def myroots(expr_1, expr_2, input_type, output_type):
from sympy import roots, sympify, latex
expr_1 = sympify(expr_1)
expr_2 = sympify(expr_2)
input_type = str(input_type)
output_type = str(output_type)
if input_type == 'LaTeX':
print('roots({}, {}) = '.format(latex(expr_1), latex(expr_2)))
elif input_type == 'SymPy':
print('roots({}, {}) = '.format(expr_1, expr_2))
else:
print("Wrong input type")
if output_type == 'LaTeX':
print(latex(roots((expr_1), expr_2)))
elif output_type == 'SymPy':
print(roots((expr_1), expr_2))
else:
print("Wrong output type")
def get_harmonic_standard_deviations_around_minima(self,
temperature=300.0 *
unit.kelvin):
kB = unit.BOLTZMANN_CONSTANT_kB * unit.AVOGADRO_CONSTANT_NA
Eo = self.parameters['Eo']
a = self.parameters['a']
b = self.parameters['b']
k = self.parameters['k']
x, y, z = sy.symbols('x y z')
xu = x * unit.nanometers
yu = y * unit.nanometers
zu = z * unit.nanometers
potential_x = Eo * ((xu / a)**4 - 2.0 * (xu / a)**2) - (b / a) * xu
potential_y = 0.5 * k * (yu**2)
potential_z = 0.5 * k * (zu**2)
g = sy.diff(potential_y, y)
gg = sy.diff(potential_y, y, y)
roots_diff = sy.roots(g, y)
root_y = None
sigma_y = None
for root in roots_diff.keys():
effective_k = gg.subs(y, root)
if effective_k > 0:
root_y = root * unit.nanometers
sigma_y = np.sqrt(kB * temperature /
(effective_k * unit.kilocalories_per_mole /
unit.nanometers**2))
g = sy.diff(potential_x, x)
gg = sy.diff(potential_x, x, x)
roots_diff = sy.roots(g, x)
roots = []
sigmas = []
for root in roots_diff.keys():
effective_k = gg.subs(x, root)
if effective_k > 0:
root_3d = np.zeros([3], dtype=float) * unit.nanometers
root_3d[0] = root * unit.nanometers
root_3d[1] = root_y
root_3d[2] = root_y
roots.append(root_3d)
sigma_3d = np.zeros([3], dtype=float) * unit.nanometers
sigma_3d[0] = np.sqrt(
kB * temperature /
(effective_k * unit.kilocalories_per_mole /
unit.nanometers**2))
sigma_3d[1] = sigma_y
sigma_3d[2] = sigma_y
sigmas.append(sigma_3d)
del (x, y, z)
return roots, sigmas
def f2zpk(function: Basic,
var: Symbol) -> Tuple[Dict[Basic, int], Dict[Basic, int], Basic]:
numer, denom = f2nd(function)
numer_lc, denom_lc = map(lambda expr: leading_coeff(expr, var),
(numer, denom))
# It's not really necesary to devide by the leading
# coefficient, but what do I know
return roots(numer/numer_lc, var), \
roots(denom/denom_lc, var), numer_lc/denom_lc
def get_small_oscillations_time_periods_around_minima(self):
Eo = self.parameters['Eo']
a = self.parameters['a']
b = self.parameters['b']
k = self.parameters['k']
mass = self.parameters['mass']
x, y, z = sy.symbols('x y z')
xu = x * unit.nanometers
yu = y * unit.nanometers
zu = z * unit.nanometers
potential_x = Eo * ((xu / a)**4 - 2.0 * (xu / a)**2) - (b / a) * xu
potential_y = 0.5 * k * (yu**2)
potential_z = 0.5 * k * (zu**2)
g = sy.diff(potential_y, y)
gg = sy.diff(potential_y, y, y)
roots_diff = sy.roots(g, y)
root_y = None
T_y = None
for root in roots_diff.keys():
effective_k = gg.subs(y, root)
if effective_k > 0:
root_y = root * unit.nanometers
T_y = 2 * np.pi * np.sqrt(
mass / (effective_k * unit.kilocalories_per_mole /
unit.nanometers**2))
g = sy.diff(potential_x, x)
gg = sy.diff(potential_x, x, x)
roots_diff = sy.roots(g, x)
roots = []
Ts = []
for root in roots_diff.keys():
effective_k = gg.subs(x, root)
if effective_k > 0:
root_3d = np.zeros([3], dtype=float) * unit.nanometers
root_3d[0] = root * unit.nanometers
root_3d[1] = root_y
root_3d[2] = root_y
roots.append(root_3d)
T_3d = np.zeros([3], dtype=float) * unit.picoseconds
T_3d[0] = 2 * np.pi * np.sqrt(
mass / (effective_k * unit.kilocalories_per_mole /
unit.nanometers**2))
T_3d[1] = T_y
T_3d[2] = T_y
Ts.append(T_3d)
del (x, y, z)
return roots, Ts
def analyze_controller(self, K):
if self.control_tfs is None:
self.get_controller_transfer_functions()
for X_r in self.X_r:
# set all others in X_r to zero
tfs = [
sp.simplify(
tf.subs([(X_r_, 0)
for X_r_ in self.X_r if X_r_ is not X_r]) / X_r)
for tf in self.control_tfs
]
for X, tf in zip(self.X, tfs):
numerator, denominator = tf.as_numer_denom()
routh_table = np.array(routh(sp.Poly(denominator, self.s)))
routh_conditions = routh_table[:, 0]
tf = self.phys.apply_substitutions(tf)
numerator = self.phys.apply_substitutions(numerator)
denominator = self.phys.apply_substitutions(denominator)
zeros = sp.roots(
numerator.subs(zip(self.k_pd_mat.flatten(), K.flatten())),
self.s)
poles = sp.roots(
denominator.subs(zip(self.k_pd_mat.flatten(),
K.flatten())), self.s)
gain = get_gain(
self.s, tf.subs(zip(self.k_pd_mat.flatten(), K.flatten())),
zeros, poles)
zeros_description = ', '.join([
str(zero.evalf(6)) if multiplicity == 1 else
f'{zero.evalf(6)} (x{multiplicity})'
for zero, multiplicity in zeros.items()
])
poles_description = ', '.join([
str(pole.evalf(6)) if multiplicity == 1 else
f'{pole.evalf(6)} (x{multiplicity})'
for pole, multiplicity in poles.items()
])
routh_conditions_description = '\n'.join([
to_string(condition.evalf(3))
for condition in routh_conditions
])
print(
f'{X}/{X_r}: Routh Conditions:\n{routh_conditions_description}\nGain: {gain.evalf(6)}\n'
f'Poles: {poles_description}\nZeros: {zeros_description}\n\n'
)
def get_world_segments(root_a, root_b, root_c, initial_t, final_t,
intersection_radius):
"""
The world consists of
three axis lines,
six circles marking the intersections,
and a single parametric curve.
@return: a collection of (p0, p1, style) triples
"""
seg_length_min = 0.1
segments = []
# add the axis line segments
d = 5
f_x = pcurve.LineSegment(np.array([-d, 0, 0]), np.array([d, 0, 0]))
f_y = pcurve.LineSegment(np.array([0, -d, 0]), np.array([0, d, 0]))
f_z = pcurve.LineSegment(np.array([0, 0, -d]), np.array([0, 0, d]))
x_axis_segs = pcurve.get_piecewise_curve(f_x, 0, 1, 10, seg_length_min)
y_axis_segs = pcurve.get_piecewise_curve(f_y, 0, 1, 10, seg_length_min)
z_axis_segs = pcurve.get_piecewise_curve(f_z, 0, 1, 10, seg_length_min)
segments.extend((p0, p1, STYLE_X) for p0, p1 in x_axis_segs)
segments.extend((p0, p1, STYLE_Y) for p0, p1 in y_axis_segs)
segments.extend((p0, p1, STYLE_Z) for p0, p1 in z_axis_segs)
# add the parametric curve
roots = (root_a, root_b, root_c)
polys = interlace.roots_to_differential_polys(roots)
f_poly = interlace.Multiplex((sympyutils.WrappedUniPoly(p) for p in polys))
poly_segs = pcurve.get_piecewise_curve(f_poly, initial_t, final_t, 10,
seg_length_min)
segments.extend((p0, p1, STYLE_CURVE) for p0, p1 in poly_segs)
# add the intersection circles
x_roots_symbolic = sympy.roots(polys[0])
y_roots_symbolic = sympy.roots(polys[1])
z_roots_symbolic = sympy.roots(polys[2])
x_roots = [float(r) for r in x_roots_symbolic]
y_roots = [float(r) for r in y_roots_symbolic]
z_roots = [float(r) for r in z_roots_symbolic]
for r in x_roots:
f = pcurve.OrthoCircle(f_poly(r), intersection_radius, 0)
segs = pcurve.get_piecewise_curve(f, 0, 1, 10, seg_length_min)
segments.extend((p0, p1, STYLE_X) for p0, p1 in segs)
for r in y_roots:
f = pcurve.OrthoCircle(f_poly(r), intersection_radius, 1)
segs = pcurve.get_piecewise_curve(f, 0, 1, 10, seg_length_min)
segments.extend((p0, p1, STYLE_Y) for p0, p1 in segs)
for r in z_roots:
f = pcurve.OrthoCircle(f_poly(r), intersection_radius, 2)
segs = pcurve.get_piecewise_curve(f, 0, 1, 10, seg_length_min)
segments.extend((p0, p1, STYLE_Z) for p0, p1 in segs)
# return the segments
return segments
def ZPK(self):
"""Convert to pole-zero-gain (PZK) form.
