def test_one_dof():
# This is for a 1 dof spring-mass-damper case.
# It is described in more detail in the Kane docstring.
q, u = dynamicsymbols('q u')
qd, ud = dynamicsymbols('q u', 1)
m, c, k = symbols('m c k')
N = ReferenceFrame('N')
P = Point('P')
P.set_vel(N, u * N.x)
kd = [qd - u]
FL = [(P, (-k * q - c * u) * N.x)]
pa = Particle()
pa.mass = m
pa.point = P
BL = [pa]
KM = Kane(N)
KM.coords([q])
KM.speeds([u])
KM.kindiffeq(kd)
KM.kanes_equations(FL, BL)
MM = KM.mass_matrix
forcing = KM.forcing
rhs = MM.inv() * forcing
assert expand(rhs[0]) == expand(-(q * k + u * c) / m)
assert KM.linearize() == (Matrix([[0, 1], [k, c]]), Matrix([]))
def test_two_dof():
# This is for a 2 d.o.f., 2 particle spring-mass-damper.
# The first coordinate is the displacement of the first particle, and the
# second is the relative displacement between the first and second
# particles. Speeds are defined as the time derivatives of the particles.
q1, q2, u1, u2 = dynamicsymbols('q1 q2 u1 u2')
q1d, q2d, u1d, u2d = dynamicsymbols('q1 q2 u1 u2', 1)
m, c1, c2, k1, k2 = symbols('m c1 c2 k1 k2')
N = ReferenceFrame('N')
P1 = Point('P1')
P2 = Point('P2')
P1.set_vel(N, u1 * N.x)
P2.set_vel(N, (u1 + u2) * N.x)
kd = [q1d - u1, q2d - u2]
# Now we create the list of forces, then assign properties to each
# particle, then create a list of all particles.
FL = [(P1, (-k1 * q1 - c1 * u1 + k2 * q2 + c2 * u2) * N.x), (P2, (-k2 *
q2 - c2 * u2) * N.x)]
pa1 = Particle('pa1', P1, m)
pa2 = Particle('pa2', P2, m)
BL = [pa1, pa2]
# Finally we create the KanesMethod object, specify the inertial frame,
# pass relevant information, and form Fr & Fr*. Then we calculate the mass
# matrix and forcing terms, and finally solve for the udots.
KM = KanesMethod(N, q_ind=[q1, q2], u_ind=[u1, u2], kd_eqs=kd)
KM.kanes_equations(FL, BL)
MM = KM.mass_matrix
forcing = KM.forcing
rhs = MM.inv() * forcing
assert expand(rhs[0]) == expand((-k1 * q1 - c1 * u1 + k2 * q2 + c2 * u2)/m)
assert expand(rhs[1]) == expand((k1 * q1 + c1 * u1 - 2 * k2 * q2 - 2 *
c2 * u2) / m)
def test_pend():
q, u = dynamicsymbols('q u')
qd, ud = dynamicsymbols('q u', 1)
m, l, g = symbols('m l g')
N = ReferenceFrame('N')
P = Point('P')
P.set_vel(N, -l * u * sin(q) * N.x + l * u * cos(q) * N.y)
kd = [qd - u]
FL = [(P, m * g * N.x)]
pa = Particle()
pa.mass = m
pa.point = P
BL = [pa]
KM = Kane(N)
KM.coords([q])
KM.speeds([u])
KM.kindiffeq(kd)
KM.kanes_equations(FL, BL)
MM = KM.mass_matrix
forcing = KM.forcing
rhs = MM.inv() * forcing
rhs.simplify()
assert expand(rhs[0]) == expand(-g / l * sin(q))
def _solve_recur(expr, rhs=S(0)):
if expr == var:
return rhs
expr = expand(expr)
if isinstance(expr, Mul):
lhs = S(1)
# Try by rank
l_coeff, var_expr, r_coeff = expr_coeff(expr, var)
lhs = var_expr
rhs = Inverse(l_coeff) * rhs * Inverse(r_coeff)
return _solve_recur(lhs, rhs)
if isinstance(expr, Add):
lhs = 0
for arg in expr.args:
if var in arg:
lhs += arg
else:
rhs -= arg
if isinstance(lhs, Add):
coeff = lhs.coeff(var)
if expand(coeff * var) == lhs:
rhs /= coeff
lhs = var
return _solve_recur(lhs, rhs)
if isinstance(expr, Transpose):
return _solve_recur(expr.args[0], Transpose(rhs))
raise NotImplementedError("Can't handle expr of type %s" % type(expr))
def test_fast_substitute(self):
f = generate_variables(2, name="f")
substitutions = {}
substitutions[Dagger(f[0]) * f[0]] = -f[0] * Dagger(f[0])
monomial = Dagger(f[0]) * f[0]
lhs = Dagger(f[0]) * f[0]
rhs = -f[0] * Dagger(f[0])
self.assertTrue(fast_substitute(monomial, lhs, rhs) == monomial.subs(lhs, rhs))
monomial = Dagger(f[0]) * f[0] ** 2
self.assertTrue(fast_substitute(monomial, lhs, rhs) == monomial.subs(lhs, rhs))
monomial = Dagger(f[0]) ** 2 * f[0]
self.assertTrue(fast_substitute(monomial, lhs, rhs) == monomial.subs(lhs, rhs))
monomial = Dagger(f[0]) ** 2 * f[0]
lhs = Dagger(f[0]) ** 2
rhs = -f[0] * Dagger(f[0])
self.assertTrue(fast_substitute(monomial, lhs, rhs) == monomial.subs(lhs, rhs))
g = generate_variables(2, name="g")
monomial = 2 * g[0] ** 3 * g[1] * Dagger(f[0]) ** 2 * f[0]
self.assertTrue(fast_substitute(monomial, lhs, rhs) == monomial.subs(lhs, rhs))
monomial = S.One
self.assertTrue(fast_substitute(monomial, lhs, rhs) == monomial.subs(lhs, rhs))
monomial = 5
self.assertTrue(fast_substitute(monomial, lhs, rhs) == monomial)
monomial = 2 * g[0] ** 3 * g[1] * Dagger(f[0]) ** 2 * f[0] + f[1]
self.assertTrue(fast_substitute(monomial, lhs, rhs) == monomial.subs(lhs, rhs))
monomial = f[1] * Dagger(f[0]) ** 2 * f[0]
lhs = f[1]
rhs = 1.0 + f[0]
self.assertTrue(fast_substitute(monomial, lhs, rhs) == expand(monomial.subs(lhs, rhs)))
monomial = f[1] ** 2 * Dagger(f[0]) ** 2 * f[0]
result = fast_substitute(fast_substitute(monomial, lhs, rhs), lhs, rhs)
self.assertTrue(result == expand(monomial.subs(lhs, rhs)))
def main():
print(symbols('a b c d'))
a, b, c, d = symbols('a b c d')
group4 = [a, b, c, d]
print(group4)
four = group_solve(group4)
print(four)
print(expand(four))
print(four.collect(a))
print(four.subs(a, 10))
print()
group4_ = add1(group4)
print(group4_)
four1 = group_solve(group4_)
print(four1)
print(expand(four1))
print(four1.collect(a))
print()
print(13717421)
for i in range(1, 10 ** 4):
sum_power(i)
def test_apply_substitutions(self):
def apply_correct_substitutions(monomial, substitutions):
if isinstance(monomial, int) or isinstance(monomial, float):
return monomial
original_monomial = monomial
changed = True
while changed:
for lhs, rhs in substitutions.items():
monomial = monomial.subs(lhs, rhs)
if original_monomial == monomial:
changed = False
original_monomial = monomial
return monomial
length, h, U, t = 2, 3.8, -6, 1
fu = generate_operators('fu', length)
fd = generate_operators('fd', length)
_b = flatten([fu, fd])
hamiltonian = 0
for j in range(length):
hamiltonian += U * (Dagger(fu[j])*Dagger(fd[j]) * fd[j]*fu[j])
hamiltonian += -h/2*(Dagger(fu[j])*fu[j] - Dagger(fd[j])*fd[j])
for k in get_neighbors(j, len(fu), width=1):
hamiltonian += -t*Dagger(fu[j])*fu[k]-t*Dagger(fu[k])*fu[j]
hamiltonian += -t*Dagger(fd[j])*fd[k]-t*Dagger(fd[k])*fd[j]
substitutions = fermionic_constraints(_b)
monomials = expand(hamiltonian).as_coeff_mul()[1][0].as_coeff_add()[1]
substituted_hamiltonian = sum([apply_substitutions(monomial,
substitutions)
for monomial in monomials])
correct_hamiltonian = sum([apply_correct_substitutions(monomial,
substitutions)
for monomial in monomials])
self.assertTrue(substituted_hamiltonian == expand(correct_hamiltonian))
def case0(f, N=3):
B = 1 - x ** 3
dBdx = sm.diff(B, x)
# Compute basis functions and their derivatives
phi = {0: [x ** (i + 1) * (1 - x) for i in range(N + 1)]}
phi[1] = [sm.diff(phi_i, x) for phi_i in phi[0]]
def integrand_lhs(phi, i, j):
return phi[1][i] * phi[1][j]
def integrand_rhs(phi, i):
return f * phi[0][i] - dBdx * phi[1][i]
Omega = [0, 1]
u_bar = solve(integrand_lhs, integrand_rhs, phi, Omega, verbose=True, numint=False)
u = B + u_bar
print "solution u:", sm.simplify(sm.expand(u))
# Calculate analytical solution
# Solve -u''=f by integrating f twice
f1 = sm.integrate(f, x)
f2 = sm.integrate(f1, x)
# Add integration constants
C1, C2 = sm.symbols("C1 C2")
u_e = -f2 + C1 * x + C2
# Find C1 and C2 from the boundary conditions u(0)=0, u(1)=1
s = sm.solve([u_e.subs(x, 0) - 1, u_e.subs(x, 1) - 0], [C1, C2])
# Form the exact solution
u_e = -f2 + s[C1] * x + s[C2]
print "analytical solution:", u_e
# print 'error:', u - u_e # many terms - which cancel
print "error:", sm.expand(u - u_e)
def _interpolate_expression(expr):
change = False
expr = sympy.expand(expr)
res = expr
if isinstance(expr, sympy.Function):# and not change:
path = np.array([])
temp, change = _interpolate_Function(_follow_path(expr, path))
res = _interchange(expr, temp, path)
for i in np.arange(len(expr.args)):
path = np.array([i])
if isinstance(_follow_path(expr, path), sympy.Function) and not change:
temp, change = _interpolate_Function(_follow_path(expr, path))
res = _interchange(expr, temp, path)
for j in np.arange(len(expr.args[i].args)):
path = np.array([i, j])
if isinstance(_follow_path(expr, path), sympy.Function) and not change:
temp, change = _interpolate_Function(_follow_path(expr, path))
res = _interchange(expr, temp, path)
if change:
res = _interpolate_expression(res)
return sympy.expand(res)
def _interpolate_Function(expr):
path = 'None'
factor = 'None'
change = False
summand_0 = 'None'
summand_1 = 'None'
res = expr
for i in np.arange(len(expr.args)):
argument = sympy.expand(expr.args[i])
if argument.func == sympy.Add:
for j in np.arange(len(argument.args)):
summand = argument.args[j]
if summand.func == sympy.Mul:
for k in np.arange(len(summand.args)):
temp = 0
if summand.args[k] == sympy.Symbol('a'):
temp = sympy.Mul(sympy.Mul(*summand.args[:k]),
sympy.Mul(*summand.args[k+1:]))
#print(temp)
if not temp == int(temp):
#print('found one')
factor = (temp)
path = np.array([i, j, k])
if not factor == 'None':
change = True
sign = np.sign(factor)
offsets = np.array([int(factor), sign * (int(sign * factor) + 1)])
weights = 1/np.abs(offsets - factor)
weights = weights/np.sum(weights)
res = ( weights[0] * _interchange(expr, offsets[0] * sympy.Symbol('a'), path[:-1])
+ weights[1] * _interchange(expr, offsets[1] * sympy.Symbol('a'), path[:-1]))
return sympy.expand(res), change
def tests():
x, y, z = symbols('x,y,z')
#print(x + x + 1)
expr = x**2 - y**2
factors = factor(expr)
print(factors, " | ", expand(factors))
pprint(expand(factors))
def gen_errs(self, a, b, n, x, y):
"""
What is the best thing to do here?
