def test_multiple_sums():
s = Sum(i * x + j, (i, a, b), (j, c, d))
l = lambdarepr(s)
assert l == "(builtins.sum(i*x + j for i in range(a, b+1) for j in range(c, d+1)))"
assert (lambdify((x, a, b, c, d), s)(2, 3, 4, 5, 6) ==
s.subs([(x, 2), (a, 3), (b, 4), (c, 5), (d, 6)]).doit())
def test_sum__2():
s = Sum(i * x, (i, a, b))
l = lambdarepr(s)
assert l == "(builtins.sum(i*x for i in range(a, b+1)))"
args = x, a, b
f = lambdify(args, s)
v = 2, 3, 8
assert f(*v) == s.subs(zip(args, v)).doit()
def test_sum__2():
s = Sum(i * x, (i, a, b))
l = lambdarepr(s)
assert l == "(builtins.sum(i*x for i in range(a, b+1)))"
args = x, a, b
f = lambdify(args, s)
v = 2, 3, 8
assert f(*v) == s.subs(zip(args, v)).doit()
def test_sum():
# In each case, test eval() the lambdarepr() to make sure that
# it evaluates to the same results as the symbolic expression
s = Sum(x**i, (i, a, b))
l = lambdarepr(s)
assert l == "(builtins.sum(x**i for i in range(a, b+1)))"
assert (lambdify((x, a, b), s)(2, 3, 8) == s.subs([(x, 2), (a, 3),
(b, 8)]).doit())
s = Sum(i * x, (i, a, b))
l = lambdarepr(s)
assert l == "(builtins.sum(i*x for i in range(a, b+1)))"
assert (lambdify((x, a, b), s)(2, 3, 8) == s.subs([(x, 2), (a, 3),
(b, 8)]).doit())
def test_sum():
# In each case, test eval() the lambdarepr() to make sure that
# it evaluates to the same results as the symbolic expression
s = Sum(x ** i, (i, a, b))
l = lambdarepr(s)
assert l == "(builtins.sum(x**i for i in range(a, b+1)))"
assert (lambdify((x, a, b), s)(2, 3, 8) ==
s.subs([(x, 2), (a, 3), (b, 8)]).doit())
s = Sum(i * x, (i, a, b))
l = lambdarepr(s)
assert l == "(builtins.sum(i*x for i in range(a, b+1)))"
assert (lambdify((x, a, b), s)(2, 3, 8) ==
s.subs([(x, 2), (a, 3), (b, 8)]).doit())
def test_sum__1():
# In each case, test eval() the lambdarepr() to make sure that
# it evaluates to the same results as the symbolic expression
s = Sum(x**i, (i, a, b))
l = lambdarepr(s)
assert l == "(builtins.sum(x**i for i in range(a, b+1)))"
args = x, a, b
f = lambdify(args, s)
v = 2, 3, 8
assert f(*v) == s.subs(zip(args, v)).doit()
def test_sum__1():
# In each case, test eval() the lambdarepr() to make sure that
# it evaluates to the same results as the symbolic expression
s = Sum(x ** i, (i, a, b))
l = lambdarepr(s)
assert l == "(builtins.sum(x**i for i in range(a, b+1)))"
args = x, a, b
f = lambdify(args, s)
v = 2, 3, 8
assert f(*v) == s.subs(zip(args, v)).doit()
def test_multiple_sums():
s = Sum(i * x + j, (i, a, b), (j, c, d))
l = lambdarepr(s)
assert l == "(builtins.sum(i*x + j for i in range(a, b+1) for j in range(c, d+1)))"
args = x, a, b, c, d
f = lambdify(args, s)
vals = 2, 3, 4, 5, 6
f_ref = s.subs(zip(args, vals)).doit()
f_res = f(*vals)
assert f_res == f_ref
def test_multiple_sums():
s = Sum(i * x + j, (i, a, b), (j, c, d))
l = lambdarepr(s)
assert l == "(builtins.sum(i*x + j for i in range(a, b+1) for j in range(c, d+1)))"
args = x, a, b, c, d
f = lambdify(args, s)
vals = 2, 3, 4, 5, 6
f_ref = s.subs(zip(args, vals)).doit()
f_res = f(*vals)
assert f_res == f_ref
def test_issue_14640():
i, n = symbols("i n", integer=True)
a, b, c = symbols("a b c")
assert Sum(a**-i/(a - b), (i, 0, n)).doit() == Sum(
1/(a*a**i - a**i*b), (i, 0, n)).doit() == Piecewise(
(n + 1, Eq(1/a, 1)),
((-a**(-n - 1) + 1)/(1 - 1/a), True))/(a - b)
assert Sum((b*a**i - c*a**i)**-2, (i, 0, n)).doit() == Piecewise(
(n + 1, Eq(a**(-2), 1)),
((-a**(-2*n - 2) + 1)/(1 - 1/a**2), True))/(b - c)**2
s = Sum(i*(a**(n - i) - b**(n - i))/(a - b), (i, 0, n)).doit()
assert not s.has(Sum)
assert s.subs({a: 2, b: 3, n: 5}) == 122
def generate_coupling_expansions(obj, verbose=True):
"""
generate expansions for coupling.
