def arc_length_explain(self, start, stop, a_type = None,
explanation_type=None, preview = None):
if a_type is None:
a_type = self.a_type
v = self.v_
u = self.u
start = sym.sympify(start)
stop = sym.sympify(stop)
if explanation_type is None:
explanation_type = self.kwargs.get('explanation_type', 'simple')
if preview is None:
preview = self.kwargs.get('preview', False)
explanation = ArcLengthProb(path='arc_length/explanation',
a_type = a_type,
explanation_type = explanation_type,
preview = preview).explain()
explanation += """
<p>
For this problem the curve is $_C$_ is given by $_%s$_
with the independent variable being $_%s$_ on the interval
$_[%s,%s]$_. So the arc length is
$$s =\\int_{%s}^{%s} \sqrt{1+\\left(\\frac{d%s}{d%s}\\right)^2}\,d%s$$
$$\\frac{d%s}{d%s} = %s = %s$$
\\begin{align}
\\sqrt{1 + \\left(%s\\right)^2} &= \\sqrt{%s} \\\\
&= \sqrt{\\left(%s\\right)^2} = %s
\\end{align}
</p>
"""%tuple(map(lambda x: sym.latex(x).replace("\\log","\\ln"),
[sym.Eq(v, self.v), u, start, stop, start, stop,
v, u, u, v, u,
self.Dv, self.dv, self.dv, 1 + (self.dv**2).expand(),
self.ds, self.ds]))
aa = [sym.latex(self.arc_length(start, stop,'integral')),
"\\left." + sym.latex(self.Ids).replace('\\log', '\\ln') +
"\\right|_{%s=%s}^{%s=%s}"%(u, start, u, stop),
sym.latex(self.arc_length(start, stop,'exact')).replace("\\log", "\\ln") +
'\\approx %.3f'%(self.arc_length(start, stop,'numeric'))]
# Add self.numeric / self.anti
explanation += """
<p>
Thus the arclength is given by
%s
</p>
"""%(tools.align(*aa))
#
return explanation
def __init__(self, name, unit_set):
"""
Create a quantity in the given unit set. The name must be listed in the
units_of_quantities dictionary."""
self.name = name
self.unit_set = unit_set
self.unit_name = units_of_quantities[self.name]
unit_expr = sm.sympify(self.unit_name)
unit_expr = unit_expr.subs(derived_units)
self.symbolic_value = sm.sympify(unit_expr)
atoms = self.symbolic_value.atoms(sm.Symbol)
self.def_units = [Unit(atom.name) for atom in atoms]
self.def_names, self.def_units = self._get_dicts(self.def_units)
self.names, self.units = self._get_dicts(self.unit_set)
self.def_coef = float(self.symbolic_value.subs(self.def_names))
coef_dict = {}
for key, val in six.iteritems(self.def_units):
coef_dict[val.name] = self.units[key].coef
self.coef_dict = coef_dict
self.raw_coef = float(self.symbolic_value.subs(self.coef_dict))
self.coef = self.raw_coef / self.def_coef
def test_issue_7840():
# daveknippers' example
C393 = sympify( \
'Piecewise((C391 - 1.65, C390 < 0.5), (Piecewise((C391 - 1.65, \
C391 > 2.35), (C392, True)), True))'
)
C391 = sympify( \
'Piecewise((2.05*C390**(-1.03), C390 < 0.5), (2.5*C390**(-0.625), True))'
)
C393 = C393.subs('C391',C391)
# simple substitution
sub = {}
sub['C390'] = 0.703451854
sub['C392'] = 1.01417794
ss_answer = C393.subs(sub)
# cse
substitutions,new_eqn = cse(C393)
for pair in substitutions:
sub[pair[0].name] = pair[1].subs(sub)
cse_answer = new_eqn[0].subs(sub)
# both methods should be the same
assert ss_answer == cse_answer
# GitRay's example
expr = sympify(
"Piecewise((Symbol('ON'), Equality(Symbol('mode'), Symbol('ON'))), \
(Piecewise((Piecewise((Symbol('OFF'), StrictLessThan(Symbol('x'), \
Symbol('threshold'))), (Symbol('ON'), true)), Equality(Symbol('mode'), \
Symbol('AUTO'))), (Symbol('OFF'), true)), true))"
)
substitutions, new_eqn = cse(expr)
# this Piecewise should be exactly the same
assert new_eqn[0] == expr
# there should not be any replacements
assert len(substitutions) < 1
def arc_length(self, start, stop, q_type = None):
"""
Parameters:
---------
q_type: string
'exact', 'integral', 'numeric'
start, stop: numeric/str
These are the limits of integration
"""
start = sym.sympify(start)
stop = sym.sympify(stop)
if q_type is None:
q_type = self.q_type
u = self.u
I = sym.Integral(self.ds, (u, start, stop))
if q_type == 'integral':
return I
elif q_type == 'exact':
return I.doit()
else:
f = make_func(str(self.ds), func_params=(self.ind_var), func_type='numpy')
value = mp.quad(f, [float(start.evalf()), float(stop.evalf())])
return value
def test_product_basic():
H, T = 'H', 'T'
unit_line = Interval(0, 1)
d6 = FiniteSet(1, 2, 3, 4, 5, 6)
d4 = FiniteSet(1, 2, 3, 4)
coin = FiniteSet(H, T)
square = unit_line * unit_line
assert (0, 0) in square
assert 0 not in square
assert (H, T) in coin ** 2
assert (.5, .5, .5) in square * unit_line
assert (H, 3, 3) in coin * d6* d6
HH, TT = sympify(H), sympify(T)
assert set(coin**2) == set(((HH, HH), (HH, TT), (TT, HH), (TT, TT)))
assert (d4*d4).is_subset(d6*d6)
assert square.complement(Interval(-oo, oo)*Interval(-oo, oo)) == Union(
(Interval(-oo, 0, True, True) +
Interval(1, oo, True, True))*Interval(-oo, oo),
Interval(-oo, oo)*(Interval(-oo, 0, True, True) +
Interval(1, oo, True, True)))
assert (Interval(-5, 5)**3).is_subset(Interval(-10, 10)**3)
assert not (Interval(-10, 10)**3).is_subset(Interval(-5, 5)**3)
assert not (Interval(-5, 5)**2).is_subset(Interval(-10, 10)**3)
assert (Interval(.2, .5)*FiniteSet(.5)).is_subset(square) # segment in square
assert len(coin*coin*coin) == 8
assert len(S.EmptySet*S.EmptySet) == 0
assert len(S.EmptySet*coin) == 0
raises(TypeError, lambda: len(coin*Interval(0, 2)))
def higher_order_diff(eqs, syms, order=2):
'''Takes higher order derivatives of a list of equations w.r.t a list of paramters'''
import numpy
eqs = list([sympy.sympify(eq) for eq in eqs])
syms = list([sympy.sympify(s) for s in syms])
neq = len(eqs)
p = len(syms)
D = [numpy.array(eqs)]
orders = []
for i in range(1,order+1):
par = D[i-1]
mat = numpy.empty([neq] + [p]*i, dtype=object) #.append( numpy.zeros(orders))
for ind in non_decreasing_series(p,i):
ind_parent = ind[:-1]
k = ind[-1]
for line in range(neq):
ii = [line] + ind
iid = [line] + ind_parent
eeq = par[ tuple(iid) ]
mat[tuple(ii)] = eeq.diff(syms[k])
D.append(mat)
return D
def test_nsimplify():
x = Symbol("x")
assert nsimplify(0) == 0
assert nsimplify(-1) == -1
assert nsimplify(1) == 1
assert nsimplify(1+x) == 1+x
assert nsimplify(2.7) == Rational(27, 10)
assert nsimplify(1-GoldenRatio) == (1-sqrt(5))/2
assert nsimplify((1+sqrt(5))/4, [GoldenRatio]) == GoldenRatio/2
assert nsimplify(2/GoldenRatio, [GoldenRatio]) == 2*GoldenRatio - 2
assert nsimplify(exp(5*pi*I/3, evaluate=False)) == sympify('1/2 - sqrt(3)*I/2')
assert nsimplify(sin(3*pi/5, evaluate=False)) == sympify('sqrt(sqrt(5)/8 + 5/8)')
assert nsimplify(sqrt(atan('1', evaluate=False))*(2+I), [pi]) == sqrt(pi) + sqrt(pi)/2*I
assert nsimplify(2 + exp(2*atan('1/4')*I)) == sympify('49/17 + 8*I/17')
assert nsimplify(pi, tolerance=0.01) == Rational(22, 7)
assert nsimplify(pi, tolerance=0.001) == Rational(355, 113)
assert nsimplify(0.33333, tolerance=1e-4) == Rational(1, 3)
assert nsimplify(2.0**(1/3.), tolerance=0.001) == Rational(635, 504)
assert nsimplify(2.0**(1/3.), tolerance=0.001, full=True) == 2**Rational(1, 3)
assert nsimplify(x + .5, rational=True) == Rational(1, 2) + x
assert nsimplify(1/.3 + x, rational=True) == Rational(10, 3) + x
assert nsimplify(log(3).n(), rational=True) == \
sympify('109861228866811/100000000000000')
assert nsimplify(Float(0.272198261287950), [pi,log(2)]) == pi*log(2)/8
assert nsimplify(Float(0.272198261287950).n(3), [pi,log(2)]) == \
-pi/4 - log(2) + S(7)/4
assert nsimplify(x/7.0) == x/7
assert nsimplify(pi/1e2) == pi/100
assert nsimplify(pi/1e2, rational=False) == pi/100.0
assert nsimplify(pi/1e-7) == 10000000*pi
def _minpoly_exp(ex, x):
"""
Returns the minimal polynomial of ``exp(ex)``
"""
c, a = ex.args[0].as_coeff_Mul()
p = sympify(c.p)
q = sympify(c.q)
if a == I*pi:
if c.is_rational:
if c.p == 1 or c.p == -1:
if q == 3:
return x**2 - x + 1
if q == 4:
return x**4 + 1
if q == 6:
return x**4 - x**2 + 1
if q == 8:
return x**8 + 1
if q == 9:
return x**6 - x**3 + 1
if q == 10:
return x**8 - x**6 + x**4 - x**2 + 1