See also canonical, general, mixedfrac, and partfrac"""
N, D, delay = self._as_ratfun_delay()
K = sym.cancel(N.LC() / D.LC())
if delay != 0:
K *= sym.exp(self.var * delay)
zeros = sym.roots(N)
poles = sym.roots(D)
return self.__class__(_zp2tf(zeros, poles, K, self.var))
def ZPK(self):
"""Convert to pole-zero-gain (PZK) form.
See also canonical, general, mixedfrac, and partfrac"""
N, D, delay = self.as_ratfun_delay()
K = sym.cancel(N.LC() / D.LC())
if delay != 0:
K *= sym.exp(self.var * delay)
zeros = sym.roots(N)
poles = sym.roots(D)
return _zp2tf(zeros, poles, K, self.var)
def get_world_segments(root_a, root_b, root_c, initial_t, final_t, intersection_radius):
"""
The world consists of
three axis lines,
six circles marking the intersections,
and a single parametric curve.
@return: a collection of (p0, p1, style) triples
"""
seg_length_min = 0.1
segments = []
# add the axis line segments
d = 5
f_x = pcurve.LineSegment(np.array([-d, 0, 0]), np.array([d, 0, 0]))
f_y = pcurve.LineSegment(np.array([0, -d, 0]), np.array([0, d, 0]))
f_z = pcurve.LineSegment(np.array([0, 0, -d]), np.array([0, 0, d]))
x_axis_segs = pcurve.get_piecewise_curve(f_x, 0, 1, 10, seg_length_min)
y_axis_segs = pcurve.get_piecewise_curve(f_y, 0, 1, 10, seg_length_min)
z_axis_segs = pcurve.get_piecewise_curve(f_z, 0, 1, 10, seg_length_min)
segments.extend((p0, p1, STYLE_X) for p0, p1 in x_axis_segs)
segments.extend((p0, p1, STYLE_Y) for p0, p1 in y_axis_segs)
segments.extend((p0, p1, STYLE_Z) for p0, p1 in z_axis_segs)
# add the parametric curve
roots = (root_a, root_b, root_c)
polys = interlace.roots_to_differential_polys(roots)
f_poly = interlace.Multiplex((sympyutils.WrappedUniPoly(p) for p in polys))
poly_segs = pcurve.get_piecewise_curve(f_poly, initial_t, final_t, 10, seg_length_min)
segments.extend((p0, p1, STYLE_CURVE) for p0, p1 in poly_segs)
# add the intersection circles
x_roots_symbolic = sympy.roots(polys[0])
y_roots_symbolic = sympy.roots(polys[1])
z_roots_symbolic = sympy.roots(polys[2])
x_roots = [float(r) for r in x_roots_symbolic]
y_roots = [float(r) for r in y_roots_symbolic]
z_roots = [float(r) for r in z_roots_symbolic]
for r in x_roots:
f = pcurve.OrthoCircle(f_poly(r), intersection_radius, 0)
segs = pcurve.get_piecewise_curve(f, 0, 1, 10, seg_length_min)
segments.extend((p0, p1, STYLE_X) for p0, p1 in segs)
for r in y_roots:
f = pcurve.OrthoCircle(f_poly(r), intersection_radius, 1)
segs = pcurve.get_piecewise_curve(f, 0, 1, 10, seg_length_min)
segments.extend((p0, p1, STYLE_Y) for p0, p1 in segs)
for r in z_roots:
f = pcurve.OrthoCircle(f_poly(r), intersection_radius, 2)
segs = pcurve.get_piecewise_curve(f, 0, 1, 10, seg_length_min)
segments.extend((p0, p1, STYLE_Z) for p0, p1 in segs)
# return the segments
return segments
def get_coordinates_maximum(self):
Eo = self.parameters['Eo']
a = self.parameters['a']
b = self.parameters['b']
k = self.parameters['k']
x, y, z = sy.symbols('x y z')
xu = x * unit.nanometers
yu = y * unit.nanometers
zu = z * unit.nanometers
potential_x = Eo * ((xu / a)**4 - 2.0 * (xu / a)**2) - (b / a) * xu
potential_y = 0.5 * k * (yu**2)
potential_z = 0.5 * k * (zu**2)
g = sy.diff(potential_x, x)
gg = sy.diff(potential_x, x, x)
roots_diff = sy.roots(g, x)
roots = []
for root in roots_diff.keys():
effective_k = gg.subs(x, root)
if effective_k < 0:
root_3d = np.zeros([3], dtype=float) * unit.nanometers
root_3d[0] = root * unit.nanometers
roots.append(root_3d)
del (x, y, z)
return roots
def test_singularities(self):
S1 = sympify(singularities(f1,x,y))
S2 = sympify(singularities(f2,x,y))
S3 = sympify(singularities(f3,x,y))
S4 = sympify(singularities(f4,x,y))
S5 = sympify(singularities(f5,x,y))
S6 = sympify(singularities(f6,x,y))
# S7 = sympify(singularities(f7,x,y))
# S8 = sympify(singularities(f8,x,y))
S9 = sympify(singularities(f9,x,y))
S10= sympify(singularities(f10,x,y))
rt5 = sympy.roots(x**2 + 1, x).keys()
S1act = sympify([
((0,0,1),(2,2,1)),
((0,1,0),(2,1,2))
])
S2act = sympify([
((0,0,1),(3,4,2)),
((0,1,0),(4,9,1))
])
S3act = sympify([
((0,0,1),(2,1,2)),
((1,-1,1),(2,1,2)),
((1,1,1),(2,1,2))
])
S4act = sympify([
((0,0,1),(2,1,2))
])
S5act = sympify([
((0,0,1),(3,3,3)),
((1,rt5[1],0),(3,3,3)),
((1,rt5[0],0),(3,3,3))
])
S6act = sympify([
((0,0,1),(2,2,2)),
((1,0,0),(2,2,2))
])
# S7act = sympify([((0,1,0),(3,6,3))])
# S8act = sympify([
# ((0,1,0),(6,21,3)),
# ((1,0,0),(3,7,2))
# ])
S9act = sympify([((0,1,0),(5,12,1))])
S10act= sympify([
((0,1,0),(3,6,1)),
((1,0,0),(4,6,4))
])
self.assertItemsEqual(S1,S1act)
self.assertItemsEqual(S2,S2act)
self.assertItemsEqual(S3,S3act)
self.assertItemsEqual(S4,S4act)
self.assertItemsEqual(S5,S5act)
self.assertItemsEqual(S6,S6act)
# self.assertItemsEqual(S7,S7act)
# self.assertItemsEqual(S8,S8act)
self.assertItemsEqual(S9,S9act)
self.assertItemsEqual(S10,S10act)
def test_singular_points(self):
S1 = singularities(f1,x,y)
S2 = singularities(f2,x,y)
S3 = singularities(f3,x,y)
S4 = singularities(f4,x,y)
S5 = singularities(f5,x,y)
S6 = singularities(f6,x,y)
S7 = singularities(f7,x,y)
S8 = singularities(f8,x,y)
S9 = singularities(f9,x,y)
S10= singularities(f10,x,y)
rt5 = [rt for rt,_ in sympy.roots(x**2 + 1, x).iteritems()]
S1act = [(0,0,1),(0,1,0)]
S2act = [(0,0,1),(0,1,0)]
S3act = [(0,0,1),(1,-1,1),(1,1,1)]
S4act = [(0,0,1)]
S5act = [(0,0,1),(rt5[0],1,0),(rt5[1],1,0)]
S6act = [(0,0,1),(1,0,0)]
S7act = [(0,1,0)]
S8act = [(0,1,0),(1,0,0)]
S9act = [(0,1,0)]
S10act= [(0,1,0),(1,0,0)]
self.assertItemsEqual(S1,S1act)
self.assertItemsEqual(S2,S2act)
self.assertItemsEqual(S3,S3act)
self.assertItemsEqual(S4,S4act)
self.assertItemsEqual(S5,S5act)
self.assertItemsEqual(S6,S6act)
self.assertItemsEqual(S7,S7act)
self.assertItemsEqual(S8,S8act)
self.assertItemsEqual(S9,S9act)
self.assertItemsEqual(S10,S10act)
def characteristic_roots(self):
"""Compute the roots of the characteristic equation.
:returns: List of roots returned by :mod:`sympy`.
