def as_subindex(self, index):
# The docstring of this method is currently on NDindex.as_subindex, as
# this is the only method that is actually implemented so far.
from .ndindex import ndindex
index = ndindex(index)
if not isinstance(index, Slice):
raise NotImplementedError(
"Slice.as_subindex is only implemented for slices")
s = self.reduce()
index = index.reduce()
if s.step < 0 or index.step < 0:
raise NotImplementedError(
"Slice.as_subindex is only implemented for slices with positive steps"
)
# After reducing, start is not None when step > 0
if index.stop is None or s.stop is None or s.start < 0 or index.start < 0 or s.stop < 0 or index.stop < 0:
raise NotImplementedError(
"Slice.as_subindex is only implemented for slices with nonnegative start and stop. Try calling reduce() with a shape first."
)
# Chinese Remainder Theorem. We are looking for a solution to
#
# x = s.start (mod s.step)
# x = index.start (mod index.step)
#
# If crt() returns None, then there are no solutions (the slices do
# not overlap).
res = crt([s.step, index.step], [s.start, index.start])
if res is None:
return Slice(0, 0, 1)
common, _ = res
lcm = ilcm(s.step, index.step)
start = max(s.start, index.start)
def _smallest(x, a, m):
"""
Gives the smallest integer >= x that equals a (mod m)
Assumes x >= 0, m >= 1, and 0 <= a < m.
"""
n = int(math.ceil((x - a) / m))
return a + n * m
# Get the smallest lcm multiple of common that is >= start
start = _smallest(start, common, lcm)
# Finally, we need to shift start so that it is relative to index
start = (start - index.start) // index.step
step = lcm // index.step # = s.step//igcd(s.step, index.step)
stop = math.ceil((min(s.stop, index.stop) - index.start) / index.step)
if stop < 0:
stop = 0
return Slice(start, stop, step)
def find_period_xyz(pos, vel, steps):
period = [0, 0, 0]
# isolate each axis...
for c in [0, 1, 2]:
cpos = []
for body in pos:
cpos_b = [0, 0, 0]
cpos_b[c] = body[c]
cpos.append(cpos_b)
cvel = []
for bodyv in vel:
cvel_b = [0, 0, 0]
cvel_b[c] = bodyv[c]
cvel.append(cvel_b)
# ...and find its period
period[c] = find_period(cpos, cvel, steps)
return ilcm(*period) # sympy least common multiple
def get_period(self):
"""
Return a number P such that G(x*exp(I*P)) == G(x).
>>> from sympy.functions.special.hyper import meijerg
>>> from sympy.abc import z
>>> from sympy import pi, S
>>> meijerg([1], [], [], [], z).get_period()
2*pi
>>> meijerg([pi], [], [], [], z).get_period()
oo
>>> meijerg([1, 2], [], [], [], z).get_period()
oo
>>> meijerg([1,1], [2], [1, S(1)/2, S(1)/3], [1], z).get_period()
12*pi
"""
# This follows from slater's theorem.
from sympy import oo, ilcm, pi, Min
from sympy.simplify.hyperexpand import Mod1
def compute(l):
# first check that no two differ by an integer
for i, b in enumerate(l):
if not b.is_Rational:
return oo
for j in range(i + 1, len(l)):
if Mod1(b) == Mod1(l[j]):
return oo
return reduce(ilcm, (x.q for x in l), 1)
beta = compute(self.bm)
alpha = compute(self.an)
p, q = len(self.ap), len(self.bq)
if p == q:
if beta == oo or alpha == oo:
return oo
return 2 * pi * ilcm(alpha, beta)
elif p < q:
return 2 * pi * beta
else:
return 2 * pi * alpha
def get_period(self):
"""
Return a number P such that G(x*exp(I*P)) == G(x).