if q.is_prime:
s = 0
for i in range(q):
s += (-x)**i
return s
# x**(2*q) = product(factors)
factors = [cyclotomic_poly(i, x) for i in divisors(2*q)]
mp = _choose_factor(factors, x, ex)
return mp
else:
raise NotAlgebraic("%s doesn't seem to be an algebraic element" % ex)
raise NotAlgebraic("%s doesn't seem to be an algebraic element" % ex)
def field_isomorphism(a, b, **args):
"""Construct an isomorphism between two number fields. """
a, b = sympify(a), sympify(b)
if not a.is_AlgebraicNumber:
a = AlgebraicNumber(a)
if not b.is_AlgebraicNumber:
b = AlgebraicNumber(b)
if a == b:
return a.coeffs()
n = a.minpoly.degree()
m = b.minpoly.degree()
if n == 1:
return [a.root]
if m % n != 0:
return None
if args.get('fast', True):
try:
result = field_isomorphism_pslq(a, b)
if result is not None:
return result
except NotImplementedError:
pass
return field_isomorphism_factor(a, b)
def _divide_constriant(s, symbols, cons_index, cons_dict, cons_import):
# Creates a CustomConstraint of the form `CustomConstraint(lambda a, x: FreeQ(a, x))`
lambda_symbols = list(set(get_free_symbols(s, symbols, [])))
r = generate_sympy_from_parsed(s)
r = sympify(r, locals={"Or": Function("Or"), "And": Function("And"), "Not":Function("Not")})
if r.has(Function('MatchQ')):
match_res = set_matchq_in_constraint(s, cons_index)
res = match_res[1]
res += '\n return {}'.format(rubi_printer(sympify(generate_sympy_from_parsed(match_res[0]), locals={"Or": Function("Or"), "And": Function("And"), "Not":Function("Not")}), sympy_integers = True))
elif contains_diff_return_type(s):
res = ' try:\n return {}\n except (TypeError, AttributeError):\n return False'.format(rubi_printer(r, sympy_integers=True))
else:
res = ' return {}'.format(rubi_printer(r, sympy_integers=True))
# First it checks if a constraint is already present in `cons_dict`, If yes, use it else create a new one.
if not res in cons_dict.values():
cons_index += 1
cons = '\n def cons_f{}({}):\n'.format(cons_index, ', '.join(lambda_symbols))
if 'x' in lambda_symbols:
cons += ' if isinstance(x, (int, Integer, float, Float)):\n return False\n'
cons += res
cons += '\n\n cons{} = CustomConstraint({})\n'.format(cons_index, 'cons_f{}'.format(cons_index))
cons_name = 'cons{}'.format(cons_index)
cons_dict[cons_name] = res
else:
cons = ''
cons_name = next(key for key, value in cons_dict.items() if value == res)
if cons_name not in cons_import:
cons_import.append(cons_name)
return (cons_name, cons, cons_index)
def test_product_basic():
H,T = 'H', 'T'
unit_line = Interval(0,1)
d6 = FiniteSet(1,2,3,4,5,6)
d4 = FiniteSet(1,2,3,4)
coin = FiniteSet(H, T)
square = unit_line * unit_line
assert (0,0) in square
assert 0 not in square
assert (H, T) in coin ** 2
assert (.5,.5,.5) in square * unit_line
assert (H, 3, 3) in coin * d6* d6
HH, TT = sympify(H), sympify(T)
assert set(coin**2) == set(((HH, HH), (HH, TT), (TT, HH), (TT, TT)))
assert (d6*d6).subset(d4*d4)
inf, neginf = S.Infinity, S.NegativeInfinity
assert square.complement == Union(
Interval(0,1) * (Interval(neginf,0,True,True)+Interval(1,inf,True,True)),
(Interval(neginf,0,True,True)+Interval(1,inf,True,True))*Interval(0,1),
((Interval(neginf,0,True,True) + Interval(1,inf, True, True))
* (Interval(neginf,0,True,True) + Interval(1,inf, True,True))))
assert (Interval(-10,10)**3).subset(Interval(-5,5)**3)
assert not (Interval(-5,5)**3).subset(Interval(-10,10)**3)
assert not (Interval(-10,10)**2).subset(Interval(-5,5)**3)
assert square.subset(Interval(.2,.5)*FiniteSet(.5)) # segment in square
def transverse_magnification(si, so):
"""
Calculates the transverse magnification, which is the ratio of the
image size to the object size.
Parameters
==========
so: sympifiable
Lens-object distance
si: sympifiable
Lens-image distance
Example
=======
>>> from sympy.physics.optics import transverse_magnification
>>> transverse_magnification(30, 15)
-2
"""
si = sympify(si)
so = sympify(so)
return (-(si/so))
def check_output(self, want, got, optionflags):
if IGNORE_OUTPUT & optionflags:
return True
# When writing tests we sometimes want to assure ourselves that the
# results are _not_ equal
wanted_tf = not (NOT_EQUAL & optionflags)
# Strip dtype
if IGNORE_DTYPE & optionflags:
want = ignore_dtype(want)
got = ignore_dtype(got)
# Strip array repr from got and want if requested
if STRIP_ARRAY_REPR & optionflags:
# STRIP_ARRAY_REPR only matches for a line containing *only* an
# array repr. Use IGNORE_DTYPE to ignore a dtype specifier embedded
# within a more complex line.
want = strip_array_repr(want)
got = strip_array_repr(got)
# If testing floating point, round to required number of digits
if optionflags & (FP_4DP | FP_6DP):
if optionflags & FP_4DP:
dp = 4
elif optionflags & FP_6DP:
dp = 6
want = round_numbers(want, dp)
got = round_numbers(got, dp)
# Are the strings equal when run through sympy?
if SYMPY_EQUAL & optionflags:
from sympy import sympify
res = sympify(want) == sympify(got)
return res == wanted_tf
# Pass tests through two-pass numpy checker
res = NumpyOutputChecker.check_output(self, want, got, optionflags)
# Return True if we wanted True and got True, or if we wanted False and
# got False
return res == wanted_tf
def _generic_mul(q1, q2):
q1 = sympify(q1)
q2 = sympify(q2)
# None is a Quaternion:
if not isinstance(q1, Quaternion) and not isinstance(q2, Quaternion):
return q1 * q2
# If q1 is a number or a sympy expression instead of a quaternion
if not isinstance(q1, Quaternion):
if q2.real_field:
if q1.is_complex:
return q2 * Quaternion(re(q1), im(q1), 0, 0)
else:
return Mul(q1, q2)
else:
return Quaternion(q1 * q2.a, q1 * q2.b, q1 * q2.c, q1 * q2.d)
# If q2 is a number or a sympy expression instead of a quaternion
if not isinstance(q2, Quaternion):
if q1.real_field:
if q2.is_complex:
return q1 * Quaternion(re(q2), im(q2), 0, 0)
else:
return Mul(q1, q2)
else:
return Quaternion(q2 * q1.a, q2 * q1.b, q2 * q1.c, q2 * q1.d)
return Quaternion(-q1.b*q2.b - q1.c*q2.c - q1.d*q2.d + q1.a*q2.a,
q1.b*q2.a + q1.c*q2.d - q1.d*q2.c + q1.a*q2.b,
-q1.b*q2.d + q1.c*q2.a + q1.d*q2.b + q1.a*q2.c,
q1.b*q2.c - q1.c*q2.b + q1.d*q2.a + q1.a * q2.d)
def hyperfocal_distance(f, N, c):
"""
Parameters
==========
f: sympifiable
Focal length of a given lens
N: sympifiable
F-number of a given lens
c: sympifiable
Circle of Confusion (CoC) of a given image format
Example
=======
>>> from sympy.physics.optics import hyperfocal_distance
>>> from sympy.abc import f, N, c
>>> round(hyperfocal_distance(f = 0.5, N = 8, c = 0.0033), 2)
9.47
"""
f = sympify(f)
N = sympify(N)
c = sympify(c)
return (1/(N * c))*(f**2)
def __init__(
self,
amplitude,
frequency=None,
phase=S.Zero,
time_period=None,
n=Symbol('n')):
frequency = sympify(frequency)
amplitude = sympify(amplitude)
phase = sympify(phase)
time_period = sympify(time_period)
n = sympify(n)
self._frequency = frequency
self._amplitude = amplitude
self._phase = phase
self._time_period = time_period
self._n = n
if time_period is not None:
self._frequency = 1/self._time_period
if frequency is not None:
self._time_period = 1/self._frequency
if time_period is not None:
if frequency != 1/time_period:
raise ValueError("frequency and time_period should be consistent.")
if frequency is None and time_period is None:
raise ValueError("Either frequency or time period is needed.")
def declare_functions():
Expression("create_preMPF", sympify("k9/(1 + k31*OBS_p53)"))
Expression("Hill_Mdm2", sympify("k24*OBS_Int^n / (k_m^n + OBS_Int^n)"))
Expression("create_intermediate", sympify("k27*OBS_p53/ (1 + k26*OBS_p53*OBS_Mdm2)"))
Expression("sig_deg", sympify("Deg_0 - k_deg*(signal-signal_damp)"))
Expression("kdamp_DDS0", sympify("k_damp*DDS_0"))
def brewster_angle(medium1, medium2):
"""
This function calculates the Brewster's angle of incidence to Medium 2 from
Medium 1 in radians.
