def ExactIntegrationOfSinus(t, a=None, b=None):
with precision(300):
if a == None and b == None:
return 0.5 * math.pi * math.sqrt(t) * (mpmath.angerj(0.5, t) -
mpmath.angerj(-0.5, t))
elif a == None and b != None:
a = 0
elif b == None:
b = t
mpmath.mp.dps = 50
mpmath.mp.pretty = True
pi = mpmath.mp.pi
pi = +pi
fcos = mpmath.fresnelc
fsin = mpmath.fresnels
arg_b = mpmath.sqrt(2 * (t - b) / pi)
if a == "MinusInfinity":
return mpmath.sqrt(
0.5 * mpmath.mp.pi) * (-2 * mpmath.sin(t) * fcos(arg_b) +
2 * mpmath.cos(t) * fsin(arg_b) +
mpmath.sin(t) - mpmath.cos(t))
else:
arg_a = mpmath.sqrt(2 * (t - a) / pi)
return mpmath.sqrt(2 * mpmath.mp.pi) * (
(fsin(arg_b) - fsin(arg_a)) * mpmath.cos(t) +
(fcos(arg_a) - fcos(arg_b)) * mpmath.sin(t))
def cos_gamma(_a: float, incl: float, tol=10e-5) -> float:
"""Calculate the cos of the angle gamma"""
if abs(incl) < tol:
return 0
else:
return mpmath.cos(_a) / mpmath.sqrt(
mpmath.cos(_a)**2 + mpmath.cot(incl)**2) # real
def _view_cone_calc(lat_geoc, lon_geoc, sat_pos, sat_vel, q_max, m):
"""Semi-private: Performs the viewing cone visibility calculation for the day defined by m.
Note: This function is based on a paper titled "rapid satellite-to-site visibility determination
based on self-adaptive interpolation technique" with some variation to account for interaction
of viewing cone with the satellite orbit.
Args:
lat_geoc (float): site location in degrees at the start of POI
lon_geoc (float): site location in degrees at the start of POI
sat_pos (Vector3D): position of satellite (at the same time as sat_vel)
sat_vel (Vector3D): velocity of satellite (at the same time as sat_pos)
q_max (float): maximum orbital radius
m (int): interval offsets (number of days after initial condition)
Returns:
Returns 4 numbers representing times at which the orbit is tangent to the viewing cone,
Raises:
ValueError: if any of the 4 formulas has a complex answer. This happens when the orbit and
viewing cone do not intersect or only intersect twice.
Note: With more analysis it should be possible to find a correct interval even in the case
where there are only two intersections but this is beyond the current scope of the project.
"""
lat_geoc = (lat_geoc * mp.pi) / 180
lon_geoc = (lon_geoc * mp.pi) / 180
# P vector (also referred to as orbital angular momentum in the paper) calculations
p_unit_x, p_unit_y, p_unit_z = cross(sat_pos, sat_vel) / (mp.norm(sat_pos) * mp.norm(sat_vel))
# Following are equations from Viewing cone section of referenced paper
r_site_magnitude = earth_radius_at_geocetric_lat(lat_geoc)
gamma1 = THETA_NAUGHT + mp.asin((r_site_magnitude * mp.sin((mp.pi / 2) + THETA_NAUGHT)) / q_max)
gamma2 = mp.pi - gamma1
# Note: atan2 instead of atan to get the correct quadrant.
arctan_term = mp.atan2(p_unit_x, p_unit_y)
arcsin_term_gamma, arcsin_term_gamma2 = [(mp.asin((mp.cos(gamma) - p_unit_z * mp.sin(lat_geoc))
/ (mp.sqrt((p_unit_x ** 2) + (p_unit_y ** 2)) *
mp.cos(lat_geoc)))) for gamma in [gamma1, gamma2]]
angle_1 = (arcsin_term_gamma - lon_geoc - arctan_term + 2 * mp.pi * m)
angle_2 = (mp.pi - arcsin_term_gamma - lon_geoc - arctan_term + 2 * mp.pi * m)
angle_3 = (arcsin_term_gamma2 - lon_geoc - arctan_term + 2 * mp.pi * m)
angle_4 = (mp.pi - arcsin_term_gamma2 - lon_geoc - arctan_term + 2 * mp.pi * m)
angles = [angle_1, angle_2, angle_3, angle_4]
# Check for complex answers
if any([not isinstance(angle, mp.mpf) for angle in angles]):
raise ValueError()
# Map all angles to 0 to 2*pi
for idx in range(len(angles)):
while angles[idx] < 0:
angles[idx] += 2 * mp.pi
while angles[idx] > 2 * mp.pi:
angles[idx] -= 2 * mp.pi
# Calculate the corresponding time for each angle and return
return [mp.nint((1 / ANGULAR_VELOCITY_EARTH) * angle) for angle in angles]
def any_f_to_poly(f, a, b, order):
# remap [a, b] to [-1, 1] for better conditioning
p = mpmath.lu_solve(
vandermonde(mpmath.matrix([a, b]), 2),
mpmath.matrix([-1., 1.])
)
# fit polynomial to [-1, 1], then compose with mapping function
p_x = mpmath.matrix(
[mpmath.cos((i + .5) * mpmath.pi / order) for i in range(order)]
)
x = mpmath.matrix(
[a + (b - a) * (.5 + .5 * p_x[i]) for i in range(order)]
)
q = utils.poly.compose(
mpmath.lu_solve(vandermonde(p_x, order), f(x)),
p
)
# checking
p_x = mpmath.matrix(
[mpmath.cos(i * mpmath.pi / order) for i in range(order + 1)]
)
x = mpmath.matrix(
[a + (b - a) * (.5 + .5 * p_x[i]) for i in range(order + 1)]
)
y = utils.poly.eval_multi(q, x) - f(x)
est_err = max([abs(y[i]) for i in range(y.rows)])
print('est_err', est_err)
return q
def integrand(theta, phi):
evals[0] += 1
qab = q * mp.sin(theta)
qa = qab * mp.cos(phi)
qb = qab * mp.sin(phi)
qc = q * mp.cos(theta)
Zq = KERNEL_MP(qa, qb, qc)**2
return Zq * mp.sin(theta)
def easom(data):
"""
Easom’s function
Whose global minimum is f∗ = −1 at x∗ = (π, π) within −100 ≤ x, y ≤ 100. It has many
local minima.
