def test_arithmetic_sums():
assert summation(1, (n, a, b)) == b-a+1
assert summation(1, (n, 1, 10)) == 10
assert summation(2*n, (n, 0, 10**10)) == 100000000010000000000
assert summation(4*n*m, (n, a, 1), (m, 1, d)).expand() == \
2*d + 2*d**2 + a*d + a*d**2 - d*a**2 - a**2*d**2
assert summation(cos(n), (n, -2, 1)) == cos(-2)+cos(-1)+cos(0)+cos(1)
def test_polynomial_sums():
assert summation(n**2, (n, 3, 8)) == 199
assert summation(n, (n, a, b)) == \
((a + b)*(b - a + 1)/2).expand()
assert summation(n**2, (n, 1, b)) == \
((2*b**3 + 3*b**2 + b)/6).expand()
assert summation(n**3, (n, 1, b)) == \
((b**4 + 2*b**3 + b**2)/4).expand()
assert summation(n**6, (n, 1, b)) == \
((6*b**7 + 21*b**6 + 21*b**5 - 7*b**3 + b)/42).expand()
def test_arithmetic_sums():
assert summation(1, (n, a, b)) == b - a + 1
assert Sum(S.NaN, (n, a, b)) is S.NaN
assert Sum(x, (n, a, a)).doit() == x
assert Sum(x, (x, a, a)).doit() == a
assert Sum(x, (n, 1, a)).doit() == a*x
assert Sum(x, (x, Range(1, 11))).doit() == 55
assert Sum(x, (x, Range(1, 11, 2))).doit() == 25
assert Sum(x, (x, Range(1, 10, 2))) == Sum(x, (x, Range(9, 0, -2)))
lo, hi = 1, 2
s1 = Sum(n, (n, lo, hi))
s2 = Sum(n, (n, hi, lo))
assert s1 != s2
assert s1.doit() == 3 and s2.doit() == 0
lo, hi = x, x + 1
s1 = Sum(n, (n, lo, hi))
s2 = Sum(n, (n, hi, lo))
assert s1 != s2
assert s1.doit() == 2*x + 1 and s2.doit() == 0
assert Sum(Integral(x, (x, 1, y)) + x, (x, 1, 2)).doit() == \
y**2 + 2
assert summation(1, (n, 1, 10)) == 10
assert summation(2*n, (n, 0, 10**10)) == 100000000010000000000
assert summation(4*n*m, (n, a, 1), (m, 1, d)).expand() == \
2*d + 2*d**2 + a*d + a*d**2 - d*a**2 - a**2*d**2
assert summation(cos(n), (n, -2, 1)) == cos(-2) + cos(-1) + cos(0) + cos(1)
assert summation(cos(n), (n, x, x + 2)) == cos(x) + cos(x + 1) + cos(x + 2)
assert isinstance(summation(cos(n), (n, x, x + S.Half)), Sum)
assert summation(k, (k, 0, oo)) == oo
assert summation(k, (k, Range(1, 11))) == 55
def test_geometric_sums():
assert summation(pi**n, (n, 0, b)) == (1-pi**(b+1)) / (1-pi)
assert summation(2 * 3**n, (n, 0, b)) == 3**(b+1) - 1
assert summation(Rational(1,2)**n, (n, 1, oo)) == 1
assert summation(2**n, (n, 0, b)) == 2**(b+1) - 1
assert summation(2**n, (n, 1, oo)) == oo
assert summation(2**(-n), (n, 1, oo)) == 1
assert summation(3**(-n), (n, 4, oo)) == Rational(1,54)
assert summation(2**(-4*n+3), (n, 1, oo)) == Rational(8,15)
assert summation(2**(n+1), (n, 1, b)).expand() == 4*(2**b-1)
def test_hypersum():
from sympy import simplify, sin, hyper
assert simplify(summation(x**n/fac(n), (n, 1, oo))) == -1 + exp(x)
assert summation((-1)**n * x**(2*n) / fac(2*n), (n, 0, oo)) == cos(x)
assert simplify(summation((-1)**n*x**(2*n+1)/factorial(2*n+1),
(n, 3, oo))) \
== -x + sin(x) + x**3/6 - x**5/120
# TODO to get this without hyper need to improve hyperexpand
assert summation(1/(n+2)**3, (n, 1, oo)) == \
hyper([3, 3, 3, 1], [4, 4, 4], 1)/27
s = summation(x**n*n, (n, -oo, 0))
assert s.is_Piecewise
assert s.args[0].args[0] == -1/(x*(1-1/x)**2)
assert s.args[0].args[1] == (abs(1/x) < 1)
def __init__(self, states, interval, differential_order):
"""
:param states: tuple of states in beginning and end of interval
:param interval: time interval (tuple)
:param differential_order: grade of differential flatness :math:`\\gamma`
"""
self.yd = states
self.t0 = interval[0]
self.t1 = interval[1]
self.dt = interval[1] - interval[0]
gamma = differential_order # + 1 # TODO check this against notes
# setup symbolic expressions
tau, k = sp.symbols('tau, k')
alpha = sp.factorial(2 * gamma + 1)
f = sp.binomial(gamma, k) * (-1) ** k * tau ** (gamma + k + 1) / (gamma + k + 1)
phi = alpha / sp.factorial(gamma) ** 2 * sp.summation(f, (k, 0, gamma))
# differentiate phi(tau), index in list corresponds to order
dphi_sym = [phi] # init with phi(tau)
for order in range(differential_order):
dphi_sym.append(dphi_sym[-1].diff(tau))
# lambdify
self.dphi_num = []
for der in dphi_sym:
self.dphi_num.append(sp.lambdify(tau, der, 'numpy'))
def test_totient():
assert [totient(k) for k in range(1, 12)] == \
[1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10]
assert totient(5005) == 2880
assert totient(5006) == 2502
assert totient(5009) == 5008
assert totient(2**100) == 2**99
raises(ValueError, lambda: totient(30.1))
raises(ValueError, lambda: totient(20.001))
m = Symbol("m", integer=True)
assert totient(m)
assert totient(m).subs(m, 3**10) == 3**10 - 3**9