Parameters
==========
medium 1 : Medium or sympifiable
Refractive index of Medium 1
medium 2 : Medium or sympifiable
Refractive index of Medium 1
Examples
========
>>> from sympy.physics.optics import brewster_angle
>>> brewster_angle(1, 1.33)
0.926093295503462
"""
if isinstance(medium1, Medium):
n1 = medium1.refractive_index
else:
n1 = sympify(medium1)
if isinstance(medium2, Medium):
n2 = medium2.refractive_index
else:
n2 = sympify(medium2)
return atan2(n2, n1)
def eval_graph(evaluator, variable):
from sympy.plotting.plot import LineOver1DRangeSeries
func = evaluator.eval("input_evaluated")
free_symbols = sympy.sympify(func).free_symbols
if len(free_symbols) != 1 or variable not in free_symbols:
raise ValueError("Cannot graph function of multiple variables")
try:
series = LineOver1DRangeSeries(func, (variable, -10, 10), nb_of_points=200)
# returns a list of [[x,y], [next_x, next_y]] pairs
series = series.get_segments()
except TypeError:
raise ValueError("Cannot graph function")
xvalues = []
yvalues = []
for point in series:
xvalues.append(point[0][0])
yvalues.append(point[0][1])
xvalues.append(series[-1][1][0])
yvalues.append(series[-1][1][1])
return {
'function': sympy.jscode(sympy.sympify(func)),
'variable': repr(variable),
'xvalues': json.dumps(xvalues),
'yvalues': json.dumps(yvalues)
}
def daubechis(N):
# make polynomial
q_y = [sm.binomial(N-1+k,k) for k in reversed(range(N))]
# get polynomial roots y[k]
y = sm.mp.polyroots(q_y, maxsteps=200, extraprec=64)
z = []
for yk in y:
# subustitute y = -1/4z + 1/2 - 1/4/z to factor f(y) = y - y[k]
f = [sm.mpf('-1/4'), sm.mpf('1/2') - yk, sm.mpf('-1/4')]
# get polynomial roots z[k]
z += sm.mp.polyroots(f)
# make polynomial using the roots within unit circle
h0z = sm.sqrt('2')
for zk in z:
if sm.fabs(zk) < 1:
h0z *= sympy.sympify('(z-zk)/(1-zk)').subs('zk',zk)
# adapt vanising moments
hz = (sympy.sympify('(1+z)/2')**N*h0z).expand()
# get scaling coefficients
return [sympy.re(hz.coeff('z',k)) for k in reversed(range(N*2))]
def __init__(self, cu,cv,la,mu):
x, y = sympy.symbols('x y')
self.cu=cu
self.cv=cv
self.L=la
self.M=mu
self.u=sympy.sympify(cu)
self.v=sympy.sympify(cv)
self.fu=eval("lambda x,y: numpy.array(["+cu[0]+","+cu[1]+"])")
self.fv=eval("lambda x,y: numpy.array(["+cv[0]+","+cv[1]+"])")
if self.u[0].is_polynomial(x,y) and self.u[1].is_polynomial(x,y):
D0=sympy.polys.Poly(self.u[0],x,y).as_dict()
D1=sympy.polys.Poly(self.u[1],x,y).as_dict()
self.du=max(max(numpy.sum(list(D0.keys()),axis=1)),max(numpy.sum(list(D1.keys()),axis=1)))
else:
self.du=-1
if self.v[0].is_polynomial(x,y) and self.v[1].is_polynomial(x,y):
D0=sympy.polys.Poly(self.v[0],x,y).as_dict()
D1=sympy.polys.Poly(self.v[1],x,y).as_dict()
self.dv=max(max(numpy.sum(list(D0.keys()),axis=1)),max(numpy.sum(list(D1.keys()),axis=1)))
else:
self.dv=-1
H=Hooke(self.L,self.M)
gU=Gamma(self.u)
gV=Gamma(self.v)
self.I=sympy.integrate(sympy.integrate(gV.T*H*gU,(x,0,1)),(y,0,1))[0,0]
def test_anonymous_expression_operators(self):
result = Expression('a + b') # Arbitrary starting expression
expr_iter = cycle(anonymous_expressions)
for op in self.ops:
if op in uniary_ops:
ss_result = op(result.rhs)
ee_result = op(result)
op_str = ("{}({})".format(op.__name__, result))
else:
expr = next(expr_iter)
if op in div_ops and expr.rhs == sympify(0.0):
expr = sympify(0.1)
ss_result = op(result.rhs, expr.rhs)
ee_result = op(result, expr)
es_result = op(result, expr.rhs)
se_result = op(result.rhs, expr)
op_str = ("{}({}, {})".format(op.__name__, result, expr))
self.assertEqual(
es_result, ss_result,
op_str + " not equal between Expression and sympy")
self.assertEqual(
se_result, ss_result,
"{} not equal between Expression ({}) and sympy ({})"
.format(op_str, se_result, ss_result))
self.assertIsInstance(es_result, SympyBaseClass,
op_str + " did not return a Expression")
self.assertIsInstance(se_result, SympyBaseClass,
op_str + " did not return a Expression")
self.assertEqual(
ee_result, ss_result,
"{} not equal between Expression ({}) and sympy ({})"
.format(op_str, ee_result, ss_result))
self.assertIsInstance(ee_result, SympyBaseClass,
op_str + " did not return a Expression")
result = Expression(ee_result)
def sympify(arg, real=False, positive=None, cache=True):
"""Create a sympy expression."""
if isinstance(arg, (sym.symbol.Symbol, sym.symbol.Expr)):
return arg
# Why doesn't sympy do this?
if isinstance(arg, complex):
re = sym.sympify(arg.real, rational=True)
im = sym.sympify(arg.imag, rational=True)
if im == 1.0:
arg = re + sym.I
else:
arg = re + sym.I * im
return arg
if isinstance(arg, str):
# Sympy considers E1 to be the generalized exponential integral.
# N is for numerical evaluation.
if symbol_pattern.match(arg) is not None:
return symbol(arg, real=real, positive=positive, cache=cache)
match = symbol_pattern2.match(arg)
if match is not None:
# Convert R_{out} to R_out for sympy to recognise.
arg = match.groups()[0] + match.groups()[1]
return symbol(arg, real=True)
# Perhaps have dictionary of functions and their replacements?
arg = arg.replace('u(t', 'Heaviside(t')
arg = arg.replace('delta(t', 'DiracDelta(t')
return sym.sympify(arg, rational=True, locals=symbols)
def __new__(cls, q0, q1, q2, q3):
q0 = sympify(q0)
q1 = sympify(q1)
q2 = sympify(q2)
q3 = sympify(q3)
parent_orient = (Matrix([[q0 ** 2 + q1 ** 2 - q2 ** 2 -
q3 ** 2,
2 * (q1 * q2 - q0 * q3),
2 * (q0 * q2 + q1 * q3)],
[2 * (q1 * q2 + q0 * q3),
q0 ** 2 - q1 ** 2 +
q2 ** 2 - q3 ** 2,
2 * (q2 * q3 - q0 * q1)],
[2 * (q1 * q3 - q0 * q2),
2 * (q0 * q1 + q2 * q3),
q0 ** 2 - q1 ** 2 -
q2 ** 2 + q3 ** 2]]))
parent_orient = parent_orient.T
obj = super(QuaternionOrienter, cls).__new__(cls, q0, q1, q2, q3)
obj._q0 = q0
obj._q1 = q1
obj._q2 = q2
obj._q3 = q3
obj._parent_orient = parent_orient
return obj
def Ruvws_from_file(filePath):
"""
Generates nested lists where each list entry contains a again a list.
The first entry is the Ruvw vector and the second is a list which contains all variables names that have this Ruvw vector.