"""
return float(-mp.cos(data[0]) * mp.cos(data[1]) *
mp.exp(-(data[0] - mp.pi)**2 - (data[1] - mp.pi)**2))
def get_dtheta_at(self, phi, covered_r):
from math import asin, cos, sqrt, pi
if covered_r >= sqrt(2) * cos(phi):
# In this case, the confidence radius is so large such that it covers the whole orbit and the north pole.
dtheta = pi
else:
dtheta = 2 * asin((.5 * covered_r) / cos(phi))
return dtheta
def sec_der(x_val: Real) -> mp.mpf:
"""Define the actual second derivative."""
x_squared = mp.power(x_val, 2)
return (
mp.cos(x_val)
+ (-4 * x_squared) * mp.cos(x_squared)
+ -2 * mp.sin(x_squared)
+ mp.sin(x_val)
)
def f2(x):
"""
Another angular integral in electron noise calculation.
"""
term1 = 2*x**3 * (2*mp.si(2*x)-mp.si(x))
term2 = 2*x**2 * (mp.cos(2*x) - mp.cos(x))
term3 = x * (mp.sin(2*x) - 2*mp.sin(x)) - (mp.cos(2*x) - 4* mp.cos(x) + 3)
return (term1 + term2 + term3)/(12 * x**2)
def get_rotation_matrix_north_pole_to(theta, phi):
from numpy import array, matmul
from math import sin, cos
Rx = array([[1, 0, 0], [0, sin(phi), cos(phi)], [0, -cos(phi), sin(phi)]])
Rz = array([[sin(theta), cos(theta), 0], [-cos(theta),
sin(theta), 0], [0, 0, 1]])
rotation_matrix = matmul(Rz, Rx)
return rotation_matrix
def FSingleDisc(x1,x2,t,a):
"""
inside of log() for connected HEE. We take single interval subsystem A=[x1,x2].
"""
t1 = T(x1,t,a)
t2 = T(x2,t,a)
tb1 = TB(x1,t,a)
tb2 = TB(x2,t,a)
num = (a**4)*( (exp(j*t1)-exp(-j*tb1))*(exp(j*t2)-exp(-j*tb2)) )**2
den = exp(j*(t1-tb1+t2-tb2))*((cos(t1)*cos(tb1)*cos(t2)*cos(tb2))**2)
return num/den
def S_single_hol_disconn(x1, x2, t, a):
t1, t2, tb1, tb2 = theta(x1, t, a), theta(x2, t,
a), thetaB(x1, t,
a), thetaB(x2, t, a)
z1, z2, zb1, zb2 = j * exp(j * t1), j * exp(j * t2), j * exp(
-j * tb1), j * exp(-j * tb2)
deriv = a**2 / (exp(j * (t1 - tb1 + t2 - tb2) / 2) * cos(t1) * cos(t2) *
cos(tb1) * cos(tb2))
disconn = (z1 - zb1) * (z2 - zb2)
return re(log(deriv * disconn)) / 6 - log((x2 - x1)**2) / 6
def getAntiprismVolumeOperator(n, k):
result = getProduct([
fdiv(
fprod([
n,
sqrt(fsub(fmul(4, cos(cos(fdiv(pi, fmul(n, 2))))), 1)),
sin(fdiv(fmul(3, pi), fmul(2, n)))
]), fmul(12, sin(sin(fdiv(pi, n))))),
sin(fdiv(fmul(3, pi), fmul(2, n))),
getPower(k, 3)
])
return result.convert('meter^3')
def Diagonalizzazione(I):
α = AngoloDiRotazione(I)
Iη = I[0]
Iξ = I[1]
Iηξ = I[2]
if α != 0:
Iy = Iη * mpmath.cos(α)**2 + Iξ * mpmath.sin(
α)**2 - 2 * Iηξ * mpmath.cos(α) * mpmath.sin(α)
Iz = Iξ * mpmath.sin(α)**2 + Iξ * mpmath.cos(
α)**2 + 2 * Iηξ * mpmath.cos(α) * mpmath.sin(α)
J = [Iy, Iz]
return J
else:
return I
def elliptic_core_g(x,y): x = x/2 y = y/2 factor = (cos(x)/sin(y) + sin(y)/cos(x) - (cos(y)/tan(y)/cos(x) + sin(x)*tan(x)/sin(y)))/pi k = tan(x)/tan(y) m = k*k n = (sin(x)/sin(y))*(sin(x)/sin(y)) u = asin(tan(y)/tan(x)) complete = ellipk(m) - ellippi(n, m) incomplete = ellipf(u,m) - ellippi(n/k/k,1/m)/k #incomplete = ellipf(u,m) - ellippi(n,u,m) return re(1.0 - factor*(incomplete + complete))
def calculate_coordinates(input_angles):
# Constants
scalling_const = np.sqrt(.3)
f = 195 * scalling_const * math.power(10, -3)
t = 375 * scalling_const * math.power(10, -3)
coord_list = []
for angle in input_angles:
T_1 = angle[0] - np.pi
T_2 = angle[1] - np.pi
x = float((f * math.cos(T_1) + t * math.cos(
(T_1 - T_2))) * math.power(10, 3))
y = float((f * math.sin(T_1) + t * math.sin(
(T_1 - T_2))) * math.power(10, 3))
coord_list.append([x, y])
return coord_list
def family_graphical_combination(t, c, angle_alpha_prime):
"""
Aberdeen family of graphical operators
"""
if not (isinstance(t, Opinion) and isinstance(c, Opinion) \
and t.check() and c.check()):
raise Exception("Two valid Opinions are required!")
if mpmath.almosteq(angle_alpha_prime, -mpmath.pi / 3, epsilon):
new_magnitude = c.get_magnitude_ratio() * (
mpmath.mpf("2") * t.getUncertainty() /
mpmath.sqrt(mpmath.mpf("3")))
#new_magnitude = 0
elif mpmath.almosteq(angle_alpha_prime,
mpmath.mpf("2/3") * mpmath.pi, epsilon):
new_magnitude = c.get_magnitude_ratio() * (
mpmath.mpf("2") * (mpmath.mpf("1") - t.getUncertainty()) /
mpmath.sqrt(mpmath.mpf("3")))
#new_magnitude = 0
elif mpmath.almosteq(angle_alpha_prime,
mpmath.mpf("1/2") * mpmath.pi, epsilon):
#new_magnitude = 1 - t.getUncertainty()
new_magnitude = 2 * t.getBelief()
else:
new_magnitude = c.get_magnitude_ratio() * (mpmath.mpf("2") * \
(mpmath.sqrt(mpmath.power(mpmath.tan(angle_alpha_prime), mpmath.mpf("2")) +1 ) /
( mpmath.absmax(mpmath.tan(angle_alpha_prime) + mpmath.sqrt(mpmath.mpf("3"))) ) ) * \
t.getBelief())
new_uncertainty = t.getUncertainty(
) + mpmath.sin(angle_alpha_prime) * new_magnitude
new_disbelief = t.getDisbelief() + (
t.getUncertainty() - new_uncertainty) * mpmath.cos(
mpmath.pi / 3) + mpmath.cos(angle_alpha_prime) * mpmath.sin(
mpmath.pi / 3) * new_magnitude
if mpmath.almosteq(new_uncertainty, 1, epsilon):
new_uncertainty = mpmath.mpf("1")
if mpmath.almosteq(new_uncertainty, 0, epsilon):
new_uncertainty = mpmath.mpf("0")
if mpmath.almosteq(new_disbelief, 1, epsilon):
new_disbelief = mpmath.mpf("1")
if mpmath.almosteq(new_disbelief, 0, epsilon):
new_disbelief = mpmath.mpf("0")
return Opinion(1 - new_disbelief - new_uncertainty, new_disbelief,
new_uncertainty, "1/2")
def cos(arg):
"""Cosine.