"""
errs = set()
# Generate som obvious candidates
errs.add(sym.expand(self.gen_prob(-a, -b, n, x, y)))
if a < 0:
errs.add(sym.expand(self.gen_prob(-a, b, n, x, y)))
if b < 0:
errs.add(sym.expand(self.gen_prob(a, -b, n, x, y)))
expr = sym.expand(self.gen_prob(a, b, n, x, y))
coeffs = sym.Poly(expr).coeffs()
for i in range(4):
expr1 = self.gen_poly(map(lambda l: random.choice([-1, 1]) * l, coeffs), x, y)
expr2 = self.gen_poly(map(lambda l: random.choice([-2, -1, 0, 1, 2]) * l, coeffs), x, y)
errs.update([expr1, expr2, expr - (expr2 - sym.LM(expr))])
errs = list(errs)
errs_ = [err for err in errs if err != expr]
random.shuffle(errs_)
return errs
def test_SVCVS_laplace_d3_n1():
"""Test VCCS with a laplace defined transfer function with second order
numerator and third order denominator
"""
pycircuit.circuit.circuit.default_toolkit = symbolic
cir = SubCircuit()
n1,n2 = cir.add_nodes('1','2')
b0,a0,a1,a2,a3,Gdc = [sympy.Symbol(symname, real=True) for
symname in 'b0,a0,a1,a2,a3,Gdc'
.split(',')]
s = sympy.Symbol('s', complex=True)
cir['VS'] = VS( n1, gnd, vac=1)
cir['VCVS'] = SVCVS( n1, gnd, n2, gnd,
denominator = [a0, a1, a2, a3],
numerator = [b0, 0, 0])
res = AC(cir, toolkit=symbolic).solve(s, complexfreq=True)
assert_equal(sympy.cancel(sympy.expand(res.v(n2,gnd))),
sympy.expand((b0*s*s)/(a0*s*s*s+a1*s*s+a2*s+a3)))
def alphaBetaToTransfer(adjMat, alphas, betas):
s = sympy.Symbol('s')
subs = {}
# iterate over the columns of the adjacency matrix
for i, col in enumerate(adjMat.getA()):
expr = sympy.S(0)
i += 1
# iterate over the elements in each column of the adjacency matrix
# this is to build the replacement expression for what were once "output to env"s
# but are now the turnover rates of each node
for j, elem in enumerate(col):
j += 1
if elem != 0:
expr += (-sympy.var('a_%d%d' % (j,i)))
subs[sympy.var('a_%d%d' % (i,i))] = expr
#print "*** VAR->EXPR SUBSTITUTIONS: "
#print subs
for pos in range(0, len(betas)):
orderedKeysBeta = getOrderedKeys(betas[pos])
for key in orderedKeysBeta:
betas[pos][s**key] = sympy.simplify(sympy.expand(betas[pos][s**key].xreplace(subs)))
#finished all the betas so add the alphas, but only once
orderedKeysAlpha = getOrderedKeys(alphas[pos])
for key in orderedKeysAlpha:
alphas[pos][s**key] = sympy.simplify(sympy.expand(alphas[pos][s**key].xreplace(subs)))
def T_exact_symbolic(verbose=False):
"""Compute the exact solution formula via sympy."""
# sol1: solution for t < t_star,
# sol2: solution for t > t_star
import sympy as sym
T0 = sym.symbols('T0')
k = sym.symbols('k', positive=True)
# Piecewise linear T_sunction
t, t_star, C0, C1 = sym.symbols('t t_star C0 C1')
T_s = C0
I = sym.integrate(sym.exp(k*t)*T_s, (t, 0, t))
sol1 = T0*sym.exp(-k*t) + k*sym.exp(-k*t)*I
sol1 = sym.simplify(sym.expand(sol1))
if verbose:
# Some debugging print
print 'solution t < t_star:', sol1
#print sym.latex(sol1)
T_s = C1
I = sym.integrate(sym.exp(k*t)*C0, (t, 0, t_star)) + \
sym.integrate(sym.exp(k*t)*C1, (t, t_star, t))
sol2 = T0*sym.exp(-k*t) + k*sym.exp(-k*t)*I
sol2 = sym.simplify(sym.expand(sol2))
if verbose:
print 'solution t > t_star:', sol2
#print sym.latex(sol2)
# Convert to numerical functions
exact0 = sym.lambdify([t, C0, k, T0],
sol1, modules='numpy')
exact1 = sym.lambdify([t, C0, C1, t_star, k, T0],
sol2, modules='numpy')
return exact0, exact1
def test_order_noncommutative():
A = Symbol("A", commutative=False)
assert Order(A + A * x, x) == Order(1, x)
assert (A + A * x) * Order(x) == Order(x)
assert (A * x) * Order(x) == Order(x ** 2, x)
assert expand((1 + Order(x)) * A * A * x) == A * A * x + Order(x ** 2, x)
assert expand((A * A + Order(x)) * x) == A * A * x + Order(x ** 2, x)
assert expand((A + Order(x)) * A * x) == A * A * x + Order(x ** 2, x)
def test_piecewise_fold_expand():
p1 = Piecewise((1,Interval(0,1,False,True)),(0,True))
p2 = piecewise_fold(expand((1-x)*p1))
assert p2 == Piecewise((1 - x, Interval(0,1,False,True)), \
(Piecewise((-x, Interval(0,1,False,True)), (0, True)), True))
p2 = expand(piecewise_fold((1-x)*p1))
assert p2 == Piecewise((1 - x, Interval(0,1,False,True)), (0, True))
def test_meijerint():
from sympy import symbols, expand, arg
s, t, mu = symbols("s t mu", real=True)
assert integrate(
meijerg([], [], [0], [], s * t) * meijerg([], [], [mu / 2], [-mu / 2], t ** 2 / 4), (t, 0, oo)
).is_Piecewise
s = symbols("s", positive=True)
assert integrate(x ** s * meijerg([[], []], [[0], []], x), (x, 0, oo)) == gamma(s + 1)
assert integrate(x ** s * meijerg([[], []], [[0], []], x), (x, 0, oo), meijerg=True) == gamma(s + 1)
assert isinstance(integrate(x ** s * meijerg([[], []], [[0], []], x), (x, 0, oo), meijerg=False), Integral)
assert meijerint_indefinite(exp(x), x) == exp(x)
# TODO what simplifications should be done automatically?
# This tests "extra case" for antecedents_1.
a, b = symbols("a b", positive=True)
assert simplify(meijerint_definite(x ** a, x, 0, b)[0]) == b ** (a + 1) / (a + 1)
# This tests various conditions and expansions:
meijerint_definite((x + 1) ** 3 * exp(-x), x, 0, oo) == (16, True)
# Again, how about simplifications?
sigma, mu = symbols("sigma mu", positive=True)
i, c = meijerint_definite(exp(-((x - mu) / (2 * sigma)) ** 2), x, 0, oo)
assert simplify(i) == sqrt(pi) * sigma * (erf(mu / (2 * sigma)) + 1)
assert c is True
i, _ = meijerint_definite(exp(-mu * x) * exp(sigma * x), x, 0, oo)
# TODO it would be nice to test the condition
assert simplify(i) == 1 / (mu - sigma)
# Test substitutions to change limits
assert meijerint_definite(exp(x), x, -oo, 2) == (exp(2), True)
assert expand(meijerint_definite(exp(x), x, 0, I)[0]) == exp(I) - 1
assert expand(meijerint_definite(exp(-x), x, 0, x)[0]) == 1 - exp(-exp(I * arg(x)) * abs(x))
# Test -oo to oo
assert meijerint_definite(exp(-x ** 2), x, -oo, oo) == (sqrt(pi), True)
assert meijerint_definite(exp(-abs(x)), x, -oo, oo) == (2, True)
assert meijerint_definite(exp(-(2 * x - 3) ** 2), x, -oo, oo) == (sqrt(pi) / 2, True)
assert meijerint_definite(exp(-abs(2 * x - 3)), x, -oo, oo) == (1, True)
assert meijerint_definite(exp(-((x - mu) / sigma) ** 2 / 2) / sqrt(2 * pi * sigma ** 2), x, -oo, oo) == (1, True)
# Test one of the extra conditions for 2 g-functinos
assert meijerint_definite(exp(-x) * sin(x), x, 0, oo) == (S(1) / 2, True)
# Test a bug
def res(n):
return (1 / (1 + x ** 2)).diff(x, n).subs(x, 1) * (-1) ** n
for n in range(6):
assert integrate(exp(-x) * sin(x) * x ** n, (x, 0, oo), meijerg=True) == res(n)
# Test trigexpand:
assert integrate(exp(-x) * sin(x + a), (x, 0, oo), meijerg=True) == sin(a) / 2 + cos(a) / 2
def _w_diff_dcm(self, otherframe):
"""Angular velocity from time differentiating the DCM. """
from sympy.physics.vector.functions import dynamicsymbols
dcm2diff = self.dcm(otherframe)
diffed = dcm2diff.diff(dynamicsymbols._t)
angvelmat = diffed * dcm2diff.T
w1 = trigsimp(expand(angvelmat[7]), recursive=True)
w2 = trigsimp(expand(angvelmat[2]), recursive=True)
w3 = trigsimp(expand(angvelmat[3]), recursive=True)
return -Vector([(Matrix([w1, w2, w3]), self)])
def __new__(cls, *args, **old_assumptions):
from sympy.physics.quantum.state import KetBase, BraBase
if len(args) != 2:
raise ValueError('2 parameters expected, got %d' % len(args))
ket_expr = expand(args[0])
bra_expr = expand(args[1])
if (isinstance(ket_expr, (KetBase, Mul)) and
isinstance(bra_expr, (BraBase, Mul))):
ket_c, kets = ket_expr.args_cnc()
bra_c, bras = bra_expr.args_cnc()
if len(kets) != 1 or not isinstance(kets[0], KetBase):
raise TypeError('KetBase subclass expected'
', got: %r' % Mul(*kets))
if len(bras) != 1 or not isinstance(bras[0], BraBase):
raise TypeError('BraBase subclass expected'
', got: %r' % Mul(*bras))
if not kets[0].dual_class() == bras[0].__class__:
raise TypeError(