Input file must be formatted like:
0,0,0
variableName
variableName
filePath ... string location to the file which conatins the Ruvw relations
returns ... list[np.array,list[string]] the information on all Ruvws
"""
with open(filePath) as file:
content = file.readlines()
allRuvw = []
entries = []
r = None
for line in content:
if ',' in line:
allRuvw.append([r,entries]) # if it is the first occurence, a dummy entry will be generated
# Parsing Ruvw
r = line.split(',')
r = np.array([sp.sympify(r[0]).evalf(), sp.sympify(r[1]).evalf(), sp.sympify(r[2]).evalf()])
entries = []
else:
entries.append(line.strip())
# applying the last entry
allRuvw.append([r,entries])
del allRuvw[0] # first one is a dummy entry
return allRuvw
def get_gamma_keq_terms(mod, sympy_terms):
model_map = pysces.ModelMap(mod) # model map to get substrates, products
# and parameters for each reaction
messages = {}
gamma_keq_terms = {}
for name, terms in sympy_terms.iteritems():
reaction_map = getattr(model_map, name)
substrates = [sympify(substrate) for substrate in
reaction_map.hasSubstrates()]
products = [sympify(product) for product in reaction_map.hasProducts()]
if len(terms) == 2: # condition for reversible reactions
# make sure negative term is second in term list
terms = sort_terms(terms)
# divide pos term by neg term and factorise
expressions = (-terms[0] / terms[1]).factor()
# get substrate, product and keq terms (and strategy)
st, pt, keq, _ = get_st_pt_keq(expressions, substrates,
products)
if all([st, pt, keq]):
gamma_keq_terms[name] = pt / (keq*st)
messages[name] = 'successful generation of gamma/keq term'
else:
messages[name] = 'generation of gamma/keq term failed'
return gamma_keq_terms, messages
def __init__(self, name=None, mult=None, num=None, den=None, repeats=None):
if mult == 1:
mult = '1'
if den == 1:
den = '1'
if num == 1:
num = '1'
self.den = den
self.mult = mult
self.num = num
self.name = name
self.denvariables = sorted(set(re.findall(FINDVARIABLES, den)))
self.numvariables = sorted(set(re.findall(FINDVARIABLES, num)))
self.denominator_eq = sympy.sympify(den)
self.numerator_eq = sympy.sympify(num)
self.denominator = sympy.lambdify(self.denvariables, self.denominator_eq)
self.numerator = sympy.lambdify(self.numvariables, self.numerator_eq)
self.repeats = None
if type(mult) == list:
self.multvariables = sorted(set([variable for x in self.mult for variable in x.variables]))
self.multiplier_eq = [str(x) for x in self.mult]
self.multiplier = mult
else:
self.multvariables = sorted(set(re.findall(FINDVARIABLES, mult)))
self.multiplier_eq = sympy.sympify(mult)
self.multiplier = sympy.lambdify(self.multvariables, self.multiplier_eq)
self.variables = self.denvariables + self.numvariables + self.multvariables
def get_sympified(instance):
if hasattr(instance, 'iteritems'):
return dict([(k,sympify(v)) for k,v in instance.iteritems()])
elif hasattr(instance, '__iter__'):
return [sympify(x) for x in instance]
else:
NotImplemented
def arctan_rule(integral):
integrand, symbol = integral
base, exp = integrand.as_base_exp()
if sympy.simplify(exp + 1) == 0:
a = sympy.Wild('a', exclude=[symbol])
b = sympy.Wild('b', exclude=[symbol])
match = base.match(a + b*symbol**2)
if match:
a, b = match[a], match[b]
if ((isinstance(a, sympy.Number) and a < 0) or (isinstance(b, sympy.Number) and b < 0)):
return
if (sympy.ask(sympy.Q.negative(a) | sympy.Q.negative(b) | sympy.Q.is_true(a <= 0) | sympy.Q.is_true(b <= 0))):
return
# / dx 1 / dx 1 / dx | | 1 1 / du
# | --------- = -- | -------------- = -- | -------------------- = | sqrt(b/a)x = u | = -- ---------- | -------
# / a + bx^2 a / 1 + (b/a)x^2 a / 1 + (sqrt(b/a)x)^2 | dx = du / sqrt(b/a) | a sqrt(b/a) / 1 + u^2
if a == 1 and b == 1:
return ArctanRule(integrand, symbol)
if a == b:
constant = 1 / a
integrand_ = 1 / (1 + symbol**2)
substep = ArctanRule(integrand_, symbol)
return ConstantTimesRule(constant, integrand_, substep, integrand, symbol)
u_var = new_symbol_(symbol)
u_func = sympy.sqrt(sympy.sympify(b) / a) * symbol
integrand_ = 1 / (1 + u_func**2)
constant = 1 / sympy.sqrt(sympy.sympify(b) / a)
substituted = 1 / (1 + u_var**2)
substep = ArctanRule(substituted, u_var)
substep = ConstantTimesRule(constant, substituted, substep, constant*substituted, u_var)
substep = URule(u_var, u_func, constant, substep, constant*substituted, integrand_, symbol)
return ConstantTimesRule(1/a, integrand_, substep, integrand, symbol)
def Conditional(cond, true_value, false_value):
"""
Declares a conditional
Arguments
---------
cond : A conditional
The conditional which should be evaluated
true_value : Any model expression
Model expression for a true evaluation of the conditional
false_value : Any model expression
Model expression for a false evaluation of the conditional
"""
cond = sp.sympify(cond)
from sympy.core.relational import Equality, Relational
from sympy.logic.boolalg import Boolean
# If the conditional is a bool it is already evaluated
if isinstance(cond, bool):
return true_value if cond else false_value
if not isinstance(cond, (Relational, Boolean)):
raise type_error("Cond %s is of type %s, but must be a Relational" \
" or Boolean." % (cond, type(cond)))
return sp.functions.Piecewise((true_value, cond), (false_value, sp.sympify(True)),
evaluate=True)
def NS(e, n=15, **options):
return sstr(sympify(e).evalf(n, **options), full_prec=True)
def _generic_mul(q1, q2):
"""Generic multiplication.
Parameters
==========
q1 : Quaternion or symbol
q2 : Quaternion or symbol
It's important to note that if neither q1 nor q2 is a Quaternion,
this function simply returns q1 * q2.
Returns
=======
Quaternion
The resultant quaternion after multiplying q1 and q2
Examples
========
>>> from sympy.algebras.quaternion import Quaternion
>>> from sympy import Symbol
>>> q1 = Quaternion(1, 2, 3, 4)
>>> q2 = Quaternion(5, 6, 7, 8)
>>> Quaternion._generic_mul(q1, q2)
(-60) + 12*i + 30*j + 24*k
>>> Quaternion._generic_mul(q1, 2)
2 + 4*i + 6*j + 8*k
>>> x = Symbol('x', real = True)
>>> Quaternion._generic_mul(q1, x)
x + 2*x*i + 3*x*j + 4*x*k
Quaternions over complex fields :
>>> from sympy.algebras.quaternion import Quaternion
>>> from sympy import I
>>> q3 = Quaternion(3 + 4*I, 2 + 5*I, 0, 7 + 8*I, real_field = False)
>>> Quaternion._generic_mul(q3, 2 + 3*I)
(2 + 3*I)*(3 + 4*I) + (2 + 3*I)*(2 + 5*I)*i + 0*j + (2 + 3*I)*(7 + 8*I)*k
"""
q1 = sympify(q1)
q2 = sympify(q2)
# None is a Quaternion:
if not isinstance(q1, Quaternion) and not isinstance(q2, Quaternion):
return q1 * q2
# If q1 is a number or a sympy expression instead of a quaternion
if not isinstance(q1, Quaternion):
if q2.real_field and q1.is_complex:
return Quaternion(re(q1), im(q1), 0, 0) * q2
elif q1.is_commutative:
return Quaternion(q1 * q2.a, q1 * q2.b, q1 * q2.c, q1 * q2.d)
else:
raise ValueError(
"Only commutative expressions can be multiplied with a Quaternion."
)
# If q2 is a number or a sympy expression instead of a quaternion
if not isinstance(q2, Quaternion):
if q1.real_field and q2.is_complex:
return q1 * Quaternion(re(q2), im(q2), 0, 0)
elif q2.is_commutative:
return Quaternion(q2 * q1.a, q2 * q1.b, q2 * q1.c, q2 * q1.d)
else:
raise ValueError(
"Only commutative expressions can be multiplied with a Quaternion."