This function is a wrapper around a lower level function. If the argument is a standard *float* or *int*,
the function from the builtin :mod:`math` module will be used. If the argument is an :mod:`mpmath`
float, the corresponding multiprecision function, if available, will be used. Otherwise, the argument is assumed
to be a series type and a function from the piranha C++ library is used.
:param arg: cosine argument
:returns: cosine of *arg*
:raises: :exc:`TypeError` if the type of *arg* is not supported, or any other exception raised by the invoked
low-level function
>>> from .types import poisson_series, polynomial, rational, short
>>> t = poisson_series(polynomial(rational,short))()
>>> cos(2 * t('x'))
cos(2*x)
>>> cos('y') # doctest: +IGNORE_EXCEPTION_DETAIL
Traceback (most recent call last):
...
TypeError: invalid argument type(s)
"""
if isinstance(arg,float) or isinstance(arg,int):
import math
return math.cos(arg)
try:
from mpmath import mpf, cos
if isinstance(arg,mpf):
return cos(arg)
except ImportError:
pass
from ._core import _cos
return _cpp_type_catcher(_cos,arg)
def calc_J(chi, En, Lz, Q, aa, slr, ecc):
"""
Schmidt's J function.
Parameters:
chi (float): radial angle
En (float): energy
Lz (float): angular momentum
Q (float): Carter constant
aa (float): spin
slr (float): semi-latus rectum
ecc (float): eccentricity
Returns:
J (float)
"""
En2 = En * En
ecc2 = ecc * ecc
aa2 = aa * aa
Lz2 = Lz * Lz
slr2 = slr * slr
eta = 1 + ecc * cos(chi)
eta2 = eta * eta
J = ((1 - ecc2) * (1 - En2) + 2 * (1 - En2 - (1 - ecc2) / slr) * eta +
(((3 + ecc2) * (1 - En2)) / (1 - ecc2) +
((1 - ecc2) * (aa2 * (1 - En2) + Lz2 + Q)) / slr2 - 4 / slr) * eta2)
return J
def CalculateCoefSum(n, m, k):
# define a[k_] as per Paul D. Beale
a_coef = (1 + x*x)**2 - b*mp.cos(np.pi*k/n)
c2_coef_sum = 0
for j in range(0, m+2, 2): # should include endpoint so m --> m+2
c2_coef_sum += sy.expand((mh.factorial(m)/mh.factorial(j)/mh.factorial(m-j))*(a_coef**2 - b*b)**(j//2) * a_coef**(m - j))
return c2_coef_sum
def cos(arg):
"""Cosine.
This function is a wrapper around a lower level function. If the argument is a standard *float* or *int*,
the function from the builtin :mod:`math` module will be used. If the argument is an :mod:`mpmath`
float, the corresponding multiprecision function, if available, will be used. Otherwise, the argument is assumed
to be a series type and a function from the piranha C++ library is used.
:param arg: cosine argument
:rtype: cosine of *arg*
:raises: :exc:`TypeError` if the type of *arg* is not supported, or any other exception raised by the invoked
low-level function
>>> from . import get_series
>>> t = get_series('poisson_series<polynomial<rational,short>>')
>>> cos(2 * t('x'))
cos(2x)
>>> cos('y') # doctest: +IGNORE_EXCEPTION_DETAIL
Traceback (most recent call last):
...
TypeError: invalid argument type(s)
"""
if isinstance(arg, float) or isinstance(arg, int):
import math
return math.cos(arg)
try:
from mpmath import mpf, cos
if isinstance(arg, mpf):
return cos(arg)
except ImportError:
pass
from ._core import _cos
return _cpp_type_catcher(_cos, arg)
def Pprime(self, z):
# A+S 18.9.
from mpmath import ellipfun, sqrt, cos, sin, mpc, mpf
Delta = self.Delta
e1, e2, e3 = self.__roots
if self.__ng3:
z = mpc(0, 1) * z
if Delta > 0:
zs = sqrt(e1 - e3) * z
m = (e2 - e3) / (e1 - e3)
retval = -2 * sqrt(
(e1 - e3)**3) * ellipfun('cn', u=zs, m=m) * ellipfun(
'dn', u=zs, m=m) / (ellipfun('sn', u=zs, m=m)**3)
elif Delta < 0:
H2 = (sqrt((e2 - e3) * (e2 - e1))).real
assert (H2 > 0)
m = mpf(1) / mpf(2) - 3 * e2 / (4 * H2)
zp = 2 * z * sqrt(H2)
retval = -4 * sqrt(H2**3) * ellipfun('sn', u=zp, m=m) * ellipfun(
'dn', u=zp, m=m) / ((1 - ellipfun('cn', u=zp, m=m))**2)
else:
g2, g3 = self.__invariants
if g2 == 0 and g3 == 0:
retval = -2 / (z**3)
else:
c = e1 / 2
A = sqrt(3 * c)
retval = -6 * c * A * cos(A * z) / (sin(A * z))**3
if self.__ng3:
return mpc(0, -1) * retval
else:
return retval
def get_angle_beta(self):
if mpmath.almosteq(self.getDisbelief(), 1, epsilon):
return mpmath.pi / 3
return mpmath.atan(
(self.getUncertainty() * mpmath.sin(mpmath.pi / mpmath.mpf("3"))) /
(mpmath.mpf("1") - (self.getDisbelief() + self.getUncertainty() *
mpmath.cos(mpmath.pi / mpmath.mpf("3")))))
def d_theta(angle,k_v,R_v,alpha_v,An):
'''Off-axis level, :math:`D_{\\theta}`, for piston in a sphere
Parameters
----------
angle : mpmath.mpf
Azimuth/elevation angle in radians
k_v : mpmath.mpf
Wavenumber
alpha: mpmath.mpf
Piston equivalent aperture in radians.