"""
return sympy.roots(self.characteristic_poly())
def polyroots(poly, var):
"""Return roots of polynomial `poly` for variable `var`."""
roots = sym.roots(poly)
num_roots = 0
for root, n in roots.items():
num_roots += n
if num_roots != poly.degree():
if poly.degree() >= 5:
warn(
'Cannot find symbolic roots for polynomial of degree %d, see Abel-Ruffini theorem'
% poly.degree())
# If the coefficients of the polynomial are numerical,
# the SymPy nroots function can be used to find
# numerical approximations to the roots.
a = set()
a.add(var)
if poly.free_symbols == a:
warn('Only %d of %d roots found, using numerical approximation' %
(num_roots, poly.degree()))
nroots = poly.nroots()
roots = {}
for root in nroots:
if root in roots:
roots[root] += 1
else:
roots[root] = 1
return roots
warn('Only %d of %d roots found' % (num_roots, poly.degree()))
return roots
def test_legendre():
raises(ValueError, lambda: legendre(-1, x))
assert legendre(0, x) == 1
assert legendre(1, x) == x
assert legendre(2, x) == ((3 * x**2 - 1) / 2).expand()
assert legendre(3, x) == ((5 * x**3 - 3 * x) / 2).expand()
assert legendre(4, x) == ((35 * x**4 - 30 * x**2 + 3) / 8).expand()
assert legendre(5, x) == ((63 * x**5 - 70 * x**3 + 15 * x) / 8).expand()
assert legendre(6, x) == ((231 * x**6 - 315 * x**4 + 105 * x**2 - 5) /
16).expand()
assert legendre(10, -1) == 1
assert legendre(11, -1) == -1
assert legendre(10, 1) == 1
assert legendre(11, 1) == 1
assert legendre(10, 0) != 0
assert legendre(11, 0) == 0
assert roots(legendre(4, x), x) == {
sqrt(Rational(3, 7) - Rational(2, 35) * sqrt(30)): 1,
-sqrt(Rational(3, 7) - Rational(2, 35) * sqrt(30)): 1,
sqrt(Rational(3, 7) + Rational(2, 35) * sqrt(30)): 1,
-sqrt(Rational(3, 7) + Rational(2, 35) * sqrt(30)): 1,
}
n = Symbol("n")
X = legendre(n, x)
assert isinstance(X, legendre)
assert legendre(-n, x) == legendre(n - 1, x)
assert legendre(n, -x) == (-1)**n * legendre(n, x)
assert diff(legendre(
n, x), x) == n * (x * legendre(n, x) - legendre(n - 1, x)) / (x**2 - 1)
assert diff(legendre(n, x), n) == Derivative(legendre(n, x), n)
def get_pane_tikz_lines(poly_pair, color_pair, t0, t1):
"""
This is called maybe three times.
@param poly_pair: the lower and higher order Poly objects
@param color_pair: the polynomial colors
@param t0: initial time
@param t1: final time
@return: a list of tikz lines
"""
lines = []
pa, pb = poly_pair
ca, cb = color_pair
# draw the segmentation corresponding to the lower order polynomial
for left, right, sign in get_segmentation(pa, t0, t1):
width = {-1 : '0.5mm', 1 : '2mm'}[sign]
line = '\\draw[color=%s,line width=%s] (%s, 0) -- (%s, 0);' % (
ca, width, left, right)
lines.append(line)
# draw the cuts corresponding to the higher order polynomial
for r in sorted(sympy.roots(pb)):
cut = float(r)
line = '\\draw[color=%s,line width=0.5mm] (%s, -3mm) -- (%s, 3mm);' % (
cb, cut, cut)
lines.append(line)
return lines
def test_legendre():
raises(ValueError, lambda: legendre(-1, x))
assert legendre(0, x) == 1
assert legendre(1, x) == x
assert legendre(2, x) == ((3*x**2 - 1)/2).expand()
assert legendre(3, x) == ((5*x**3 - 3*x)/2).expand()
assert legendre(4, x) == ((35*x**4 - 30*x**2 + 3)/8).expand()
assert legendre(5, x) == ((63*x**5 - 70*x**3 + 15*x)/8).expand()
assert legendre(6, x) == ((231*x**6 - 315*x**4 + 105*x**2 - 5)/16).expand()
assert legendre(10, -1) == 1
assert legendre(11, -1) == -1
assert legendre(10, 1) == 1
assert legendre(11, 1) == 1
assert legendre(10, 0) != 0
assert legendre(11, 0) == 0
assert roots(legendre(4, x), x) == {
sqrt(Rational(3, 7) - Rational(2, 35)*sqrt(30)): 1,
-sqrt(Rational(3, 7) - Rational(2, 35)*sqrt(30)): 1,
sqrt(Rational(3, 7) + Rational(2, 35)*sqrt(30)): 1,
-sqrt(Rational(3, 7) + Rational(2, 35)*sqrt(30)): 1,
}
n = Symbol("n")
X = legendre(n, x)
assert isinstance(X, legendre)
assert legendre(-n, x) == legendre(n - 1, x)
assert legendre(n, -x) == (-1)**n*legendre(n, x)
assert diff(legendre(n, x), x) == \
n*(x*legendre(n, x) - legendre(n - 1, x))/(x**2 - 1)
assert diff(legendre(n, x), n) == Derivative(legendre(n, x), n)
def factor_linear_quadratic(f, x, as_list=False):
"""
Factors a polynomial f(x) with real coefficients
into linear and quadratic terms.
Will not work if f depends on variables other than x.
Returns a tuple with the constant coefficient and a list of the factors
if as_list is True.
"""
factors = []
origin_roots = 0
for root, multiplicity in sp.roots(f, x).items():
if root == 0:
factors.append((x, multiplicity))
origin_roots = multiplicity
elif root.is_real:
factors.append((x - root, multiplicity))
elif sp.im(root) > 0:
factors.append((sp.expand(
(x - root) * (x - sp.conjugate(root))), multiplicity))
coefficient = (f / x ** origin_roots).subs(x, 0) / \
sp.prod([factor.subs(x, 0) ** multiplicity
for factor, multiplicity in factors if factor != x])
if as_list:
return coefficient, factors
return coefficient * sp.prod(
[factor**multiplicity for factor, multiplicity in factors])
def _roots(poly, var):
""" like roots, but works on higher-order polynomials. """
r = roots(poly, var, multiple=True)
n = degree(poly)
if len(r) != n:
r = [rootof(poly, var, k) for k in range(n)]
return r
def as_ZPK(self):
"""Decompose expression into zeros, poles, gain, undef where
expression = K * (prod_n (var - z_n) / (prod_n (var - p_n)) * undef
"""
Bpoly = self.Bpoly
Apoly = self.Apoly
K = sym.cancel(Bpoly.LC() / Apoly.LC())
if self.delay != 0:
K *= sym.exp(self.var * self.delay)
zeros = sym.roots(Bpoly)
poles = sym.roots(Apoly)
return zeros, poles, K, self.undef
def find_roots(equation):
poly_values = []
for key, value in equation.items():
# poly_values.append(float(value.split("*r")[0]))
poly_values.append(parse_expr(value.split("*r")[0]))
poly_roots = sy.roots(poly_values)
print("roots:", poly_roots)
return poly_roots
def get_poles_from_matrix(a, show_steps=False):
'''Takes the A matrix of a system and returns the poles, format is {pole: multiplicity} can be used to determine stability'''
ss = sy.symbols('s')
char_mtx = ss * sy.eye(a.rows) - a
char_eq = char_mtx.det()
if show_steps:
display(char_mtx)
display(char_eq)
return (sy.roots(char_eq))
def d1c_roots(self, h_):
# substitute U_0**2 into eq. 2.8 for given value of h and
# simplify to create a polynomial in d1c
d1cp = WHBase.eq28().subs(U0 ** 2, u_squared())
d1cp = d1cp.subs({h: h_, S: self.S, d0: self.d0})
d1cp = sp.fraction(d1cp.simplify())[0]
# get all roots of this polynomial
d1c_roots = sp.roots(d1cp, filter=None, multiple=True)
return np.array(d1c_roots, dtype=complex)
def test_legendre():
assert legendre(0, x) == 1
assert legendre(1, x) == x
assert legendre(2, x) == ((3 * x**2 - 1) / 2).expand()
assert legendre(3, x) == ((5 * x**3 - 3 * x) / 2).expand()
assert legendre(4, x) == ((35 * x**4 - 30 * x**2 + 3) / 8).expand()
assert legendre(5, x) == ((63 * x**5 - 70 * x**3 + 15 * x) / 8).expand()
assert (legendre(6, x) == ((231 * x**6 - 315 * x**4 + 105 * x**2 - 5) /
16).expand())
assert legendre(10, -1) == 1
assert legendre(11, -1) == -1
assert legendre(10, 1) == 1
assert legendre(11, 1) == 1
assert legendre(10, 0) != 0
assert legendre(11, 0) == 0
assert legendre(-1, x) == 1
k = Symbol("k")
assert legendre(5 - k, x).subs(k, 2) == ((5 * x**3 - 3 * x) / 2).expand()
assert roots(legendre(4, x), x) == {
sqrt(Rational(3, 7) - Rational(2, 35) * sqrt(30)): 1,
-sqrt(Rational(3, 7) - Rational(2, 35) * sqrt(30)): 1,
sqrt(Rational(3, 7) + Rational(2, 35) * sqrt(30)): 1,
-sqrt(Rational(3, 7) + Rational(2, 35) * sqrt(30)): 1,
}
n = Symbol("n")
X = legendre(n, x)
assert isinstance(X, legendre)
assert unchanged(legendre, n, x)
assert legendre(n,
0) == sqrt(pi) / (gamma(S.Half - n / 2) * gamma(n / 2 + 1))
assert legendre(n, 1) == 1
assert legendre(n, oo) is oo
assert legendre(-n, x) == legendre(n - 1, x)
assert legendre(n, -x) == (-1)**n * legendre(n, x)
assert unchanged(legendre, -n + k, x)
assert conjugate(legendre(n, x)) == legendre(n, conjugate(x))
assert diff(legendre(
n, x), x) == n * (x * legendre(n, x) - legendre(n - 1, x)) / (x**2 - 1)
assert diff(legendre(n, x), n) == Derivative(legendre(n, x), n)
_k = Dummy("k")
assert (legendre(n, x).rewrite("polynomial").dummy_eq(
Sum(
(-1)**_k * (S.Half - x / 2)**_k * (x / 2 + S.Half)**(-_k + n) *
binomial(n, _k)**2,
(_k, 0, n),
)))
raises(ArgumentIndexError, lambda: legendre(n, x).fdiff(1))
raises(ArgumentIndexError, lambda: legendre(n, x).fdiff(3))
def rivnynnya(equt):
"""
Function to find all roots (returns a dict of them)
Here: key - root; value - number of its repetitions
>>> 'r**2 - 2* r + 1'
{1: 2}
"""
result = roots(equt)
return result
def getSols(poly, sym, orderDiff):
df = poly
polyDiffs = []
diffRoots = []
for i in range(1 + orderDiff):
polyDiffs += [df]
roots = sympy.roots(df)
diffRoots += [roots]
df = df.diff()
return polyDiffs, diffRoots
def ZPK(self):
"""Convert to zero-pole-gain (ZPK) form.