'ket and bra are not dual classes: %r, %r' %
(kets[0].__class__, bras[0].__class__)
)
# TODO: make sure the hilbert spaces of the bra and ket are
# compatible
obj = Expr.__new__(cls, *(kets[0], bras[0]), **old_assumptions)
obj.hilbert_space = kets[0].hilbert_space
return Mul(*(ket_c + bra_c)) * obj
op_terms = []
if isinstance(ket_expr, Add) and isinstance(bra_expr, Add):
for ket_term in ket_expr.args:
for bra_term in bra_expr.args:
op_terms.append(OuterProduct(ket_term, bra_term,
**old_assumptions))
elif isinstance(ket_expr, Add):
for ket_term in ket_expr.args:
op_terms.append(OuterProduct(ket_term, bra_expr,
**old_assumptions))
elif isinstance(bra_expr, Add):
for bra_term in bra_expr.args:
op_terms.append(OuterProduct(ket_expr, bra_term,
**old_assumptions))
else:
raise TypeError(
'Expected ket and bra expression, got: %r, %r' %
(ket_expr, bra_expr)
)
return Add(*op_terms)
def test_input_format():
# 1 dof problem from test_one_dof
q, u = dynamicsymbols('q u')
qd, ud = dynamicsymbols('q u', 1)
m, c, k = symbols('m c k')
N = ReferenceFrame('N')
P = Point('P')
P.set_vel(N, u * N.x)
kd = [qd - u]
FL = [(P, (-k * q - c * u) * N.x)]
pa = Particle('pa', P, m)
BL = [pa]
KM = KanesMethod(N, [q], [u], kd)
# test for input format kane.kanes_equations((body1, body2, particle1))
assert KM.kanes_equations(BL)[0] == Matrix([0])
# test for input format kane.kanes_equations(bodies=(body1, body 2), loads=(load1,load2))
assert KM.kanes_equations(bodies=BL, loads=None)[0] == Matrix([0])
# test for input format kane.kanes_equations(bodies=(body1, body 2), loads=None)
assert KM.kanes_equations(BL, loads=None)[0] == Matrix([0])
# test for input format kane.kanes_equations(bodies=(body1, body 2))
assert KM.kanes_equations(BL)[0] == Matrix([0])
# test for error raised when a wrong force list (in this case a string) is provided
from sympy.utilities.pytest import raises
raises(ValueError, lambda: KM._form_fr('bad input'))
# 2 dof problem from test_two_dof
q1, q2, u1, u2 = dynamicsymbols('q1 q2 u1 u2')
q1d, q2d, u1d, u2d = dynamicsymbols('q1 q2 u1 u2', 1)
m, c1, c2, k1, k2 = symbols('m c1 c2 k1 k2')
N = ReferenceFrame('N')
P1 = Point('P1')
P2 = Point('P2')
P1.set_vel(N, u1 * N.x)
P2.set_vel(N, (u1 + u2) * N.x)
kd = [q1d - u1, q2d - u2]
FL = ((P1, (-k1 * q1 - c1 * u1 + k2 * q2 + c2 * u2) * N.x), (P2, (-k2 *
q2 - c2 * u2) * N.x))
pa1 = Particle('pa1', P1, m)
pa2 = Particle('pa2', P2, m)
BL = (pa1, pa2)
KM = KanesMethod(N, q_ind=[q1, q2], u_ind=[u1, u2], kd_eqs=kd)
# test for input format
# kane.kanes_equations((body1, body2), (load1, load2))
KM.kanes_equations(BL, FL)
MM = KM.mass_matrix
forcing = KM.forcing
rhs = MM.inv() * forcing
assert expand(rhs[0]) == expand((-k1 * q1 - c1 * u1 + k2 * q2 + c2 * u2)/m)
assert expand(rhs[1]) == expand((k1 * q1 + c1 * u1 - 2 * k2 * q2 - 2 *
c2 * u2) / m)
def hermitiana_1D(x, n):
resultado1 = lagrangiana_1D(x, n)
resultado2 = []
puntos2 = np.linspace(-1, 1, n)
for i in range(np.size(puntos2)):
A = x - puntos2[i]
B = (resultado1[i].diff(x)).subs(x, puntos2[i])
C = resultado1[i]**2
D = (1 - (A*B))*C
resultado2.append(expand(D))
resultado2.append(expand(A*C))
return resultado2
def lagrange(nodes):
tmp_x_nodes = [node['X']for node in nodes]
tmp_y_nodes = [node['Y']for node in nodes]
x_nodes = np.asarray(tmp_x_nodes,dtype=np.float)
y_nodes = np.asarray(tmp_y_nodes,dtype=np.float)
x = sp.symbols('x')
y = sp.symbols('y')
pol_base = []
pol_lag = []
for i in range(len(x_nodes)):
LUp = []
LDown = []
for j in range(len(nodes)):
if j != i:
express=(x-x_nodes[j])
express_1=(x_nodes[i]-x_nodes[j])
LUp.append("{" + str(express) + " \over " + str(express_1) + "}")
LDown.append(express/express_1)
pol_base.append([i, LUp])
pol_lag.append(LDown)
#print pol_base
P=0
product_result = []
for n1 in range(len(pol_lag)):
test = 1
for n2 in range(len(pol_lag[n1])):
test = test * pol_lag[n1][n2]
ptoria = sp.expand(test)
product_result.append([n1, sp.latex(ptoria), y_nodes[n1]])
P = P + (test * y_nodes[n1])
expt = sp.expand(P)
latx = sp.latex(expt)
response_data = {
'result': latx,
'basePol': pol_base,
'sumProduct': product_result,
}
return response_data
def run_taylor_leastsq_parabola_illconditioning(N=2):
"""
Test Taylor approx to a parabola and exact symbolics vs
ill-conditioned numerical approaches.
"""
f = 10*(x-1)**2 - 1
u = least_squares(f, psi=[x**i for i in range(N+1)], Omega=[1, 2])
# Note: in least_squares there is extra code for numerical solution
# of the systems
print 'f:', sm.expand(f)
print 'u:', sm.expand(u)
comparison_plot(f, u, [1, 2], 'parabola_ls_taylor%d.pdf' % N)
def case3():
a, c, x1, x2 = symbols('a c x1 x2') # Define symbols
b = 3 * (a - 1) # Relation between a and b
y = a * x1 + b * x2 + 5 # Given function y
# Error of x1 and x2
Sigx1 = 0.8
Sigx2 = 1.5
print "In the case that x1 and x2 are partial dependent:\n"
Sigx1x2 = c # Correlation coefficient equals to an unknown parameter c
SigmaXX = np.matrix([[Sigx1**2, Sigx1x2], # Define covariance
[Sigx1x2, Sigx2**2]]) # matrix of x1 and x2
# Jacobian matrix of y function with respect to x1 and x2
Jyx = np.matrix([diff(y, x1), diff(y, x2)])
SigmaYY = Jyx * SigmaXX * Jyx.T # Compute covariance matrix of y function
# Compute solution in witch the SigmaYY is minimum
dc = diff(expand(SigmaYY[0, 0]), c) # Derivative SigmaYY with respect
# to c
a1, a2 = solve(dc) # Get two solutions of a
# Compute results with a = 0
print "Parameter a has two solutions [%d, %d]\nIf a = %d:" % (a1, a2, a1)
# Get solution of c with a = 0
c1 = solve(diff(expand(SigmaYY[0, 0]), a), c)[0].evalf(subs={'a': a1})
b1 = round(b.evalf(subs={'a': a1}), 8)
# Compute sigma y with a = 0
SigmaY1 = sqrt(SigmaYY[0, 0].evalf(subs={'a': a1, 'b': b1, 'c': c1}))
print "SigmaXX =\n"
pprint(Matrix(SigmaXX).evalf(subs={'c': round(c1, 4)}))
print
print "a = %.4f\nb = %.4f\nc = %.4f" % (a1, b1, c1)
print "SigmaY = %.4f\n" % float(SigmaY1)
# Compute results with a = 1
print "If a = %d:" % a2
# Get solution of c with a = 1
c2 = solve(diff(expand(SigmaYY[0, 0]), a), c)[0].evalf(subs={'a': a2})
b2 = round(b.evalf(subs={'a': a2}), 8)
# Compute sigma y with a = 1
SigmaY2 = sqrt(SigmaYY[0, 0].evalf(subs={'a': a2, 'b': b2, 'c': c2}))
print "SigmaXX =\n"
pprint(Matrix(SigmaXX).evalf(subs={'c': round(c2, 4)}))
print
print "a = %.4f\nb = %.4f\nc = %.4f" % (a2, b2, c2)
print "SigmaY = %.4f" % float(SigmaY2)
def adj(self):
self_adj = []
self.Ga.dot_mode = '|'
for e_j in self.Ga.basis:
s = S(0)
for (e_i, er_i) in zip(self.Ga.basis, self.Ga.r_basis):
s += er_i * self.Ga.dot(e_j, self(e_i, obj=True))
if self.Ga.is_ortho:
self_adj.append(expand(s))
else:
self_adj.append(expand(s) / self.Ga.inorm)
return Lt(self_adj, ga=self.Ga)
def main():
x, y = symbols("x y")
print(x + y - x)
print(x + 2 * y - 6 * x - 5 * x)
expr = (x + x * y) / x
print(simplify(expr))
expr = x + y
expr = x * expr
print(expr)
print(expand(expr))
print(factor(expand(expr)))
def main(argv):
_expression = None
_filePath = None
try:
opts, args = getopt.getopt(argv, "he:f:", ["help", "expression=" ,"filePath="])
except getopt.GetoptError:
usage()
sys.exit(2)
for opt, arg in opts:
if opt in ("-h", "--help"):
usage()
sys.exit(2)
elif opt in ("-e", "--expression"):
_expression = arg
elif opt in ("-f", "--filePath"):
_filePath = arg
if _expression:
try:
s = normalize(_expression)
except ValueError as errorMessage:
print(errorMessage)
return -1
try:
s = sympy.expand(s)
except SyntaxError:
print("Выражение не может быть приведено к стандартному виду.")
return -2
print(prettify(str(s)))
if _filePath:
with open(_filePath, 'r') as file:
for line in file:
try:
s = normalize(line)
except ValueError as errorMessage:
print(errorMessage)
else:
try:
s = sympy.expand(s)
except SyntaxError:
print("Выражение не может быть приведено к стандартному виду.")
