where t is an unrelated symbol and
filtering out solution that does not contain t.
For more information on the implemented algorithm refer to:
[1] D. Cox, J. Little, D. O'Shea, Ideals, Varieties and
Algorithms, Springer, Second Edition, 1997, pp. 187
"""
if not isinstance(f, Poly):
f = Poly(f, *symbols)
elif symbols:
raise SymbolsError("Redundant symbols were given")
f, g = f.unify_with(g)
symbols, flags = f.symbols, f.flags
if f.is_monomial and g.is_monomial:
monom = monomial_lcm(f.LM, g.LM)
fc, gc = f.LC, g.LC
if fc.is_Rational and gc.is_Rational:
coeff = Integer(ilcm(fc.p, gc.p))
else:
coeff = S.One
return Poly((coeff, monom), *symbols, **flags)
fc, f = f.as_primitive()
gc, g = g.as_primitive()
lcm = ilcm(int(fc), int(gc))
if f.is_multivariate:
t = Symbol('t', dummy=True)
lex = {'order': 'lex'}
f_monoms = [(1, ) + monom for monom in f.monoms]
F = Poly((f.coeffs, f_monoms), t, *symbols, **lex)
g_monoms = [ (0,) + monom for monom in g.monoms ] + \
[ (1,) + monom for monom in g.monoms ]
g_coeffs = list(g.coeffs) + [-coeff for coeff in g.coeffs]
G = Poly(dict(zip(g_monoms, g_coeffs)), t, *symbols, **lex)
def independent(h):
return all(not monom[0] for monom in h.monoms)
H = [h for h in poly_groebner((F, G)) if independent(h)]
if lcm != 1:
h_coeffs = [coeff * lcm for coeff in H[0].coeffs]
else:
h_coeffs = H[0].coeffs
h_monoms = [monom[1:] for monom in H[0].monoms]
return Poly(dict(zip(h_monoms, h_coeffs)), *symbols, **flags)
else:
h = poly_div(f * g, poly_gcd(f, g))[0]
if lcm != 1:
return h.mul_term(lcm / h.LC)
else:
return h.as_monic()
def _(a, b):
# Check that there are no symbolic range arguments
if not all(all(v.is_number for v in r.args) for r in [a, b]):
return None
# non-overlap quick exits
if not b:
return S.EmptySet
if not a:
return S.EmptySet
if b.sup < a.inf:
return S.EmptySet
if b.inf > a.sup:
return S.EmptySet
# work with finite end at the start
r1 = a
if r1.start.is_infinite:
r1 = r1.reversed
r2 = b
if r2.start.is_infinite:
r2 = r2.reversed
# If both ends are infinite then it means that one Range is just the set
# of all integers (the step must be 1).
if r1.start.is_infinite:
return b
if r2.start.is_infinite:
return a
from sympy.solvers.diophantine.diophantine import diop_linear
# this equation represents the values of the Range;
# it's a linear equation
eq = lambda r, i: r.start + i * r.step
# we want to know when the two equations might
# have integer solutions so we use the diophantine
# solver
va, vb = diop_linear(eq(r1, Dummy('a')) - eq(r2, Dummy('b')))
# check for no solution
no_solution = va is None and vb is None
if no_solution:
return S.EmptySet
# there is a solution
# -------------------
# find the coincident point, c
a0 = va.as_coeff_Add()[0]
c = eq(r1, a0)
# find the first point, if possible, in each range
# since c may not be that point
def _first_finite_point(r1, c):
if c == r1.start:
return c
# st is the signed step we need to take to
# get from c to r1.start
st = sign(r1.start - c) * step
# use Range to calculate the first point:
# we want to get as close as possible to
# r1.start; the Range will not be null since
# it will at least contain c
s1 = Range(c, r1.start + st, st)[-1]
if s1 == r1.start:
pass
else:
# if we didn't hit r1.start then, if the
# sign of st didn't match the sign of r1.step
# we are off by one and s1 is not in r1
if sign(r1.step) != sign(st):
s1 -= st
if s1 not in r1:
return
return s1
# calculate the step size of the new Range
step = abs(ilcm(r1.step, r2.step))
s1 = _first_finite_point(r1, c)
if s1 is None:
return S.EmptySet
s2 = _first_finite_point(r2, c)
if s2 is None:
return S.EmptySet