)
return Quaternion(
-q1.b * q2.b - q1.c * q2.c - q1.d * q2.d + q1.a * q2.a,
q1.b * q2.a + q1.c * q2.d - q1.d * q2.c + q1.a * q2.b,
-q1.b * q2.d + q1.c * q2.a + q1.d * q2.b + q1.a * q2.c,
q1.b * q2.c - q1.c * q2.b + q1.d * q2.a + q1.a * q2.d)
def __rtruediv__(self, other):
return sympify(other) * self**-1
def __truediv__(self, other):
return self * sympify(other)**-1
def test_issue_9448():
tmp = sympify(
"1/(1 - (-1)**(2/3) - (-1)**(1/3)) + 1/(1 + (-1)**(2/3) + (-1)**(1/3))"
)
assert nsimplify(tmp) == S.Half
def test_nsimplify():
x = Symbol("x")
assert nsimplify(0) == 0
assert nsimplify(-1) == -1
assert nsimplify(1) == 1
assert nsimplify(1 + x) == 1 + x
assert nsimplify(2.7) == Rational(27, 10)
assert nsimplify(1 - GoldenRatio) == (1 - sqrt(5)) / 2
assert nsimplify((1 + sqrt(5)) / 4, [GoldenRatio]) == GoldenRatio / 2
assert nsimplify(2 / GoldenRatio, [GoldenRatio]) == 2 * GoldenRatio - 2
assert nsimplify(exp(pi*I*Rational(5, 3), evaluate=False)) == \
sympify('1/2 - sqrt(3)*I/2')
assert nsimplify(sin(pi*Rational(3, 5), evaluate=False)) == \
sympify('sqrt(sqrt(5)/8 + 5/8)')
assert nsimplify(sqrt(atan('1', evaluate=False))*(2 + I), [pi]) == \
sqrt(pi) + sqrt(pi)/2*I
assert nsimplify(2 + exp(2 * atan('1/4') * I)) == sympify('49/17 + 8*I/17')
assert nsimplify(pi, tolerance=0.01) == Rational(22, 7)
assert nsimplify(pi, tolerance=0.001) == Rational(355, 113)
assert nsimplify(0.33333, tolerance=1e-4) == Rational(1, 3)
assert nsimplify(2.0**(1 / 3.), tolerance=0.001) == Rational(635, 504)
assert nsimplify(2.0**(1/3.), tolerance=0.001, full=True) == \
2**Rational(1, 3)
assert nsimplify(x + .5, rational=True) == S.Half + x
assert nsimplify(1 / .3 + x, rational=True) == Rational(10, 3) + x
assert nsimplify(log(3).n(), rational=True) == \
sympify('109861228866811/100000000000000')
assert nsimplify(Float(0.272198261287950), [pi, log(2)]) == pi * log(2) / 8
assert nsimplify(Float(0.272198261287950).n(3), [pi, log(2)]) == \
-pi/4 - log(2) + Rational(7, 4)
assert nsimplify(x / 7.0) == x / 7
assert nsimplify(pi / 1e2) == pi / 100
assert nsimplify(pi / 1e2, rational=False) == pi / 100.0
assert nsimplify(pi / 1e-7) == 10000000 * pi
assert not nsimplify(
factor(-3.0 * z**2 * (z**2)**(-2.5) + 3 * (z**2)**(-1.5))).atoms(Float)
e = x**0.0
assert e.is_Pow and nsimplify(x**0.0) == 1
assert nsimplify(3.333333, tolerance=0.1, rational=True) == Rational(10, 3)
assert nsimplify(3.333333, tolerance=0.01,
rational=True) == Rational(10, 3)
assert nsimplify(3.666666, tolerance=0.1, rational=True) == Rational(11, 3)
assert nsimplify(3.666666, tolerance=0.01,
rational=True) == Rational(11, 3)
assert nsimplify(33, tolerance=10, rational=True) == Rational(33)
assert nsimplify(33.33, tolerance=10, rational=True) == Rational(30)
assert nsimplify(37.76, tolerance=10, rational=True) == Rational(40)
assert nsimplify(-203.1) == Rational(-2031, 10)
assert nsimplify(.2, tolerance=0) == Rational(1, 5)
assert nsimplify(-.2, tolerance=0) == Rational(-1, 5)
assert nsimplify(.2222, tolerance=0) == Rational(1111, 5000)
assert nsimplify(-.2222, tolerance=0) == Rational(-1111, 5000)
# issue 7211, PR 4112
assert nsimplify(S(2e-8)) == Rational(1, 50000000)
# issue 7322 direct test
assert nsimplify(1e-42, rational=True) != 0
# issue 10336
inf = Float('inf')
infs = (-oo, oo, inf, -inf)
for zi in infs:
ans = sign(zi) * oo
assert nsimplify(zi) == ans
assert nsimplify(zi + x) == x + ans
assert nsimplify(0.33333333, rational=True,
rational_conversion='exact') == Rational(0.33333333)
# Make sure nsimplify on expressions uses full precision
assert nsimplify(
pi.evalf(100) * x,
rational_conversion='exact').evalf(100) == pi.evalf(100) * x
def test_simplify_expr():
x, y, z, k, n, m, w, s, A = symbols('x,y,z,k,n,m,w,s,A')
f = Function('f')
assert all(simplify(tmp) == tmp for tmp in [I, E, oo, x, -x, -oo, -E, -I])
e = 1 / x + 1 / y
assert e != (x + y) / (x * y)
assert simplify(e) == (x + y) / (x * y)
e = A**2 * s**4 / (4 * pi * k * m**3)
assert simplify(e) == e
e = (4 + 4 * x - 2 * (2 + 2 * x)) / (2 + 2 * x)
assert simplify(e) == 0
e = (-4 * x * y**2 - 2 * y**3 - 2 * x**2 * y) / (x + y)**2
assert simplify(e) == -2 * y
e = -x - y - (x + y)**(-1) * y**2 + (x + y)**(-1) * x**2
assert simplify(e) == -2 * y
e = (x + x * y) / x
assert simplify(e) == 1 + y
e = (f(x) + y * f(x)) / f(x)
assert simplify(e) == 1 + y
e = (2 * (1 / n - cos(n * pi) / n)) / pi
assert simplify(e) == (-cos(pi * n) + 1) / (pi * n) * 2
e = integrate(1 / (x**3 + 1), x).diff(x)
assert simplify(e) == 1 / (x**3 + 1)
e = integrate(x / (x**2 + 3 * x + 1), x).diff(x)
assert simplify(e) == x / (x**2 + 3 * x + 1)
f = Symbol('f')
A = Matrix([[2 * k - m * w**2, -k], [-k, k - m * w**2]]).inv()
assert simplify((A * Matrix([0, f]))[1] - (-f * (2 * k - m * w**2) /
(k**2 - (k - m * w**2) *
(2 * k - m * w**2)))) == 0
f = -x + y / (z + t) + z * x / (z + t) + z * a / (z + t) + t * x / (z + t)
assert simplify(f) == (y + a * z) / (z + t)
# issue 10347
expr = -x * (y**2 - 1) * (
2 * y**2 * (x**2 - 1) / (a * (x**2 - y**2)**2) + (x**2 - 1) /
(a * (x**2 - y**2))) / (a * (x**2 - y**2)) + x * (
-2 * x**2 * sqrt(-x**2 * y**2 + x**2 + y**2 - 1) * sin(z) /
(a * (x**2 - y**2)**2) -
x**2 * sqrt(-x**2 * y**2 + x**2 + y**2 - 1) * sin(z) /
(a * (x**2 - 1) * (x**2 - y**2)) +
(x**2 * sqrt((-x**2 + 1) * (y**2 - 1)) *
sqrt(-x**2 * y**2 + x**2 + y**2 - 1) * sin(z) / (x**2 - 1) + sqrt(
(-x**2 + 1) * (y**2 - 1)) *
(x * (-x * y**2 + x) / sqrt(-x**2 * y**2 + x**2 + y**2 - 1) +
sqrt(-x**2 * y**2 + x**2 + y**2 - 1)) * sin(z)) / (a * sqrt(
(-x**2 + 1) * (y**2 - 1)) * (x**2 - y**2))
) * sqrt(-x**2 * y**2 + x**2 + y**2 - 1) * sin(z) / (
a * (x**2 - y**2)) + x * (
-2 * x**2 * sqrt(-x**2 * y**2 + x**2 + y**2 - 1) * cos(z) /
(a * (x**2 - y**2)**2) -
x**2 * sqrt(-x**2 * y**2 + x**2 + y**2 - 1) * cos(z) /
(a * (x**2 - 1) * (x**2 - y**2)) +
(x**2 * sqrt((-x**2 + 1) *
(y**2 - 1)) * sqrt(-x**2 * y**2 + x**2 + y**2 - 1)
* cos(z) / (x**2 - 1) + x * sqrt(
(-x**2 + 1) * (y**2 - 1)) * (-x * y**2 + x) * cos(z) /
sqrt(-x**2 * y**2 + x**2 + y**2 - 1) + sqrt(
(-x**2 + 1) *
(y**2 - 1)) * sqrt(-x**2 * y**2 + x**2 + y**2 - 1) *
cos(z)) / (a * sqrt((-x**2 + 1) * (y**2 - 1)) * (x**2 - y**2))
) * sqrt(-x**2 * y**2 + x**2 + y**2 - 1) * cos(z) / (
a *
(x**2 - y**2)) - y * sqrt((-x**2 + 1) * (y**2 - 1)) * (
-x * y * sqrt(-x**2 * y**2 + x**2 + y**2 - 1) * sin(z) /
(a * (x**2 - y**2) *
(y**2 - 1)) +
2 * x * y * sqrt(-x**2 * y**2 + x**2 + y**2 - 1) * sin(z) /
(a * (x**2 - y**2)**2) +
(x * y * sqrt((-x**2 + 1) * (y**2 - 1)) *
sqrt(-x**2 * y**2 + x**2 + y**2 - 1) * sin(z) /
(y**2 - 1) + x * sqrt(
(-x**2 + 1) * (y**2 - 1)) * (-x**2 * y + y) * sin(z) /
sqrt(-x**2 * y**2 + x**2 + y**2 - 1)) / (a * sqrt(
(-x**2 + 1) * (y**2 - 1)) * (x**2 - y**2))
) * sin(z) / (a * (x**2 - y**2)) + y * (x**2 - 1) * (
-2 * x * y *
(x**2 - 1) /
(a * (x**2 - y**2)**2) + 2 * x * y / (a * (x**2 - y**2))
) / (a *
(x**2 - y**2)) + y * (x**2 - 1) * (y**2 - 1) * (
-x * y * sqrt(-x**2 * y**2 + x**2 + y**2 - 1) *
cos(z) / (a * (x**2 - y**2) *
(y**2 - 1)) + 2 * x * y *
sqrt(-x**2 * y**2 + x**2 + y**2 - 1) * cos(z) /
(a *
(x**2 - y**2)**2) +
(x * y * sqrt((-x**2 + 1) * (y**2 - 1)) *
sqrt(-x**2 * y**2 + x**2 + y**2 - 1) * cos(z) /
(y**2 - 1) + x * sqrt(
(-x**2 + 1) * (y**2 - 1)) * (-x**2 * y + y) *
cos(z) / sqrt(-x**2 * y**2 + x**2 + y**2 - 1)) /
(a * sqrt((-x**2 + 1) * (y**2 - 1)) *
(x**2 - y**2))) * cos(z) / (a * sqrt(
(-x**2 + 1) *
(y**2 - 1)) * (x**2 - y**2)) - x * sqrt(
(-x**2 + 1) * (y**2 - 1)
) * sqrt(-x**2 * y**2 + x**2 + y**2 - 1) * sin(
z)**2 / (a**2 * (x**2 - 1) * (x**2 - y**2) *
(y**2 - 1)) - x * sqrt(
(-x**2 + 1) *
(y**2 - 1)) * sqrt(
-x**2 * y**2 + x**2 + y**2 -
1) * cos(z)**2 / (
a**2 * (x**2 - 1) *
(x**2 - y**2) *
(y**2 - 1))
assert simplify(expr) == 2 * x / (a**2 * (x**2 - y**2))
#issue 17631
assert simplify('((-1/2)*Boole(True)*Boole(False)-1)*Boole(True)') == \
Mul(sympify('(2 + Boole(True)*Boole(False))'), sympify('-Boole(True)/2'))
A, B = symbols('A,B', commutative=False)
assert simplify(A * B - B * A) == A * B - B * A
assert simplify(A / (1 + y / x)) == x * A / (x + y)
assert simplify(A * (1 / x + 1 / y)) == A / x + A / y #(x + y)*A/(x*y)
assert simplify(log(2) + log(3)) == log(6)
assert simplify(log(2 * x) - log(2)) == log(x)
assert simplify(hyper([], [], x)) == exp(x)
def domain(self):
return SingleContinuousDomain(sympify(self.symbol), self.set)
def gen_lobatto(max_order):
assert max_order > 2
x = sm.symbols('x')
lobs = [0, 1]
lobs[0] = (1 - x) / 2
lobs[1] = (1 + x) / 2
dlobs = [lob.diff('x') for lob in lobs]
legs = [sm.legendre(0, 'y')]
clegs = [sm.ccode(legs[0])]
dlegs = [sm.legendre(0, 'y').diff('y')]
cdlegs = [sm.ccode(dlegs[0])]
clobs = [sm.ccode(lob) for lob in lobs]
cdlobs = [sm.ccode(dlob) for dlob in dlobs]
denoms = [] # for lobs.