else:
print(prettify(str(s)))
return 0
def compute_cg(self):
"""
Compute center of gravity
- return a sympy array with x and y coordinates
"""
x_cg = sum([self.g.node[n]["area"] * self.g.node[n]["ip"][0] for n in self.g.nodes()]) / sum(
[self.g.node[n]["area"] for n in self.g.nodes()]
)
y_cg = sum([self.g.node[n]["area"] * self.g.node[n]["ip"][1] for n in self.g.nodes()]) / sum(
[self.g.node[n]["area"] for n in self.g.nodes()]
)
return sympy.Matrix([sympy.simplify(sympy.expand(x_cg)), sympy.simplify(sympy.expand(y_cg))])
for k, v in kde_map.items():
kde_map[k.diff(t)] = v.diff(t)
# f1, f2 are forces the panes of glass exert on P1, P2 respectively
R1 = f1*B.z + C*E.x - m1*g*B.y
R2 = f2*B.z - C*E.x - m2*g*B.y
forces = [(pP1, R1), (pP2, R2)]
system = [Particle('P1', pP1, m1), Particle('P2', pP2, m2)]
partials = partial_velocities([pP1, pP2], u, A, kde_map)
Fr, _ = generalized_active_forces(partials, forces)
Fr_star, _ = generalized_inertia_forces(partials, system, kde_map)
# dynamical equations
dyn_eq = [x + y for x, y in zip(Fr, Fr_star)]
u1d, u2d, u3d = ud = [x.diff(t) for x in u]
dyn_eq_map = solve(dyn_eq, ud)
for x in ud:
print('{0} = {1}'.format(msprint(x),
msprint(cancel(trigsimp(dyn_eq_map[x])))))
u1d_expected = (-g*sin(q3) + omega**2*q1*cos(q3) + u2*u3 +
(omega**2*cos(q3)**2 + u3**2)*L*m2/(m1 + m2))
u2d_expected = -g*cos(q3) - (omega**2*q1*sin(q3) + u3*u1)
u3d_expected = -omega**2*sin(q3)*cos(q3)
assert expand(cancel(expand_trig(dyn_eq_map[u1d] - u1d_expected))) == 0
assert expand(cancel(expand_trig(dyn_eq_map[u2d] - u2d_expected))) == 0
assert expand(expand_trig(dyn_eq_map[u3d] - u3d_expected)) == 0
def expand(self):
self.fvalue = expand(self.fvalue)
return self
def LSLF(equation, fit_par, ind_var, x_data_point, f_data_point):
par_sym = [sy.Symbol(fit_par[k]) for k in range(len(fit_par))]
var_sym = [sy.Symbol(ind_var[k]) for k in range(len(ind_var))]
f = parse_expr(equation)
f_true = sy.Symbol('f_true')
def QE(f):
return 1 / 2 * (f - f_true)**2
E = QE(f)
dE = [sy.diff(E, par_sym[k]) for k in range(len(par_sym))]
dE_ex = [sy.expand(dE[k]) for k in range(len(par_sym))]
dE_ex_arg = [dE_ex[k].args for k in range(len(par_sym))]
A_sym = sy.eye(len(par_sym))
B_sym = sy.zeros(len(par_sym), 1)
for k in range(len(par_sym)):
for i in range(len(dE_ex_arg[k])):
comp = dE_ex_arg[k][i]
kk = 0
for l in range(len(par_sym)):
if par_sym[l] in comp.free_symbols:
A_sym[k, l] = comp / par_sym[l]
else:
kk = kk + 1
if kk == len(par_sym):
B_sym[k, 0] = -comp
A = np.zeros(len(A_sym))
for i in range(len(A_sym)):
A_temp2 = 0
for ii in range(len(x_data_point[0, :])):
A_temp = A_sym[i]
for k in range(len(var_sym)):
A_temp = A_temp.subs(var_sym[k], x_data_point[k, ii])
A_temp2 = A_temp2 + A_temp
A[i] = np.float64(A_temp2.evalf())
A = np.reshape(A, A_sym.shape)
B = np.zeros(len(B_sym))
for i in range(len(B_sym)):
B_temp2 = 0
for ii in range(len(x_data_point[0, :])):
B_temp = B_sym[i]
for k in range(len(var_sym)):
B_temp = B_temp.subs(var_sym[k], x_data_point[k, ii])
B_temp2 = B_temp2 + B_temp.subs(f_true, f_data_point[ii])
B[i] = np.float64(B_temp2.evalf())
B = np.reshape(B, B_sym.shape)
C = np.linalg.solve(A, B)
f_fit = np.zeros_like(f_data_point)
for ii in range(len(f_data_point)):
A_temp = f
for k in range(len(par_sym)):
A_temp = A_temp.subs(par_sym[k], C[k])
for k in range(len(var_sym)):
A_temp = A_temp.subs(var_sym[k], x_data_point[k, ii])
f_fit[ii] = np.float64(A_temp.evalf())
Err = np.linalg.norm(f_data_point - f_fit, ord=2) / np.linalg.norm(
f_data_point, ord=2)
return C, f_fit, Err
# -*- coding: utf-8 -*-
# Deducción de las funciones de forma del elemento de barra de 2 nodos
import sympy as sp
# Defino las variables simbólicas
u1, u2, x, x1, x2, a1, a0 = sp.symbols('u1 u2 x x1 x2 a1 a0')
r = sp.solve((sp.Eq(u1, a1*x1 + a0),
sp.Eq(u2, a1*x2 + a0)), (a0, a1))
print('a0 = '); sp.pprint(r[a0])
print('a1 = '); sp.pprint(r[a1])
u = sp.expand(r[a1]*x + r[a0]) # Se define ahora u(x) ya que conocemos a1 y a0
u = sp.collect(u, (u1,u2)) # Se factoriza u2
print('u = '); sp.pprint(u) # Observe aquí las funciones de forma
def generate(self):
x = symbols('x')
a, b, c, d, e, f = symbols('a b c d e f')
substitutions, eqn, LQ = [], None, None
# HCF
if self.difficulty == 1:
expr = expand(random.choice([
x * (x + a),
x * (x - a),
]))
if random.random() < 0.3:
expr *= b
if random.random() < 0.3:
expr *= -1
eqn = Eq(expr, 0)
substitutions = [
(a, random.randint(2, 10)),
(b, random.randint(2, 10)),
(c, random.randint(2, 10)),
(d, random.randint(1, 10)),
]
# Cross
elif self.difficulty == 2:
expr = expand(
random.choice([
(x + a) * (x + b),
(x - a) * (x + b),
(x + a) * (x - b),
(x - a) * (x - b),
]))
if random.random() < 0.5:
expr *= c
eqn = Eq(expr, 0)
substitutions = [
(a, random.randint(2, 10)),
(b, random.randint(1, 6)),
(c, random.randint(2, 5)),
(d, random.randint(1, 10)),
]
# DOPS
elif self.difficulty == 3:
expr = expand(
random.choice([
(x + a) * (x - a),
(a * x + b) * (a * x - b),
]))
if random.random() < 0.3:
expr *= c
eqn = Eq(expr, 0)
substitutions = [
(a, random.randint(2, 10)),
(b, random.randint(1, 10)),
(c, random.randint(2, 5)),
(d, random.randint(1, 10)),
]
# Non-monic
elif self.difficulty == 4:
expr = expand(
random.choice([
(a * x + b) * (x + d),
(a * x + b) * (c * x + d),
]))
if random.random() < 0.3:
expr *= a
eqn = Eq(expr, 0)
substitutions = [
(a, random.randint(2, 4)),
(b, random.randint(2, 10)),
(c, random.randint(2, 5)),
(d, random.randint(2, 5)),
]
for old, new in substitutions:
with evaluate(True):
eqn = eqn.replace(old, new)
try:
soln = solve(eqn, x)[0], solve(eqn, x)[1]
except IndexError:
# no solution
soln = 'No solution'
a = substitutions[0][1]
b = substitutions[1][1]
c = substitutions[2][1]
d = substitutions[3][1]
Q = getPretty(eqn)
A = getPretty(soln)
LQ = '$\displaystyle {}$'.format(latex(eqn))
LA = '$\displaystyle x={}, \ {}$'.format(soln[0], soln[1])
return Q, A, LQ, LA
def setup(base,
n=None,
metric=None,
coords=None,
curv=(None, None),
debug=False):
"""
Generate basis of vector space as tuple of vectors and
associated metric tensor as Matrix. See str_array(base,n) for
usage of base and n and str_array(metric) for usage of metric.
To overide elements in the default metric use the character '#'
in the metric string. For example if one wishes the diagonal
elements of the metric tensor to be zero enter metric = '0 #,# 0'.
If the basis vectors are e1 and e2 then the default metric -
Vector.metric = ((dot(e1,e1),dot(e1,e2)),dot(e2,e1),dot(e2,e2))
becomes -
Vector.metric = ((0,dot(e1,e2)),(dot(e2,e1),0)).
The function dot returns a Symbol and is symmetric.
The functions 'Bases' calculates the global quantities: -
Vector.basis
tuple of basis vectors
Vector.base_to_index
dictionary to convert base to base inded
Vector.metric
metric tensor represented as a matrix of symbols and numbers
"""
Vector.is_orthogonal = False
Vector.coords = coords
Vector.subscripts = []
base_name_lst = base.split(' ')
# Define basis vectors
if '*' in base:
base_lst = base.split('*')
base = base_lst[0]
Vector.subscripts = base_lst[1].split('|')
base_name_lst = []
for subscript in Vector.subscripts:
base_name_lst.append(base + '_' + subscript)
else:
if len(base_name_lst) > 1:
Vector.subscripts = []
for base_name in base_name_lst:
tmp = base_name.split('_')
Vector.subscripts.append(tmp[-1])
elif len(base_name_lst) == 1 and Vector.coords is not None:
base_name_lst = []
for coord in Vector.coords:
Vector.subscripts.append(str(coord))
base_name_lst.append(base + '_' + str(coord))
else:
raise TypeError("'%s' does not define basis vectors" % base)
basis = []
base_to_index = {}
index = 0
for base_name in base_name_lst:
basis_vec = Vector(base_name)
basis.append(basis_vec)
base_to_index[basis_vec.obj] = index
index += 1
Vector.base_to_index = base_to_index
Vector.basis = tuple(basis)
# define metric tensor
default_metric = []
for bv1 in Vector.basis:
row = []
for bv2 in Vector.basis:
row.append(Vector.basic_dot(bv1, bv2))
default_metric.append(row)
Vector.metric = Matrix(default_metric)
if metric is not None:
if metric[0] == '[' and metric[-1] == ']':
Vector.is_orthogonal = True
metric_str_lst = metric[1:-1].split(',')
Vector.metric = []
for g_ii in metric_str_lst:
Vector.metric.append(S(g_ii))
Vector.metric = Matrix(Vector.metric)
else:
metric_str_lst = flatten(str_array(metric))
for index in range(len(metric_str_lst)):
if metric_str_lst[index] != '#':
Vector.metric[index] = S(metric_str_lst[index])
Vector.metric_dict = {} # Used to calculate dot product
N = range(len(Vector.basis))
if Vector.is_orthogonal:
for ii in N:
Vector.metric_dict[Vector.basis[ii].obj] = Vector.metric[ii]
else:
for irow in N:
for icol in N:
Vector.metric_dict[(
Vector.basis[irow].obj,
Vector.basis[icol].obj)] = Vector.metric[irow, icol]
# calculate tangent vectors and metric for curvilinear basis
if curv != (None, None):
X = S.Zero
for (coef, base) in zip(curv[0], Vector.basis):
X += coef * base.obj
Vector.tangents = []
for (coord, norm) in zip(Vector.coords, curv[1]):
tau = diff(X, coord)
tau = trigsimp(tau)
tau /= norm
tau = expand(tau)
Vtau = Vector()
Vtau.obj = tau
Vector.tangents.append(Vtau)
metric = []
for tv1 in Vector.tangents:
row = []
for tv2 in Vector.tangents:
row.append(tv1 * tv2)
metric.append(row)
metric = Matrix(metric)
metric = metric.applyfunc(TrigSimp)
Vector.metric_dict = {}
if metric.is_diagonal:
Vector.is_orthogonal = True
tmp_metric = []
for ii in N:
tmp_metric.append(metric[ii, ii])
Vector.metric_dict[Vector.basis[ii].obj] = metric[ii, ii]
Vector.metric = Matrix(tmp_metric)
else:
Vector.is_orthogonal = False
Vector.metric = metric
for irow in N:
for icol in N:
Vector.metric_dict[(
Vector.basis[irow].obj,
Vector.basis[icol].obj)] = Vector.metric[irow,
icol]
Vector.norm = curv[1]
if debug:
oprint('Tangent Vectors',
Vector.tangents,
'Metric',
Vector.metric,
'Metric Dictionary',
Vector.metric_dict,
'Normalization',
Vector.norm,
dict_mode=True)
# calculate derivatives of tangent vectors
Vector.dtau_dict = None