assert summation(totient(m), (m, 1, 11)) == 42
n = Symbol("n", integer=True, positive=True)
assert totient(n).is_integer
x=Symbol("x", integer=False)
raises(ValueError, lambda: totient(x))
y=Symbol("y", positive=False)
raises(ValueError, lambda: totient(y))
z=Symbol("z", positive=True, integer=True)
raises(ValueError, lambda: totient(2**(-z)))
def expectation(self, expr, var, evaluate=True, **kwargs):
""" Expectation of expression over distribution """
# TODO: support discrete sets with non integer stepsizes
if evaluate:
try:
p = poly(expr, var)
t = Dummy('t', real=True)
mgf = self.moment_generating_function(t)
deg = p.degree()
taylor = poly(series(mgf, t, 0, deg + 1).removeO(), t)
result = 0
for k in range(deg+1):
result += p.coeff_monomial(var ** k) * taylor.coeff_monomial(t ** k) * factorial(k)
return result
except PolynomialError:
return summation(expr * self.pdf(var),
(var, self.set.inf, self.set.sup), **kwargs)
else:
return Sum(expr * self.pdf(var),
(var, self.set.inf, self.set.sup), **kwargs)
def repeated(n, i, e, a, b):
# let n_a = n; n_{a+k+1} = e(n=n_{a+k}, i=a+k+1)
# return n_b
if e.has(i):
if e.has(n):
if not (e - n).has(n):
term = e - n
return n + sympy.summation(term, (i, a, b))
raise NotImplementedError("has i and n")
else:
return e.subs(i, b)
else:
if e.has(n):
if not (e - n).has(n):
term = e - n
return n + term * (b - a + 1)
c, args = e.as_coeff_add(n)
arg, = args
if not (arg / n).simplify().has(n):
coeff = arg / n
# print("Coefficient is %s, iterations is %s" %
# (coeff, (b-a+1)))
return n * coeff ** (b - a + 1)
raise NotImplementedError
else:
return e
def to_sympy(self, expr, **kwargs):
if expr.has_form('Sum', 2) and expr.leaves[1].has_form('List', 3):
index = expr.leaves[1]
result = sympy.summation(expr.leaves[0].to_sympy(), (
index.leaves[0].to_sympy(), index.leaves[1].to_sympy(),
index.leaves[2].to_sympy()))
return result
def to_sympy(self, expr, **kwargs):
if expr.has_form('Sum', 2) and expr.leaves[1].has_form('List', 3):
index = expr.leaves[1]
arg = expr.leaves[0].to_sympy()
bounds = (index.leaves[0].to_sympy(), index.leaves[1].to_sympy(), index.leaves[2].to_sympy())
if arg is not None and None not in bounds:
return sympy.summation(arg, bounds)
def test_Sum_doit():
assert Sum(n*Integral(a**2), (n, 0, 2)).doit() == a**3
assert Sum(n*Integral(a**2), (n, 0, 2)).doit(deep=False) == \
3*Integral(a**2)
assert summation(n*Integral(a**2), (n, 0, 2)) == 3*Integral(a**2)
# test nested sum evaluation
s = Sum( Sum( Sum(2,(z,1,n+1)), (y,x+1,n)), (x,1,n))
assert 0 == (s.doit() - n*(n+1)*(n-1)).factor()
assert Sum(Sum(KroneckerDelta(m, n), (m, 1, 3)), (n, 1, 3)).doit() == 3
assert Sum(Sum(KroneckerDelta(k, m), (m, 1, 3)), (n, 1, 3)).doit() == \
3*Piecewise((1, And(S(1) <= k, k <= 3)), (0, True))
assert Sum(f(n)*Sum(KroneckerDelta(m, n), (m, 0, oo)), (n, 1, 3)).doit() == \
f(1) + f(2) + f(3)
assert Sum(f(n)*Sum(KroneckerDelta(m, n), (m, 0, oo)), (n, 1, oo)).doit() == \
Sum(Piecewise((f(n), n >= 0), (0, True)), (n, 1, oo))
l = Symbol('l', integer=True, positive=True)
assert Sum(f(l)*Sum(KroneckerDelta(m, l), (m, 0, oo)), (l, 1, oo)).doit() == \
Sum(f(l), (l, 1, oo))
# issue 2597
nmax = symbols('N', integer=True, positive=True)
pw = Piecewise((1, And(S(1) <= n, n <= nmax)), (0, True))
assert Sum(pw, (n, 1, nmax)).doit() == Sum(pw, (n, 1, nmax))
def test_Sum_doit():
assert Sum(n * Integral(a ** 2), (n, 0, 2)).doit() == a ** 3
assert Sum(n * Integral(a ** 2), (n, 0, 2)).doit(deep=False) == 3 * Integral(a ** 2)
assert summation(n * Integral(a ** 2), (n, 0, 2)) == 3 * Integral(a ** 2)
# test nested sum evaluation
S = Sum(Sum(Sum(2, (z, 1, n + 1)), (y, x + 1, n)), (x, 1, n))
assert 0 == (S.doit() - n * (n + 1) * (n - 1)).factor()
def expectation(self, expr, var, evaluate=True, **kwargs):
""" Expectation of expression over distribution """
# TODO: support discrete sets with non integer stepsizes
if evaluate:
return summation(expr * self.pdf(var),
(var, self.set.inf, self.set.sup), **kwargs)
else:
return Sum(expr * self.pdf(var),
(var, self.set.inf, self.set.sup), **kwargs)
def compute_characteristic_function(self, **kwargs):
""" Compute the characteristic function from the PDF
Returns a Lambda
"""
x, t = symbols('x, t', real=True, finite=True, cls=Dummy)
pdf = self.pdf(x)
cf = summation(exp(I*t*x)*pdf, (x, self.set.inf, self.set.sup))
return Lambda(t, cf)
def test_Sum_interface():
