def director_field_vectorized(self, k, x, jac=False):
""" Returns the value of the vector field for a nonlinear class at a given point.
args:
k (int) : indexes which nonlinear class
x (numpy.array) : (2,N)
kwargs:
jac: If True then returns the Jacobian of the vector-field too.
returns:
out1: ndarray (2,N)
out2: ndarray (2,2,N) if jac==True
"""
from numpy.polynomial.legendre import legval2d, legder
theta = legval2d(x[0] / self.width, x[1] / self.height,
self.theta_coeffs[k])
out1 = np.array([np.cos(theta), np.sin(theta)])
if jac:
dtheta_dx = legval2d(x[0] / self.width, x[1] / self.height,
legder(self.theta_coeffs[k],
axis=0)) / self.width
dtheta_dy = legval2d(x[0] / self.width, x[1] / self.height,
legder(self.theta_coeffs[k],
axis=1)) / self.height
N = x.shape[1]
out2 = np.zeros((2, 2, N))
out2[0, 0, :] = -out1[1] * dtheta_dx
out2[0, 1, :] = -out1[1] * dtheta_dy
out2[1, 0, :] = out1[0] * dtheta_dx
out2[1, 1, :] = out1[0] * dtheta_dy
return out1, out2
return out1
def director_field(self, k, x, jac=False):
""" Returns the value of the vector field for a nonlinear class.
args:
x: ndarray (2,)
k: int
kwargs:
jac: If True then returns the Jacobian of the vector-field too.
returns:
out1: ndarray (2,)
out2: ndarray (2,2) if jac==True
"""
from numpy.polynomial.legendre import legval2d, legder
theta = legval2d(x[0] / self.width, x[1] / self.height,
self.theta_coeffs[k])
#NOTE: Yes, I know this scaling seems off by a factor of 2. At the moment, this is correct. However, this should be refactored so that we use a scaling convention that is consistent with the rest of the code-base (e.g. posteriors.x_given_k
out1 = np.array([np.cos(theta), np.sin(theta)])
if jac:
dtheta_dx = legval2d(x[0] / self.width, x[1] / self.height,
legder(self.theta_coeffs[k],
axis=0)) / self.width
dtheta_dy = legval2d(x[0] / self.width, x[1] / self.height,
legder(self.theta_coeffs[k],
axis=1)) / self.height
out2 = np.zeros((2, 2))
out2[0, 0] = -out1[1] * dtheta_dx
out2[0, 1] = -out1[1] * dtheta_dy
out2[1, 0] = out1[0] * dtheta_dx
out2[1, 1] = out1[0] * dtheta_dy
return out1, out2
return out1
def __gradient(self, point_grid: ndarray) -> (NUMPY_TYPE, NUMPY_TYPE):
coeffs_of_grad_x = legder(self.__c.mat / self.__scale.mat, axis=0)
coeffs_of_grad_y = legder(self.__c.mat / self.__scale.mat, axis=1)
sqrt_p = legval2d(*point_grid, self.__c.mat/self.__scale.mat)
factor = 2.0 * self.__controller.N
grad_x = factor * sqrt_p * legval2d(*point_grid, coeffs_of_grad_x)
grad_y = factor * sqrt_p * legval2d(*point_grid, coeffs_of_grad_y)
return self.__map.out(grad_x), self.__map.out(grad_y)
def hamiltonian(coefs, func=BASIS_FUNC, V=V, C=C):
'''hamiltonian operator acting on wavefunction'''
if func == 'legendre':
ham = -C * L.legder(L.legder(coefs)) + V * L.Legendre(coefs)
return ham
elif func == 'fourier':
ham = [V] + [(i**2) * C * coefs[i] for i in range(1, len(coefs))]
return ham
def parallels_transitions(turbine,
joints_parallel,
joints_vel_parallel,
joints_acc_parallel,
times_parallel,
step=.1,
number_of_points=8.):
""" Given all parallels solutions, i.e., joints solutions for all waypoints split in parallels (lists),
this method computes cubic polynomials to interpolate the two points: end of a parallel - begin of the next
parallel, in joint space, considering velocities. It will return the parallels and the computed transition
paths between them.
Obs.: This is taking too long. No improvements are being made since plates can be used on the transition part.
Args:
turbine: (@ref Turbine) turbine object
joints_parallel: (float[m][n<SUB>i</SUB>][nDOF]) list of joints for all parallels
times_parallel: (float[m][n<SUB>i</SUB>]) list of deltatimes for all parallels
step: distance (meters) between points
number_of_points: minimal number of points
max_acc: maximum permitted acceleration (percentage)
Returns:
Modifies the current joints_parallel and times_parallel with the computed transitions.