dtau_dict = {}
for x in Vector.coords:
for (tau, base) in zip(Vector.tangents, Vector.basis):
dtau = tau.diff(x).applyfunc(TrigSimp)
result = S.Zero
for (t, b) in zip(Vector.tangents, Vector.basis):
t_dtau = TrigSimp(t * dtau)
result += t_dtau * b.obj
dtau_dict[(base.obj, x)] = result
Vector.dtau_dict = dtau_dict
if debug:
oprint('Basis Derivatives', Vector.dtau_dict, dict_mode=True)
return tuple(Vector.basis)
def test_inverse_mellin_transform():
from sympy import (sin, simplify, Max, Min, expand,
powsimp, exp_polar, cos, cot)
IMT = inverse_mellin_transform
assert IMT(gamma(s), s, x, (0, oo)) == exp(-x)
assert IMT(gamma(-s), s, x, (-oo, 0)) == exp(-1/x)
assert simplify(IMT(s/(2*s**2 - 2), s, x, (2, oo))) == \
(x**2 + 1)*Heaviside(1 - x)/(4*x)
# test passing "None"
assert IMT(1/(s**2 - 1), s, x, (-1, None)) == \
-x*Heaviside(-x + 1)/2 - Heaviside(x - 1)/(2*x)
assert IMT(1/(s**2 - 1), s, x, (None, 1)) == \
-x*Heaviside(-x + 1)/2 - Heaviside(x - 1)/(2*x)
# test expansion of sums
assert IMT(gamma(s) + gamma(s - 1), s, x, (1, oo)) == (x + 1)*exp(-x)/x
# test factorisation of polys
r = symbols('r', real=True)
assert IMT(1/(s**2 + 1), s, exp(-x), (None, oo)
).subs(x, r).rewrite(sin).simplify() \
== sin(r)*Heaviside(1 - exp(-r))
# test multiplicative substitution
_a, _b = symbols('a b', positive=True)
assert IMT(_b**(-s/_a)*factorial(s/_a)/s, s, x, (0, oo)) == exp(-_b*x**_a)
assert IMT(factorial(_a/_b + s/_b)/(_a + s), s, x, (-_a, oo)) == x**_a*exp(-x**_b)
def simp_pows(expr):
return simplify(powsimp(expand_mul(expr, deep=False), force=True)).replace(exp_polar, exp)
# Now test the inverses of all direct transforms tested above
# Section 8.4.2
nu = symbols('nu', real=True, finite=True)
assert IMT(-1/(nu + s), s, x, (-oo, None)) == x**nu*Heaviside(x - 1)
assert IMT(1/(nu + s), s, x, (None, oo)) == x**nu*Heaviside(1 - x)
assert simp_pows(IMT(gamma(beta)*gamma(s)/gamma(s + beta), s, x, (0, oo))) \
== (1 - x)**(beta - 1)*Heaviside(1 - x)
assert simp_pows(IMT(gamma(beta)*gamma(1 - beta - s)/gamma(1 - s),
s, x, (-oo, None))) \
== (x - 1)**(beta - 1)*Heaviside(x - 1)
assert simp_pows(IMT(gamma(s)*gamma(rho - s)/gamma(rho), s, x, (0, None))) \
== (1/(x + 1))**rho
assert simp_pows(IMT(d**c*d**(s - 1)*sin(pi*c)
*gamma(s)*gamma(s + c)*gamma(1 - s)*gamma(1 - s - c)/pi,
s, x, (Max(-re(c), 0), Min(1 - re(c), 1)))) \
== (x**c - d**c)/(x - d)
assert simplify(IMT(1/sqrt(pi)*(-c/2)*gamma(s)*gamma((1 - c)/2 - s)
*gamma(-c/2 - s)/gamma(1 - c - s),
s, x, (0, -re(c)/2))) == \
(1 + sqrt(x + 1))**c
assert simplify(IMT(2**(a + 2*s)*b**(a + 2*s - 1)*gamma(s)*gamma(1 - a - 2*s)
/gamma(1 - a - s), s, x, (0, (-re(a) + 1)/2))) == \
b**(a - 1)*(sqrt(1 + x/b**2) + 1)**(a - 1)*(b**2*sqrt(1 + x/b**2) +
b**2 + x)/(b**2 + x)
assert simplify(IMT(-2**(c + 2*s)*c*b**(c + 2*s)*gamma(s)*gamma(-c - 2*s)
/ gamma(-c - s + 1), s, x, (0, -re(c)/2))) == \
b**c*(sqrt(1 + x/b**2) + 1)**c
# Section 8.4.5
assert IMT(24/s**5, s, x, (0, oo)) == log(x)**4*Heaviside(1 - x)
assert expand(IMT(6/s**4, s, x, (-oo, 0)), force=True) == \
log(x)**3*Heaviside(x - 1)
assert IMT(pi/(s*sin(pi*s)), s, x, (-1, 0)) == log(x + 1)
assert IMT(pi/(s*sin(pi*s/2)), s, x, (-2, 0)) == log(x**2 + 1)
assert IMT(pi/(s*sin(2*pi*s)), s, x, (-S(1)/2, 0)) == log(sqrt(x) + 1)
assert IMT(pi/(s*sin(pi*s)), s, x, (0, 1)) == log(1 + 1/x)
# TODO
def mysimp(expr):
from sympy import expand, logcombine, powsimp
return expand(
powsimp(logcombine(expr, force=True), force=True, deep=True),
force=True).replace(exp_polar, exp)
assert mysimp(mysimp(IMT(pi/(s*tan(pi*s)), s, x, (-1, 0)))) in [
log(1 - x)*Heaviside(1 - x) + log(x - 1)*Heaviside(x - 1),
log(x)*Heaviside(x - 1) + log(1 - 1/x)*Heaviside(x - 1) + log(-x +
1)*Heaviside(-x + 1)]
# test passing cot
assert mysimp(IMT(pi*cot(pi*s)/s, s, x, (0, 1))) in [
log(1/x - 1)*Heaviside(1 - x) + log(1 - 1/x)*Heaviside(x - 1),
-log(x)*Heaviside(-x + 1) + log(1 - 1/x)*Heaviside(x - 1) + log(-x +
1)*Heaviside(-x + 1), ]
# 8.4.14
assert IMT(-gamma(s + S(1)/2)/(sqrt(pi)*s), s, x, (-S(1)/2, 0)) == \
erf(sqrt(x))
# 8.4.19
assert simplify(IMT(gamma(a/2 + s)/gamma(a/2 - s + 1), s, x, (-re(a)/2, S(3)/4))) \
== besselj(a, 2*sqrt(x))
assert simplify(IMT(2**a*gamma(S(1)/2 - 2*s)*gamma(s + (a + 1)/2)
/ (gamma(1 - s - a/2)*gamma(1 - 2*s + a)),
s, x, (-(re(a) + 1)/2, S(1)/4))) == \
sin(sqrt(x))*besselj(a, sqrt(x))
assert simplify(IMT(2**a*gamma(a/2 + s)*gamma(S(1)/2 - 2*s)
/ (gamma(S(1)/2 - s - a/2)*gamma(1 - 2*s + a)),
s, x, (-re(a)/2, S(1)/4))) == \
cos(sqrt(x))*besselj(a, sqrt(x))
# TODO this comes out as an amazing mess, but simplifies nicely
assert simplify(IMT(gamma(a + s)*gamma(S(1)/2 - s)
/ (sqrt(pi)*gamma(1 - s)*gamma(1 + a - s)),
s, x, (-re(a), S(1)/2))) == \
besselj(a, sqrt(x))**2
assert simplify(IMT(gamma(s)*gamma(S(1)/2 - s)
/ (sqrt(pi)*gamma(1 - s - a)*gamma(1 + a - s)),
s, x, (0, S(1)/2))) == \
besselj(-a, sqrt(x))*besselj(a, sqrt(x))
assert simplify(IMT(4**s*gamma(-2*s + 1)*gamma(a/2 + b/2 + s)
/ (gamma(-a/2 + b/2 - s + 1)*gamma(a/2 - b/2 - s + 1)
*gamma(a/2 + b/2 - s + 1)),
s, x, (-(re(a) + re(b))/2, S(1)/2))) == \
besselj(a, sqrt(x))*besselj(b, sqrt(x))
# Section 8.4.20
# TODO this can be further simplified!
assert simplify(IMT(-2**(2*s)*cos(pi*a/2 - pi*b/2 + pi*s)*gamma(-2*s + 1) *
gamma(a/2 - b/2 + s)*gamma(a/2 + b/2 + s) /
(pi*gamma(a/2 - b/2 - s + 1)*gamma(a/2 + b/2 - s + 1)),
s, x,
(Max(-re(a)/2 - re(b)/2, -re(a)/2 + re(b)/2), S(1)/2))) == \
besselj(a, sqrt(x))*-(besselj(-b, sqrt(x)) -
besselj(b, sqrt(x))*cos(pi*b))/sin(pi*b)
# TODO more
# for coverage
assert IMT(pi/cos(pi*s), s, x, (0, S(1)/2)) == sqrt(x)/(x + 1)
def getPolynomialCoefficient(coordinates):
x, y, a = sympy.symbols('x, y, a')
numberOfPoints = len(coordinates)
setOfPoints = {}
#Creates a dictionary setOfPoints of the form
# {0: [[1,1], [2,0], [3,0], [4,0]], 1: [[1,0], [2,8], [3,0], [4,0]], 2: ...}
for i in range(0, numberOfPoints):
for j in range(0, numberOfPoints):
if (j == i):
setOfPoints[i] = setOfPoints.get(
i, []) + [[coordinates[i][0], coordinates[i][1]]]
elif (j != i):
setOfPoints[i] = setOfPoints.get(i,
[]) + [[coordinates[j][0], 0]]
#Creates a polynomial for each of the coordinates of the previous dictionary where y is non-zero,
#subs in the other coordinate to find the constant a
setOfEquations = {}
setOfSubbedEquations = {}
for i in range(0, len(setOfPoints)):
#print ""
#print "setOfPoints[{0}]: {1}".format(i, setOfPoints[i])
setOfEquations[i] = a
for j in range(0, len(setOfPoints[i])):
#Constructs initial polynomial
if (setOfPoints[i][j][1] == 0):
setOfEquations[i] = setOfEquations[i] * (x -
setOfPoints[i][j][0])
#print ""
#print "setOfPoints[{0}]: {1}".format(i, setOfPoints[i])
#print "setOfEquations[{0}]: {1}".format(i, setOfEquations[i])
allZeroBoolean = True
for j in range(0, len(setOfPoints[i])):
if (setOfPoints[i][j][1] != 0):
allZeroBoolean = False
break
else:
continue
if (allZeroBoolean):
setOfSubbedEquations[i] = 0 * x
else:
#Subs in the non-zero coordinates
unSubbedEquation = sympy.Eq(y, setOfEquations[i])
subbedEquation = unSubbedEquation.subs([(y, setOfPoints[i][i][1]),
(x, setOfPoints[i][i][0])])
#Solves each polynomial to find the constant a, and subs into the final set
solutionForA = sympy.solve(subbedEquation, a)[0]
setOfSubbedEquations[i] = setOfEquations[i].subs(a, solutionForA)
#print "solutionForA:", solutionForA
#print "setOfSubbedEquations[{0}]: {1}".format(i, setOfSubbedEquations[i])
summedEquations = sum(setOfSubbedEquations.itervalues())
#print ""
#print "setOfSubbedEquations:", setOfSubbedEquations
#print "summedEquations:", summedEquations
#Sums each polynomial and expands it into a simpler form
finalPolynomial = sympy.expand(summedEquations)
coefficients = sympy.Poly(finalPolynomial, x).coeffs()
constant = coefficients[len(coefficients) - 1]
moddedConstant = constant % MODULUS
print "finalPolynomial:", finalPolynomial
return moddedConstant
import sympy as sym
from sympy import symbols
from xlrd import open_workbook
from sympy.plotting import plot
from sympy.plotting import plot3d
# Exercise 1 ___________________________________________________
x = sym.symbols('x')
y, i, n, a, b, z = sym.symbols('y i n a b z')
# a)
expr = x**2 + x**3 + 21 * x**4 + 10 * x + 1
print(expr.subs(x, 7))
# b)
print(sym.expand((x + y)**2))
#c)
simp = 4 * x**3 + 21 * x**2 + 10 * x + 12
print(sym.simplify(simp))
#d
#e)
print(sym.summation(2 * i + i - 1, (i, 5, n)))
#f
print(sym.integrate(sin(x) + exp(x) * cos(x) + tan(x), x))
#g
print(sym.factor(x**3 + 12 * x * y * z + 3 * y**2 * z))
def _check(expr, simplified, deep=True, matrix=True):
assert nc_simplify(expr, deep=deep) == simplified
assert expand(expr) == expand(simplified)
if matrix:
m_simp = _to_matrix(simplified).doit(inv_expand=False)
assert nc_simplify(_to_matrix(expr), deep=deep) == m_simp
#centroid ==> the intersection of asympotoes with the real-axis
centroid = ny.real(poles_summation / asymptotes)
plt.plot(centroid, 0, 'ro')
print("Centroid: ", end="")
print(centroid)
for angle in angles:
endX = 72 * math.cos(angle) + centroid
endY = 72 * math.sin(angle)
asmX = [centroid, endX]
asm_Y = [0, endY]
plt.plot(asmX, asm_Y, '--', lineWidth=1)
s = sy.symbols('s')
eq = 1
for i in range(len(poles)):
eq *= (s - complex(poles[i][0], poles[i][1]))
eq = sy.expand(eq)
print("Characteristic Equation: ", end="")
print(eq)
expDiff = sy.Derivative(eq, s).doit()
print("Derivative of the Characteristic Equation: ", end="")
print(expDiff)
roots = sy.solve(expDiff, s)
print("Roots of the Derivative of the Characteristic Equation: ", end="")
print(roots)
realPoles = []
for i in range(len(poles)):
if poles[i][1] == 0:
realPoles.append(poles[i][0])
realPoles.sort()
realPoles.reverse()
break_away_points = []
display(h)
# h.subs(x,2) dersem x yerine 2 yazar ve bize sonucu verir
# h.subs(x,z) dersem x yerine z yazar ve bize z türünden sonucu verir
print("H Fonksiyoun sonucu: " + str(h.subs(x, 1.5)))
display(h.subs(x, z))
#sp.simplify(h) idafeyi sadeleştirir.