for ii in range(2, max_order + 1):
coef = sm.sympify('sqrt(2 * (2 * %s - 1)) / 2' % ii)
leg = sm.legendre(ii - 1, 'y')
pleg = leg.as_poly()
coefs = pleg.all_coeffs()
denom = max(sm.denom(val) for val in coefs)
cleg = sm.ccode(sm.horner(leg * denom) / denom)
dleg = leg.diff('y')
cdleg = sm.ccode(sm.horner(dleg * denom) / denom)
lob = sm.simplify(coef * sm.integrate(leg, ('y', -1, x)))
lobnc = sm.simplify(sm.integrate(leg, ('y', -1, x)))
plobnc = lobnc.as_poly()
coefs = plobnc.all_coeffs()
denom = sm.denom(coef) * max(sm.denom(val) for val in coefs)
clob = sm.ccode(sm.horner(lob * denom) / denom)
dlob = lob.diff('x')
cdlob = sm.ccode(sm.horner(dlob * denom) / denom)
legs.append(leg)
clegs.append(cleg)
dlegs.append(dleg)
cdlegs.append(cdleg)
lobs.append(lob)
clobs.append(clob)
dlobs.append(dlob)
cdlobs.append(cdlob)
denoms.append(denom)
coef = sm.sympify('sqrt(2 * (2 * %s - 1)) / 2' % (max_order + 1))
leg = sm.legendre(max_order, 'y')
pleg = leg.as_poly()
coefs = pleg.all_coeffs()
denom = max(sm.denom(val) for val in coefs)
cleg = sm.ccode(sm.horner(leg * denom) / denom)
dleg = leg.diff('y')
cdleg = sm.ccode(sm.horner(dleg * denom) / denom)
legs.append(leg)
clegs.append(cleg)
dlegs.append(dleg)
cdlegs.append(cdleg)
kerns = []
ckerns = []
dkerns = []
cdkerns = []
for ii, lob in enumerate(lobs[2:]):
kern = sm.simplify(lob / (lobs[0] * lobs[1]))
dkern = kern.diff('x')
denom = denoms[ii] / 4
ckern = sm.ccode(sm.horner(kern * denom) / denom)
cdkern = sm.ccode(sm.horner(dkern * denom) / denom)
kerns.append(kern)
ckerns.append(ckern)
dkerns.append(dkern)
cdkerns.append(cdkern)
return (legs, clegs, dlegs, cdlegs, lobs, clobs, dlobs, cdlobs, kerns,
ckerns, dkerns, cdkerns, denoms)
def mathematica(s):
return sympify(parse(s))
def NS(e, n=15, **options):
return str(sympify(e).evalf(n, **options))
SI = States()
SI.name = ['X', 'Y', 'Psi', 'Psidot', 'V', 'Vdot']
numStates = 6
SI.states = numStates * [None]
#### TESTING PURPOSES
Stest = States()
Stest.states = 10 * [0]
lf = 1.3314298821150170049065764033003
lr = 1.5185105279655697341212317041936
#base = [30.6286298219,-96.4821284934]
#### LINEARIZE EQUATIONS USING JACOBIAN
sym.init_printing(use_latex=True)
myvarsF = sym.sympify('x(t), y(t), psi(t), diff(psi(t),t), v(t), diff(v(t),t)')
myvarsG = sym.sympify('delta_f(t), a(t)')
bet = sym.sympify('atan(l_r/(l_f+l_r)*tan(delta_f(t)))')
f1 = sym.sympify('v(t)*cos(psi(t)+bet)') # xdot
f1 = f1.subs({'bet': bet})
f2 = sym.sympify('v(t)*sin(psi(t)+bet)') # ydot
f2 = f2.subs({'bet': bet})
f3 = sym.sympify('v(t)/l_r*sin(bet)') # psidot
f3 = f3.subs({'bet': bet}) #substitute bet expression
f4 = sym.diff(f3, sym.symbols('t')) #psidotdot
f4 = f4.subs({'bet': bet}) #sub bet expression
f5 = sym.sympify('a(t)') #vdot
f6 = f5.diff('t') #vdotdot
J = sym.Matrix([f1, f2, f3, f4, f5, f6]) #define vector-valued function
F = J.jacobian(myvarsF) # Perform jacobiam wrt to myvars
G = J.jacobian(myvarsG)
def __new__(cls, mat):
mat = sympify(mat)
return Basic.__new__(cls, mat)
# The Derivative of a function y = f(x) expresses the rate of change in the dependent variable
# y with respect to the independent variable , x. It's denoted as either f'x or dy/dx.
#Derivative Calculator
from sympy import Symbol, Derivative, sympify, pprint
from sympy.core.sympify import SympifyError
def derivative(f, var):
var = Symbol(var)
d = Derivative(f, var).doit()
pprint(d)
if __name__ == '__main__':
f = input('Enter a function: ')
var = input('Enter the variable to diffrentiate with respect to: ')
try:
f = sympify(f)
except SympifyError:
print('Invalid input')
else:
derivative(f, var)
def residue(expr, x, x0):
"""
Finds the residue of ``expr`` at the point x=x0.
The residue is defined as the coefficient of 1/(x-x0) in the power series
expansion about x=x0.
Examples
========
>>> from sympy import Symbol, residue, sin
>>> x = Symbol("x")
>>> residue(1/x, x, 0)
1
>>> residue(1/x**2, x, 0)
0
>>> residue(2/sin(x), x, 0)
2
This function is essential for the Residue Theorem [1].
References
==========
1. http://en.wikipedia.org/wiki/Residue_theorem
"""
# The current implementation uses series expansion to
# calculate it. A more general implementation is explained in
# the section 5.6 of the Bronstein's book {M. Bronstein:
# Symbolic Integration I, Springer Verlag (2005)}. For purely
# rational functions, the algorithm is much easier. See
# sections 2.4, 2.5, and 2.7 (this section actually gives an
# algorithm for computing any Laurent series coefficient for
# a rational function). The theory in section 2.4 will help to
# understand why the resultant works in the general algorithm.
# For the definition of a resultant, see section 1.4 (and any
# previous sections for more review).
from sympy import collect, Mul, Order, S
expr = sympify(expr)
if x0 != 0:
expr = expr.subs(x, x + x0)
for n in [0, 1, 2, 4, 8, 16, 32]:
if n == 0:
s = expr.series(x, n=0)
else:
s = expr.nseries(x, n=n)
if not s.has(Order) or s.getn() >= 0:
break
s = collect(s.removeO(), x)
if s.is_Add:
args = s.args
else:
args = [s]
res = S(0)
for arg in args:
c, m = arg.as_coeff_mul(x)
m = Mul(*m)
if not (m == 1 or m == x or (m.is_Pow and m.exp.is_Integer)):
raise NotImplementedError('term of unexpected form: %s' % m)
if m == 1 / x:
res += c
return res
def _calculate_price_expression(self, price_expr, context):
local_context = {}
for p in context.keys():
local_context[p] = context[p]
p = sympy.sympify(price_expr, locals=local_context)
return Decimal(float(p))
def convertir_numero(valor):
return lambdify((), sympify(valor))()
check_arg(old, sp.Basic, 1)
check_arg(new, sp.Basic, 2)
for ind, (old0, new0) in enumerate(subs):
if old0 == old:
subs.pop(ind)
break
subs.append((old, new))
# Create a sympy evaulation namespace
sp_namespace = {}
sp_namespace.update((name, op) for name, op in sp.functions.__dict__.items() \
if name[0] != "_")
sp_namespace["Conditional"] = Conditional
sp_namespace["ContinuousConditional"] = ContinuousConditional
sp_namespace.update((name, op) for name, op in _relational.__dict__.items() \
if name in ["Eq", "Ne", "Gt", "Ge", "Lt", "Le"])
sp_namespace.update((name, getattr(sp, name)) for name in ["And", "Or"])
sp_namespace["pi"] = sp.numbers.pi
sp_namespace["E"] = sp.numbers.E
sp_namespace["one"] = sp.sympify(1)
sp_namespace["two"] = sp.sympify(2)
sp_namespace["three"] = sp.sympify(3)
sp_namespace["four"] = sp.sympify(4)
sp_namespace["five"] = sp.sympify(5)
sp_namespace["ten"] = sp.sympify(10)
__all__ = [_name for _name in globals().keys() if _name[0] != "_"]
def _eval_power(self, other):
other = sympify(other)
return Dimension(self.name**other)
def __init__(self, name, abbrev, exponent, base=sympify(10)):
self.name = name
self.abbrev = abbrev
self.factor = base**exponent
def split_expression(expr):
'''
Split an expression into a part containing the function ``f`` and another
one containing the function ``g``. Returns a tuple of the two expressions
(as sympy expressions).