assert isinstance(Sum(0, (n, 0, 2)), Sum)
assert Sum(nan, (n, 0, 2)) is nan
assert Sum(nan, (n, 0, oo)) is nan
assert Sum(0, (n, 0, 2)).doit() == 0
assert isinstance(Sum(0, (n, 0, oo)), Sum)
assert Sum(0, (n, 0, oo)).doit() == 0
raises(ValueError, lambda: Sum(1))
raises(ValueError, lambda: summation(1))
def test_composite_sums():
f = Rational(1, 2)*(7 - 6*n + Rational(1, 7)*n**3)
s = summation(f, (n, a, b))
assert not isinstance(s, Sum)
A = 0
for i in range(-3, 5):
A += f.subs(n, i)
B = s.subs(a, -3).subs(b, 4)
assert A == B
def _eval_product(self, a, n, term):
from sympy import summation, Sum
k = self.index
if not term.has(k):
return term**(n-a+1)
elif term.is_polynomial(k):
poly = term.as_poly(k)
A = B = Q = S.One
C_= poly.LC()
all_roots = roots(poly, multiple=True)
for r in all_roots:
A *= C.RisingFactorial(a-r, n-a+1)
Q *= n - r
if len(all_roots) < poly.degree():
B = Product(quo(poly, Q.as_poly(k)), (k, a, n))
return poly.LC()**(n-a+1) * A * B
elif term.is_Add:
p, q = term.as_numer_denom()
p = self._eval_product(a, n, p)
q = self._eval_product(a, n, q)
return p / q
elif term.is_Mul:
exclude, include = [], []
for t in term.args:
p = self._eval_product(a, n, t)
if p is not None:
exclude.append(p)
else:
include.append(t)
if not exclude:
return None
else:
A, B = Mul(*exclude), term._new_rawargs(*include)
return A * Product(B, (k, a, n))
elif term.is_Pow:
if not term.base.has(k):
s = summation(term.exp, (k, a, n))
if not isinstance(s, Sum):
return term.base**s
elif not term.exp.has(k):
p = self._eval_product(a, n, term.base)
if p is not None:
return p**term.exp
def grow_function_sym(N = Symbol("N"),k = 0):
"""
@type k int
"""
from sympy import factorial,summation,combsimp
ksym = Symbol("ksym")
def C(k,n):
return factorial(n) / (factorial(k)*factorial(n-k))
return combsimp(summation(C(ksym,N),(ksym,0,k-1)))
def compute_quantile(self, **kwargs):
""" Compute the Quantile from the PDF
Returns a Lambda
"""
x = symbols('x', integer=True, finite=True, cls=Dummy)
p = symbols('p', real=True, finite=True, cls=Dummy)
left_bound = self.set.inf
pdf = self.pdf(x)
cdf = summation(pdf, (x, left_bound, x), **kwargs)
set = ((x, p <= cdf), )
return Lambda(p, Piecewise(*set))
def test_totient():
assert [totient(k) for k in range(1, 12)] == \
[1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10]
assert totient(5005) == 2880
assert totient(5006) == 2502
assert totient(5009) == 5008
assert totient(2**100) == 2**99
m = Symbol("m", integer=True)
assert totient(m)
assert totient(m).subs(m, 3**10) == 3**10 - 3**9
assert summation(totient(m), (m, 1, 11)) == 42
def test_primeomega():
assert primeomega(2) == 1
assert primeomega(2 * 2) == 2
assert primeomega(2 * 2 * 3) == 3
assert primeomega(3 * 25) == primeomega(3) + primeomega(25)
assert [primeomega(p) for p in primerange(1, 10)] == [1, 1, 1, 1]
assert primeomega(fac(50)) == 108
assert primeomega(2 ** 9941 - 1) == 1
n = Symbol('n', integer=True)
assert primeomega(n)
assert primeomega(n).subs(n, 2 ** 31 - 1) == 1
assert summation(primeomega(n), (n, 2, 30)) == 59
def compute_cdf(self, **kwargs):
""" Compute the CDF from the PDF
Returns a Lambda
"""
x, z = symbols('x, z', integer=True, finite=True, cls=Dummy)
left_bound = self.set.inf
# CDF is integral of PDF from left bound to z
pdf = self.pdf(x)
cdf = summation(pdf, (x, left_bound, z), **kwargs)
# CDF Ensure that CDF left of left_bound is zero
cdf = Piecewise((cdf, z >= left_bound), (0, True))
return Lambda(z, cdf)
def test_reduced_totient():
assert [reduced_totient(k) for k in range(1, 16)] == \
[1, 1, 2, 2, 4, 2, 6, 2, 6, 4, 10, 2, 12, 6, 4]
assert reduced_totient(5005) == 60
assert reduced_totient(5006) == 2502
assert reduced_totient(5009) == 5008
assert reduced_totient(2**100) == 2**98
m = Symbol("m", integer=True)
assert reduced_totient(m)
assert reduced_totient(m).subs(m, 2**3*3**10) == 3**10 - 3**9
assert summation(reduced_totient(m), (m, 1, 16)) == 68
n = Symbol("n", integer=True, positive=True)
assert reduced_totient(n).is_integer
def test_divisor_sigma():
assert [divisor_sigma(k) for k in range(1, 12)] == [1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12]
assert [divisor_sigma(k, 2) for k in range(1, 12)] == [1, 5, 10, 21, 26, 50, 50, 85, 91, 130, 122]
assert divisor_sigma(23450) == 50592
assert divisor_sigma(23450, 0) == 24
assert divisor_sigma(23450, 1) == 50592
assert divisor_sigma(23450, 2) == 730747500
assert divisor_sigma(23450, 3) == 14666785333344