An : mpmath.matrix
An is a required pre-calculated matrix which results from the
satisfying the conditions of the model - specific to each parameter
set.
Returns
-------
abs(rel_level()) : mpmath.mpf
The off-axis value at angle :math:`\theta`
'''
num = 4
N_v = An.rows
denom = (k_v**2)*(R_v**2)*mpmath.sin(alpha_v)**2
part1 = num/denom
jn_matrix = mpmath.matrix([I**f for f in range(N_v)])
legendre_matrix = mpmath.matrix([legendre(n_v, mpmath.cos(angle)) for n_v in range(N_v)])
Anjn = HP(An,jn_matrix).doit()
part2_matrix = HP(Anjn,legendre_matrix).doit()
part2 = sum(part2_matrix)
rel_level = lambdify([], -part1*part2, 'mpmath')
return abs(rel_level())
def rotate_point(x, y, degrees):
radians = math.radians(degrees)
cos_theta = math.cos(radians)
sin_theta = math.sin(radians)
return x * cos_theta - y * sin_theta, \
x * sin_theta + y * cos_theta
def test_rationalize():
from mpmath import cos, pi, mpf
assert rationalize(cos(pi / 3)) == S.Half
assert rationalize(mpf("0.333333333333333")) == Rational(1, 3)
assert rationalize(mpf("-0.333333333333333")) == Rational(-1, 3)
assert rationalize(pi, maxcoeff=250) == Rational(355, 113)
def get_angle_alpha(self):
if (mpmath.almosteq(self.getBelief(), 1, epsilon)):
return mpmath.mpf("0")
return mpmath.atan(
(self.getUncertainty() * mpmath.sin(mpmath.pi / mpmath.mpf("3"))) /
(self.getDisbelief() +
self.getUncertainty() * mpmath.cos(math.pi / mpmath.mpf("3"))))
def Pprime(self,z):
# A+S 18.9.
from mpmath import ellipfun, sqrt, cos, sin, mpc, mpf
Delta = self.Delta
e1, e2, e3 = self.__roots
if self.__ng3:
z = mpc(0,1) * z
if Delta > 0:
zs = sqrt(e1 - e3) * z
m = (e2 - e3) / (e1 - e3)
retval = -2 * sqrt((e1 - e3)**3) * ellipfun('cn',u=zs,m=m) * ellipfun('dn',u=zs,m=m) / (ellipfun('sn',u=zs,m=m)**3)
elif Delta < 0:
H2 = (sqrt((e2 - e3) * (e2 - e1))).real
assert(H2 > 0)
m = mpf(1) / mpf(2) - 3 * e2 / (4 * H2)
zp = 2 * z * sqrt(H2)
retval = -4 * sqrt(H2**3) * ellipfun('sn',u=zp,m=m) * ellipfun('dn',u=zp,m=m) / ((1 - ellipfun('cn',u=zp,m=m))**2)
else:
g2, g3 = self.__invariants
if g2 == 0 and g3 == 0:
retval = -2 / (z**3)
else:
c = e1 / 2
A = sqrt(3 * c)
retval = -6 * c * A * cos(A * z) / (sin(A * z))**3
if self.__ng3:
return mpc(0,-1) * retval
else:
return retval
def invcdf(p):
"""
Inverse of the CDF of the raised cosine distribution.
"""
with mpmath.extradps(5):
p = mpmath.mpf(p)
if p < 0 or p > 1:
return mpmath.nan
if p == 0:
return -mpmath.pi
if p == 1:
return mpmath.pi
if p < 0.094:
x = _poly_approx(mpmath.cbrt(12 * mpmath.pi * p)) - mpmath.pi
elif p > 0.906:
x = mpmath.pi - _poly_approx(mpmath.cbrt(12 * mpmath.pi * (1 - p)))
else:
y = mpmath.pi * (2 * p - 1)
y2 = y**2
x = y * _p2(y2) / _q2(y2)
solver = 'mnewton'
x = mpmath.findroot(f=lambda t: cdf(t) - p,
x0=x,
df=lambda t: (1 + mpmath.cos(t)) / (2 * mpmath.pi),
df2=lambda t: -mpmath.sin(t) / (2 * mpmath.pi),
solver=solver)
return x
def test_odefun_harmonic():
mp.dps = 15
# Harmonic oscillator
f = odefun(lambda x, y: [-y[1], y[0]], 0, [1, 0])
for x in [0, 1, 2.5, 8, 3.7]: # we go back to 3.7 to check caching
c, s = f(x)
assert c.ae(cos(x))
assert s.ae(sin(x))
def getAntiprismVolume( n, k ):
if real( n ) < 3:
raise ValueError( 'the number of sides of the prism cannot be less than 3,' )
if not isinstance( k, RPNMeasurement ):
return getAntiprismVolume( n, RPNMeasurement( real( k ), 'meter' ) )
if k.getDimensions( ) != { 'length' : 1 }:
raise ValueError( '\'antiprism_volume\' argument 2 must be a length' )
result = getProduct( [ fdiv( fprod( [ n, sqrt( fsub( fmul( 4, cos( cos( fdiv( pi, fmul( n, 2 ) ) ) ) ), 1 ) ),
sin( fdiv( fmul( 3, pi ), fmul( 2, n ) ) ) ] ),
fmul( 12, sin( sin( fdiv( pi, n ) ) ) ) ),
sin( fdiv( fmul( 3, pi ), fmul( 2, n ) ) ),
getPower( k, 3 ) ] )
return result.convert( 'meter^3' )
def test_odefun_harmonic():
mp.dps = 15
# Harmonic oscillator
f = odefun(lambda x, y: [-y[1], y[0]], 0, [1, 0])
for x in [0, 1, 2.5, 8, 3.7]: # we go back to 3.7 to check caching
c, s = f(x)
assert c.ae(cos(x))
assert s.ae(sin(x))
def _root_approx(n, k):
# Tricomi approximation
with mpmath.extradps(5):
n = mpmath.mpf(n)
k = mpmath.mpf(k)
c = 1 + (1 / (8 * n**2)) * (-1 + 1 / n)
a = mpmath.pi * (4 * k - 1) / (4 * n + 2)
return c * mpmath.cos(a)
def invFourier(x,F):
k1 = r2*mpm.pi
f = []
for xi in x:
kx = k1*(xi/Lx)
f.append( mpm.fsum( F[ki]*mpm.cos(kx*mpm.mpf(ki)) \
for ki in xrange(len(F)) ) )
return f
def error(coefs, progress=True):
(a, b) = coefs
xs = (x * mp.pi / mp.mpf(4096) for x in range(-4096, 4097))