>>> from sympy.functions.special.hyper import meijerg
>>> from sympy.abc import z
>>> from sympy import pi, S
>>> meijerg([1], [], [], [], z).get_period()
2*pi
>>> meijerg([pi], [], [], [], z).get_period()
oo
>>> meijerg([1, 2], [], [], [], z).get_period()
oo
>>> meijerg([1,1], [2], [1, S(1)/2, S(1)/3], [1], z).get_period()
12*pi
"""
# This follows from slater's theorem.
from sympy import oo, ilcm, pi, Min, Mod
def compute(l):
# first check that no two differ by an integer
for i, b in enumerate(l):
if not b.is_Rational:
return oo
for j in range(i + 1, len(l)):
if not Mod((b - l[j]).simplify(), 1):
return oo
return reduce(ilcm, (x.q for x in l), 1)
beta = compute(self.bm)
alpha = compute(self.an)
p, q = len(self.ap), len(self.bq)
if p == q:
if beta == oo or alpha == oo:
return oo
return 2 * pi * ilcm(alpha, beta)
elif p < q:
return 2 * pi * beta
else:
return 2 * pi * alpha
def answer(num1, num2, lcm): return lcm - ilcm(num1, num2)
def distractor1(num1, num2, gcd): return gcd - ilcm(num1, num2)
def diop_quadratic(var, coeff, t):
"""
Solves quadratic diophantine equations, i.e equations of the form
Ax**2 + Bxy + Cy**2 + Dx + Ey + F = 0. Returns a set containing
the tuples (x, y) which contains the solutions.
Usage
=====
diop_quadratic(var, coeff) -> var is a list of variables and
coeff is a dictionary containing coefficients of the symbols.
Details
=======
``var`` a list which contains two variables x and y.
``coeff`` a dict which generally contains six key value pairs.
The set of keys is {x**2, y**2, x*y, x, y, Integer(1)}.
``t`` the parameter to be used in the solution.
Examples
========
>>> from sympy.abc import x, y, t
>>> from sympy import Integer
>>> from sympy.solvers.diophantine import diop_quadratic
>>> diop_quadratic([x, y], {x**2: 1, y**2: 1, x*y: 0, x: 2, y: 2, Integer(1): 2}, t)
set([(-1, -1)])
References
==========
.. [1] http://www.alpertron.com.ar/METHODS.HTM
"""
x = var[0]
y = var[1]
for term in [x**2, y**2, x * y, x, y, Integer(1)]:
if term not in coeff.keys():
coeff[term] = Integer(0)
A = coeff[x**2]
B = coeff[x * y]
C = coeff[y**2]
D = coeff[x]
E = coeff[y]
F = coeff[Integer(1)]
d = igcd(A, igcd(B, igcd(C, igcd(D, igcd(E, F)))))
A = A // d
B = B // d
C = C // d
D = D // d
E = E // d
F = F // d
# (1) Linear case: A = B = C = 0 -> considered under linear diophantine equations
# (2) Simple-Hyperbolic case:A = C = 0, B != 0
# In this case equation can be converted to (Bx + E)(By + D) = DE - BF
# We consider two cases; DE - BF = 0 and DE - BF != 0
# More details, http://www.alpertron.com.ar/METHODS.HTM#SHyperb
l = set([])
if A == 0 and C == 0 and B != 0:
if D * E - B * F == 0:
if divisible(int(E), int(B)):
l.add((-E / B, t))
if divisible(int(D), int(B)):
l.add((t, -D / B))
else:
div = divisors(D * E - B * F)
div = div + [-term for term in div]
for d in div:
if divisible(int(d - E), int(B)):
x0 = (d - E) // B
if divisible(int(D * E - B * F), int(d)):
if divisible(int((D * E - B * F) // d - D), int(B)):
y0 = ((D * E - B * F) // d - D) // B
l.add((x0, y0))
# (3) Elliptical case: B**2 - 4AC < 0
# More Details, http://www.alpertron.com.ar/METHODS.HTM#Ellipse
# In this case x should lie between the roots of
# (B**2 - 4AC)x**2 + 2(BE - 2CD)x + (E**2 - 4CF) = 0
elif B**2 - 4 * A * C < 0:
z = symbols("z", real=True)
roots = solve((B**2 - 4 * A * C) * z**2 + 2 * (B * E - 2 * C * D) * z +
E**2 - 4 * C * F)
solve_y = lambda x, e: (-(B * x + E) + e * sqrt(
(B * x + E)**2 - 4 * C * (A * x**2 + D * x + F))) / (2 * C)
if len(roots) == 1 and isinstance(roots[0], Integer):
x_vals = [roots[0]]
elif len(roots) == 2:
x_vals = [
i for i in range(ceiling(min(roots)), ceiling(max(roots)))
] # ceiling = floor +/- 1
else:
x_vals = []
for x0 in x_vals:
if isinstance(
sqrt((B * x0 + E)**2 - 4 * C * (A * x0**2 + D * x0 + F)),
Integer):
if isinstance(solve_y(x0, 1), Integer):
l.add((Integer(x0), solve_y(x0, 1)))
if isinstance(solve_y(x0, -1), Integer):
l.add((Integer(x0), solve_y(x0, -1)))