m = Symbol("m", integer=True)
k = Symbol("k", integer=True)
assert divisor_sigma(m)
assert divisor_sigma(m, k)
assert divisor_sigma(m).subs(m, 3 ** 10) == 88573
assert divisor_sigma(m, k).subs([(m, 3 ** 10), (k, 3)]) == 213810021790597
assert summation(divisor_sigma(m), (m, 1, 11)) == 99
def test_udivisor_sigma():
assert [udivisor_sigma(k) for k in range(1, 12)] == [1, 3, 4, 5, 6, 12, 8, 9, 10, 18, 12]
assert [udivisor_sigma(k, 3) for k in range(1, 12)] == [1, 9, 28, 65, 126, 252, 344, 513, 730, 1134, 1332]
assert udivisor_sigma(23450) == 42432
assert udivisor_sigma(23450, 0) == 16
assert udivisor_sigma(23450, 1) == 42432
assert udivisor_sigma(23450, 2) == 702685000
assert udivisor_sigma(23450, 4) == 321426961814978248
m = Symbol("m", integer=True)
k = Symbol("k", integer=True)
assert udivisor_sigma(m)
assert udivisor_sigma(m, k)
assert udivisor_sigma(m).subs(m, 4 ** 9) == 262145
assert udivisor_sigma(m, k).subs([(m, 4 ** 9), (k, 2)]) == 68719476737
assert summation(udivisor_sigma(m), (m, 2, 15)) == 169
def eval_prob(self, _domain):
if isinstance(_domain, Range):
n = symbols('n')
inf, sup, step = (r for r in _domain.args)
summand = ((self.pdf).replace(
self.symbol, inf + n*step))
rv = summation(summand,
(n, 0, floor((sup - inf)/step - 1))).doit()
return rv
elif isinstance(_domain, FiniteSet):
pdf = Lambda(self.symbol, self.pdf)
rv = sum(pdf(x) for x in _domain)
return rv
elif isinstance(_domain, Union):
rv = sum(self.eval_prob(x) for x in _domain.args)
return rv
def eval_prob(self, _domain):
sym = list(self.symbols)[0]
if isinstance(_domain, Range):
n = symbols('n', integer=True, finite=True)
inf, sup, step = (r for r in _domain.args)
summand = ((self.pdf).replace(
sym, n*step))
rv = summation(summand,
(n, inf/step, (sup)/step - 1)).doit()
return rv
elif isinstance(_domain, FiniteSet):
pdf = Lambda(sym, self.pdf)
rv = sum(pdf(x) for x in _domain)
return rv
elif isinstance(_domain, Union):
rv = sum(self.eval_prob(x) for x in _domain.args)
return rv
def integrate(self, expr, rvs=None, **kwargs):
rvs = rvs or (self.value,)
if self.value not in rvs:
return expr
expr = expr.xreplace(dict((rv, rv.symbol) for rv in rvs))
x = self.value.symbol
try:
return self.distribution.expectation(expr, x, **kwargs)
except:
evaluate = kwargs.pop('evaluate', True)
if evaluate:
return summation(expr * self.pdf, (x, self.set.inf, self.set.sup),
**kwargs)
else:
return Sum(expr * self.pdf, (x, self.set.inf, self.set.sup),
**kwargs)
def test_hypersum():
from sympy import simplify, sin, hyper
assert simplify(summation(x**n/fac(n), (n, 1, oo))) == -1 + exp(x)
assert summation((-1)**n * x**(2*n) / fac(2*n), (n, 0, oo)) == cos(x)
assert simplify(summation((-1)**n*x**(2*n + 1) /
factorial(2*n + 1), (n, 3, oo))) == -x + sin(x) + x**3/6 - x**5/120
assert summation(1/(n + 2)**3, (n, 1, oo)) == -S(9)/8 + zeta(3)
assert summation(1/n**4, (n, 1, oo)) == pi**4/90
s = summation(x**n*n, (n, -oo, 0))
assert s.is_Piecewise
assert s.args[0].args[0] == -1/(x*(1 - 1/x)**2)
assert s.args[0].args[1] == (abs(1/x) < 1)
m = Symbol('n', integer=True, positive=True)
assert summation(binomial(m, k), (k, 0, m)) == 2**m
def add_single_resonance(self, j, k, l, indexIn=1, indexOut=2):
"""
Add a single term associated the j:j-k MMR between planets 'indexIn' and 'indexOut'.
Inputs:
indexIn - index of the inner planet
indexOut - index of the outer planet
j - together with k specifies the MMR j:j-k
k - order of the resonance
l - picks out the eIn^(l) * eOut^(k-l) subterm
"""
# Canonical variables
assert indexOut == indexIn + 1, "Only resonances for adjacent pairs are currently supported"
G = symbols('G')
mIn, MIn, LambdaIn, lambdaIn, XIn, YIn = symbols(
'm{0},M{0},Lambda{0},lambda{0},X{0},Y{0}'.format(indexIn))
mOut, MOut, LambdaOut, lambdaOut, XOut, YOut = symbols(
'm{0},M{0},Lambda{0},lambda{0},X{0},Y{0}'.format(indexOut))
#mIn, MIn, muIn, LambdaIn,lambdaIn,XIn,YIn = self._get_symbols(indexIn)
#mOut, MOut, muOut, LambdaOut,lambdaOut,XOut,YOut = self._get_symbols(indexOut)
# Resonance index
assert l <= k, "Invalid resonance term, l must be less than or equal to k."
alpha = self.particles[indexIn].a / self.state.particles[indexOut].a
# Resonance components
#
Cjkl = symbols("C_{0}\,{1}\,{2}".format(j, k, l))
self.Hparams[Cjkl] = general_order_coefficient(j, k, l, alpha)
#
i = symbols('i')
lBy2 = int(np.floor(l / 2.))
k_l = k - l
k_lBy2 = int(np.floor((k_l) / 2.))