err = max(fabs(cos(x) - f(x, a, b)) for x in xs)
if progress:
print('(a, b, c): ({}, {}, {})'.format(a, b, c(a, b)))
print('evaluated error: ', err)
print()
return float(err)
def spin_vector(t):
import numpy
ht = self.__ht_time(t)
H = self.__H_time(t)
G = d_eval['G']
Gt = d_eval['\\tilde{G}']
Hts = d_eval['\\tilde{H}_\\ast']
Gtxys = sqrt(Gt**2-(Hts-H)**2)
return numpy.array([x.real for x in [Gtxys*sin(ht),-Gtxys*cos(ht),Hts-H]])
def rotate_point_around_anchor(self, x, y, anchor_x, anchor_y, angle):
anglefrac = angle.as_integer_ratio()
radian_qty = radians(anglefrac[0] / anglefrac[1])
old_x = x - anchor_x
old_y = y - anchor_y
coord = matrix([float(old_x), float(old_y)])
transform = matrix([[cos(radian_qty), -sin(radian_qty)], [sin(radian_qty), cos(radian_qty)]])
result = transform * coord
new_x = Decimal(str(result[0]))
new_y = Decimal(str(result[1]))
new_x += anchor_x
new_y += anchor_y
return new_x, new_y
def test_odefun_sinc_large():
mp.dps = 15
# Sinc function; test for large x
f = sinc
g = odefun(lambda x, y: [(cos(x) - y[0]) / x],
1, [f(1)],
tol=0.01,
degree=5)
assert abs(f(100) - g(100)[0]) / f(100) < 0.01
def orbit_vector(t): import numpy ht = self.__ht_time(t) hs = self.__hs_time(t) h = hs + ht H = self.__H_time(t) G = d_eval['G'] Gxy = sqrt(G**2-H**2) return numpy.array([x.real for x in [Gxy*sin(h),-Gxy*cos(h),H]])
def polar_cubic(a, q, r):
'''
The polar coordinate solution of the cubic function; for use when
Q^3+R^2 < 0
This version formulated to work with mpmath for arbitrary floating point
precision and numpy arrays.
'''
theta = mpmath.acos(r / numpy.power(mpf('-1') * numpy.power(q, mpf('3')), mpf('0.5')))
return mpmath.cos(theta / mpf('3')) * numpy.power(mpf('-1') * q, mpf('0.5')) * mpf('2') - a / mpf('3')
def sph_yn_exact(n, z):
"""Return the value of y_n computed using the exact formula.
The expression used is http://dlmf.nist.gov/10.49.E4 .
"""
zm = mpmathify(z)
s1 = sum((-1)**k*_a(2*k, n)/zm**(2*k+1) for k in xrange(0, int(n/2) + 1))
s2 = sum((-1)**k*_a(2*k+1, n)/zm**(2*k+2) for k in xrange(0, int((n-1)/2) + 1))
return -cos(zm - n*pi/2)*s1 + sin(zm - n*pi/2)*s2
def Fourier(N,f):
k1 = r2*mpm.pi
x = [-r1/r2+mpm.mpf(i)/mpm.mpf(2*N) for i in xrange(2*N)]
fx = [f(Lx*xi) for xi in x]
F = []
for ki in xrange(N):
kx = k1*mpm.mpf(ki)
F.append( mpm.fsum( fi*mpm.cos(kx*xi) for xi,fi in zip(x,fx) ) )
F[-1] /= mpm.mpf(N)
F[0] /= r2 # scale the first coefficient (mean value)
return F
def obliquity(t):
from mpmath import acos, cos, sqrt
H = self.__H_time(t)
hs = self.__hs_time(t)
G = d_eval['G']
Gt = d_eval['\\tilde{G}']
Hts = d_eval['\\tilde{H}_\\ast']
Gxy = sqrt(G**2-H**2)
Gtxys = sqrt(Gt**2-(Hts-H)**2)
retval = (Gxy*Gtxys*cos(hs)+H*(Hts-H))/(G*Gt)
return acos(retval)
def __init__(self,params):
import sympy
from mpmath import mpf, sqrt, cos
from copy import deepcopy
self.__set_original_H()
# Set up constants.
self.__eps_val = (1./mpf(299792458.))**2
self.__GG_val = mpf(6.673E-11)
# Setup names.
names = ['m2','r2','rot2','r1','rot1','i_a','ht','a','e','i','h']
if not all([s in params for s in names]):
raise ValueError('invalid set of parameters')
# Convert all the values to mpf and introduce convenience shortcuts.
m2, r2, rot2, r1, rot1, i_a, ht_in, a, e, i, h_in = [mpf(params[s]) for s in names]
L_in = sqrt(self.__GG_val * m2 * a)
G_in = L_in * sqrt(1. - e**2)
H_in = G_in * cos(i)
Gt_in = (2 * r1**2 * rot1) / 5
Ht_in = Gt_in * cos(i_a)
Hts_in = H_in + Ht_in
hs_in = h_in - ht_in
Gxy_in = sqrt(G_in**2 - H_in**2)
Gtxys_in = sqrt(Gt_in**2 - Ht_in**2)
J2 = (2 * m2 * r2**2 * rot2) / 5
II_1 = mpf(5) / (2 * r1**2)
# Evaluation dictionary.
eval_names = [sympy.Symbol(s) for s in ['L','G','H','\\tilde{G}','\\tilde{H}_\\ast','h_\\ast','m_2','\\mathcal{G}','J_2',\
'\\varepsilon','\\mathcal{I}_1','\\tilde{h}','g']]
eval_values = [sympy.Float(x) for x in [L_in,G_in,H_in,Gt_in,Hts_in,hs_in,m2,self.__GG_val,J2,self.__eps_val,II_1,ht_in,0]]
d_eval = dict(zip(eval_names,eval_values))
self.__init_d_eval = deepcopy(d_eval)
def diff_f(x,_):
from sympy import Symbol
H = x[0]
hs = x[2]
d_eval[Symbol('H')] = H
d_eval[Symbol('h_\\ast')] = hs
return [d.evalf(subs = d_eval) for d in self.__diffs]
self.__diff_f = diff_f
def ExactIntegrationOfSinusKernel(t, a = None, b = None):
with precision(300):
if a == None and b == None:
return 0.5 * math.pi * math.sqrt(t) * (mpmath.angerj(0.5, t) - mpmath.angerj(-0.5, t))
if a == None and b != None:
a = t
elif b == None:
b = 0.