Parameters
----------
expr : str
An expression containing references to functions ``f`` and ``g``.
Returns
-------
(non_stochastic, stochastic) : tuple of sympy expressions
A pair of expressions representing the non-stochastic (containing
function-independent terms and terms involving ``f``) and the
stochastic part of the expression (terms involving ``g`` and/or ``dW``).
Examples
--------
>>> split_expression('dt * __f(__x, __t)')
(dt*__f(__x, __t), None)
>>> split_expression('dt * __f(__x, __t) + __dW * __g(__x, __t)')
(dt*__f(__x, __t), __dW*__g(__x, __t))
>>> split_expression('1/(2*dt**.5)*(__g_support - __g(__x, __t))*(__dW**2)')
(0, __dW**2*__g_support*dt**(-0.5)/2 - __dW**2*dt**(-0.5)*__g(__x, __t)/2)
'''
f = SYMBOLS['__f']
g = SYMBOLS['__g']
dW = SYMBOLS['__dW']
# Arguments of the f and g functions
x_f = sympy.Wild('x_f', exclude=[f, g], real=True)
t_f = sympy.Wild('t_f', exclude=[f, g], real=True)
x_g = sympy.Wild('x_g', exclude=[f, g], real=True)
t_g = sympy.Wild('t_g', exclude=[f, g], real=True)
# Reorder the expression so that f(x,t) and g(x,t) are factored out
sympy_expr = sympy.sympify(expr, locals=SYMBOLS).expand()
sympy_expr = sympy.collect(sympy_expr, f(x_f, t_f))
sympy_expr = sympy.collect(sympy_expr, g(x_g, t_g))
# Constant part, contains neither f, g nor dW
independent = sympy.Wild('independent', exclude=[f, g, dW], real=True)
# The exponent of the random number
dW_exponent = sympy.Wild('dW_exponent', exclude=[f, g, dW, 0], real=True)
# The factor for the random number, not containing the g function
independent_dW = sympy.Wild('independent_dW',
exclude=[f, g, dW],
real=True)
# The factor for the f function
f_factor = sympy.Wild('f_factor', exclude=[f, g], real=True)
# The factor for the g function
g_factor = sympy.Wild('g_factor', exclude=[f, g], real=True)
match_expr = (independent + f_factor * f(x_f, t_f) +
independent_dW * dW**dW_exponent + g_factor * g(x_g, t_g))
matches = sympy_expr.match(match_expr)
if matches is None:
raise ValueError(('Expression "%s" in the state updater description '
'could not be parsed.' % sympy_expr))
# Non-stochastic part
if x_f in matches:
# Includes the f function
non_stochastic = matches[independent] + (matches[f_factor] *
f(matches[x_f], matches[t_f]))
else:
# Does not include f, might be 0
non_stochastic = matches[independent]
# Stochastic part
if independent_dW in matches and matches[independent_dW] != 0:
# includes a random variable term with a non-zero factor
stochastic = (matches[g_factor] * g(matches[x_g], matches[t_g]) +
matches[independent_dW] * dW**matches[dW_exponent])
elif x_g in matches:
# Does not include a random variable but the g function
stochastic = matches[g_factor] * g(matches[x_g], matches[t_g])
else:
# Contains neither random variable nor g function --> empty
stochastic = None
return (non_stochastic, stochastic)
def graficar(expr, intervalo, metodo, particiones):
expr = sympify(expr, evaluate=False)
func = lambdify(abc.x, expr)
particiones = convertir_numero(particiones)
intervalo = a, b = list(
sorted([convertir_numero(limite) for limite in intervalo]))
fig, ax = plt.subplots()
def establecer_interfaz():
# Establecer título
fig.canvas.set_window_title('{}, {}'.format(expr, intervalo))
# Escribir función
plt.text((a + b) / 2,
func((a + b) / 2),
'${}$'.format(latex(expr)),
horizontalalignment='center',
fontsize=20)
def graficar_funcion():
x = np.arange(a, b, 0.001)
y = func(x)
plt.plot(x, y, '#FE4A49')
def graficar_trapecio():
x = np.linspace(a, b, particiones + 1)
y = func(x)
puntos = list(zip(x, y))
for i in range(particiones):
x1, _ = puntos[i]
x2, _ = puntos[i + 1]
vertices = [(x1, 0), puntos[i], puntos[i + 1], (x2, 0)]
poligono = Polygon(vertices,
facecolor='#F26430' if i %
2 == 0 else '#F48055',
edgecolor='#F26430')
ax.add_patch(poligono)
def graficar_simpson():
def graficar_parabola(p1, p2, p3, obscuro):
inicio = p1[0]
fin = p3[0]
func = parabola_puntos(p1, p2, p3)
x = np.arange(inicio, fin, 0.001)
y = func(x)
vertices = [(inicio, 0)] + list(zip(x, y)) + [(fin, 0)]
poligono = Polygon(vertices,
facecolor='#009FB7' if obscuro else '#7FC0CA',
edgecolor='#009FB7')
ax.add_patch(poligono)
x = np.linspace(a, b, particiones + 1)
y = func(x)
puntos = list(zip(x, y))
for i in range(particiones // 2):
graficar_parabola(puntos[2 * i], puntos[2 * i + 1],
puntos[2 * i + 2], i % 2 == 0)
def graficar_area_bajo_curva():
x = np.arange(a, b, 0.001)
y = func(x)
vertices = [(a, 0)] + list(zip(x, y)) + [(b, 0)]
poligono = Polygon(vertices, facecolor='#009FB7', edgecolor='#009FB7')
ax.add_patch(poligono)
establecer_interfaz()
graficar_funcion()
if particiones > 1000:
graficar_area_bajo_curva()
else:
if metodo == 'trapecio':
graficar_trapecio()
else:
graficar_simpson()
plt.show()
def print_cooling(self, assign_to):
eq = sympy.sympify("0")
for term in self.cooling_actions:
eq += self.cooling_actions[term].equation
return ccode(eq, assign_to = assign_to)
def prob_of(self, elem):
elem = sympify(elem)
density = self._density
if isinstance(list(density.keys())[0], FiniteSet):
return density.get(elem, S.Zero)
return density.get(tuple(elem)[0][1], S.Zero)
import matplotlib.pyplot as plt import math import matplotlib as mpl import numpy as np import sympy from sympy.abc import x from sympy import sin, cos, pi x1 = np.arange(-10, 9, 0.01) # start,stop,step y1 = sympy.sympify(-(x**3) / (x - 9)) y1 = sympy.lambdify(x, y1, 'numpy') x2 = np.arange(4, 10, 0.01) y2 = sympy.sympify(-1 / (x - 4) + 2) y2 = sympy.lambdify(x, y2, 'numpy') plt.grid(True) plt.axhline(y=0, lw=1, color='black') plt.axvline(x=0, lw=1, color='black') plt.xticks(np.arange(-100, 100, 1)) plt.yticks(np.arange(-100, 100, 1)) plot_y1 = y1(x1) plot_y2 = y2(x2) plt.plot(x1, plot_y1, color='black', linewidth=1) plt.plot(x2, plot_y2, color='black', linewidth=1) plt.ylim(-10, 10) plt.xlim(-10, 10)
def _Complement__new__ (cls, a, b, evaluate = True): # sets.Complement patched to sympify args if evaluate: return Complement.reduce (sympify (a), sympify (b)) return Basic.__new__ (cls, a, b)
def primitive_element(extension, x=None, **args):
"""Construct a common number field for all extensions. """
if not extension:
raise ValueError("can't compute primitive element for empty extension")
if x is not None:
x, cls = sympify(x), Poly
else:
x, cls = Dummy('x'), PurePoly
if not args.get('ex', False):
extension = [ AlgebraicNumber(ext, gen=x) for ext in extension ]
g, coeffs = extension[0].minpoly.replace(x), [1]
for ext in extension[1:]:
s, _, g = sqf_norm(g, x, extension=ext)
coeffs = [ s*c for c in coeffs ] + [1]
if not args.get('polys', False):
return g.as_expr(), coeffs
else:
return cls(g), coeffs
generator = numbered_symbols('y', cls=Dummy)
F, Y = [], []
for ext in extension:
y = next(generator)
if ext.is_Poly:
if ext.is_univariate:
f = ext.as_expr(y)
else:
raise ValueError("expected minimal polynomial, got %s" % ext)
else:
f = minpoly(ext, y)
F.append(f)
Y.append(y)
coeffs_generator = args.get('coeffs', _coeffs_generator)
for coeffs in coeffs_generator(len(Y)):
f = x - sum([ c*y for c, y in zip(coeffs, Y)])
G = groebner(F + [f], Y + [x], order='lex', field=True)
H, g = G[:-1], cls(G[-1], x, domain='QQ')
for i, (h, y) in enumerate(zip(H, Y)):
try:
H[i] = Poly(y - h, x,
domain='QQ').all_coeffs() # XXX: composite=False
except CoercionFailed: # pragma: no cover
break # G is not a triangular set
else:
break
else: # pragma: no cover
raise RuntimeError("run out of coefficient configurations")
_, g = g.clear_denoms()
if not args.get('polys', False):
return g.as_expr(), coeffs, H
else:
return g, coeffs, H
def species_total(self, species):
eq = sympy.sympify("0")
for rn, rxn in sorted(self.reactions.items()):
eq += rxn.species_equation(species)
return eq
def minimal_polynomial(ex, x=None, **args):
"""
Computes the minimal polynomial of an algebraic element.