if l > 0:
z_to_p_In_re = summation(
binomial(l, (2 * i)) * XIn**(l - (2 * i)) * YIn**(2 * i) *
(-1)**(i), (i, 0, lBy2))
z_to_p_In_im = summation(
binomial(l, (2 * i + 1)) * XIn**(l - (2 * i + 1)) *
YIn**(2 * i + 1) * (-1)**(i), (i, 0, lBy2))
else:
z_to_p_In_re = 1
z_to_p_In_im = 0
if k_l > 0:
z_to_p_Out_re = summation(
binomial(k_l, (2 * i)) * XOut**(k_l - (2 * i)) *
YOut**(2 * i) * (-1)**(i), (i, 0, k_lBy2))
z_to_p_Out_im = summation(
binomial(k_l, (2 * i + 1)) * XOut**(k_l - (2 * i + 1)) *
YOut**(2 * i + 1) * (-1)**(i), (i, 0, k_lBy2))
else:
z_to_p_Out_re = 1
z_to_p_Out_im = 0
reFactor = (z_to_p_In_re * z_to_p_Out_re - z_to_p_In_im *
z_to_p_Out_im) / sqrt(LambdaIn)**l / sqrt(LambdaOut)**k_l
imFactor = (z_to_p_In_im * z_to_p_Out_re + z_to_p_In_re *
z_to_p_Out_im) / sqrt(LambdaIn)**l / sqrt(LambdaOut)**k_l
# Keep track of resonances
self.resonance_indices.append((indexIn, indexOut, (j, k, l)))
# Update internal Hamiltonian
prefactor = -G**2 * MOut**2 * mOut**3 * (mIn / MIn) / (LambdaOut**2)
costerm = cos(j * lambdaOut - (j - k) * lambdaIn)
sinterm = sin(j * lambdaOut - (j - k) * lambdaIn)
self.H += prefactor * Cjkl * (reFactor * costerm - imFactor * sinterm)
self._update()
# update polar Hamiltonian
GammaIn, gammaIn, GammaOut, gammaOut = symbols(
'Gamma{0},gamma{0},Gamma{1},gamma{1}'.format(indexIn, indexOut))
eccIn = sqrt(2 * GammaIn / LambdaIn)
eccOut = sqrt(2 * GammaOut / LambdaOut)
#
costerm = cos(j * lambdaOut - (j - k) * lambdaIn + l * gammaIn +
(k - l) * gammaOut)
#
self.Hpolar += prefactor * Cjkl * (eccIn**l) * (eccOut
**(k - l)) * costerm
import sympy as sp
from sympy.core.sympify import SympifyError
def input_reader():
try:
expr1 = sp.sympify(input('Enter the n-th term of a series: '))
n = int(input('Enter n: '))
return expr1, n
except (SympifyError, ValueError) as e:
print(e)
exit(1)
if __name__ == '__main__':
exp, num = input_reader()
assert (num > 0)
n = sp.Symbol('n')
series = sp.summation(exp, (n, 1, num))
sp.pprint(series)
def _eval_product(self, term, limits):
from sympy import summation
(k, a, n) = limits
if k not in term.free_symbols:
return term**(n - a + 1)
if a == n:
return term.subs(k, a)
dif = n - a
if dif.is_Integer:
return Mul(*[term.subs(k, a + i) for i in xrange(dif + 1)])
elif term.is_polynomial(k):
poly = term.as_poly(k)
A = B = Q = S.One
all_roots = roots(poly, multiple=True)
for r in all_roots:
A *= C.RisingFactorial(a - r, n - a + 1)
Q *= n - r
if len(all_roots) < poly.degree():
arg = quo(poly, Q.as_poly(k))
B = Product(arg, (k, a, n)).doit()
return poly.LC()**(n - a + 1) * A * B
elif term.is_Add:
p, q = term.as_numer_denom()
p = self._eval_product(p, (k, a, n))
q = self._eval_product(q, (k, a, n))
return p / q
elif term.is_Mul:
exclude, include = [], []
for t in term.args:
p = self._eval_product(t, (k, a, n))
if p is not None:
exclude.append(p)
else:
include.append(t)
if not exclude:
return None
else:
arg = term._new_rawargs(*include)
A = Mul(*exclude)
B = Product(arg, (k, a, n)).doit()
return A * B
elif term.is_Pow:
if not term.base.has(k):
s = summation(term.exp, (k, a, n))
return term.base**s
elif not term.exp.has(k):
p = self._eval_product(term.base, (k, a, n))
if p is not None:
return p**term.exp
elif isinstance(term, Product):
evaluated = term.doit()
f = self._eval_product(evaluated, limits)
if f is None:
return Product(evaluated, limits)
else:
return f
def bench_sum():
x, i = symbols('x i')
summation(x**i / i, (i, 1, 400))
arr = np.array(l)
print(sum(arr))
#list2
from numpy import inf
l = []
for i in range(1, 100000):
a = (1 - 1 / i)**(2 * i)
l.append(a)
print(l[99998])
#ndarray2
import numpy as np
from numpy import inf
l = []
for i in range(1, 100000):
a = (1 - 1 / i)**(2 * i)
l.append(a)
arr = np.array(l)
print(arr[99998])
#sympy
import sympy
from sympy import *
n = Symbol('n')
s = (1 - (1 / n))**(2 * n)
print(limit(s, n, oo))
i = sympy.symbols('i', integer=True)
a = float(
sympy.summation((((-1)**(i + 1)) / i) * ((6.0 / 7.0)**i),
(i, 1, sympy.oo)))
print(a)
print(degree(result))
syntelestes = result.coeffs()
syntelestes = syntelestes[::-1]
Lista2 = list()
for i in range(0, degree(result)):
temp = syntelestes[i] * ((ff(b + 1, i + 1)) - ff(a, i + 1))
Lista2.append(temp)
p3 = 0
for i in range(0, len(Lista2)):
p3 = p3 + Lista2[i]
print('\n' '\n')
p3 = Poly(p3, a, b)
print('\n', p3)
c = p3.coeffs()
c = np.array(c)
for i in range(0, len(c)):
c[i] = Rational(c[i]).limit_denominator()
##
# Εκτυπώνουμε ξανά τους συντελεστές στη σωστή μορφή όπως θα έδινε η summation,
# ο όρος μηδέν αντιπροσωπεύει το συντελεστή του a^3 και b^3 ο οποίος λύπει απο τη summation
# και παραθέτουμε και την επαλήθευσή της
##
print('\n', c)
print('\n', simplify(summation(p, (x, a, b))))
z = S.Symbol('z')
a = S.Symbol('a')
b = S.Symbol('b')
c = S.Symbol('c')
fx = -4 * x**2 + 2 * x + 1
# 解一元二次方程
print(S.solve(fx, x))
# 平方根
print(S.sqrt(2) * S.sqrt(3))
# 去括号
print(S.expand((a + b)**2))
# 方程组
print(S.solve([x + y - 1, x - y - 3]), [x, y])
# 求和式
i = S.Symbol('i', integer=True)
fx = S.summation(x, (i, 1, 5)) + 10 * x - 15
print(S.solve(fx, x))
# 解一元二次方程
print(S.solve(x**2 - 2, x))
f = S.solve(x * S.log(4, 3) - 2, x)
print(f)
#print(S.log(S.E))
#print(S.log(1000,10))
print(S.expand_log(S.log(x * y), force=True))
#约分
fx = (x**2 + 3 * x + 2) / (x**2 + x)
print(S.cancel(fx))
#三角函数
print(S.solve([S.sin(x - y), S.cos(x + y)], [x, y]))
def test_other_sums():
f = m**2 + m*exp(m)
g = 3*exp(S(3)/2)/2 + exp(S(1)/2)/2 - exp(-S(1)/2)/2 - 3*exp(-S(3)/2)/2 + 5
assert summation(f, (m, -S(3)/2, S(3)/2)).expand() == g
assert summation(f, (m, -1.5, 1.5)).evalf().epsilon_eq(g.evalf(), 1e-10)
def test_issue_4171():