mpmath.mp.pretty = True
pi = mpmath.mp.pi
pi = +pi
fcos = mpmath.fresnelc
fsin = mpmath.fresnels
arg_a = mpmath.sqrt(2 * (t - a) / mpmath.mp.pi)
arg_b = mpmath.sqrt(2 * (t - b) / mpmath.mp.pi)
return mpmath.sqrt(2 * mpmath.mp.pi) * ((fsin(arg_b) - fsin(arg_a)) * mpmath.cos(t) + (fcos(arg_a) - fcos(arg_b)) * mpmath.sin(t))
def rotate(self, identifier, angles, step=1):
ang_num = 2*math.pi/360
rotateZ = [math.cos(angles[0]*ang_num),-math.sin(angles[0]*ang_num),0,math.sin(angles[0]*ang_num),math.cos(angles[0]*ang_num),0,0,0,1]
rotateY = [math.cos(angles[1]*ang_num),0,-math.sin(angles[1]*ang_num),0,1,0,math.sin(angles[1]*ang_num),0,math.cos(angles[1]*ang_num)]
rotateX = [1,0,0,0,math.cos(angles[2]*ang_num),-math.sin(angles[2]*ang_num),0,math.sin(angles[2]*ang_num),math.cos(angles[2]*ang_num)]
antirotateZ = [math.cos(-angles[0]*ang_num),-math.sin(-angles[0]*ang_num),0,math.sin(-angles[0]*ang_num),math.cos(-angles[0]*ang_num),0,0,0,1]
antirotateY = [math.cos(-angles[1]*ang_num),0,-math.sin(-angles[1]*ang_num),0,1,0,math.sin(-angles[1]*ang_num),0,math.cos(-angles[1]*ang_num)]
antirotateX = [1,0,0,0,math.cos(-angles[2]*ang_num),-math.sin(-angles[2]*ang_num),0,math.sin(-angles[2]*ang_num),math.cos(-angles[2]*ang_num)]
#self.command("transform "+identifier+""+"{ {%s %s %s} {%s %s %s} {%s %s %s} }"%tuple(rotateZ))
self.command("transform "+identifier+"."+str(step+1)+"{ {%s %s %s} {%s %s %s} {%s %s %s} }"%tuple(antirotateZ))
#self.command("transform "+identifier+""+"{ {%s %s %s} {%s %s %s} {%s %s %s} }"%tuple(rotateY))
self.command("transform "+identifier+"."+str(step+1)+"{ {%s %s %s} {%s %s %s} {%s %s %s} }"%tuple(antirotateY))
#self.command("transform "+identifier+""+"{ {%s %s %s} {%s %s %s} {%s %s %s} }"%tuple(rotateX))
self.command("transform "+identifier+"."+str(step+1)+"{ {%s %s %s} {%s %s %s} {%s %s %s} }"%tuple(antirotateX))
for i in range(0,3):
self.orientation[i] += angles[i]
T0 = time.clock()
THk = []
N2 , a2 = N**2 , a**2
k1 = r2*mpm.pi/Lx
for ki in range(Nk):
k = k1*mpm.mpf(ki)
k2 = k**2
PHI = a2*(k2+l**2) + N2
tmp = mpm.sqrt(PHI**2 - r4*a2*k2*N2)
PSIs2 = PHI + tmp
PSIg2 = PHI - tmp
den = r2*tmp
Os = mpm.sqrt(PSIs2/r2)
Og = mpm.sqrt(PSIg2/r2)
theta = THk0[ki] \
* ( (mpm.mpf(2)*N2-PSIg2)/den * mpm.cos(Os*t) \
+(PSIs2-mpm.mpf(2)*N2)/den * mpm.cos(Og*t) )
THk.append(theta)
TH = invFourier((xi-U*t+(Lx/r2-xc) for xi in x),THk)
T1 = time.clock()
print('Computation time: %f'%(T1-T0))
py.figure(3)
py.plot( \
[float(xi+xc) for xi in x] , \
[float(ti) for ti in TH0], \
'k-', label='initial condition')
py.plot( \
[float(xi+Lx/r2) for xi in x] , \
def test_rationalize():
from mpmath import cos, pi, mpf
assert rationalize(cos(pi/3)) == Rational(1, 2)
assert rationalize(mpf("0.333333333333333")) == Rational(1, 3)
assert rationalize(mpf("-0.333333333333333")) == Rational(-1, 3)
assert rationalize(pi, maxcoeff = 250) == Rational(355, 113)
def ftot(t):
return mp.cos(mp.cos(t)) * mp.sin(t) * mp.exp(1j * omega * mp.sin(t))
def test_rationalize():
from mpmath import cos, pi, mpf
assert rationalize(cos(pi/3)) == sympify("1/2")
assert rationalize(mpf("0.333333333333333")) == sympify("1/3")
assert rationalize(mpf("-0.333333333333333")) == sympify("-1/3")
assert rationalize(pi, maxcoeff = 250) == sympify("355/113")
# generate the cordic luts
import mpmath, operator
mpmath.mp.prec = 100
tangents = [mpmath.mpf(0.5) ** i for i in range(0, 61)]
angles = [mpmath.atan(tan) for tan in tangents]
circle_fracs = [angle / (mpmath.pi / 2) for angle in angles]
# To speed up the cordic function, we can ignore fractions that are too
# small
circle_fracs = [cf for cf in circle_fracs if cf >= mpmath.mpf(0.5) ** (frac_bits + 2)]
cordic_lut = [decimal.Decimal(str(c)) for c in circle_fracs]
ps = [mpmath.cos(angle) for angle in angles]
p = decimal.Decimal(str(reduce(operator.mul, ps)))
cordic_p = decimal_to_fix_extrabits(p, internal_frac_bits)
with args["lutfile"] as f:
lutc = "#ifndef LUT_H\n"
lutc += "#define LUT_H\n"
lutc += "\n"
lutc += '#include "base.h"\n'
lutc += '#include "internal.h"\n'
lutc += "\n"
lutc += make_c_internal_define_lut(internal_inv_integer_lut, "INT_INV_LUT", "LUT_int_inv_integer")
lutc += "\n"
lutc += make_c_internal_defines(ln_coef_lut, "FIX_LN_COEF")
lutc += "\n"
lutc += make_c_internal_defines(log2_coef_lut, "FIX_LOG2_COEF")
lutc += "\n"
TEST_YELLOW = pygame.Color(255, 0, 255, 255)#-16711681
TEST_BLUE = pygame.Color(0, 0, 255, 255)#-65536
#Visibility flags
SHOW_ALL = 0
ONLY_SHOW_EXPOSED = 1
#135 degrees
THETA = 3 * mp.pi / 4
#Isometric Unprojection
UNPROJECT = mp.matrix([[1, 0, 0],
[0, -1, 0], #NOTE: the y coordinate has to be inverted because my
[0, 0, 1]]) #grid coordinates are a bit weird
UNPROJECT *= mp.matrix([[ mp.cos(THETA), mp.sin(THETA), 0],
[-mp.sin(THETA), mp.cos(THETA), 0],