# (4) Parabolic case: B**2 - 4*A*C = 0
# There are two subcases to be considered in this case.
# sqrt(c)D - sqrt(a)E = 0 and sqrt(c)D - sqrt(a)E != 0
# More Details, http://www.alpertron.com.ar/METHODS.HTM#Parabol
elif B**2 - 4 * A * C == 0:
g = igcd(A, C)
g = abs(g) * sign(A)
a = A // g
b = B // g
c = C // g
e = sign(B / A)
if e * sqrt(c) * D - sqrt(a) * E == 0:
z = symbols("z", real=True)
roots = solve(sqrt(a) * g * z**2 + D * z + sqrt(a) * F)
for root in roots:
if isinstance(root, Integer):
l.add(
(diop_solve(sqrt(a) * x + e * sqrt(c) * y - root)[x],
diop_solve(sqrt(a) * x + e * sqrt(c) * y - root)[y]))
elif isinstance(e * sqrt(c) * D - sqrt(a) * E, Integer):
solve_x = lambda u: e*sqrt(c)*g*(sqrt(a)*E - e*sqrt(c)*D)*t**2 - (E + 2*e*sqrt(c)*g*u)*t\
- (e*sqrt(c)*g*u**2 + E*u + e*sqrt(c)*F) // (e*sqrt(c)*D - sqrt(a)*E)
solve_y = lambda u: sqrt(a)*g*(e*sqrt(c)*D - sqrt(a)*E)*t**2 + (D + 2*sqrt(a)*g*u)*t \
+ (sqrt(a)*g*u**2 + D*u + sqrt(a)*F) // (e*sqrt(c)*D - sqrt(a)*E)
for z0 in range(0, abs(e * sqrt(c) * D - sqrt(a) * E)):
if divisible(
sqrt(a) * g * z0**2 + D * z0 + sqrt(a) * F,
e * sqrt(c) * D - sqrt(a) * E):
l.add((solve_x(z0), solve_y(z0)))
# (5) B**2 - 4*A*C > 0
elif B**2 - 4 * A * C > 0:
# Method used when B**2 - 4*A*C is a square, is descibed in p. 6 of the below paper
# by John P. Robertson.
# http://www.jpr2718.org/ax2p.pdf
if isinstance(sqrt(B**2 - 4 * A * C), Integer):
if A != 0:
r = sqrt(B**2 - 4 * A * C)
u, v = symbols("u, v", integer=True)
eq = simplify(4 * A * r * u * v + 4 * A * D *
(B * v + r * u + r * v - B * u) +
2 * A * 4 * A * E * (u - v) +
4 * A * r * 4 * A * F)
sol = diop_solve(eq, t)
sol = list(sol)
for solution in sol:
s0 = solution[0]
t0 = solution[1]
x_0 = S(B * t0 + r * s0 + r * t0 - B * s0) / (4 * A * r)
y_0 = S(s0 - t0) / (2 * r)
if isinstance(s0, Symbol) or isinstance(t0, Symbol):
if check_param(x_0, y_0, 4 * A * r, t) != (None, None):
l.add((check_param(x_0, y_0, 4 * A * r, t)[0],
check_param(x_0, y_0, 4 * A * r, t)[1]))
elif divisible(B * t0 + r * s0 + r * t0 - B * s0,
4 * A * r):
if divisible(s0 - t0, 2 * r):
if is_solution_quad(var, coeff, x_0, y_0):
l.add((x_0, y_0))
else:
var[0], var[1] = var[1], var[0] # Interchange x and y
s = diop_quadratic(var, coeff, t)
while len(s) > 0:
sol = s.pop()
l.add((sol[1], sol[0]))
else:
# In this case equation can be transformed into a Pell equation
A, B = _transformation_to_pell(var, coeff)
D, N = _find_DN(var, coeff)
solns_pell = diop_pell(D, N)
n = symbols("n", integer=True)
a = diop_pell(D, 1)
T = a[0][0]
U = a[0][1]
if (isinstance(A[0], Integer) and isinstance(A[1], Integer)
and isinstance(A[2], Integer)
and isinstance(A[3], Integer)