assert summation(factorial(2*k + 1)/factorial(2*k), (k, 0, oo)) == oo
assert summation(2*k + 1, (k, 0, oo)) == oo
def test_issue_7097():
assert sum(x**n/n for n in range(1, 401)) == summation(x**n/n, (n, 1, 400))
def test_issue_4170():
assert summation(1 / factorial(k), (k, 0, oo)) == E
def test_hypergeometric_sums():
assert summation(binomial(2 * k, k) / 4**k,
(k, 0, n)) == (1 + 2 * n) * binomial(2 * n, n) / 4**n
def test_harmonic_sums():
assert summation(1 / k, (k, 0, n)) == Sum(1 / k, (k, 0, n))
assert summation(1 / k, (k, 1, n)) == harmonic(n)
assert summation(n / k, (k, 1, n)) == n * harmonic(n)
assert summation(1 / k, (k, 5, n)) == harmonic(n) - harmonic(4)
def plot_and_save(name):
tmp_file = TmpFileManager.tmp_file
x = Symbol('x')
y = Symbol('y')
z = Symbol('z')
###
# Examples from the 'introduction' notebook
###
p = plot(x)
p = plot(x * sin(x), x * cos(x))
p.extend(p)
p[0].line_color = lambda a: a
p[1].line_color = 'b'
p.title = 'Big title'
p.xlabel = 'the x axis'
p[1].label = 'straight line'
p.legend = True
p.aspect_ratio = (1, 1)
p.xlim = (-15, 20)
p.save(tmp_file('%s_basic_options_and_colors' % name))
p.extend(plot(x + 1))
p.append(plot(x + 3, x**2)[1])
p.save(tmp_file('%s_plot_extend_append' % name))
p[2] = plot(x**2, (x, -2, 3))
p.save(tmp_file('%s_plot_setitem' % name))
p = plot(sin(x), (x, -2 * pi, 4 * pi))
p.save(tmp_file('%s_line_explicit' % name))
p = plot(sin(x))
p.save(tmp_file('%s_line_default_range' % name))
p = plot((x**2, (x, -5, 5)), (x**3, (x, -3, 3)))
p.save(tmp_file('%s_line_multiple_range' % name))
raises(ValueError, lambda: plot(x, y))
#parametric 2d plots.
#Single plot with default range.
plot_parametric(sin(x), cos(x)).save(tmp_file())
#Single plot with range.
p = plot_parametric(sin(x), cos(x), (x, -5, 5))
p.save(tmp_file('%s_parametric_range' % name))
#Multiple plots with same range.
p = plot_parametric((sin(x), cos(x)), (x, sin(x)))
p.save(tmp_file('%s_parametric_multiple' % name))
#Multiple plots with different ranges.
p = plot_parametric((sin(x), cos(x), (x, -3, 3)), (x, sin(x), (x, -5, 5)))
p.save(tmp_file('%s_parametric_multiple_ranges' % name))
#depth of recursion specified.
p = plot_parametric(x, sin(x), depth=13)
p.save(tmp_file('%s_recursion_depth' % name))
#No adaptive sampling.
p = plot_parametric(cos(x), sin(x), adaptive=False, nb_of_points=500)
p.save(tmp_file('%s_adaptive' % name))
#3d parametric plots
p = plot3d_parametric_line(sin(x), cos(x), x)
p.save(tmp_file('%s_3d_line' % name))
p = plot3d_parametric_line((sin(x), cos(x), x, (x, -5, 5)),
(cos(x), sin(x), x, (x, -3, 3)))
p.save(tmp_file('%s_3d_line_multiple' % name))
p = plot3d_parametric_line(sin(x), cos(x), x, nb_of_points=30)
p.save(tmp_file('%s_3d_line_points' % name))
# 3d surface single plot.
p = plot3d(x * y)
p.save(tmp_file('%s_surface' % name))
# Multiple 3D plots with same range.
p = plot3d(-x * y, x * y, (x, -5, 5))
p.save(tmp_file('%s_surface_multiple' % name))
# Multiple 3D plots with different ranges.
p = plot3d((x * y, (x, -3, 3), (y, -3, 3)),
(-x * y, (x, -3, 3), (y, -3, 3)))
p.save(tmp_file('%s_surface_multiple_ranges' % name))
# Single Parametric 3D plot
p = plot3d_parametric_surface(sin(x + y), cos(x - y), x - y)
p.save(tmp_file('%s_parametric_surface' % name))
# Multiple Parametric 3D plots.
p = plot3d_parametric_surface(
(x * sin(z), x * cos(z), z, (x, -5, 5), (z, -5, 5)),
(sin(x + y), cos(x - y), x - y, (x, -5, 5), (y, -5, 5)))
p.save(tmp_file('%s_parametric_surface' % name))
###
# Examples from the 'colors' notebook
###
p = plot(sin(x))
p[0].line_color = lambda a: a
p.save(tmp_file('%s_colors_line_arity1' % name))
p[0].line_color = lambda a, b: b
p.save(tmp_file('%s_colors_line_arity2' % name))
p = plot(x * sin(x), x * cos(x), (x, 0, 10))
p[0].line_color = lambda a: a
p.save(tmp_file('%s_colors_param_line_arity1' % name))
p[0].line_color = lambda a, b: a
p.save(tmp_file('%s_colors_param_line_arity2a' % name))
p[0].line_color = lambda a, b: b
p.save(tmp_file('%s_colors_param_line_arity2b' % name))
p = plot3d_parametric_line(
sin(x) + 0.1 * sin(x) * cos(7 * x),
cos(x) + 0.1 * cos(x) * cos(7 * x), 0.1 * sin(7 * x), (x, 0, 2 * pi))
p[0].line_color = lambda a: sin(4 * a)
p.save(tmp_file('%s_colors_3d_line_arity1' % name))
p[0].line_color = lambda a, b: b
p.save(tmp_file('%s_colors_3d_line_arity2' % name))
p[0].line_color = lambda a, b, c: c
p.save(tmp_file('%s_colors_3d_line_arity3' % name))
p = plot3d(sin(x) * y, (x, 0, 6 * pi), (y, -5, 5))
p[0].surface_color = lambda a: a
p.save(tmp_file('%s_colors_surface_arity1' % name))
p[0].surface_color = lambda a, b: b
p.save(tmp_file('%s_colors_surface_arity2' % name))
p[0].surface_color = lambda a, b, c: c
p.save(tmp_file('%s_colors_surface_arity3a' % name))
p[0].surface_color = lambda a, b, c: sqrt((a - 3 * pi)**2 + b**2)
p.save(tmp_file('%s_colors_surface_arity3b' % name))
p = plot3d_parametric_surface(x * cos(4 * y), x * sin(4 * y), y,
(x, -1, 1), (y, -1, 1))
p[0].surface_color = lambda a: a
p.save(tmp_file('%s_colors_param_surf_arity1' % name))
p[0].surface_color = lambda a, b: a * b
p.save(tmp_file('%s_colors_param_surf_arity2' % name))
p[0].surface_color = lambda a, b, c: sqrt(a**2 + b**2 + c**2)
p.save(tmp_file('%s_colors_param_surf_arity3' % name))
###
# Examples from the 'advanced' notebook
###
i = Integral(log((sin(x)**2 + 1) * sqrt(x**2 + 1)), (x, 0, y))
p = plot(i, (y, 1, 5))
p.save(tmp_file('%s_advanced_integral' % name))
s = summation(1 / x**y, (x, 1, oo))
p = plot(s, (y, 2, 10))
p.save(tmp_file('%s_advanced_inf_sum' % name))
p = plot(summation(1 / x, (x, 1, y)), (y, 2, 10), show=False)
p[0].only_integers = True
p[0].steps = True
p.save(tmp_file('%s_advanced_fin_sum' % name))