h = (x**2 - x - 6) / (x**2 - 3 * x)
#print(h.simplify(1)) tercihi kullanım aşağıda
print(sp.simplify(h))
# Verilen Çarpmım ifadesini açar sp.expand(f)
f = (x + 1)**3 * (x - 2)**2
display(sp.expand(f))
# sp.factor(f) Dağıtılmış şeklinde verilen ifadeyi çarpanlara ayırır.
f = 3 * x**4 - 36 * x**3 + 99 * x**2 - 6 * x - 144
display(f)
display("ÇARPANLARA AYRILMIŞ HALİ: ", sp.factor(f))
# Örnek: x + x**2/2 + x**3/3 + x**4/4 + .... x**n/n
x = sp.Symbol('x')
# simdilik n = 10 olsun
n = 10
series = x
for i in range(2, n + 1):
series = series + (x**i) / i
display(series)
sonuc = float(series.subs(x, 2))
#PROCEDIMIENTO
n = len(xi)
x = sym.Symbol('x')
polinomio = 0
for i in range(0,n,1):
numerador = 1
denominador = 1
for j in range(0,n,1):
if (i!=j):
numerador = numerador*(x-xi[j])
denominador = denominador*(xi[i]-xi[j])
termino = (numerador/denominador)*fi[i]
polinomio = polinomio + termino
polisimple = sym.expand(polinomio)
px = sym.lambdify(x,polinomio)
#VECTORES PARA GRAFICAS
muestras = 51
a = np.min(xi)
b = np.max(xi)
p_xi = np.linspace(a,b,muestras)
pfi = px(p_xi)
#SALIDA
print('polinomio: ')
print(polinomio)
prob:
max f(x1,x2) = -(x1-2)^2-x1-x2^2
begin at the point (2.5,1.5)
"""
import sympy as sp
point = [2.5, 1.5]
x1, x2, t0 = sp.symbols("x1 x2 t0")
x = [x1, x2]
f_expr = -(x1 - 2)**2 - x1 - x2**2
f_gradient = [sp.diff(f_expr, x1), sp.diff(f_expr, x2)]
print("f_gradient:", f_gradient)
f_gradient_value = [f_gradient[i].subs(x[i], point[i]) for i in range(2)]
print('gradient:', f_gradient_value)
while f_gradient_value != [0, 0]:
point = [(x[i] + f_gradient_value[i] * t0).subs([(x1, point[0]),
(x2, point[1])])
for i in range(2)]
f_t0 = sp.expand(f_expr.subs([(x1, point[0]), (x2, point[1])]))
t0_value = sp.solve(sp.diff(f_t0, t0), t0)
points = [point[i].subs(t0, t0_value[0]) for i in range(2)]
f_gradient_value = [f_gradient[i].subs(x[i], points[i]) for i in range(2)]
print("point:", points)
print("gradient:", f_gradient_value)
f_gradient: [-2 * x1 + 3, -2 * x2]
gradient: [-2.0, -3.0]
point: [1.5, 0]
gradient: [0, 0]
def noneuclidian_distance_calculation():
from sympy import solve,sqrt
metric = '0 # #,# 0 #,# # 1'
(X,Y,e) = MV.setup('X Y e',metric)
print('g_{ij} =',MV.metric)
print('(X^Y)**2 =',(X^Y)*(X^Y))
L = X^Y^e
B = L*e # D&L 10.152
print('B =',B)
Bsq = B*B
print('B**2 =',Bsq)
Bsq = Bsq.scalar()
print('#L = X^Y^e is a non-euclidian line')
print('B = L*e =',B)
BeBr =B*e*B.rev()
print('B*e*B.rev() =',BeBr)
print('B**2 =',B*B)
print('L**2 =',L*L) # D&L 10.153
(s,c,Binv,M,S,C,alpha,XdotY,Xdote,Ydote) = symbols('s c (1/B) M S C alpha (X.Y) (X.e) (Y.e)')
Bhat = Binv*B # D&L 10.154
R = c+s*Bhat # Rotor R = exp(alpha*Bhat/2)
print('s = sinh(alpha/2) and c = cosh(alpha/2)')
print('exp(alpha*B/(2*|B|)) =',R)
Z = R*X*R.rev() # D&L 10.155
Z.obj = expand(Z.obj)
Z.obj = Z.obj.collect([Binv,s,c,XdotY])
Z.Fmt(3,'R*X*R.rev()')
W = Z|Y # Extract scalar part of multivector
# From this point forward all calculations are with sympy scalars
print('Objective is to determine value of C = cosh(alpha) such that W = 0')
W = W.scalar()
print('Z|Y =',W)
W = expand(W)
W = simplify(W)
W = W.collect([s*Binv])
M = 1/Bsq
W = W.subs(Binv**2,M)
W = simplify(W)
Bmag = sqrt(XdotY**2-2*XdotY*Xdote*Ydote)
W = W.collect([Binv*c*s,XdotY])
#Double angle substitutions
W = W.subs(2*XdotY**2-4*XdotY*Xdote*Ydote,2/(Binv**2))
W = W.subs(2*c*s,S)
W = W.subs(c**2,(C+1)/2)
W = W.subs(s**2,(C-1)/2)
W = simplify(W)
W = W.subs(1/Binv,Bmag)
W = expand(W)
print('S = sinh(alpha) and C = cosh(alpha)')
print('W =',W)
Wd = collect(W,[C,S],exact=True,evaluate=False)
Wd_1 = Wd[ONE]
Wd_C = Wd[C]
Wd_S = Wd[S]
print('Scalar Coefficient =',Wd_1)
print('Cosh Coefficient =',Wd_C)
print('Sinh Coefficient =',Wd_S)
print('|B| =',Bmag)
Wd_1 = Wd_1.subs(Bmag,1/Binv)
Wd_C = Wd_C.subs(Bmag,1/Binv)
Wd_S = Wd_S.subs(Bmag,1/Binv)
lhs = Wd_1+Wd_C*C
rhs = -Wd_S*S
lhs = lhs**2
rhs = rhs**2
W = expand(lhs-rhs)
W = expand(W.subs(1/Binv**2,Bmag**2))
W = expand(W.subs(S**2,C**2-1))
W = W.collect([C,C**2],evaluate=False)
a = simplify(W[C**2])
b = simplify(W[C])
c = simplify(W[ONE])
print('Require a*C**2+b*C+c = 0')
print('a =',a)
print('b =',b)
print('c =',c)
x = Symbol('x')
C = solve(a*x**2+b*x+c,x)[0]
print('cosh(alpha) = C = -b/(2*a) =',expand(simplify(expand(C))))
return
def mysimp(expr):
from sympy import expand, logcombine, powsimp
return expand(
powsimp(logcombine(expr, force=True), force=True, deep=True),
force=True).replace(exp_polar, exp)
print('E3 = (e1^e2)*E =', E3)
w = (E1 | e2)
w = w.expand()
print('E1|e2 =', w)
w = (E1 | e3)
w = w.expand()
print('E1|e3 =', w)
w = (E2 | e1)
w = w.expand()
print('E2|e1 =', w)
w = (E2 | e3)
w = w.expand()
print('E2|e3 =', w)
w = (E3 | e1)
w = w.expand()
print('E3|e1 =', w)
w = (E3 | e2)
w = w.expand()
print('E3|e2 =', w)
w = (E1 | e1)
w = (w.expand()).scalar()
Esq = expand(Esq)
print('%(E1\\cdot e1)/E^{2} =', simplify(w / Esq))
w = (E2 | e2)
w = (w.expand()).scalar()
print('%(E2\\cdot e2)/E^{2} =', simplify(w / Esq))
w = (E3 | e3)
w = (w.expand()).scalar()
print('%(E3\\cdot e3)/E^{2} =', simplify(w / Esq))
xpdf(paper='letter', prog=True)
def test_expand_function():
assert expand(x+y) == x + y
assert expand(x+y, complex=True) == I*im(x) + I*im(y) + re(x) + re(y)
assert expand((x + y)**11, modulus=11) == x**11 + y**11
#動点P1，P2がそれぞれC1，C2上を一定の速さで反時計回りに回転している．
#時刻t=0でO,A,P1,P2がこの順に一直線上に並び，また，P1の角速度はP2の2倍とする．
#△A P1 P2 の面積の最大値を求めよ．
###########################################################################
import sympy
#P2の角度をΘ（Θ>0）として，ベクトルAP1とAP2を定義
theta = sympy.Symbol('theta', positive=True)
AP1 = [5 * sympy.cos(2 * theta) - 2, 5 * sympy.sin(2 * theta)]
AP2 = [10 * sympy.cos(theta) - 2, 10 * sympy.sin(theta)]
#△A P1 P2 の面積（後で絶対値を取る）
S = (AP1[0] * AP2[1] - AP1[1] * AP2[0]) / 2
#三角関数を展開して，2ΘをΘで書き換え
S = sympy.expand(S, trig=True)
#SをΘで微分
dS = sympy.diff(S, theta)
#dS=0を解いて，解の候補として，Sが極値を取るときの0を求める
cand = sympy.solve(sympy.diff(S, theta))
#候補についてSを計算し（ここで絶対値を取る），Sの最大値とその時のΘを求める
S_max = 0.0
max_theta = []
for value in cand:
tmp = float(abs(S.subs(theta, value)))
if (tmp >= S_max):
if (tmp > S_max):
S_max = tmp
def find_linear_recurrence(self, n, d=None, gfvar=None):
r"""
Finds the shortest linear recurrence that satisfies the first n
terms of sequence of order `\leq` n/2 if possible.
If d is specified, find shortest linear recurrence of order
`\leq` min(d, n/2) if possible.
Returns list of coefficients ``[b(1), b(2), ...]`` corresponding to the
recurrence relation ``x(n) = b(1)*x(n-1) + b(2)*x(n-2) + ...``
Returns ``[]`` if no recurrence is found.
If gfvar is specified, also returns ordinary generating function as a
function of gfvar.