and isinstance(B[0], Integer)
and isinstance(B[1], Integer)):
for sol in solns_pell:
r = sol[0]
s = sol[1]
x_n = S((r + s * sqrt(D)) * (T + U * sqrt(D))**n +
(r - s * sqrt(D)) * (T - U * sqrt(D))**n) / 2
y_n = S((r + s * sqrt(D)) * (T + U * sqrt(D))**n -
(r - s * sqrt(D)) *
(T - U * sqrt(D))**n) / (2 * sqrt(D))
x_n, y_n = (A * Matrix([x_n, y_n]) +
B)[0], (A * Matrix([x_n, y_n]) + B)[1]
l.add((x_n, y_n))
else:
L = ilcm(
S(A[0]).q,
ilcm(
S(A[1]).q,
ilcm(
S(A[2]).q,
ilcm(S(A[3]).q, ilcm(S(B[0]).q,
S(B[1]).q)))))
k = 0
done = False
T_k = T
U_k = U
while not done:
k = k + 1
if (T_k - 1) % L == 0 and U_k % L == 0:
done = True
T_k, U_k = T_k * T + D * U_k * U, T_k * U + U_k * T
for soln in solns_pell:
x_0 = soln[0]
y_0 = soln[1]
x_i = x_0
y_i = y_0
for i in range(k):
X = (A * Matrix([x_i, y_i]) + B)[0]
Y = (A * Matrix([x_i, y_i]) + B)[1]
if isinstance(X, Integer) and isinstance(Y, Integer):
if is_solution_quad(var, coeff, X, Y):
x_n = S((x_i + sqrt(D) * y_i) *
(T + sqrt(D) * U)**(n * L) +
(x_i - sqrt(D) * y_i) *
(T - sqrt(D) * U)**(n * L)) / 2
y_n = S((x_i + sqrt(D) * y_i) *
(T + sqrt(D) * U)**(n * L) -
(x_i - sqrt(D) * y_i) *
(T - sqrt(D) * U)**(n * L)) / (2 *
sqrt(D))
x_n, y_n = (A * Matrix([x_n, y_n]) +
B)[0], (A * Matrix([x_n, y_n]) +
B)[1]
l.add((x_n, y_n))
x_i = x_i * T + D * U * y_i
y_i = x_i * U + y_i * T
return l
def diop_quadratic(var, coeff, t):
"""
Solves quadratic diophantine equations, i.e equations of the form
Ax**2 + Bxy + Cy**2 + Dx + Ey + F = 0. Returns a set containing
the tuples (x, y) which contains the solutions.
Usage
=====
diop_quadratic(var, coeff) -> var is a list of variables and
coeff is a dictionary containing coefficients of the symbols.
Details
=======
``var`` a list which contains two variables x and y.
``coeff`` a dict which generally contains six key value pairs.
The set of keys is {x**2, y**2, x*y, x, y, Integer(1)}.
``t`` the parameter to be used in the solution.
Examples
========
>>> from sympy.abc import x, y, t
>>> from sympy import Integer
>>> from sympy.solvers.diophantine import diop_quadratic
>>> diop_quadratic([x, y], {x**2: 1, y**2: 1, x*y: 0, x: 2, y: 2, Integer(1): 2}, t)
set([(-1, -1)])
References
==========
.. [1] http://www.alpertron.com.ar/METHODS.HTM
"""
x = var[0]
y = var[1]
for term in [x**2, y**2, x*y, x, y, Integer(1)]:
if term not in coeff.keys():
coeff[term] = Integer(0)
A = coeff[x**2]
B = coeff[x*y]
C = coeff[y**2]
D = coeff[x]
E = coeff[y]
F = coeff[Integer(1)]
d = igcd(A, igcd(B, igcd(C, igcd(D, igcd(E, F)))))
A = A // d
B = B // d
C = C // d
D = D // d
E = E // d
F = F // d