###
# Test expressions that can not be translated to np and generate complex
# results.
###
plot(sin(x) + I * cos(x)).save(tmp_file())
plot(sqrt(sqrt(-x))).save(tmp_file())
plot(LambertW(x)).save(tmp_file())
plot(sqrt(LambertW(x))).save(tmp_file())
#Characteristic function of a StudentT distribution with nu=10
plot((meijerg(
((1 / 2, ), ()),
((5, 0, 1 / 2), ()), 5 * x**2 * exp_polar(-I * pi) / 2) + meijerg(
((1 / 2, ), ()),
((5, 0, 1 / 2),
()), 5 * x**2 * exp_polar(I * pi) / 2)) / (48 * pi),
(x, 1e-6, 1e-2)).save(tmp_file())
def _eval_rewrite_as_factorial(self, arg, **kwargs):
from sympy import summation
i = Dummy('i')
f = S.NegativeOne**i / factorial(i)
return factorial(arg) * summation(f, (i, 0, arg))
def test_issue_15852():
assert summation(x**y*y, (y, -oo, oo)).doit() == Sum(x**y*y, (y, -oo, oo))
def test_geometric_sums():
assert summation(pi**n, (n, 0, b)) == (1 - pi**(b + 1)) / (1 - pi)
assert summation(2 * 3**n, (n, 0, b)) == 3**(b + 1) - 1
assert summation(Rational(1, 2)**n, (n, 1, oo)) == 1
assert summation(2**n, (n, 0, b)) == 2**(b + 1) - 1
assert summation(2**n, (n, 1, oo)) == oo
assert summation(2**(-n), (n, 1, oo)) == 1
assert summation(3**(-n), (n, 4, oo)) == Rational(1, 54)
assert summation(2**(-4 * n + 3), (n, 1, oo)) == Rational(8, 15)
assert summation(2**(n + 1), (n, 1, b)).expand() == 4 * (2**b - 1)
# issue 6664:
assert summation(x**n, (n, 0, oo)) == \
Piecewise((1/(-x + 1), Abs(x) < 1), (Sum(x**n, (n, 0, oo)), True))
assert summation(-2**n, (n, 0, oo)) == -oo
assert summation(I**n, (n, 0, oo)) == Sum(I**n, (n, 0, oo))
# issue 6802:
assert summation((-1)**(2 * x + 2), (x, 0, n)) == n + 1
assert summation((-2)**(2 * x + 2),
(x, 0, n)) == 4 * 4**(n + 1) / S(3) - S(4) / 3
assert summation((-1)**x, (x, 0, n)) == -(-1)**(n + 1) / S(2) + S(1) / 2
assert summation(y**x, (x, a, b)) == \
Piecewise((-a + b + 1, Eq(y, 1)), ((y**a - y**(b + 1))/(-y + 1), True))
assert summation((-2)**(y*x + 2), (x, 0, n)) == \
4*Piecewise((n + 1, Eq((-2)**y, 1)),
((-(-2)**(y*(n + 1)) + 1)/(-(-2)**y + 1), True))
def test_Sum():
assert str(summation(cos(3 * z), (z, x, y))) == "Sum(cos(3*z), (z, x, y))"
assert str(Sum(x*y**2, (x, -2, 2), (y, -5, 5))) == \
"Sum(x*y**2, (x, -2, 2), (y, -5, 5))"
print(sym.diff((6*x**3) + (5*x**2)+(4)))
f1 = diff(sin(x), x)
print(f1)
f2 = diff(sin(2*x), x)
print(f2)
SeriesExp1 = cos(x).series(x,0,15)
print(SeriesExp1)
print(summation(2*i - 1, (i, 1, n)))
print(summation(i,(i, 0, 2)))
integral1 = integrate(7*x**5)
print(integral1)
integral2 = integrate(log(x))
print(integral2)
f = Function('f')
f(x).diff(x,x)+f(x)
print(dsolve(f(x).diff(x,x) + f(x), f(x)))
def compute_moment_generating_function(self, **kwargs):
t = Dummy('t', real=True)
x = Dummy('x', integer=True)
pdf = self.pdf(x)
mgf = summation(exp(t*x)*pdf, (x, self.set.inf, self.set.sup))
return Lambda(t, mgf)
def find_sum(n_term, num_terms):
n = Symbol("n")
s = summation(n_term, (n, 1, num_terms))
pprint(s)
"""
from sympy.plotting import plot
import sympy as sym
import xlsxwriter
from xlrd import open_workbook
from sympy.plotting import plot3d
#Exercise one :
x = sym.Symbol('x')
y, i, n, a, b, z = sym.symbols('y i n a b z')
expr = x**2 + x**3 + 21 * x**4 + 10 * x + 1
print(expr.subs(x, 7))
print(sym.expand(x + y)**2)
print(sym.simplify(4 * x**3 + 21 * x**2 + 10 * x + 12))
print(sym.limit(1 / (x**2), x, sym.oo))
print(sym.summation(2 * i + i - 1, (i, 5, n)))
print(sym.integrate(sym.sin(x) + sym.exp(x) * sym.cos(x) + sym.tan(x), x))
print(sym.factor(x**3 + 12 * x * y * z + 3 * y**2 * z))
print(sym.solveset(x - 4, x))
m1 = sym.Matrix([[5, 12, 40], [30, 70, 2]])
m2 = sym.Matrix([2, 1, 0])
print(m1 * m2)
plot(x**3 + 3, (x, -10, 10))
f = x**2 * y**3
plot3d(f, (x, -6, 6), (y, -6, 6))
#==============================================================================
#Exercise two :
workbook = xlsxwriter.Workbook('test1.xlsx')
worksheet = workbook.add_worksheet()
worksheet.autofilter('A1:A5')
data = ["This is Example", "My first export example", 1, 2, 3]
n = sym.Symbol("n")
x = sym.Symbol("x")
formula_p = input("n回目のゲームに勝つ確率は？ ※サンクトペテルブルクのパラドックスなら (1/2)**n：")
formula_x = input("n回目のゲームに勝ったときの賞金Xは？ ※サンクトペテルブルクのパラドックスなら 2**n：")
formula_p = sym.sympify(formula_p)
formula_x = sym.sympify(formula_x)
formula_n = sym.solve(formula_x - x, n)
formula_n = sym.sympify(formula_n[0])
formula = formula_p.subs(n, formula_n)
def temp(x1):
return formula_x.subs(n, x1)
print(
f"求める確率分布の式は：{formula}\nただし、X=({temp(1)},{temp(2)},{temp(3)},...,{formula_x},...)"