Examples
========
>>> from sympy import sequence, sqrt, oo, lucas
>>> from sympy.abc import n, x, y
>>> sequence(n**2).find_linear_recurrence(10, 2)
[]
>>> sequence(n**2).find_linear_recurrence(10)
[3, -3, 1]
>>> sequence(2**n).find_linear_recurrence(10)
[2]
>>> sequence(23*n**4+91*n**2).find_linear_recurrence(10)
[5, -10, 10, -5, 1]
>>> sequence(sqrt(5)*(((1 + sqrt(5))/2)**n - (-(1 + sqrt(5))/2)**(-n))/5).find_linear_recurrence(10)
[1, 1]
>>> sequence(x+y*(-2)**(-n), (n, 0, oo)).find_linear_recurrence(30)
[1/2, 1/2]
>>> sequence(3*5**n + 12).find_linear_recurrence(20,gfvar=x)
([6, -5], 3*(5 - 21*x)/((x - 1)*(5*x - 1)))
>>> sequence(lucas(n)).find_linear_recurrence(15,gfvar=x)
([1, 1], (x - 2)/(x**2 + x - 1))
"""
from sympy.matrices import Matrix
x = [simplify(expand(t)) for t in self[:n]]
lx = len(x)
if d is None:
r = lx // 2
else:
r = min(d, lx // 2)
coeffs = []
for l in range(1, r + 1):
l2 = 2 * l
mlist = []
for k in range(l):
mlist.append(x[k:k + l])
m = Matrix(mlist)
if m.det() != 0:
y = simplify(m.LUsolve(Matrix(x[l:l2])))
if lx == l2:
coeffs = flatten(y[::-1])
break
mlist = []
for k in range(l, lx - l):
mlist.append(x[k:k + l])
m = Matrix(mlist)
if m * y == Matrix(x[l2:]):
coeffs = flatten(y[::-1])
break
if gfvar is None:
return coeffs
else:
l = len(coeffs)
if l == 0:
return [], None
else:
n, d = x[l - 1] * gfvar**(l - 1), 1 - coeffs[l - 1] * gfvar**l
for i in range(l - 1):
n += x[i] * gfvar**i
for j in range(l - i - 1):
n -= coeffs[i] * x[j] * gfvar**(i + j + 1)
d -= coeffs[i] * gfvar**(i + 1)
return coeffs, simplify(factor(n) / factor(d))
from sympy import Symbol
from sympy import expand
from sympy import pprint
from sympy import factor
x = Symbol('x')
y = Symbol('y')
print(factor(expr))
factors = factor(expr)
expand = expand(factors)
print(factors,expand)
expr = x**3 + 3*x**2*y + 3*x*y**2 + y**3
factors = factor(expr)
print(factors)
pprint(factors)
x = Symbol('x')
series = x
n = 5
for i in range(2, n+1):
series = series + (x**i)/i
pprint(series)
expr = x*x + x*y + x*y + y*y
res = expr.subs({x:1,y:2})
print(res)
r = expr.subs({x:1-y})
print(r)
x = Symbol('x')
series = x
def generator(self):
x = sympy.symbols("x")
self.expression = sympy.expand(self.a * (x + self.h)**2 + self.k)
def expand(self, *args, **kwargs):
return SeqFormula(expand(self.formula, *args, **kwargs), self.args[1])
g = int(u[1])
gg = lis[g]
ui[ii] = gg
ii += 1
user_input = ' '.join(map(str, ui))
# condition that handles symbolic computation
if any(x in user_input for x in syms):
if not any(x in user_input for x in mychar_n) and len(neg) == 0:
if 'expand' in user_input:
user_input = user_input.replace('expand', '')
lis.append(sym.expand(user_input))
print(
str(len(lis) - 1) + ': ' + str(sym.expand(user_input)))
else:
li = eval(str(user_input))
lis.append(li)
print(str(len(lis) - 1) + ': ' + str(li))
# condition that handles basic arithmetic
else:
if (x in user_input for x in bsop) and not any(
x in user_input for x in mychar) and len(neg) == 0:
li = eval(user_input)
if user_input.count('/') > 1:
def test_Abs():
raises(TypeError, lambda: Abs(Interval(2, 3))) # issue 8717
x, y = symbols('x,y')
assert sign(sign(x)) == sign(x)
assert sign(x*y).func is sign
assert Abs(0) == 0
assert Abs(1) == 1
assert Abs(-1) == 1
assert Abs(I) == 1
assert Abs(-I) == 1
assert Abs(nan) is nan
assert Abs(zoo) is oo
assert Abs(I * pi) == pi
assert Abs(-I * pi) == pi
assert Abs(I * x) == Abs(x)
assert Abs(-I * x) == Abs(x)
assert Abs(-2*x) == 2*Abs(x)
assert Abs(-2.0*x) == 2.0*Abs(x)
assert Abs(2*pi*x*y) == 2*pi*Abs(x*y)
assert Abs(conjugate(x)) == Abs(x)
assert conjugate(Abs(x)) == Abs(x)
assert Abs(x).expand(complex=True) == sqrt(re(x)**2 + im(x)**2)
a = Symbol('a', positive=True)
assert Abs(2*pi*x*a) == 2*pi*a*Abs(x)
assert Abs(2*pi*I*x*a) == 2*pi*a*Abs(x)
x = Symbol('x', real=True)
n = Symbol('n', integer=True)
assert Abs((-1)**n) == 1
assert x**(2*n) == Abs(x)**(2*n)
assert Abs(x).diff(x) == sign(x)
assert abs(x) == Abs(x) # Python built-in
assert Abs(x)**3 == x**2*Abs(x)
assert Abs(x)**4 == x**4
assert (
Abs(x)**(3*n)).args == (Abs(x), 3*n) # leave symbolic odd unchanged
assert (1/Abs(x)).args == (Abs(x), -1)
assert 1/Abs(x)**3 == 1/(x**2*Abs(x))
assert Abs(x)**-3 == Abs(x)/(x**4)
assert Abs(x**3) == x**2*Abs(x)
assert Abs(I**I) == exp(-pi/2)
assert Abs((4 + 5*I)**(6 + 7*I)) == 68921*exp(-7*atan(Rational(5, 4)))
y = Symbol('y', real=True)
assert Abs(I**y) == 1
y = Symbol('y')
assert Abs(I**y) == exp(-pi*im(y)/2)
x = Symbol('x', imaginary=True)
assert Abs(x).diff(x) == -sign(x)
eq = -sqrt(10 + 6*sqrt(3)) + sqrt(1 + sqrt(3)) + sqrt(3 + 3*sqrt(3))
# if there is a fast way to know when you can and when you cannot prove an
# expression like this is zero then the equality to zero is ok
assert abs(eq).func is Abs or abs(eq) == 0
# but sometimes it's hard to do this so it's better not to load
# abs down with tests that will be very slow
q = 1 + sqrt(2) - 2*sqrt(3) + 1331*sqrt(6)
p = expand(q**3)**Rational(1, 3)
d = p - q
assert abs(d).func is Abs or abs(d) == 0
assert Abs(4*exp(pi*I/4)) == 4
assert Abs(3**(2 + I)) == 9
assert Abs((-3)**(1 - I)) == 3*exp(pi)
assert Abs(oo) is oo
assert Abs(-oo) is oo
assert Abs(oo + I) is oo
assert Abs(oo + I*oo) is oo
a = Symbol('a', algebraic=True)
t = Symbol('t', transcendental=True)
x = Symbol('x')
assert re(a).is_algebraic
assert re(x).is_algebraic is None
assert re(t).is_algebraic is False
assert Abs(x).fdiff() == sign(x)
raises(ArgumentIndexError, lambda: Abs(x).fdiff(2))
# doesn't have recursion error
arg = sqrt(acos(1 - I)*acos(1 + I))
assert abs(arg) == arg
# special handling to put Abs in denom
assert abs(1/x) == 1/Abs(x)
e = abs(2/x**2)
assert e.is_Mul and e == 2/Abs(x**2)
assert unchanged(Abs, y/x)
assert unchanged(Abs, x/(x + 1))
assert unchanged(Abs, x*y)
p = Symbol('p', positive=True)
assert abs(x/p) == abs(x)/p
# coverage
assert unchanged(Abs, Symbol('x', real=True)**y)
def test_sign():
assert sign(1.2) == 1
assert sign(-1.2) == -1
assert sign(3*I) == I
assert sign(-3*I) == -I
assert sign(0) == 0
assert sign(nan) == nan
assert sign(2 + 2*I).doit() == sqrt(2)*(2 + 2*I)/4
assert sign(2 + 3*I).simplify() == sign(2 + 3*I)
assert sign(2 + 2*I).simplify() == sign(1 + I)
assert sign(im(sqrt(1 - sqrt(3)))) == 1
assert sign(sqrt(1 - sqrt(3))) == I
x = Symbol('x')
assert sign(x).is_bounded is True
assert sign(x).is_complex is True
assert sign(x).is_imaginary is None
assert sign(x).is_integer is None
assert sign(x).is_real is None
assert sign(x).is_zero is None
assert sign(x).doit() == sign(x)
assert sign(1.2*x) == sign(x)
assert sign(2*x) == sign(x)
assert sign(I*x) == I*sign(x)
assert sign(-2*I*x) == -I*sign(x)
assert sign(conjugate(x)) == conjugate(sign(x))
p = Symbol('p', positive=True)
n = Symbol('n', negative=True)
m = Symbol('m', negative=True)
assert sign(2*p*x) == sign(x)
assert sign(n*x) == -sign(x)
assert sign(n*m*x) == sign(x)
x = Symbol('x', imaginary=True)
assert sign(x).is_imaginary is True
assert sign(x).is_integer is False
assert sign(x).is_real is False
assert sign(x).is_zero is False
assert sign(x).diff(x) == 2*DiracDelta(-I*x)
assert sign(x).doit() == x / Abs(x)
assert conjugate(sign(x)) == -sign(x)
x = Symbol('x', real=True)
assert sign(x).is_imaginary is False
assert sign(x).is_integer is True
assert sign(x).is_real is True
assert sign(x).is_zero is None
assert sign(x).diff(x) == 2*DiracDelta(x)
assert sign(x).doit() == sign(x)
assert conjugate(sign(x)) == sign(x)
x = Symbol('x', nonzero=True)
assert sign(x).is_imaginary is None
assert sign(x).is_integer is None
assert sign(x).is_real is None
assert sign(x).is_zero is False
assert sign(x).doit() == x / Abs(x)
assert sign(Abs(x)) == 1
assert Abs(sign(x)) == 1
x = Symbol('x', positive=True)
assert sign(x).is_imaginary is False
assert sign(x).is_integer is True
assert sign(x).is_real is True
assert sign(x).is_zero is False
assert sign(x).doit() == x / Abs(x)
assert sign(Abs(x)) == 1
assert Abs(sign(x)) == 1
x = 0
assert sign(x).is_imaginary is False
assert sign(x).is_integer is True
assert sign(x).is_real is True
assert sign(x).is_zero is True
assert sign(x).doit() == 0
assert sign(Abs(x)) == 0
assert Abs(sign(x)) == 0
nz = Symbol('nz', nonzero=True, integer=True)
assert sign(nz).is_imaginary is False
assert sign(nz).is_integer is True
assert sign(nz).is_real is True
assert sign(nz).is_zero is False
assert sign(nz)**2 == 1
assert (sign(nz)**3).args == (sign(nz), 3)
assert sign(Symbol('x', nonnegative=True)).is_nonnegative
assert sign(Symbol('x', nonnegative=True)).is_nonpositive is None
assert sign(Symbol('x', nonpositive=True)).is_nonnegative is None
assert sign(Symbol('x', nonpositive=True)).is_nonpositive
assert sign(Symbol('x', real=True)).is_nonnegative is None
assert sign(Symbol('x', real=True)).is_nonpositive is None
assert sign(Symbol('x', real=True, zero=False)).is_nonpositive is None
x, y = Symbol('x', real=True), Symbol('y')
assert sign(x).rewrite(Piecewise) == \
Piecewise((1, x > 0), (-1, x < 0), (0, True))
assert sign(y).rewrite(Piecewise) == sign(y)
# evaluate what can be evaluated
assert sign(exp_polar(I*pi)*pi) is S.NegativeOne