Parameters
==========
ex : algebraic element expression
x : independent variable of the minimal polynomial
Options
=======
compose : if ``True`` ``_minpoly_compose`` is used, if ``False`` the ``groebner`` algorithm
polys : if ``True`` returns a ``Poly`` object
domain : ground domain
Notes
=====
By default ``compose=True``, the minimal polynomial of the subexpressions of ``ex``
are computed, then the arithmetic operations on them are performed using the resultant
and factorization.
If ``compose=False``, a bottom-up algorithm is used with ``groebner``.
The default algorithm stalls less frequently.
If no ground domain is given, it will be generated automatically from the expression.
Examples
========
>>> from sympy import minimal_polynomial, sqrt, solve, QQ
>>> from sympy.abc import x, y
>>> minimal_polynomial(sqrt(2), x)
x**2 - 2
>>> minimal_polynomial(sqrt(2), x, domain=QQ.algebraic_field(sqrt(2)))
x - sqrt(2)
>>> minimal_polynomial(sqrt(2) + sqrt(3), x)
x**4 - 10*x**2 + 1
>>> minimal_polynomial(solve(x**3 + x + 3)[0], x)
x**3 + x + 3
>>> minimal_polynomial(sqrt(y), x)
x**2 - y
"""
from sympy.polys.polytools import degree
from sympy.polys.domains import FractionField
from sympy.core.basic import preorder_traversal
compose = args.get('compose', True)
polys = args.get('polys', False)
dom = args.get('domain', None)
ex = sympify(ex)
for expr in preorder_traversal(ex):
if expr.is_AlgebraicNumber:
compose = False
break
if x is not None:
x, cls = sympify(x), Poly
else:
x, cls = Dummy('x'), PurePoly
if not dom:
dom = FractionField(QQ, list(ex.free_symbols)) if ex.free_symbols else QQ
if hasattr(dom, 'symbols') and x in dom.symbols:
raise GeneratorsError("the variable %s is an element of the ground domain %s" % (x, dom))
if compose:
result = _minpoly_compose(ex, x, dom)
result = result.primitive()[1]
c = result.coeff(x**degree(result, x))
if c.is_negative:
result = expand_mul(-result)
return cls(result, x, field=True) if polys else result.collect(x)
if not dom.is_QQ:
raise NotImplementedError("groebner method only works for QQ")
result = _minpoly_groebner(ex, x, cls)
return cls(result, x, field=True) if polys else result.collect(x)
def test_linearize_rolling_disc_kane():
# Symbols for time and constant parameters
t, r, m, g, v = symbols("t r m g v")
# Configuration variables and their time derivatives
q1, q2, q3, q4, q5, q6 = q = dynamicsymbols("q1:7")
q1d, q2d, q3d, q4d, q5d, q6d = qd = [qi.diff(t) for qi in q]
# Generalized speeds and their time derivatives
u = dynamicsymbols("u:6")
u1, u2, u3, u4, u5, u6 = u = dynamicsymbols("u1:7")
u1d, u2d, u3d, u4d, u5d, u6d = [ui.diff(t) for ui in u]
# Reference frames
N = ReferenceFrame("N") # Inertial frame
NO = Point("NO") # Inertial origin
A = N.orientnew("A", "Axis", [q1, N.z]) # Yaw intermediate frame
B = A.orientnew("B", "Axis", [q2, A.x]) # Lean intermediate frame
C = B.orientnew("C", "Axis", [q3, B.y]) # Disc fixed frame
CO = NO.locatenew("CO", q4 * N.x + q5 * N.y + q6 * N.z) # Disc center
# Disc angular velocity in N expressed using time derivatives of coordinates
w_c_n_qd = C.ang_vel_in(N)
w_b_n_qd = B.ang_vel_in(N)
# Inertial angular velocity and angular acceleration of disc fixed frame
C.set_ang_vel(N, u1 * B.x + u2 * B.y + u3 * B.z)
# Disc center velocity in N expressed using time derivatives of coordinates
v_co_n_qd = CO.pos_from(NO).dt(N)
# Disc center velocity in N expressed using generalized speeds
CO.set_vel(N, u4 * C.x + u5 * C.y + u6 * C.z)
# Disc Ground Contact Point
P = CO.locatenew("P", r * B.z)
P.v2pt_theory(CO, N, C)
# Configuration constraint
f_c = Matrix([q6 - dot(CO.pos_from(P), N.z)])
# Velocity level constraints
f_v = Matrix([dot(P.vel(N), uv) for uv in C])
# Kinematic differential equations
kindiffs = Matrix(
[dot(w_c_n_qd - C.ang_vel_in(N), uv) for uv in B]
+ [dot(v_co_n_qd - CO.vel(N), uv) for uv in N]
)
qdots = solve(kindiffs, qd)
# Set angular velocity of remaining frames
B.set_ang_vel(N, w_b_n_qd.subs(qdots))
C.set_ang_acc(N, C.ang_vel_in(N).dt(B) + cross(B.ang_vel_in(N), C.ang_vel_in(N)))
# Active forces
F_CO = m * g * A.z
# Create inertia dyadic of disc C about point CO
I = (m * r ** 2) / 4
J = (m * r ** 2) / 2
I_C_CO = inertia(C, I, J, I)
Disc = RigidBody("Disc", CO, C, m, (I_C_CO, CO))
BL = [Disc]
FL = [(CO, F_CO)]
KM = KanesMethod(
N,
[q1, q2, q3, q4, q5],
[u1, u2, u3],
kd_eqs=kindiffs,
q_dependent=[q6],
configuration_constraints=f_c,
u_dependent=[u4, u5, u6],
velocity_constraints=f_v,
)
with warns_deprecated_sympy():
(fr, fr_star) = KM.kanes_equations(FL, BL)
# Test generalized form equations
linearizer = KM.to_linearizer()
assert linearizer.f_c == f_c
assert linearizer.f_v == f_v
assert linearizer.f_a == f_v.diff(t).subs(KM.kindiffdict())
sol = solve(linearizer.f_0 + linearizer.f_1, qd)
for qi in qdots.keys():
assert sol[qi] == qdots[qi]
assert simplify(linearizer.f_2 + linearizer.f_3 - fr - fr_star) == Matrix([0, 0, 0])
# Perform the linearization
# Precomputed operating point
q_op = {q6: -r * cos(q2)}
u_op = {
u1: 0,
u2: sin(q2) * q1d + q3d,
u3: cos(q2) * q1d,
u4: -r * (sin(q2) * q1d + q3d) * cos(q3),
u5: 0,
u6: -r * (sin(q2) * q1d + q3d) * sin(q3),
}
qd_op = {
q2d: 0,
q4d: -r * (sin(q2) * q1d + q3d) * cos(q1),
q5d: -r * (sin(q2) * q1d + q3d) * sin(q1),
q6d: 0,
}
ud_op = {
u1d: 4 * g * sin(q2) / (5 * r)
+ sin(2 * q2) * q1d ** 2 / 2
+ 6 * cos(q2) * q1d * q3d / 5,
u2d: 0,
u3d: 0,
u4d: r * (sin(q2) * sin(q3) * q1d * q3d + sin(q3) * q3d ** 2),
u5d: r
* (
4 * g * sin(q2) / (5 * r)
+ sin(2 * q2) * q1d ** 2 / 2
+ 6 * cos(q2) * q1d * q3d / 5
),
u6d: -r * (sin(q2) * cos(q3) * q1d * q3d + cos(q3) * q3d ** 2),
}
A, B = linearizer.linearize(
op_point=[q_op, u_op, qd_op, ud_op], A_and_B=True, simplify=True
)
upright_nominal = {q1d: 0, q2: 0, m: 1, r: 1, g: 1}
# Precomputed solution
A_sol = Matrix(
[
[0, 0, 0, 0, 0, 0, 0, 1],
[0, 0, 0, 0, 0, 1, 0, 0],
[0, 0, 0, 0, 0, 0, 1, 0],
[sin(q1) * q3d, 0, 0, 0, 0, -sin(q1), -cos(q1), 0],
[-cos(q1) * q3d, 0, 0, 0, 0, cos(q1), -sin(q1), 0],
[0, Rational(4, 5), 0, 0, 0, 0, 0, 6 * q3d / 5],
[0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, -2 * q3d, 0, 0],
]
)
B_sol = Matrix([])
# Check that linearization is correct
assert A.subs(upright_nominal) == A_sol
assert B.subs(upright_nominal) == B_sol
# Check eigenvalues at critical speed are all zero:
assert sympify(A.subs(upright_nominal).subs(q3d, 1 / sqrt(3))).eigenvals() == {0: 8}