# (1) Linear case: A = B = C = 0 -> considered under linear diophantine equations
# (2) Simple-Hyperbolic case:A = C = 0, B != 0
# In this case equation can be converted to (Bx + E)(By + D) = DE - BF
# We consider two cases; DE - BF = 0 and DE - BF != 0
# More details, http://www.alpertron.com.ar/METHODS.HTM#SHyperb
l = set([])
if A == 0 and C == 0 and B != 0:
if D*E - B*F == 0:
if divisible(int(E), int(B)):
l.add((-E/B, t))
if divisible(int(D), int(B)):
l.add((t, -D/B))
else:
div = divisors(D*E - B*F)
div = div + [-term for term in div]
for d in div:
if divisible(int(d - E), int(B)):
x0 = (d - E) // B
if divisible(int(D*E - B*F), int(d)):
if divisible(int((D*E - B*F)// d - D), int(B)):
y0 = ((D*E - B*F) // d - D) // B
l.add((x0, y0))
# (3) Elliptical case: B**2 - 4AC < 0
# More Details, http://www.alpertron.com.ar/METHODS.HTM#Ellipse
# In this case x should lie between the roots of
# (B**2 - 4AC)x**2 + 2(BE - 2CD)x + (E**2 - 4CF) = 0
elif B**2 - 4*A*C < 0:
z = symbols("z", real=True)
roots = solve((B**2 - 4*A*C)*z**2 + 2*(B*E - 2*C*D)*z + E**2 - 4*C*F)
solve_y = lambda x, e: (-(B*x + E) + e*sqrt((B*x + E)**2 - 4*C*(A*x**2 + D*x + F)))/(2*C)
if len(roots) == 1 and isinstance(roots[0], Integer):
x_vals = [roots[0]]
elif len(roots) == 2:
x_vals = [i for i in range(ceiling(min(roots)), ceiling(max(roots)))] # ceiling = floor +/- 1
else:
x_vals = []
for x0 in x_vals:
if isinstance(sqrt((B*x0 + E)**2 - 4*C*(A*x0**2 + D*x0 + F)), Integer):
if isinstance(solve_y(x0, 1), Integer):
l.add((Integer(x0), solve_y(x0, 1)))
if isinstance(solve_y(x0, -1), Integer):
l.add((Integer(x0), solve_y(x0, -1)))
# (4) Parabolic case: B**2 - 4*A*C = 0
# There are two subcases to be considered in this case.
# sqrt(c)D - sqrt(a)E = 0 and sqrt(c)D - sqrt(a)E != 0
# More Details, http://www.alpertron.com.ar/METHODS.HTM#Parabol
elif B**2 - 4*A*C == 0:
g = igcd(A, C)
g = abs(g) * sign(A)
a = A // g
b = B // g
c = C // g
e = sign(B/A)
if e*sqrt(c)*D - sqrt(a)*E == 0:
z = symbols("z", real=True)
roots = solve(sqrt(a)*g*z**2 + D*z + sqrt(a)*F)
for root in roots:
if isinstance(root, Integer):
l.add((diop_solve(sqrt(a)*x + e*sqrt(c)*y - root)[x], diop_solve(sqrt(a)*x + e*sqrt(c)*y - root)[y]))
elif isinstance(e*sqrt(c)*D - sqrt(a)*E, Integer):
solve_x = lambda u: e*sqrt(c)*g*(sqrt(a)*E - e*sqrt(c)*D)*t**2 - (E + 2*e*sqrt(c)*g*u)*t\
- (e*sqrt(c)*g*u**2 + E*u + e*sqrt(c)*F) // (e*sqrt(c)*D - sqrt(a)*E)
solve_y = lambda u: sqrt(a)*g*(e*sqrt(c)*D - sqrt(a)*E)*t**2 + (D + 2*sqrt(a)*g*u)*t \
+ (sqrt(a)*g*u**2 + D*u + sqrt(a)*F) // (e*sqrt(c)*D - sqrt(a)*E)
for z0 in range(0, abs(e*sqrt(c)*D - sqrt(a)*E)):
if divisible(sqrt(a)*g*z0**2 + D*z0 + sqrt(a)*F, e*sqrt(c)*D - sqrt(a)*E):
l.add((solve_x(z0), solve_y(z0)))
# (5) B**2 - 4*A*C > 0
elif B**2 - 4*A*C > 0:
# Method used when B**2 - 4*A*C is a square, is descibed in p. 6 of the below paper
# by John P. Robertson.
# http://www.jpr2718.org/ax2p.pdf
if isinstance(sqrt(B**2 - 4*A*C), Integer):
if A != 0:
r = sqrt(B**2 - 4*A*C)
u, v = symbols("u, v", integer=True)
eq = simplify(4*A*r*u*v + 4*A*D*(B*v + r*u + r*v - B*u) + 2*A*4*A*E*(u - v) + 4*A*r*4*A*F)
sol = diop_solve(eq, t)
sol = list(sol)
for solution in sol:
s0 = solution[0]
t0 = solution[1]
x_0 = S(B*t0 + r*s0 + r*t0 - B*s0)/(4*A*r)
y_0 = S(s0 - t0)/(2*r)
if isinstance(s0, Symbol) or isinstance(t0, Symbol):
if check_param(x_0, y_0, 4*A*r, t) != (None, None):
l.add((check_param(x_0, y_0, 4*A*r, t)[0], check_param(x_0, y_0, 4*A*r, t)[1]))
elif divisible(B*t0 + r*s0 + r*t0 - B*s0, 4*A*r):
if divisible(s0 - t0, 2*r):
if is_solution_quad(var, coeff, x_0, y_0):
l.add((x_0, y_0))
else:
var[0], var[1] = var[1], var[0] # Interchange x and y
s = diop_quadratic(var, coeff, t)
while len(s) > 0:
sol = s.pop()
l.add((sol[1], sol[0]))
else:
# In this case equation can be transformed into a Pell equation
A, B = _transformation_to_pell(var, coeff)
D, N = _find_DN(var, coeff)
solns_pell = diop_pell(D, N)
n = symbols("n", integer=True)
a = diop_pell(D, 1)
T = a[0][0]
U = a[0][1]
if (isinstance(A[0], Integer) and isinstance(A[1], Integer) and isinstance(A[2], Integer)
and isinstance(A[3], Integer) and isinstance(B[0], Integer) and isinstance(B[1], Integer)):
for sol in solns_pell:
r = sol[0]
s = sol[1]
x_n = S((r + s*sqrt(D))*(T + U*sqrt(D))**n + (r - s*sqrt(D))*(T - U*sqrt(D))**n)/2
y_n = S((r + s*sqrt(D))*(T + U*sqrt(D))**n - (r - s*sqrt(D))*(T - U*sqrt(D))**n)/(2*sqrt(D))
x_n, y_n = (A*Matrix([x_n, y_n]) + B)[0], (A*Matrix([x_n, y_n]) + B)[1]
l.add((x_n, y_n))
else:
L = ilcm(S(A[0]).q, ilcm(S(A[1]).q, ilcm(S(A[2]).q, ilcm(S(A[3]).q, ilcm(S(B[0]).q, S(B[1]).q)))))
k = 0
done = False
T_k = T
U_k = U
while not done:
k = k + 1
if (T_k - 1) % L == 0 and U_k % L == 0:
done = True
T_k, U_k = T_k*T + D*U_k*U, T_k*U + U_k*T
for soln in solns_pell:
x_0 = soln[0]
y_0 = soln[1]
x_i = x_0
y_i = y_0
for i in range(k):
X = (A*Matrix([x_i, y_i]) + B)[0]
Y = (A*Matrix([x_i, y_i]) + B)[1]
if isinstance(X, Integer) and isinstance(Y, Integer):
if is_solution_quad(var, coeff, X, Y):
x_n = S( (x_i + sqrt(D)*y_i)*(T + sqrt(D)*U)**(n*L) + (x_i - sqrt(D)*y_i)*(T - sqrt(D)*U)**(n*L) )/ 2
y_n = S( (x_i + sqrt(D)*y_i)*(T + sqrt(D)*U)**(n*L) - (x_i - sqrt(D)*y_i)*(T - sqrt(D)*U)**(n*L) )/ (2*sqrt(D))