eq = -sqrt(10 + 6*sqrt(3)) + sqrt(1 + sqrt(3)) + sqrt(3 + 3*sqrt(3))
# if there is a fast way to know when and when you cannot prove an
# expression like this is zero then the equality to zero is ok
assert sign(eq).func is sign or sign(eq) == 0
# but sometimes it's hard to do this so it's better not to load
# abs down with tests that will be very slow
q = 1 + sqrt(2) - 2*sqrt(3) + 1331*sqrt(6)
p = expand(q**3)**Rational(1, 3)
d = p - q
assert sign(d).func is sign or sign(d) == 0
def test_minimal_polynomial():
assert minimal_polynomial(-7, x) == x + 7
assert minimal_polynomial(-1, x) == x + 1
assert minimal_polynomial(0, x) == x
assert minimal_polynomial(1, x) == x - 1
assert minimal_polynomial(7, x) == x - 7
assert minimal_polynomial(sqrt(2), x) == x**2 - 2
assert minimal_polynomial(sqrt(5), x) == x**2 - 5
assert minimal_polynomial(sqrt(6), x) == x**2 - 6
assert minimal_polynomial(2 * sqrt(2), x) == x**2 - 8
assert minimal_polynomial(3 * sqrt(5), x) == x**2 - 45
assert minimal_polynomial(4 * sqrt(6), x) == x**2 - 96
assert minimal_polynomial(2 * sqrt(2) + 3, x) == x**2 - 6 * x + 1
assert minimal_polynomial(3 * sqrt(5) + 6, x) == x**2 - 12 * x - 9
assert minimal_polynomial(4 * sqrt(6) + 7, x) == x**2 - 14 * x - 47
assert minimal_polynomial(2 * sqrt(2) - 3, x) == x**2 + 6 * x + 1
assert minimal_polynomial(3 * sqrt(5) - 6, x) == x**2 + 12 * x - 9
assert minimal_polynomial(4 * sqrt(6) - 7, x) == x**2 + 14 * x - 47
assert minimal_polynomial(sqrt(1 + sqrt(6)), x) == x**4 - 2 * x**2 - 5
assert minimal_polynomial(sqrt(I + sqrt(6)), x) == x**8 - 10 * x**4 + 49
assert minimal_polynomial(2 * I + sqrt(2 + I),
x) == x**4 + 4 * x**2 + 8 * x + 37
assert minimal_polynomial(sqrt(2) + sqrt(3), x) == x**4 - 10 * x**2 + 1
assert minimal_polynomial(sqrt(2) + sqrt(3) + sqrt(6),
x) == x**4 - 22 * x**2 - 48 * x - 23
a = 1 - 9 * sqrt(2) + 7 * sqrt(3)
assert minimal_polynomial(
1 / a, x) == 392 * x**4 - 1232 * x**3 + 612 * x**2 + 4 * x - 1
assert minimal_polynomial(
1 / sqrt(a), x) == 392 * x**8 - 1232 * x**6 + 612 * x**4 + 4 * x**2 - 1
raises(NotAlgebraic, lambda: minimal_polynomial(oo, x))
raises(NotAlgebraic, lambda: minimal_polynomial(2**y, x))
raises(NotAlgebraic, lambda: minimal_polynomial(sin(1), x))
assert minimal_polynomial(sqrt(2)).dummy_eq(x**2 - 2)
assert minimal_polynomial(sqrt(2), x) == x**2 - 2
assert minimal_polynomial(sqrt(2), polys=True) == Poly(x**2 - 2)
assert minimal_polynomial(sqrt(2), x, polys=True) == Poly(x**2 - 2)
assert minimal_polynomial(sqrt(2), x, polys=True,
compose=False) == Poly(x**2 - 2)
a = AlgebraicNumber(sqrt(2))
b = AlgebraicNumber(sqrt(3))
assert minimal_polynomial(a, x) == x**2 - 2
assert minimal_polynomial(b, x) == x**2 - 3
assert minimal_polynomial(a, x, polys=True) == Poly(x**2 - 2)
assert minimal_polynomial(b, x, polys=True) == Poly(x**2 - 3)
assert minimal_polynomial(sqrt(a / 2 + 17),
x) == 2 * x**4 - 68 * x**2 + 577
assert minimal_polynomial(sqrt(b / 2 + 17),
x) == 4 * x**4 - 136 * x**2 + 1153
a, b = sqrt(2) / 3 + 7, AlgebraicNumber(sqrt(2) / 3 + 7)
f = 81*x**8 - 2268*x**6 - 4536*x**5 + 22644*x**4 + 63216*x**3 - \
31608*x**2 - 189648*x + 141358
assert minimal_polynomial(sqrt(a) + sqrt(sqrt(a)), x) == f
assert minimal_polynomial(sqrt(b) + sqrt(sqrt(b)), x) == f
assert minimal_polynomial(a**Q(3, 2),
x) == 729 * x**4 - 506898 * x**2 + 84604519
# issue 5994
eq = S('''
-1/(800*sqrt(-1/240 + 1/(18000*(-1/17280000 +
sqrt(15)*I/28800000)**(1/3)) + 2*(-1/17280000 +
sqrt(15)*I/28800000)**(1/3)))''')
assert minimal_polynomial(eq, x) == 8000 * x**2 - 1
ex = 1 + sqrt(2) + sqrt(3)
mp = minimal_polynomial(ex, x)
assert mp == x**4 - 4 * x**3 - 4 * x**2 + 16 * x - 8
ex = 1 / (1 + sqrt(2) + sqrt(3))
mp = minimal_polynomial(ex, x)
assert mp == 8 * x**4 - 16 * x**3 + 4 * x**2 + 4 * x - 1
p = (expand((1 + sqrt(2) - 2 * sqrt(3) + sqrt(7))**3))**Rational(1, 3)
mp = minimal_polynomial(p, x)
assert mp == x**8 - 8 * x**7 - 56 * x**6 + 448 * x**5 + 480 * x**4 - 5056 * x**3 + 1984 * x**2 + 7424 * x - 3008
p = expand((1 + sqrt(2) - 2 * sqrt(3) + sqrt(7))**3)
mp = minimal_polynomial(p, x)
assert mp == x**8 - 512 * x**7 - 118208 * x**6 + 31131136 * x**5 + 647362560 * x**4 - 56026611712 * x**3 + 116994310144 * x**2 + 404854931456 * x - 27216576512
assert minimal_polynomial(S("-sqrt(5)/2 - 1/2 + (-sqrt(5)/2 - 1/2)**2"),
x) == x - 1
a = 1 + sqrt(2)
assert minimal_polynomial((a * sqrt(2) + a)**3, x) == x**2 - 198 * x + 1
p = 1 / (1 + sqrt(2) + sqrt(3))
assert minimal_polynomial(
p, x, compose=False) == 8 * x**4 - 16 * x**3 + 4 * x**2 + 4 * x - 1
p = 2 / (1 + sqrt(2) + sqrt(3))
assert minimal_polynomial(
p, x, compose=False) == x**4 - 4 * x**3 + 2 * x**2 + 4 * x - 2
assert minimal_polynomial(1 + sqrt(2) * I, x,
compose=False) == x**2 - 2 * x + 3
assert minimal_polynomial(1 / (1 + sqrt(2)) + 1, x,
compose=False) == x**2 - 2
assert minimal_polynomial(sqrt(2) * I + I * (1 + sqrt(2)),
x,
compose=False) == x**4 + 18 * x**2 + 49
# minimal polynomial of I
assert minimal_polynomial(I, x, domain=QQ.algebraic_field(I)) == x - I
K = QQ.algebraic_field(I * (sqrt(2) + 1))
assert minimal_polynomial(I, x, domain=K) == x - I
assert minimal_polynomial(I, x, domain=QQ) == x**2 + 1
assert minimal_polynomial(I, x, domain='QQ(y)') == x**2 + 1
def regression(equation,
fit_par,
ind_var,
x_data_point,
f_data_point,
mode,
add_par=[None, None]):
par_sym = [sy.Symbol(fit_par[k]) for k in range(len(fit_par))]
var_sym = [sy.Symbol(ind_var[k]) for k in range(len(ind_var))]
f = parse_expr(equation)
f_ex = sy.expand(f)
f_args = f_ex.args
a_sym = sy.zeros(1, len(par_sym))
for i in range(len(f_args)):
comp = f_args[i]
for l in range(len(par_sym)):
if par_sym[l] in comp.free_symbols:
a_sym[l] = comp / par_sym[l]
A = np.zeros((len(x_data_point[0, :]), len(par_sym)))
for ii in range(len(x_data_point[0, :])):
A_temp = a_sym
for k in range(len(var_sym)):
A_temp = A_temp.subs(var_sym[k], x_data_point[k, ii])
A[ii, :] = np.float64(A_temp.evalf())
#B = (np.matrix(f_data_point)).T
B = f_data_point
if mode == 'pinv':
C = np.linalg.pinv(A) @ B
elif mode == 'lstsq':
C = np.linalg.lstsq(A, B, rcond=None)[0]
elif mode == 'lasso':
if add_par[0] != None:
alpha = add_par[0]
else:
alpha = 1.0
regr = linear_model.Lasso(alpha=alpha,
copy_X=True,
max_iter=10**5,
random_state=0)
regr.fit(A, B)
C = regr.coef_
elif mode == 'ridge':
if add_par[0] != None:
alpha = add_par[0]
else:
alpha = 1.0
regr = linear_model.Ridge(alpha=alpha,
copy_X=True,
max_iter=10**5,
random_state=0)
regr.fit(A, B)
C = regr.coef_
elif mode == 'elastic':
if add_par[0] != None:
alpha = add_par[0]
else:
alpha = 1.0
if add_par[1] != None:
rho = add_par[1]
else:
rho = 0.5
regr = linear_model.ElasticNet(alpha=alpha,
l1_ratio=rho,
copy_X=True,
max_iter=10**5,
random_state=0)
regr.fit(A, B)
C = regr.coef_
elif mode == 'huber':
regr = linear_model.HuberRegressor()
regr.fit(A, B)
C = regr.coef_
f_fit = A @ C
Err = np.linalg.norm(f_data_point - f_fit, ord=2) / np.linalg.norm(
f_data_point, ord=2)
return C, f_fit, Err
def simp(x):
return simplify(expand_trig(expand_complex(expand(x))))
def reciprocal_frame_test():
metric = '1 # #,'+ \
'# 1 #,'+ \
'# # 1'
(e1,e2,e3) = MV.setup('e1 e2 e3',metric)
print('g_{ij} =\n',MV.metric)
E = e1^e2^e3
Esq = (E*E).scalar()
print('E =',E)
print('E**2 =',Esq)
Esq_inv = 1/Esq
E1 = (e2^e3)*E
E2 = (-1)*(e1^e3)*E
E3 = (e1^e2)*E
print('E1 = (e2^e3)*E =',E1)
print('E2 =-(e1^e3)*E =',E2)
print('E3 = (e1^e2)*E =',E3)
w = (E1|e2)
w = w.expand()
print('E1|e2 =',w)
w = (E1|e3)
w = w.expand()
print('E1|e3 =',w)
w = (E2|e1)
w = w.expand()
print('E2|e1 =',w)
w = (E2|e3)
w = w.expand()
print('E2|e3 =',w)
w = (E3|e1)
w = w.expand()
print('E3|e1 =',w)
w = (E3|e2)
w = w.expand()
print('E3|e2 =',w)
w = (E1|e1)
w = (w.expand()).scalar()
Esq = expand(Esq)
print('(E1|e1)/E**2 =',simplify(w/Esq))
w = (E2|e2)
w = (w.expand()).scalar()
print('(E2|e2)/E**2 =',simplify(w/Esq))
w = (E3|e3)
w = (w.expand()).scalar()
print('(E3|e3)/E**2 =',simplify(w/Esq))
return
def test_one_dof():
# This is for a 1 dof spring-mass-damper case.
# It is described in more detail in the KanesMethod docstring.
q, u = dynamicsymbols('q u')
qd, ud = dynamicsymbols('q u', 1)
m, c, k = symbols('m c k')
N = ReferenceFrame('N')
P = Point('P')
P.set_vel(N, u * N.x)
kd = [qd - u]
FL = [(P, (-k * q - c * u) * N.x)]
pa = Particle('pa', P, m)
BL = [pa]
KM = KanesMethod(N, [q], [u], kd)
KM.kanes_equations(FL, BL)
MM = KM.mass_matrix
forcing = KM.forcing
rhs = MM.inv() * forcing
assert expand(rhs[0]) == expand(-(q * k + u * c) / m)
assert (KM.linearize(A_and_B=True, new_method=True)[0] ==
Matrix([[0, 1], [-k/m, -c/m]]))
# Ensure that the old linearizer still works and that the new linearizer
# gives the same results. The old linearizer is deprecated and should be
# removed in >= 0.7.7.
M_old = KM.mass_matrix_full
# The old linearizer raises a deprecation warning, so catch it here so
# it doesn't cause py.test to fail.
with warnings.catch_warnings():
warnings.filterwarnings("ignore", category=SymPyDeprecationWarning)
F_A_old, F_B_old, r_old = KM.linearize()
M_new, F_A_new, F_B_new, r_new = KM.linearize(new_method=True)
assert simplify(M_new.inv() * F_A_new - M_old.inv() * F_A_old) == zeros(2)