"""
if verbose:
print('* Generating coupling expansions...')
i = sym.symbols('i_sym') # summation index
psi = obj.psi
eps = obj.eps
ruleA = {'psi': obj.pA['expand']}
ruleB = {'psi': obj.pB['expand']}
rule_trunc = {}
for k in range(obj.miter, obj.miter + 200):
rule_trunc.update({obj.eps**k: 0})
for key in obj.var_names:
if verbose:
print('key in coupling expand', key)
gA = Sum(psi**i * Indexed('g' + key + 'A', i),
(i, 1, obj.miter)).doit()
gB = Sum(psi**i * Indexed('g' + key + 'B', i),
(i, 1, obj.miter)).doit()
iA = Sum(psi**i * Indexed('i' + key + 'A', i),
(i, 0, obj.miter)).doit()
iB = Sum(psi**i * Indexed('i' + key + 'B', i),
(i, 0, obj.miter)).doit()
tmp = gA.subs(ruleA)
tmp = sym.expand(tmp,
basic=False,
deep=True,
power_base=False,
power_exp=False,
mul=False,
log=False,
multinomial=True)
tmp = tmp.subs(rule_trunc)
gA_collected = collect(expand(tmp).subs(rule_trunc), eps)
if verbose:
print('completed gA collected')
tmp = gB.subs(ruleB)
tmp = sym.expand(tmp,
basic=False,
deep=True,
power_base=False,
power_exp=False,
mul=False,
log=False,
multinomial=True)
tmp = tmp.subs(rule_trunc)
gB_collected = collect(expand(tmp).subs(rule_trunc), eps)
if verbose:
print('completed gB collected')
tmp = iA.subs(ruleA)
tmp = sym.expand(tmp,
basic=False,
deep=True,
power_base=False,
power_exp=False,
mul=False,
log=False,
multinomial=True)
tmp = tmp.subs(rule_trunc)
iA_collected = collect(expand(tmp).subs(rule_trunc), eps)
if verbose:
print('completed iA collected')
tmp = iB.subs(ruleB)
tmp = sym.expand(tmp,
basic=False,
deep=True,
power_base=False,
power_exp=False,
mul=False,
log=False,
multinomial=True)
tmp = tmp.subs(rule_trunc)
iB_collected = collect(expand(tmp).subs(rule_trunc), eps)
if verbose:
print('completed iB collected')
obj.g[key + '_epsA'] = 0
obj.g[key + '_epsB'] = 0
obj.i[key + '_epsA'] = 0
obj.i[key + '_epsB'] = 0
for j in range(obj.miter):
obj.g[key + '_epsA'] += eps**j * gA_collected.coeff(eps, j)
obj.g[key + '_epsB'] += eps**j * gB_collected.coeff(eps, j)
obj.i[key + '_epsA'] += eps**j * iA_collected.coeff(eps, j)
obj.i[key + '_epsB'] += eps**j * iB_collected.coeff(eps, j)
return obj.g, obj.i
## cf. Wilf and Zeilberger (1992)
x, y = symbols("x y")
f2 = (x + y) ** 2
f2.expand()
f3 = (x + y) ** 3
f3.expand()
n = Symbol("n", positive=True, integer=True)
k = Symbol("k", integer=True)
# binomialpoly = implemented_function(Function('(x+y)^n'), lambda n: (x+y)**n)
# binomialpoly_n = lambdify(n, binomialpoly(n))
binomialpoly = (x + y) ** n
binomialpolyexpand = Sum(binomial(n, k) * x ** k * y ** (n - k), (k, 0, n))
binomialpoly.subs(n, 1) == binomialpolyexpand.subs(n, 1).doit() # True
binomialpoly.subs(n, 2).expand() == binomialpolyexpand.subs(n, 2).doit() # True
binomialpoly.subs(n, 3).expand() == binomialpolyexpand.subs(n, 3).doit() # True
binomialpoly.subs(n, 4).expand() == binomialpolyexpand.subs(n, 4).doit() # True
# rising factorial
a, q = symbols("a q")
# a_n = Product(a+k,(k,0,n-1))
a_n = rf(a, n)
aq_n = Product(Rat(1) - a * q ** k, (k, 0, n - 1))
# e.g. aq_n.subs(n,3).doit()
# Hermite polynomials
HermiteF = factorial(n) * (Rat(-1)) ** k * (Rat(2) * x) ** (n - Rat(2) * k) / (factorial(n - Rat(2) * k) * factorial(k))
HermiteP = Sum(HermiteF, (k, 0, floor(n / 2)))
q = Sum(i**k * (1-z)**i, (i, 0, n-1))
u = s * (z-1)**(k+1) - n**k*z**n*(z-1)**k
'''
t = ans.euler_maclaurin()
(0**k/2 + z**(n - 1)*(n - 1)**k/2 + Integral(_x**k*z**_x, (_x, 0, n - 1)),
Abs(-oo*0**k*k - 0**k*log(z)/12 + k*z**(n - 1)*(n - 1)**k/(12*(n - 1)) + z**(n - 1)*(n - 1)**k*log(z)/12))
'''
if 0:
t = lambda _k = 0: ans.subs(k, _k).doit()
print(ans)
for i in range(5):
print('S(k={k}, z, n) = {s}'.format(k=i, s=t(i)))
elif 0:
t = lambda _k = 0: q.subs(k, _k).doit()
print(q)
for i in range(5):
print('S(k={k}, 1-z, n) = {s}'.format(k=i, s=t(i)))
elif 0:
#T d z = sum z**i C(i,d) {~i}
## test = k
## ans = test.subs(k,k-1)
## print(ans)
## assert ans == k-1
T = Sum(ff(i,k) * z**i, (i,0,n))
K = T - ff(n,k)*z**n + k*z* T.subs(k,k-1)
#ans = K.doit() fail
for i in range(1,5):
a = K.subs(n,i).doit().expand(force=True)
print(a)