x_n, y_n = (A*Matrix([x_n, y_n]) + B)[0], (A*Matrix([x_n, y_n]) + B)[1]
l.add((x_n, y_n))
x_i = x_i*T + D*U*y_i
y_i = x_i*U + y_i*T
return l
def test_multiple_parameters():
assert_equal("\\lcm(830,450)", lcm(830, 450))
assert_equal("\\lcm(6,321,429)", ilcm(6, 321, 429))
assert_equal("\\lcm(14,2324)", lcm(14, 2324))
assert_equal("\\lcm(3, 6, 2)", ilcm(3, 6, 2))
assert_equal("\\lcm(8, 9, 21)", ilcm(8, 9, 21))
assert_equal("\\lcm(144, 2988, 37116)", ilcm(144, 2988, 37116))
assert_equal("\\lcm(144,2988,37116,18,72)", ilcm(144, 2988, 37116, 18, 72))
assert_equal("\\lcm(144, 2988, 37116, 18, 72, 12, 6)",
ilcm(144, 2988, 37116, 18, 72, 12, 6))
assert_equal("\\lcm(32)", lcm(32, 32))
assert_equal("\\lcm(-8, 4, -2)", lcm(-8, lcm(4, -2)))
assert_equal("\\lcm(x, y, z)", lcm(x, lcm(y, z)), symbolically=True)
assert_equal("\\lcm(6*4, 48, 3)", ilcm(6 * 4, 48, 3))
assert_equal("\\lcm(2.4, 3.6, 0.6)",
lcm(Rational('2.4'), lcm(Rational('3.6'), Rational('0.6'))))
assert_equal("\\lcm(\\sqrt{3}, \\sqrt{2},\\sqrt{100})",
lcm(sqrt(3), lcm(sqrt(2), sqrt(100))))
assert_equal("\\lcm(1E12, 1E6, 1E3, 10)",
ilcm(Rational('1E12'), Rational('1E6'), Rational('1E3'), 10))
assert_equal("\\operatorname{lcm}(830,450)", lcm(830, 450))
assert_equal("\\operatorname{lcm}(6,321,429)", ilcm(6, 321, 429))
assert_equal("\\operatorname{lcm}(14,2324)", lcm(14, 2324))
assert_equal("\\operatorname{lcm}(3, 6, 2)", ilcm(3, 6, 2))
assert_equal("\\operatorname{lcm}(8, 9, 21)", ilcm(8, 9, 21))
assert_equal("\\operatorname{lcm}(144, 2988, 37116)",
ilcm(144, 2988, 37116))
assert_equal("\\operatorname{lcm}(144,2988,37116,18,72)",
ilcm(144, 2988, 37116, 18, 72))
assert_equal("\\operatorname{lcm}(144, 2988, 37116, 18, 72, 12, 6)",
ilcm(144, 2988, 37116, 18, 72, 12, 6))
assert_equal("\\operatorname{lcm}(32)", lcm(32, 32))
assert_equal("\\operatorname{lcm}(-8, 4, -2)", lcm(-8, lcm(4, -2)))
assert_equal("\\operatorname{lcm}(x, y, z)",
lcm(x, lcm(y, z)),
symbolically=True)
assert_equal("\\operatorname{lcm}(6*4,48, 3)", ilcm(6 * 4, 48, 3))
assert_equal("\\operatorname{lcm}(2.4, 3.6,0.6)",
lcm(Rational('2.4'), lcm(Rational('3.6'), Rational('0.6'))))
assert_equal("\\operatorname{lcm}(\\sqrt{3}, \\sqrt{2},\\sqrt{100})",
lcm(sqrt(3), lcm(sqrt(2), sqrt(100))))
assert_equal("\\operatorname{lcm}(1E12,1E6, 1E3, 10)",
ilcm(Rational('1E12'), Rational('1E6'), Rational('1E3'), 10))
def __initialize(self):
"""
Method for actually initializing the de Bruijn sequence generator.
Users need not to run this method.