See also canonical, general, standard, timeconst, and partfrac"""
N, D, delay, undef = self.as_ratfun_delay_undef()
var = self.var
Npoly = sym.Poly(N, var)
Dpoly = sym.Poly(D, var)
K = sym.cancel(Npoly.LC() / Dpoly.LC())
if delay != 0:
K *= sym.exp(self.var * delay)
zeros = sym.roots(Npoly)
poles = sym.roots(Dpoly)
return _zp2tf(zeros, poles, K, self.var) * undef
def getSols(poly, sym, orderDiff):
df = poly
polyDiffs = []
diffRoots = []
for i in range(1+orderDiff):
polyDiffs += [df]
roots = sympy.roots(df)
diffRoots += [roots]
df = df.diff()
return polyDiffs, diffRoots
def test_roots_to_poly(self):
roots = (1.0, 4.0, 5.0)
p = sympyutils.roots_to_poly(roots)
self.assertTrue(p.is_monic)
root_to_count = sympy.roots(p)
self.assertEqual(set(root_to_count.values()), set([1]))
observed = set(root_to_count.keys())
expected = set(roots)
for r_observed, r_expected in zip(sorted(observed), sorted(expected)):
self.assertAlmostEqual(r_observed, r_expected)
def test_roots_to_poly(self):
roots = (1.0, 4.0, 5.0)
p = sympyutils.roots_to_poly(roots)
self.assertTrue(p.is_monic)
root_to_count = sympy.roots(p)
self.assertEqual(set(root_to_count.values()), set([1]))
observed = set(root_to_count.keys())
expected = set(roots)
for r_observed, r_expected in zip(sorted(observed), sorted(expected)):
self.assertAlmostEqual(r_observed, r_expected)
def ZPK(self):
"""Convert to pole-zero-gain (PZK) form.
See also canonical, general, mixedfrac, and partfrac"""
N, D, delay, undef = self.as_ratfun_delay_undef()
var = self.var
Npoly = sym.Poly(N, var)
Dpoly = sym.Poly(D, var)
K = sym.cancel(Npoly.LC() / Dpoly.LC())
if delay != 0:
K *= sym.exp(self.var * delay)
zeros = sym.roots(Npoly)
poles = sym.roots(Dpoly)
return _zp2tf(zeros, poles, K, self.var) * undef
def singular_term(F,X,Y,L,I,version):
"""
Computes a single set of singular terms of the Puiseux expansions.
For `I=1`, the function computes the first term of each finite
`\mathbb{K}` expansion. For `I=2`, it computes the other terms of the
expansion.
"""
T = []
# if the curve is singular then compute the singular tuples
# otherwise, use the standard newton polygon method
if is_singular(F,X,Y):
for (q,m,l,Phi) in desingularize(F,X,Y):
for eta in Phi.all_roots(radicals=True):
tau = (q,1,m,1,sympy.together(eta))
T.append((tau,0,1))
else:
# each side of the newton polygon corresponds to a K-term.
for (q,m,l,Phi) in polygon(F,X,Y,I):
# the rational method
if version == 'rational':
u,v = _bezout(q,m)
# each newton polygon side has a characteristic
# polynomial. For each square-free factor, each root
# corresponds to a K-term
for (Psi,r) in _square_free(Phi,_Z):
Psi = sympy.Poly(Psi,_Z)
# compute the roots of Psi. Use the RootOf construct if
# possible. In the case when Psi is over EX (i.e. when
# RootOf doesn't work) then compute symbolic roots.
try:
roots = Psi.all_roots(radicals=True)
except NotImplementedError:
roots = sympy.roots(Psi,_Z).keys()
for xi in roots:
# the classical version returns the "raw" roots
if version == 'classical':
P = sympy.Poly(_U**q-xi,_U)
beta = RootOf(P,0,radicals=True)
tau = (q,1,m,beta,1)
T.append((tau,l,r))
# the rational version rescales parameters so as
# to include ony rational terms in the Puiseux
# expansions.
if version == 'rational':
mu = xi**(-v)
beta = sympy.together(xi**u)
tau = (q,mu,m,beta,1)
T.append((tau,l,r))
return T
def run_copper_smith(f, N, h, k):
X = s.S(s.ceiling((s.Pow(2, - 1 / 2) * s.Pow(h * k, -1 / (h * k - 1))) *
s.Pow(N, (h - 1) / (h * k - 1))) - 1)
gen = lll(generate(f, N, h, k, X), s.S(0.75))
final_poly = [gen[0][i] / s.Pow(X, i) for i in range(len(gen[0]))][::-1]
roots = s.roots(final_poly)
for r in roots:
print(r)
return roots
def timeconst(self):
"""Convert rational function to time constant form with unity
lowest power of denominator.
See also canonical, general, partfrac, standard, and ZPK"""
N, D, delay, undef = self.as_ratfun_delay_undef()
var = self.var
Npoly = sym.Poly(N, var)
Dpoly = sym.Poly(D, var)
K = sym.cancel(Npoly.LC() / Dpoly.LC())
if delay != 0:
K *= sym.exp(self.var * delay)
zeros = sym.roots(Npoly)
poles = sym.roots(Dpoly)
return _tc2tf(zeros, poles, K, self.var) * undef
def _singular_points_finite(f,x,y):
"""
Returns the finite singular points of f.
"""
S = []
# compute the finite singularities: use the resultant
p = sympy.Poly(f,[x,y])
n = p.degree(y)
res = sympy.Poly(sympy.resultant(p,p.diff(y),y),x)
for xk,deg in sympy.roots(res,x).iteritems():
if deg > 1:
fxk = sympy.Poly(f.subs({x:xk,y:_Z}), _Z)
for ykj,_ in sympy.roots(fxk, _Z).iteritems():
fx = f.diff(x)
fy = f.diff(y)
subs = {x:xk,y:ykj}
if (fx.subs(subs) == 0) and (fy.subs(subs) == 0):
S.append((xk,ykj,1))
return S
def fund_ries(drc):
a2,a1,a0 = drc
r = roots(a2*t**2 + a1*t +a0,t)
kl = list(r.keys())
if len(kl) == 2: # jednoduche korene
r1,r2 = kl
re_k,im_k = r1.as_real_imag()
if im_k == 0: # jednoduche realne korene
return exp(r1*t), exp(r2*t)
else: # komplexne zdruz.korene
return exp(re_k*t)*sin(im_k*t), exp(re_k*t)*cos(im_k*t)
else: # dvojnasobny koren
return exp(kl[0]*t), t*exp(kl[0]*t)
def solve_eq(init_conditions, associated):
degree = len(init_conditions)
eq = util.characteristic_eq(degree, associated)
rts = roots(parse_expr(eq))
nr_of_a = util.nr_of_alphas(rts)
if len(rts) == 1:
alphas = util.find_alphas_sol2(rts, init_conditions, nr_of_a)
return util.build_solution2(rts, alphas)
else:
alphas = util.find_alphas_sol1(rts, init_conditions, nr_of_a)
return util.build_solution1(rts, alphas)
def _singular_points_infinite(f,x,y):
"""
Returns the singular points of f at infinity.