# replace the corresponding start or stop in
# the original Ranges with these points; the
# result must have at least one point since
# we know that s1 and s2 are in the Ranges
def _updated_range(r, first):
st = sign(r.step) * step
if r.start.is_finite:
rv = Range(first, r.stop, st)
else:
rv = Range(r.start, first + st, st)
return rv
r1 = _updated_range(a, s1)
r2 = _updated_range(b, s2)
# work with them both in the increasing direction
if sign(r1.step) < 0:
r1 = r1.reversed
if sign(r2.step) < 0:
r2 = r2.reversed
# return clipped Range with positive step; it
# can't be empty at this point
start = max(r1.start, r2.start)
stop = min(r1.stop, r2.stop)
return Range(start, stop, step)
def _intersect(self, other):
from sympy.functions.elementary.integers import ceiling, floor
from sympy.functions.elementary.complexes import sign
if other is S.Naturals:
return self._intersect(Interval(1, S.Infinity))
if other is S.Integers:
return self
if other.is_Interval:
if not all(i.is_number for i in other.args[:2]):
return
# trim down to self's size, and represent
# as a Range with step 1
start = ceiling(max(other.inf, self.inf))
if start not in other:
start += 1
end = floor(min(other.sup, self.sup))
if end not in other:
end -= 1
return self.intersect(Range(start, end + 1))
if isinstance(other, Range):
from sympy.solvers.diophantine import diop_linear
from sympy.core.numbers import ilcm
# non-overlap quick exits
if not other:
return S.EmptySet
if not self:
return S.EmptySet
if other.sup < self.inf:
return S.EmptySet
if other.inf > self.sup:
return S.EmptySet
# work with finite end at the start
r1 = self
if r1.start.is_infinite:
r1 = r1.reversed
r2 = other
if r2.start.is_infinite:
r2 = r2.reversed
# this equation represents the values of the Range;
# it's a linear equation
eq = lambda r, i: r.start + i*r.step
# we want to know when the two equations might
# have integer solutions so we use the diophantine
# solver
a, b = diop_linear(eq(r1, Dummy()) - eq(r2, Dummy()))
# check for no solution
no_solution = a is None and b is None
if no_solution:
return S.EmptySet
# there is a solution
# -------------------
# find the coincident point, c
a0 = a.as_coeff_Add()[0]
c = eq(r1, a0)
# find the first point, if possible, in each range
# since c may not be that point
def _first_finite_point(r1, c):
if c == r1.start:
return c
# st is the signed step we need to take to
# get from c to r1.start
st = sign(r1.start - c)*step
# use Range to calculate the first point:
# we want to get as close as possible to
# r1.start; the Range will not be null since
# it will at least contain c
s1 = Range(c, r1.start + st, st)[-1]
if s1 == r1.start:
pass
else:
# if we didn't hit r1.start then, if the
# sign of st didn't match the sign of r1.step
# we are off by one and s1 is not in r1
if sign(r1.step) != sign(st):
s1 -= st
if s1 not in r1:
return
return s1
# calculate the step size of the new Range
step = abs(ilcm(r1.step, r2.step))
s1 = _first_finite_point(r1, c)
if s1 is None:
return S.EmptySet
s2 = _first_finite_point(r2, c)
if s2 is None:
return S.EmptySet
# replace the corresponding start or stop in
# the original Ranges with these points; the
# result must have at least one point since
# we know that s1 and s2 are in the Ranges
def _updated_range(r, first):
st = sign(r.step)*step
if r.start.is_finite:
rv = Range(first, r.stop, st)
else:
rv = Range(r.start, first + st, st)
return rv
r1 = _updated_range(self, s1)
r2 = _updated_range(other, s2)
# work with them both in the increasing direction
if sign(r1.step) < 0:
r1 = r1.reversed
if sign(r2.step) < 0:
r2 = r2.reversed
# return clipped Range with positive step; it
# can't be empty at this point
start = max(r1.start, r2.start)
stop = min(r1.stop, r2.stop)
return Range(start, stop, step)
else:
return