[ 0, 0, 1]])
UNPROJECT *= mp.matrix([[1, 0, 0],
[0, 2, 0],
[0, 0, 1]])
class OutOfIt(Exception):
def __init__(self, msg, value):
self.msg = msg
self.value = value
def __repr__(self):
# coding: utf-8
# In[1]:
import mpmath
import random
# In[4]:
scurve_1=lambda x:x
scurve_inf=lambda x:0.5*(1-mpmath.cos(x*mpmath.pi))
class WrapNoise:
def __init__(self, kappa=0.001, frequency=1.0,
scurve=scurve_inf, seed=0):
self._kappa = mpmath.mpf(kappa)
self._period = 1.0/self._kappa
self.frequency = frequency
self._tau = 2*mpmath.pi
self._seed = seed
self._rng = random.Random(self._seed)
self.scurve = scurve
self.initialise_cache()
def scale_t(self,t):
return t*self.frequency
def tau(self):
return self._tau
def initialise_cache(self):
self._angle_cache = {}
self._wrap_cache = {}
def cart_state(self,tau): from mpmath import cos, sin xi,eta,phi = self.state(tau) return xi*eta*cos(phi),xi*eta*sin(phi),(xi**2 - eta**2)/2
def cos(op):
op = convertToRadians(op)
return mpmath.cos(op)
def __set_params(self,d):
from copy import deepcopy
from mpmath import mpf, sqrt, polyroots, cos, acos, pi, mp
from weierstrass_elliptic import weierstrass_elliptic as we
names = ['m2','r2','rot2','r1','rot1','i_a','ht','a','e','i','h']
if not all([s in d for s in names]):
raise ValueError('invalid set of parameters')
# Convert all the values to mpf and introduce convenience shortcuts.
m2, r2, rot2, r1, rot1, i_a, ht_in, a, e, i, h_in = [mpf(d[s]) for s in names]
L_in = sqrt(self.__GG_val * m2 * a)
G_in = L_in * sqrt(1. - e**2)
H_in = G_in * cos(i)
Gt_in = (2 * r1**2 * rot1) / 5
Ht_in = Gt_in * cos(i_a)
Hts_in = H_in + Ht_in
hs_in = h_in - ht_in
Gxy_in = sqrt(G_in**2 - H_in**2)
Gtxys_in = sqrt(Gt_in**2 - Ht_in**2)
J2 = (2 * m2 * r2**2 * rot2) / 5
II_1 = mpf(5) / (2 * r1**2)
# Evaluation dictionary.
eval_names = ['L','G','H','\\tilde{G}','\\tilde{H}_\\ast','h_\\ast','\\tilde{G}_{xy\\ast}','m_2','\\mathcal{G}','J_2','G_{xy}',\
'\\varepsilon','\\mathcal{I}_1']
eval_values = [L_in,G_in,H_in,Gt_in,Hts_in,hs_in,Gtxys_in,m2,self.__GG_val,J2,Gxy_in,self.__eps_val,II_1]
d_eval = dict(zip(eval_names,eval_values))
# Evaluate Hamiltonian with initial conditions.
HHp_val = self.__HHp.trim().evaluate(d_eval)
# Add the value of the Hamiltonian to the eval dictionary.
d_eval['\\mathcal{H}^\\prime'] = HHp_val
# Evaluate g2 and g3.
g2_val, g3_val = self.__g2.trim().evaluate(d_eval), self.__g3.trim().evaluate(d_eval)
# Create the Weierstrass object.
wp = we(g2_val,g3_val)
# Store the period.
self.__wp_period = wp.periods[0]
# Now let's find the roots of the quartic polynomial.
# NOTE: in theory here we could use the exact solution for the quartic.
p4coeffs = [t[0] * t[1].trim().evaluate(d_eval) for t in zip([1,4,6,4,1],self.__f4_cf)]
p4roots, err = polyroots(p4coeffs,error = True, maxsteps = 1000)
# Find a reachable root.
Hr, H_max, n_lobes, lobe_idx = self.__reachable_root(p4roots,H_in)
# Determine t_r
t_r = self.__compute_t_r(n_lobes,lobe_idx,H_in,Hr,d_eval,p4roots,p4coeffs[0])
# Now evaluate the derivatives of the polynomial. We will need to replace H_in with Hr in the eval dict.
d_eval['H'] = Hr
_, f4Hp, f4Hpp, _, _ = self.__f4
f4p_eval = f4Hp.trim().evaluate(d_eval)
f4pp_eval = f4Hpp.trim().evaluate(d_eval)
# Build and store the expression for H(t).
self.__H_time = lambda t: Hr + f4p_eval / (4 * (wp.P(t - t_r) - f4pp_eval / 24))
# H will not be needed any more, replace with H_r
del d_eval['H']
d_eval['H_r'] = Hr
# Inject the invariants and the other two constants into the evaluation dictionary.
d_eval['g_2'] = g2_val
d_eval['g_3'] = g3_val
d_eval['A'] = f4p_eval / 4
d_eval['B'] = f4pp_eval / 24
# Verify the formulae in solutions.py
self.__verify_solutions(d_eval)
# Assuming g = 0 as initial angle.
self.__g_time = spin_gr_theory.__get_g_time(d_eval,t_r,0.)
self.__hs_time = spin_gr_theory.__get_hs_time(d_eval,t_r,hs_in)
self.__ht_time = spin_gr_theory.__get_ht_time(d_eval,t_r,ht_in)
def obliquity(t):
from mpmath import acos, cos, sqrt
H = self.__H_time(t)
hs = self.__hs_time(t)
G = d_eval['G']
Gt = d_eval['\\tilde{G}']
Hts = d_eval['\\tilde{H}_\\ast']
Gxy = sqrt(G**2-H**2)
Gtxys = sqrt(Gt**2-(Hts-H)**2)
retval = (Gxy*Gtxys*cos(hs)+H*(Hts-H))/(G*Gt)
return acos(retval)
self.__obliquity_time = obliquity
def spin_vector(t):
import numpy
ht = self.__ht_time(t)
H = self.__H_time(t)
G = d_eval['G']
Gt = d_eval['\\tilde{G}']
Hts = d_eval['\\tilde{H}_\\ast']
Gtxys = sqrt(Gt**2-(Hts-H)**2)
return numpy.array([x.real for x in [Gtxys*sin(ht),-Gtxys*cos(ht),Hts-H]])
self.__spin_vector_time = spin_vector
def orbit_vector(t):
import numpy
ht = self.__ht_time(t)
hs = self.__hs_time(t)
h = hs + ht
H = self.__H_time(t)
G = d_eval['G']
Gxy = sqrt(G**2-H**2)
return numpy.array([x.real for x in [Gxy*sin(h),-Gxy*cos(h),H]])