"""
S = []
# compute homogenous polynomial
F, d = homogenize(f,x,y,_z)
# find singular points at infinity
F0 = sympy.Poly(F.subs([(_z,0)]),[x,y])
domain = sympy.QQ[sympy.I]
solsX1 = sympy.roots(F0.subs({x:1,y:_Z}),_Z).keys()
solsY1 = sympy.roots(F0.subs({y:1,x:_Z}),_Z).keys()
all_sols = [ (1,yi,0) for yi in solsX1 ]
all_sols.extend( [ (xi,1,0) for xi in solsY1 ] )
# these points are in projective space, so filter out equal points
# such as (1,I,0) == (-I,1,0). We normalize these projective
# points such that 1 appears in either the x- or y- coordinate
sols = []
for xi,yi,zi in all_sols:
normalized = (1,yi/xi,0) if xi != sympy.S(0) else (xi/yi,1,0)
if not normalized in sols:
sols.append(normalized)
# Filter out any points that are not singular by checking if the
# gradient vanishes.
grad = [F.diff(var) for var in [x,y,_z]]
for xi,yi,zi in sols:
fsub = lambda e,x=x,y=y,_z=_z,xi=xi,yi=yi,zi=zi: \
e.subs({x:xi,y:yi,_z:zi}).simplify() != sympy.S(0)
if not any(map(fsub,grad)):
S.append((xi,yi,zi))
return S
def tustin(tf, sRate):
s = getMapping(tf)['s']
T = 1 / sRate
poles = sympy.roots(sympy.denom(tf), s, multiple=True)
centroid = np.mean(np.abs(poles))
invz = sympy.Symbol('invz', real=True)
# normalized center frequency derived from poles of filter SISO transfer function
w0 = 2 * np.pi * centroid / (sRate * 2 * np.pi)
# modified bilinear transform w/ frequency warping
bt = w0 / sympy.tan(w0 * T / 2) * ((1 - invz) / (1 + invz))
dt = sympy.simplify(tf.subs({s: bt}))
b = sympy.Poly(sympy.numer(dt)).all_coeffs()[::-1]
a = sympy.Poly(sympy.denom(dt)).all_coeffs()[::-1]
normalize = lambda x: float(x / a[0])
return (map(normalize, b), map(normalize, a))
def test_ndiff_roots_symbolic(self):
roots = (1.25, 4.5, 5.75)
a, b, c = roots
polys = roots_to_differential_polys(roots)
# compute linear root manually
r = float(a + b + c) / 3
# check linear root
observed = sorted(float(r) for r in sympy.roots(polys[0]))
expected = sorted([r])
self.assertTrue(np.allclose(observed, expected))
# compute quadratic roots manually
A = a * a + b * b + c * c
B = a * b + a * c + b * c
S = a + b + c
r0 = float(S + math.sqrt(A - B)) / 3
r1 = float(S - math.sqrt(A - B)) / 3
# check quadratic roots
observed = sorted(float(r) for r in sympy.roots(polys[1]))
expected = sorted([r0, r1])
self.assertTrue(np.allclose(observed, expected))
# check cubic roots
observed = sorted(float(r) for r in sympy.roots(polys[2]))
expected = sorted(roots)
self.assertTrue(np.allclose(observed, expected))
def tustin(tf, sRate):
s = getMapping(tf)['s']
T = 1/sRate
poles = sympy.roots(sympy.denom(tf), s, multiple=True)
centroid = np.mean(np.abs(poles))
invz = sympy.Symbol('invz', real=True)
# normalized center frequency derived from poles of filter SISO transfer function
w0 = 2*np.pi*centroid/(sRate*2*np.pi)
# modified bilinear transform w/ frequency warping
bt = w0/sympy.tan(w0*T/2) * ((1-invz)/(1+invz))
dt = sympy.simplify(tf.subs({s: bt}))
b = sympy.Poly(sympy.numer(dt)).all_coeffs()[::-1]
a = sympy.Poly(sympy.denom(dt)).all_coeffs()[::-1]
normalize = lambda x: float(x/a[0])
return (map(normalize, b), map(normalize, a))
def test_ndiff_roots_symbolic(self):
roots = (1.25, 4.5, 5.75)
a, b, c = roots
polys = roots_to_differential_polys(roots)
# compute linear root manually
r = float(a + b + c) / 3
# check linear root
observed = sorted(float(r) for r in sympy.roots(polys[0]))
expected = sorted([r])
self.assertTrue(np.allclose(observed, expected))
# compute quadratic roots manually
A = a*a + b*b + c*c
B = a*b + a*c + b*c
S = a + b + c
r0 = float(S + math.sqrt(A - B)) / 3
r1 = float(S - math.sqrt(A - B)) / 3
# check quadratic roots
observed = sorted(float(r) for r in sympy.roots(polys[1]))
expected = sorted([r0, r1])
self.assertTrue(np.allclose(observed, expected))
# check cubic roots
observed = sorted(float(r) for r in sympy.roots(polys[2]))
expected = sorted(roots)
self.assertTrue(np.allclose(observed, expected))
def _solve(self):
"""
Solve the recurrence relation into a closed form
Returns:
String: The solved recurrence relation in string format
"""
logging.info("Started solving recurrence relation: %s" % str(self._recurrence))
self._degree, homogenous, nonHomogenous, linear = self._analyseExpression()
msg = "homogenous" if nonHomogenous == 0 else "nonhomogenous"
logging.info("Analyzation complete, It is a %s recurrence relation with degree %d" % (msg, self._degree))
if nonHomogenous != 0:
logging.info("The part that makes the recurrence nonhomogenous is: %s" % str(nonHomogenous))
if not linear:
raise RecurrenceSolveFailed("The equation is not linear")
characteristicEq = self._getCharacteristicEquation(homogenous)
logging.info("The characteristic equation is: %s" % str(characteristicEq))
# get roots of characteristic equations and remove
# the complex roots
realRoots = { s:m for (s,m) in sympy.roots(characteristicEq).items() if sympy.I not in s.atoms() }
logging.info("With roots: multiplicities: %s" % str(realRoots))
# the sum of the multiplicity must be the same as the degree
# else we can't solve the equation
if sum(realRoots.values()) != self._degree:
msg = "The characteristic equation \"%s\" has the following real roots: %s, and the multiplicities is not the same as the degree" % (str(characteristicEq), str(realRoots))
raise RecurrenceSolveFailed(msg)
generalSolution, ctx = self._getGeneralSolution(realRoots)
logging.info("The general solution has the form: %s" % str(generalSolution))
solved = sympy.sympify("0")
if nonHomogenous == 0:
solved = self._calculateClosedFromGeneralSolution(generalSolution, ctx)
else:
solved = self._solveNonHomogeneous(realRoots, homogenous, nonHomogenous, generalSolution, ctx)
logging.info("Solved raw: %s" % str(solved))
solved = solved.simplify()
logging.info("Solved simplified: %s" % str(solved))
return solved
def ballpath(xinit=0, yinit=0, vxinit=1, vyinit=1):
x0val = xinit
y0val = yinit
v0xval = vxinit
v0yval = vyinit
# create symbolic exprs for ball path and intersection polynomial
x0, y0, v0x, v0y, t = symbols('x0 y0 v0x v0y t')
xt = x0 + v0x * t
yt = y0 + v0y * t - (1 / 2) * 9.8 * (t**2)
pgen = poly(xt**2 + yt**2 - 1, t)
# create actual functions from symbolic exprs
xtf = lambdify((t, x0, v0x), xt)
ytf = lambdify((t, y0, v0y), yt)
vytf = lambdify((t, v0y), diff(yt, t))
while True:
# subst current conditions into intersect poly
p = pgen.subs({x0: x0val, y0: y0val, v0x: v0xval, v0y: v0yval})
# solve for t
rts = roots(p, t)
rts = np.array([complex(r) for r in rts.keys()])
tcol = np.min(rts[np.logical_and(rts > 0, np.imag(rts) == 0)]).real
# find intersect point and speed at the moment
xcol = xtf(tcol, x0val, v0xval)
ycol = ytf(tcol, y0val, v0yval)
vxcol = v0xval
vycol = vytf(tcol, v0yval)
vcol = np.array([[vxcol], [vycol]])
# calculate speed vector after bounce
alpha = np.arctan2(ycol, xcol) - np.arctan2(vycol, vxcol)
R = np.array([[np.cos(2*alpha), -np.sin(2*alpha)], \
[np.sin(2*alpha), np.cos(2*alpha)]])
newv0 = R @ -vcol
tvals = np.linspace(0, tcol, 100)
xvals = xtf(tvals, x0val, v0xval)
yvals = ytf(tvals, y0val, v0yval)
yield (xvals.tolist(), yvals.tolist())
x0val = xcol
y0val = ycol
v0xval = newv0[0].item()
v0yval = newv0[1].item()
def get_segmentation(p, t0, t1):
"""
A segmentation is a sequence of triples (left, right, sign).