)
dis_x = [temp(i) for i in range(1, 11, 1)]
dis_y = [formula.subs(x, j) for j in dis_x]
plt.scatter(dis_x, dis_y)
# plot(formula,(x,1,10),title="Expected value distribution")
plt.show()
formula_cul = formula_p * formula_x
result = sym.summation(formula_cul, (n, 1, oo))
print(f"期待値は、一般項{formula_cul}の無限級数の総和を求めて、{result}")
def find_sum(nth_term,num_terms):
n=symbols('n')
s=summation(nth_term,(n,1,num_terms))
pprint(s)
def discrete_fourier_sympy(expr, n, k, N):
foo = expr * sym.exp(-2 * j * pi * n * k / N)
result = sym.summation(foo, (n, 0, N - 1))
return result
def compute_moment_generating_function(self, **kwargs):
x, t = symbols('x, t', real=True, finite=True, cls=Dummy)
pdf = self.pdf(x)
mgf = summation(exp(t * x) * pdf, (x, self.set.inf, self.set.sup))
return Lambda(t, mgf)
def test_issue_1072():
k = Symbol("k")
assert summation(factorial(2 * k + 1) / factorial(2 * k), (k, 0, oo)) == oo
def test_Sum_doit():
assert Sum(n * Integral(a**2), (n, 0, 2)).doit() == a**3
assert Sum(n*Integral(a**2), (n, 0, 2)).doit(deep = False) == \
3*Integral(a**2)
assert summation(n * Integral(a**2), (n, 0, 2)) == 3 * Integral(a**2)
def test_geometric_sums():
assert summation(pi**n, (n, 0, b)) == (1 - pi**(b + 1)) / (1 - pi)
assert summation(2 * 3**n, (n, 0, b)) == 3**(b + 1) - 1
assert summation(Rational(1, 2)**n, (n, 1, oo)) == 1
assert summation(2**n, (n, 0, b)) == 2**(b + 1) - 1
assert summation(2**n, (n, 1, oo)) == oo
assert summation(2**(-n), (n, 1, oo)) == 1
assert summation(3**(-n), (n, 4, oo)) == Rational(1, 54)
assert summation(2**(-4 * n + 3), (n, 1, oo)) == Rational(8, 15)
assert summation(2**(n + 1), (n, 1, b)).expand() == 4 * (2**b - 1)
# issue 6664:
assert summation(x**n, (n, 0, oo)) == \
Piecewise((1/(-x + 1), Abs(x) < 1), (Sum(x**n, (n, 0, oo)), True))
assert summation(-2**n, (n, 0, oo)) == -oo
assert summation(I**n, (n, 0, oo)) == Sum(I**n, (n, 0, oo))
# issue 6802:
assert summation((-1)**(2 * x + 2), (x, 0, n)) == n + 1
assert summation((-2)**(2 * x + 2),
(x, 0, n)) == 4 * 4**(n + 1) / S(3) - S(4) / 3
assert summation((-1)**x, (x, 0, n)) == -(-1)**(n + 1) / S(2) + S(1) / 2
assert summation(y**x, (x, a, b)) == \
Piecewise((-a + b + 1, Eq(y, 1)), ((y**a - y**(b + 1))/(-y + 1), True))
assert summation((-2)**(y*x + 2), (x, 0, n)) == \
4*Piecewise((n + 1, Eq((-2)**y, 1)),
((-(-2)**(y*(n + 1)) + 1)/(-(-2)**y + 1), True))
# issue 8251:
assert summation((1 / (n + 1)**2) * n**2, (n, 0, oo)) == oo
#issue 9908:
assert Sum(1 / (n**3 - 1),
(n, -oo, -2)).doit() == summation(1 / (n**3 - 1), (n, -oo, -2))
#issue 11642:
result = Sum(0.5**n, (n, 1, oo)).doit()
assert result == 1
assert result.is_Float
result = Sum(0.25**n, (n, 1, oo)).doit()
assert result == S(1) / 3
assert result.is_Float
result = Sum(0.99999**n, (n, 1, oo)).doit()
assert result == 99999
assert result.is_Float
result = Sum(Rational(1, 2)**n, (n, 1, oo)).doit()
assert result == 1
assert not result.is_Float
result = Sum(Rational(3, 5)**n, (n, 1, oo)).doit()
assert result == S(3) / 2
assert not result.is_Float
assert Sum(1.0**n, (n, 1, oo)).doit() == oo
assert Sum(2.43**n, (n, 1, oo)).doit() == oo
def test_issue_4668():
assert summation(1/n, (n, 2, oo)) == oo