self.__orbit_vector_time = orbit_vector
# Store the params of the system.
self.__params = dict(zip(names,[mpf(d[s]) for s in names]))
# Final report.
rad_conv = 360 / (2 * pi())
print("\x1b[31mAccuracy in the identification of the poly roots:\x1b[0m")
print(err)
print("\n\x1b[31mPeriod (years):\x1b[0m")
print(wp.periods[0] / (3600*24*365))
print("\n\x1b[31mMin and max orbital inclination (deg):\x1b[0m")
print(acos(Hr/G_in) * rad_conv,acos(H_max/G_in) * rad_conv)
print("\n\x1b[31mMin and max axial inclination (deg):\x1b[0m")
print(acos((Hts_in - Hr)/Gt_in) * rad_conv,acos((Hts_in-H_max)/Gt_in) * rad_conv)
print("\n\x1b[31mNumber of lobes:\x1b[0m " + str(n_lobes))
print("\n\x1b[31mLobe idx:\x1b[0m " + str(lobe_idx))
# Report the known results for simplified system for comparison.
H = H_in
HHp,G,L,GG,eps,m2,Hts,Gt,J2 = [d_eval[s] for s in ['\\mathcal{H}^\\prime','G','L','\\mathcal{G}',\
'\\varepsilon','m_2','\\tilde{H}_\\ast','\\tilde{G}','J_2']]
print("\n\x1b[31mEinstein (g):\x1b[0m " + str((3 * eps * GG**4 * m2**4/(G**2*L**3))))
print("\n\x1b[31mLense-Thirring (g):\x1b[0m " + str(((eps * ((-6*H*J2*GG**4*m2**3)/(G**4*L**3)+3*GG**4*m2**4/(G**2*L**3))))))
print("\n\x1b[31mLense-Thirring (h):\x1b[0m " + str((2*eps*J2*GG**4*m2**3/(G**3*L**3))))
# These are the Delta_ constants of quasi-periodicity.
f_period = self.wp_period
print("\n\x1b[31mDelta_g:\x1b[0m " + str(self.g_time(f_period) - self.g_time(0)))
print("\n\x1b[31mg_rate:\x1b[0m " + str((self.g_time(f_period) - self.g_time(0))/f_period))
Delta_hs = self.hs_time(f_period) - self.hs_time(0)
print("\n\x1b[31mDelta_hs:\x1b[0m " + str(Delta_hs))
print("\n\x1b[31mhs_rate:\x1b[0m " + str(Delta_hs/f_period))
Delta_ht = self.ht_time(f_period) - self.ht_time(0)
print("\n\x1b[31mDelta_ht:\x1b[0m " + str(Delta_ht))
print("\n\x1b[31mht_rate:\x1b[0m " + str(Delta_ht/f_period))
print("\n\x1b[31mDelta_h:\x1b[0m " + str(Delta_ht+Delta_hs))
print("\n\x1b[31mh_rate:\x1b[0m " + str((Delta_ht+Delta_hs)/f_period))
print("\n\n")
def test_odefun_sinc_large():
mp.dps = 15
# Sinc function; test for large x
f = sinc
g = odefun(lambda x, y: [(cos(x)-y[0])/x], 1, [f(1)], tol=0.01, degree=5)
assert abs(f(100) - g(100)[0])/f(100) < 0.01
def ftot(t):
cos2t = mp.cos(t) ** 2
return mp.exp(-z1 / cos2t) / cos2t * mp.exp(1j * omega * mp.cos(t - beta) / cos2t)
def solveCubicPolynomial( a, b, c, d ):
if mp.dps < 50:
mp.dps = 50
if a == 0:
return solveQuadraticPolynomial( b, c, d )
f = fdiv( fsub( fdiv( fmul( 3, c ), a ), fdiv( power( b, 2 ), power( a, 2 ) ) ), 3 )
g = fdiv( fadd( fsub( fdiv( fmul( 2, power( b, 3 ) ), power( a, 3 ) ),
fdiv( fprod( [ 9, b, c ] ), power( a, 2 ) ) ),
fdiv( fmul( 27, d ), a ) ), 27 )
h = fadd( fdiv( power( g, 2 ), 4 ), fdiv( power( f, 3 ), 27 ) )
# all three roots are the same
if h == 0:
x1 = fneg( root( fdiv( d, a ), 3 ) )
x2 = x1
x3 = x2
# two imaginary and one real root
elif h > 0:
r = fadd( fneg( fdiv( g, 2 ) ), sqrt( h ) )
if r < 0:
s = fneg( root( fneg( r ), 3 ) )
else:
s = root( r, 3 )
t = fsub( fneg( fdiv( g, 2 ) ), sqrt( h ) )
if t < 0:
u = fneg( root( fneg( t ), 3 ) )
else:
u = root( t, 3 )
x1 = fsub( fadd( s, u ), fdiv( b, fmul( 3, a ) ) )
real = fsub( fdiv( fneg( fadd( s, u ) ), 2 ), fdiv( b, fmul( 3, a ) ) )
imaginary = fdiv( fmul( fsub( s, u ), sqrt( 3 ) ), 2 )
x2 = mpc( real, imaginary )
x3 = mpc( real, fneg( imaginary ) )
# all real roots
else:
j = sqrt( fsub( fdiv( power( g, 2 ), 4 ), h ) )
k = acos( fneg( fdiv( g, fmul( 2, j ) ) ) )
if j < 0:
l = fneg( root( fneg( j ), 3 ) )
else:
l = root( j, 3 )
m = cos( fdiv( k, 3 ) )
n = fmul( sqrt( 3 ), sin( fdiv( k, 3 ) ) )
p = fneg( fdiv( b, fmul( 3, a ) ) )
x1 = fsub( fmul( fmul( 2, l ), cos( fdiv( k, 3 ) ) ), fdiv( b, fmul( 3, a ) ) )
x2 = fadd( fmul( fneg( l ), fadd( m, n ) ), p )
x3 = fadd( fmul( fneg( l ), fsub( m, n ) ), p )
return [ chop( x1 ), chop( x2 ), chop( x3 ) ]
def eval(self, z):
return mpmath.cos(z)