@param p: a sympy Poly
@param t0: initial time
@param t1: final time
@return: a segmentation
"""
roots = sorted(float(r) for r in sympy.roots(p))
points = [t0] + roots + [t1]
segmentation = []
for left, right in iterutils.pairwise(points):
mid = (left + right) / 2
sign = -1 if p.eval(mid) <= 0 else 1
seg = (left, right, sign)
segmentation.append(seg)
return segmentation
def get_segmentation(p, t0, t1):
"""
A segmentation is a sequence of triples (left, right, sign).
@param p: a sympy Poly
@param t0: initial time
@param t1: final time
@return: a segmentation
"""
roots = sorted(float(r) for r in sympy.roots(p))
points = [t0] + roots + [t1]
segmentation = []
for left, right in iterutils.pairwise(points):
mid = (left + right) / 2
sign = -1 if p.eval(mid) <= 0 else 1
seg = (left, right, sign)
segmentation.append(seg)
return segmentation
def test_poly():
output("""\
poly_:{
r:{$[prec>=abs(x:reim x)1;x 0;x]} each .qml.poly x;
r iasc {x:reim x;(abs x 1;neg x 1;x 0)} each r};""")
for A in poly_subjects:
A = sp.Poly(A, sp.Symbol("x"))
if A.degree() <= 3 and all(a.is_real for a in A.all_coeffs()):
R = []
for r in sp.roots(A, multiple=True):
r = sp.simplify(sp.expand_complex(r))
R.append(r)
R.sort(key=lambda x: (abs(sp.im(x)), -sp.im(x), sp.re(x)))
else:
R = mp.polyroots([mp.mpc(*(a.evalf(mp.mp.dps)).as_real_imag())
for a in A.all_coeffs()])
R.sort(key=lambda x: (abs(x.imag), -x.imag, x.real))
assert len(R) == A.degree()
test("poly_", A.all_coeffs(), R, complex_pair=True)
def given():
xp0 = randrange(7, 10)
pmax = randrange(7, 10)
Ep = R(-pmax, xp0) * x + pmax
E = sympy.integrate(Ep, x)
b = randrange(2, xp0 - 2)
cmax = int(Ep.subs(x, b).n())
c = randrange(1, cmax)
a = R(randrange(1, pmax - c), b * b)
Kp = a * (x - b) ** 2 + c
Gp = Ep - Kp
rG = [r.n() for r in sympy.roots(Gp) if r > 0][0]
G = sympy.integrate(Gp, x)
mG = G.subs(x, rG)
Ko = int(mG - 1) / 2
#K = sympy.integrate(Kp,x)+Ko
#G = E.n()-K.n()
#rts = sorted([r.n() for r in sympy.roots(G) if r > 0])
g = Struct(pmax=pmax, xp0=xp0, Ko=Ko, Kp=sympy.sstr(Kp))
return g
def test_legendre():
assert legendre(0, x) == 1
assert legendre(1, x) == x
assert legendre(2, x) == ((3*x**2-1)/2).expand()
assert legendre(3, x) == ((5*x**3-3*x)/2).expand()
assert legendre(4, x) == ((35*x**4-30*x**2+3)/8).expand()
assert legendre(5, x) == ((63*x**5-70*x**3+15*x)/8).expand()
assert legendre(6, x) == ((231*x**6-315*x**4+105*x**2-5)/16).expand()
assert legendre(10, -1) == 1
assert legendre(11, -1) == -1
assert legendre(10, 1) == 1
assert legendre(11, 1) == 1
assert legendre(10, 0) != 0
assert legendre(11, 0) == 0
assert roots(legendre(4,x), x) == {
sqrt(Rational(3, 7) - Rational(2, 35)*sqrt(30)): 1,
-sqrt(Rational(3, 7) - Rational(2, 35)*sqrt(30)): 1,
sqrt(Rational(3, 7) + Rational(2, 35)*sqrt(30)): 1,
-sqrt(Rational(3, 7) + Rational(2, 35)*sqrt(30)): 1,
}
def get_pane_tikz_lines(poly_pair, color_pair, t0, t1):
"""
This is called maybe three times.
@param poly_pair: the lower and higher order Poly objects
@param color_pair: the polynomial colors
@param t0: initial time
@param t1: final time
@return: a list of tikz lines
"""
lines = []
pa, pb = poly_pair
ca, cb = color_pair
# draw the segmentation corresponding to the lower order polynomial
for left, right, sign in get_segmentation(pa, t0, t1):
width = {-1: "0.5mm", 1: "2mm"}[sign]
line = "\\draw[color=%s,line width=%s] (%s, 0) -- (%s, 0);" % (ca, width, left, right)
lines.append(line)
# draw the cuts corresponding to the higher order polynomial
for r in sorted(sympy.roots(pb)):
cut = float(r)
line = "\\draw[color=%s,line width=0.5mm] (%s, -3mm) -- (%s, 3mm);" % (cb, cut, cut)
lines.append(line)
return lines
def calc(g):
Kp = S(g.Kp)
Ep = R(-g.pmax, g.xp0) * x + g.pmax
E = sympy.integrate(Ep, x)
K = sympy.integrate(Kp, x) + S(g.Ko)
Gp = Ep - Kp
rG = [r.n() for r in sympy.roots(Gp) if r > 0][0]
G = E - K
mG = G.subs(x, rG)
kk = sympy.Wild('kk')
dd = sympy.Wild('dd')
kd = Ep.match(kk * x + dd)
p = kd[kk] * x / 2 + kd[dd]
mp = p.subs(x, rG)
el = S(1 + kd[dd] / (kd[kk] * x / 2))
return [
sympy.sstr(Ep),
sympy.sstr(E),
sympy.sstr(K),
sympy.sstr(p),
sympy.sstr(el),
rG,
mp]
def coefficient(polinom_in):
"""
:param polinom_in:
:return:
"""
edge_dot = []
polinom_antispace = re.sub(r'\s', '', polinom_in)
edge_y = test_eval(0, polinom_antispace) # нахождение максимального y при x=0
edge_y = edge_y * 1.03
r = roots(polinom_antispace) #получаю словарь возможных корней(пересечение с x)
edge_list_x = []
for k in r.keys():
if k.__class__.__name__ == "Float":
if float(k) > 0:
edge_list_x.append(float(k))
edge_x = min(edge_list_x)
edge_x = edge_x * 1.03
print edge_x
edge_dot.append(edge_x)
edge_dot.append(edge_y)
return edge_dot # возвращает список с точкой пересечения оси икс и игрик
phi_sum = np.zeros(np.diag(Phi[:,0]).shape)
for i in xrange(Phi.shape[1]):
phi_sum+= alpha[i] * np.diag(Phi[:,i])
(lambda_2 * phi_sum)**(-.5)
#
# implementing the set-level optimization
# alternate between alpha and sigma optimization
from sympy import roots,symbols,I
from sympy.utilities.lambdify import lambdify
x,a_rootvar,b_rootvar,d_rootvar = symbols('x,a_rootvar,b_rootvar,d_rootvar')
root_list = roots(a_rootvar*x**3 +b_rootvar*x**2 +d_rootvar,x).keys()
root_exprs = [ root_expr.as_real_imag() for root_expr in root_list]
def get_real_root(a,b,d):
""" Get the real solution
to the equation in x
a*x**3 + b*x**2 + d = 0
"""
real_soln_idx = np.argmin(np.abs([root_exprs[n][1].evalf(subs={a_rootvar:a,b_rootvar:b,d_rootvar:d}) for n in xrange(len(root_exprs))]))
return root_exprs[real_soln_idx][0].evalf(subs={a_rootvar:a,b_rootvar:b,d_rootvar:d})
# now we are going to
def log_to_real(h, q, x, t):
"""Convert complex logarithms to real functions.
Given real field K and polynomials h in K[t,x] and q in K[t],
returns real function f such that:
___
df d \ `
-- = -- ) a log(h(a, x))
dx dx /__,
a | q(a) = 0
"""
u, v = symbols('u v')
H = h.subs({t:u+I*v}).as_basic().expand()
Q = q.subs({t:u+I*v}).as_basic().expand()
H_map = collect(H, I, evaluate=False)
Q_map = collect(Q, I, evaluate=False)
a, b = H_map.get(S(1), S(0)), H_map.get(I, S(0))
c, d = Q_map.get(S(1), S(0)), Q_map.get(I, S(0))
R = Poly(resultant(c, d, v), u)
R_u = roots(R, domain='R')
if len(R_u) != number_of_real_roots(R):
return None
result = S(0)
for r_u in R_u.iterkeys():
C = Poly(c.subs({u:r_u}), v)
R_v = roots(C, domain='R')
if len(R_v) != number_of_real_roots(C):
return None
for r_v in R_v:
if not r_v.is_positive:
continue
D = d.subs({u:r_u, v:r_v})
if D.evalf(chop=True) != 0:
continue
A = Poly(a.subs({u:r_u, v:r_v}), x)
B = Poly(b.subs({u:r_u, v:r_v}), x)
AB = (A**2 + B**2).as_basic()
result += r_u*log(AB) + r_v*log_to_atan(A, B)
R_q = roots(q, domain='R')
if len(R_q) != number_of_real_roots(q):
return None
for r in R_q.iterkeys():
result += r*log(h.subs(t, r).as_basic())
return result
def log_to_real(h, q, x, t):
"""
Convert complex logarithms to real functions.