"""
# populate states
self._associates = _get_associate_poly(self._polys)
for entry in self._associates:
entry_state = []
degree = _sympy.degree(entry['associate'])
init_state = [1] * degree
for i in xrange(entry['order']):
entry_state.append(
_seq_decimation(entry['associate'], entry['order'], i,
init_state)[:degree])
entry_state.append([0] * degree)
self._states.append(entry_state)
# find special state
p_matrix = []
for poly in self._polys:
degree = poly.degree()
for i in xrange(degree):
state = [0] * degree
state[i] = 1
for j in xrange(self._order - degree):
state.append(_lfsr_from_poly(poly, state[-degree:])[-1])
p_matrix += state
p_matrix = _sympy.Matrix(self._order, self._order, p_matrix)
self._p_matrix = p_matrix
special_state = map(
int,
_sympy.Matrix(1, self._order, [1] + [0] * (self._order - 1)) *
p_matrix.inv_mod(2))
special_states = []
i = 0
for j, poly in enumerate(self._polys):
check_state = special_state[i:i + _sympy.degree(poly)]
cur_states = self._states[j]
for cur_shift in xrange(self._associates[j]['period']):
try:
which = cur_states.index(check_state)
if which == len(cur_states) - 1:
special_states.append(0)
else:
special_states.append(self._associates[j]['order'] *
cur_shift + which)
break
except ValueError:
cur_states = [_lfsr_from_poly(poly, s) for s in cur_states]
continue
i += _sympy.degree(poly)
# get zech logs
zech_logs = [
_retrieve_zech_log(e['associate']) for e in self._associates
]
# find conjugate pairs and construct adjacency graph
# in here, -1 represents -Inf (in Zech's logarithm)
graph = self._graph
for z1_vals in _iters.product(*[
range(-1, 2**(e['associate'].degree()) - 1)
for e in self._associates
]):
t_vals = [
self._associates[i]['order'] for i in xrange(len(z1_vals))
]
z2_vals = []
for i, z in enumerate(z1_vals):
if z == special_states[i]:
z2_vals.append(-1)
elif z == -1:
z2_vals.append(special_states[i])
else:
conj_val = special_states[i] + zech_logs[i][
z - special_states[i]]
conj_val %= 2**self._polys[i].degree() - 1
z2_vals.append(conj_val)
param_1, param_2 = [], []
shift_1, shift_2 = [], []
for i, z1, z2 in zip(range(len(z1_vals)), z1_vals, z2_vals):
if z2 == -1:
p1, p2 = z1 % t_vals[i], t_vals[i]
s1, s2 = z1 / t_vals[i], 0
elif z1 == -1:
p1, p2 = t_vals[i], z2 % t_vals[i]
s1, s2 = 0, z2 / t_vals[i]
else:
p1, p2 = z1 % t_vals[i], z2 % t_vals[i]
s1, s2 = z1 / t_vals[i], z2 / t_vals[i]
param_1.append(p1)
param_2.append(p2)
shift_1.append(s1)
shift_2.append(s2)
f1_vals = [
1 if s == t_vals[i] else self._associates[i]['period']
for i, s in enumerate(param_1)
]
f2_vals = [
1 if s == t_vals[i] else self._associates[i]['period']
for i, s in enumerate(param_2)
]
for i, s1, s2 in zip(range(len(z1_vals)), shift_1, shift_2):
modulus1 = _sympy.gcd(f1_vals[i],
_sympy.ilcm(*([1, 1] + f1_vals[:i])))
modulus2 = _sympy.gcd(f2_vals[i],
_sympy.ilcm(*([1, 1] + f2_vals[:i])))
param_1.append((s1 - shift_1[0]) % modulus1)
param_2.append((s2 - shift_2[0]) % modulus2)
# lazy check if it's actually a conjugate pair
# state_1 = []
# state_2 = []
# for i, p in enumerate(self._polys):
# cur_state_1 = self._states[i][param_1[i]]
# cur_state_2 = self._states[i][param_2[i]]
# for j in range(shift_1[i]):
# cur_state_1 = _lfsr_from_poly(p, cur_state_1)
# for j in range(shift_2[i]):
# cur_state_2 = _lfsr_from_poly(p, cur_state_2)
# state_1 += cur_state_1
# state_2 += cur_state_2
# state_1 = (_sympy.Matrix(1, self._order, state_1) * self._p_matrix).applyfunc(lambda x: x % 2)
# state_2 = (_sympy.Matrix(1, self._order, state_2) * self._p_matrix).applyfunc(lambda x: x % 2)
# state_1[0] = 1 - state_1[0]
if param_1 > param_2: # and state_1 == state_2
param_1 = tuple(param_1)
param_2 = tuple(param_2)
graph.add_edge(param_1,
param_2,
shift={
param_1: shift_1,
param_2: shift_2
})
self._adjacency_matrix = -_sympy.Matrix(
_nx.to_numpy_matrix(self._graph)).applyfunc(int)
for i in range(self._adjacency_matrix.rows):
self._adjacency_matrix[i, i] -= sum(self._adjacency_matrix[i, :])
simple_graph = _nx.Graph(self._graph)
self._param_generator = self.__param_generator(
_spanning_trees(simple_graph))
# if auto_arm:
anf = self.__get_algebraic_normal_form(*self._param_generator.next())
self._fsr = _FSR(anf, order=self._order, init_state=self._state)