Given real field K and polynomials h in K[t,x] and q in K[t],
returns real function f such that:
___
df d \ `
-- = -- ) a log(h(a, x))
dx dx /__,
a | q(a) = 0
Examples
========
>>> from sympy.integrals.rationaltools import log_to_real
>>> from sympy.abc import x, y
>>> from sympy import Poly, sqrt, S
>>> log_to_real(Poly(x + 3*y/2 + S(1)/2, x, domain='QQ[y]'),
... Poly(3*y**2 + 1, y, domain='ZZ'), x, y)
2*sqrt(3)*atan(2*sqrt(3)*x/3 + sqrt(3)/3)/3
>>> log_to_real(Poly(x**2 - 1, x, domain='ZZ'),
... Poly(-2*y + 1, y, domain='ZZ'), x, y)
log(x**2 - 1)/2
See Also
========
log_to_atan
"""
u, v = symbols('u,v', cls=Dummy)
H = h.as_expr().subs({t: u + I*v}).expand()
Q = q.as_expr().subs({t: u + I*v}).expand()
H_map = collect(H, I, evaluate=False)
Q_map = collect(Q, I, evaluate=False)
a, b = H_map.get(S(1), S(0)), H_map.get(I, S(0))
c, d = Q_map.get(S(1), S(0)), Q_map.get(I, S(0))
R = Poly(resultant(c, d, v), u)
R_u = roots(R, filter='R')
if len(R_u) != R.count_roots():
return None
result = S(0)
for r_u in R_u.keys():
C = Poly(c.subs({u: r_u}), v)
R_v = roots(C, filter='R')
if len(R_v) != C.count_roots():
return None
for r_v in R_v:
if not r_v.is_positive:
continue
D = d.subs({u: r_u, v: r_v})
if D.evalf(chop=True) != 0:
continue
A = Poly(a.subs({u: r_u, v: r_v}), x)
B = Poly(b.subs({u: r_u, v: r_v}), x)
AB = (A**2 + B**2).as_expr()
result += r_u*log(AB) + r_v*log_to_atan(A, B)
R_q = roots(q, filter='R')
if len(R_q) != q.count_roots():
return None
for r in R_q.keys():
result += r*log(h.as_expr().subs(t, r))
return result
def _rewrite_gamma(f, s, a, b):
"""
Try to rewrite the product f(s) as a product of gamma functions,
so that the inverse mellin transform of f can be expressed as a meijer
G function.
Return (an, ap), (bm, bq), arg, exp, fac such that
G((an, ap), (bm, bq), arg/z**exp)*fac is the inverse mellin transform of f(s).
Raises IntegralTransformError or MellinTransformStripError on failure.
It is asserted that f has no poles in the fundamental strip designated by
(a, b). One of a and b is allowed to be None. The fundamental strip is
important, because it determines the inversion contour.
This function can handle exponentials, linear factors, trigonometric
functions.
This is a helper function for inverse_mellin_transform that will not
attempt any transformations on f.
>>> from sympy.integrals.transforms import _rewrite_gamma
>>> from sympy.abc import s
>>> from sympy import oo
>>> _rewrite_gamma(s*(s+3)*(s-1), s, -oo, oo)
(([], [-3, 0, 1]), ([-2, 1, 2], []), 1, 1, -1)
>>> _rewrite_gamma((s-1)**2, s, -oo, oo)
(([], [1, 1]), ([2, 2], []), 1, 1, 1)
Importance of the fundamental strip:
>>> _rewrite_gamma(1/s, s, 0, oo)
(([1], []), ([], [0]), 1, 1, 1)
>>> _rewrite_gamma(1/s, s, None, oo)
(([1], []), ([], [0]), 1, 1, 1)
>>> _rewrite_gamma(1/s, s, 0, None)
(([1], []), ([], [0]), 1, 1, 1)
>>> _rewrite_gamma(1/s, s, -oo, 0)
(([], [1]), ([0], []), 1, 1, -1)
>>> _rewrite_gamma(1/s, s, None, 0)
(([], [1]), ([0], []), 1, 1, -1)
>>> _rewrite_gamma(1/s, s, -oo, None)
(([], [1]), ([0], []), 1, 1, -1)
>>> _rewrite_gamma(2**(-s+3), s, -oo, oo)
(([], []), ([], []), 1/2, 1, 8)
"""
from itertools import repeat
from sympy import (Poly, gamma, Mul, re, RootOf, exp as exp_, E, expand,
roots, ilcm, pi, sin, cos, tan, cot, igcd, exp_polar)
# Our strategy will be as follows:
# 1) Guess a constant c such that the inversion integral should be
# performed wrt s'=c*s (instead of plain s). Write s for s'.
# 2) Process all factors, rewrite them independently as gamma functions in
# argument s, or exponentials of s.
# 3) Try to transform all gamma functions s.t. they have argument
# a+s or a-s.
# 4) Check that the resulting G function parameters are valid.
# 5) Combine all the exponentials.
a_, b_ = S([a, b])
def left(c, is_numer):
"""
Decide whether pole at c lies to the left of the fundamental strip.
"""
# heuristically, this is the best chance for us to solve the inequalities
c = expand(re(c))
if a_ is None:
return c < b_
if b_ is None:
return c <= a_
if (c >= b_) is True:
return False
if (c <= a_) is True:
return True
if is_numer:
return None
if a_.free_symbols or b_.free_symbols or c.free_symbols:
return None # XXX
#raise IntegralTransformError('Inverse Mellin', f,
# 'Could not determine position of singularity %s'
# ' relative to fundamental strip' % c)
raise MellinTransformStripError('Pole inside critical strip?')
# 1)
s_multipliers = []
for g in f.atoms(gamma):
if not g.has(s):
continue
arg = g.args[0]
if arg.is_Add:
arg = arg.as_independent(s)[1]
coeff, _ = arg.as_coeff_mul(s)
s_multipliers += [coeff]
for g in f.atoms(sin, cos, tan, cot):
if not g.has(s):
continue
arg = g.args[0]
if arg.is_Add:
arg = arg.as_independent(s)[1]
coeff, _ = arg.as_coeff_mul(s)
s_multipliers += [coeff/pi]
s_multipliers = [abs(x) for x in s_multipliers if x.is_real]
common_coefficient = S(1)
for x in s_multipliers:
if not x.is_Rational:
common_coefficient = x
break
s_multipliers = [x/common_coefficient for x in s_multipliers]
if any(not x.is_Rational for x in s_multipliers):
raise NotImplementedError
s_multiplier = common_coefficient/reduce(ilcm, [S(x.q) for x in s_multipliers], S(1))
if s_multiplier == common_coefficient:
if len(s_multipliers) == 0:
s_multiplier = common_coefficient
else:
s_multiplier = common_coefficient \
*reduce(igcd, [S(x.p) for x in s_multipliers])
exponent = S(1)
fac = S(1)
f = f.subs(s, s/s_multiplier)
fac /= s_multiplier
exponent = 1/s_multiplier
if a_ is not None:
a_ *= s_multiplier
if b_ is not None:
b_ *= s_multiplier
# 2)
numer, denom = f.as_numer_denom()
numer = Mul.make_args(numer)
denom = Mul.make_args(denom)
args = zip(numer, repeat(True)) + zip(denom, repeat(False))
facs = []
dfacs = []
# *_gammas will contain pairs (a, c) representing Gamma(a*s + c)
numer_gammas = []
denom_gammas = []
# exponentials will contain bases for exponentials of s
exponentials = []
def exception(fact):
return IntegralTransformError("Inverse Mellin", f, "Unrecognised form '%s'." % fact)
while args:
fact, is_numer = args.pop()
if is_numer:
ugammas, lgammas = numer_gammas, denom_gammas
ufacs, lfacs = facs, dfacs
else:
ugammas, lgammas = denom_gammas, numer_gammas
ufacs, lfacs = dfacs, facs
def linear_arg(arg):
""" Test if arg is of form a*s+b, raise exception if not. """
if not arg.is_polynomial(s):
raise exception(fact)
p = Poly(arg, s)
if p.degree() != 1:
raise exception(fact)
return p.all_coeffs()
# constants
if not fact.has(s):
ufacs += [fact]
# exponentials
elif fact.is_Pow or isinstance(fact, exp_):
if fact.is_Pow:
base = fact.base
exp = fact.exp
else:
base = exp_polar(1)
exp = fact.args[0]
if exp.is_Integer:
cond = is_numer
if exp < 0:
cond = not cond
args += [(base, cond)]*abs(exp)
continue
elif not base.has(s):
a, b = linear_arg(exp)
if not is_numer:
base = 1/base
exponentials += [base**a]
facs += [base**b]
else:
raise exception(fact)
# linear factors
elif fact.is_polynomial(s):
p = Poly(fact, s)
if p.degree() != 1:
# We completely factor the poly. For this we need the roots.
# Now roots() only works in some cases (low degree), and RootOf
# only works without parameters. So try both...
coeff = p.LT()[1]
rs = roots(p, s)
if len(rs) != p.degree():
rs = RootOf.all_roots(p)
ufacs += [coeff]
args += [(s - c, is_numer) for c in rs]
continue
a, c = p.all_coeffs()
ufacs += [a]
c /= -a
# Now need to convert s - c
if left(c, is_numer):
ugammas += [(S(1), -c + 1)]
lgammas += [(S(1), -c)]
else:
ufacs += [-1]
ugammas += [(S(-1), c + 1)]
lgammas += [(S(-1), c)]
elif isinstance(fact, gamma):
a, b = linear_arg(fact.args[0])
if is_numer:
if (a > 0 and (left(-b/a, is_numer) is False)) or \
(a < 0 and (left(-b/a, is_numer) is True)):
raise NotImplementedError('Gammas partially over the strip.')
ugammas += [(a, b)]
elif isinstance(fact, sin):
# We try to re-write all trigs as gammas. This is not in
# general the best strategy, since sometimes this is impossible,
# but rewriting as exponentials would work. However trig functions
# in inverse mellin transforms usually all come from simplifying
# gamma terms, so this should work.
a = fact.args[0]
if is_numer:
# No problem with the poles.
gamma1, gamma2, fac_ = gamma(a/pi), gamma(1 - a/pi), pi
else:
gamma1, gamma2, fac_ = _rewrite_sin(linear_arg(a), s, a_, b_)
args += [(gamma1, not is_numer), (gamma2, not is_numer)]
ufacs += [fac_]
elif isinstance(fact, tan):
a = fact.args[0]
args += [(sin(a, evaluate=False), is_numer),
(sin(pi/2 - a, evaluate=False), not is_numer)]
elif isinstance(fact, cos):
a = fact.args[0]
args += [(sin(pi/2 - a, evaluate=False), is_numer)]
elif isinstance(fact, cot):
a = fact.args[0]
args += [(sin(pi/2 - a, evaluate=False), is_numer),
(sin(a, evaluate=False), not is_numer)]
else:
raise exception(fact)
fac *= Mul(*facs)/Mul(*dfacs)
# 3)
an, ap, bm, bq = [], [], [], []
for gammas, plus, minus, is_numer in [(numer_gammas, an, bm, True),
(denom_gammas, bq, ap, False)]:
while gammas:
a, c = gammas.pop()
if a != -1 and a != +1:
# We use the gamma function multiplication theorem.
p = abs(S(a))
newa = a/p
newc = c/p
assert a.is_Integer
for k in range(p):
gammas += [(newa, newc + k/p)]
if is_numer:
fac *= (2*pi)**((1 - p)/2) * p**(c - S(1)/2)
exponentials += [p**a]
else:
fac /= (2*pi)**((1 - p)/2) * p**(c - S(1)/2)
exponentials += [p**(-a)]
continue
if a == +1:
plus.append(1 - c)
else:
minus.append(c)
# 4)
# TODO
# 5)
arg = Mul(*exponentials)
# for testability, sort the arguments
an.sort()
ap.sort()
bm.sort()
bq.sort()
return (an, ap), (bm, bq), arg, exponent, fac
def eval(cls, f, *gens):
from sympy import roots
rootslist = list(roots(f, *gens, filter='R').keys())
rootslist.sort(key=customized_sort_key)
return TupleNoParen(*rootslist)
def integral_basis(f,x,y):
"""
Compute the integral basis {b1, ..., bg} of the algebraic function
field C[x,y] / (f).
"""
# If the curve is not monic then map y |-> y/lc(x) where lc(x)
# is the leading coefficient of f
T = sympy.Dummy('T')
d = sympy.degree(f,y)
lc = sympy.LC(f,y)
if x in lc:
f = sympy.ratsimp( f.subs(y,y/lc)*lc**(d-1) )
else:
f = f/lc
lc = 1
#
# Compute df
#
p = sympy.Poly(f,[x,y])
n = p.degree(y)
res = sympy.resultant(p,p.diff(y),y)
factors = sympy.factor_list(res)[1]
df = [k for k,deg in factors if (deg > 1) and (sympy.LC(k) == 1)]
#
# Compute series truncations at appropriate x points
#
alpha = []
r = []
for l in range(len(df)):
k = df[l]
alphak = sympy.roots(k).keys()[0]
rk = compute_series_truncations(f,x,y,alphak,T)
alpha.append(alphak)
r.append(rk)
#
# Main Loop
#
a = [sympy.Dummy('a%d'%k) for k in xrange(n)]
b = [1]
for d in range(1,n):
bd = y*b[-1]
for l in range(len(df)):
k = df[l]
alphak = alpha[l]
rk = r[l]
found_something = True
while found_something:
# construct system of equations consisting of the coefficients
# of negative powers of (x-alphak) in the substitutions
# A(r_{k,1}),...,A(r_{k,n})
A = (sum(ak*bk for ak,bk in zip(a,b)) + bd) / (x - alphak)
coeffs = []
for rki in rk:
# substitute and extract coefficients
A_rki = A.subs(y,rki)
coeffs.extend(_negative_power_coeffs(A_rki, x, alphak))
# solve the coefficient equations for a0,...,a_{d-1}
coeffs = [coeff.as_numer_denom()[0] for coeff in coeffs]
sols = sympy.solve_poly_system(coeffs, a[:d])
if sols is None or sols == []:
found_something = False
else:
sol = sols[0]
bdm1 = sum( sol[i]*bk for i,bk in zip(range(d),b) )
bd = (bdm1 + bd) / k
# bd found. Append to list of basis elements
b.append( bd )
# finally, convert back to singularized curve if necessary
for i in xrange(1,len(b)):
b[i] = b[i].subs(y,y*lc).ratsimp()
return b
def build_series(pis,x,y,T,a,parametric):
"""
Builds all Puiseux series from pi data.
`\pi_i=(q,\mu,m,\beta,\eta), X \mapsto\mu X^q,Y \mapsto X^m(\beta+\eta Y)`
Algorithm:
Uses the formulas from p. 135 "Rational puiseux expandsions" by
D. Duval. The indexing is adjusted: m,beta,mu,eta are all shifted
down by one. However, the intexing on qh should match that of
D. Duval.
"""
series = []
# build singular part of expansion series
for pi in pis:
q,mu,m,beta,eta = zip(*pi)
R = len(pi)
qh = build_qh_dict(q)
# compute X = lam*T**e
e = qh[(0,R)]
lam = reduce(lambda z1,z2: z1*z2,
(mu[k]**qh[(0,k)] for k in xrange(R)))
X = sympy.simplify(lam)*T**e + a
# compute Y = \sum_{h=1}^R alpha_h * T**n_h.
if parametric:
Y = sympy.S(0)
else:
lams = sympy.roots(lam*_Z**e - 1,_Z,multiple=True)
Y = [sympy.S(0) for _ in xrange(e)]
n_h = sympy.S(0)
eta_h = sympy.S(1)
for h in xrange(0,R):
eta_h *= eta[h]
n_h += m[h] * qh[(h+1,R)]
alpha_h = sympy.S(1)
for i in xrange(0,h):
alpha_h *= mu[i+1]**sum(m[j]*qh[(j+1,i)]
for j in xrange(0,i))
for i in xrange(h,R-1):
alpha_h *= mu[i+1]**sum(m[j]*qh[(j+1,i)]
for j in xrange(0,h))
alpha_h *= eta_h*beta[h]
if parametric:
Y += sympy.simplify(alpha_h)*T**n_h
else:
s_h = sympy.Rational(n_h,e)
for i in xrange(e):
alpha_h_i = sympy.simplify(alpha_h*(lams[i]**n_h))
Y[i] += alpha_h_i*(x-a)**s_h
# All puiseux series associated with this place are computed.
# Add to series list.
if parametric:
series.append((X,Y))
else:
series.append(Y)
return series
# In[8]:
invL(G)
# The characteristic equation is the denominator of the transfer function
# In[5]:
ce = sympy.Eq(sympy.denom(G), 0)
ce
# In[6]:
roots = sympy.roots(ce, s)
roots
# The shape of the inverse Laplace depends on the nature of the roots, which depends on the value of $\zeta$
# Overdamped: $\zeta>1$. Two distinct real roots
# In[3]:
G
# In[7]:
invL(G.subs({zeta: 2}))