"""
robot = turbine.robot
for i in range(len(joints_parallel) - 1):
joint_0 = joints_parallel[2 * i][-1]
vel_0 = joints_vel_parallel[2 * i][-1]
joint_1 = joints_parallel[2 * i + 1][0]
vel_1 = joints_vel_parallel[2 * i + 1][0]
c = mathtools.legn_path(3, [joint_0, joint_1], [vel_0, vel_1])
joints = []
joints_vel = []
joints_acc = []
times = []
dt = min([step, 1. / number_of_points])
for t in linspace(0, 1., max([1. / step, number_of_points])):
joints.append(legendre.legval(t, c))
joints_vel.append(legendre.legval(t, legendre.legder(c, 1)))
joints_acc.append(legendre.legval(t, legendre.legder(c, 2)))
times.append(dt)
joints = joints[1:-1]
joints_vel = joints_vel[1:-1]
joints_acc = joints_acc[1:-1]
times_parallel[2 * i + 1][0] = times[-1]
times = times[1:-1]
joints_parallel.insert(2 * i + 1, joints)
joints_vel_parallel.insert(2 * i + 1, joints_vel)
joints_acc_parallel.insert(2 * i + 1, joints_acc)
times_parallel.insert(2 * i + 1, times)
return joints_parallel, joints_vel_parallel, joints_acc_parallel, times_parallel
def test_legder_axis(self):
# check that axis keyword works
c2d = np.random.random((3, 4))
tgt = np.vstack([leg.legder(c) for c in c2d.T]).T
res = leg.legder(c2d, axis=0)
assert_almost_equal(res, tgt)
tgt = np.vstack([leg.legder(c) for c in c2d])
res = leg.legder(c2d, axis=1)
assert_almost_equal(res, tgt)
def test_legder_axis(self):
# check that axis keyword works
c2d = np.random.random((3, 4))
tgt = np.vstack([leg.legder(c) for c in c2d.T]).T
res = leg.legder(c2d, axis=0)
assert_almost_equal(res, tgt)
tgt = np.vstack([leg.legder(c) for c in c2d])
res = leg.legder(c2d, axis=1)
assert_almost_equal(res, tgt)
def make_grid(self):
import numpy as np
from numpy.polynomial.legendre import legder, legroots, legval
N = self._N
self.NN = N + 1
L = self.L
factor = L / 2
d1 = np.zeros((N + 1, N + 1))
cp = legder([0] * N + [1])
zg = np.hstack([-1.0, legroots(cp), 1.0])
P_N = legval(zg, [0] * N + [1])
with np.errstate(divide='ignore'):
d1 = P_N[:, None] / (P_N[None, :] * (zg[:, None] - zg[None, :]))
d1[np.diag_indices(N + 1)] = 0.0
d1[0, 0] = -N * (N + 1) / 4
d1[N, N] = +N * (N + 1) / 4
d2 = np.dot(d1, d1)
self.zg = (zg + 1) * L / 2 + self.zmin
self.d0 = np.eye(self.NN)
self.d1 = d1 / factor
self.d2 = d2 / factor**2
# Call other objects that depend on the grid
for callback in self._observers:
callback()
def Stormer_Verlet(x0, y0, x1, y1, n_steps, theta, V_scale, Delta_t=1.0):
from numpy.polynomial.legendre import legder,legval2d
theta_x = legder( theta, axis=0, m=1)
theta_y = legder( theta, axis=1, m=1)
x_pred = np.zeros(n_steps)
y_pred = np.zeros(n_steps)
x_pred[0],x_pred[1] = (x0,x1)
y_pred[0],y_pred[1] = (y0,y1)
for k in range(n_steps-2):
x1,y1 = (x_pred[k+1],y_pred[k+1])
x0,y0 = (x_pred[k],y_pred[k])
V_x = legval2d( x1/V_scale[0], y1/V_scale[1], theta_x )/V_scale[0]
V_y = legval2d( x1/V_scale[0], y1/V_scale[1], theta_y )/V_scale[1]
x_pred[k+2] = 2*x1 - x0 - Delta_t**2 * V_x
y_pred[k+2] = 2*y1 - y0 - Delta_t**2 * V_y
return x_pred, y_pred
def Stormer_Verlet(x0, y0, x1, y1, n_steps, theta, V_scale, Delta_t=1.0):
from numpy.polynomial.legendre import legder, legval2d
theta_x = legder(theta, axis=0, m=1)
theta_y = legder(theta, axis=1, m=1)
x_pred = np.zeros(n_steps)
y_pred = np.zeros(n_steps)
x_pred[0], x_pred[1] = (x0, x1)
y_pred[0], y_pred[1] = (y0, y1)
for k in range(n_steps - 2):
x1, y1 = (x_pred[k + 1], y_pred[k + 1])
x0, y0 = (x_pred[k], y_pred[k])
V_x = legval2d(x1 / V_scale[0], y1 / V_scale[1], theta_x) / V_scale[0]
V_y = legval2d(x1 / V_scale[0], y1 / V_scale[1], theta_y) / V_scale[1]
x_pred[k + 2] = 2 * x1 - x0 - Delta_t**2 * V_x
y_pred[k + 2] = 2 * y1 - y0 - Delta_t**2 * V_y
return x_pred, y_pred
def _getNodes(self):
"""
Computes Gauss-Lobatto integration nodes.
Calculates the Gauss-Lobatto integration nodes via a root calculation of derivatives of the legendre
polynomials. Note that the precision of float 64 is not guarantied.
Copyright by Dieter Moser, 2014
Returns:
np.ndarray: array of Gauss-Lobatto nodes
"""
M = self.num_nodes
a = self.tleft
b = self.tright
roots = leg.legroots(
leg.legder(np.array([0] * (M - 1) + [1], dtype=np.float64)))
nodes = np.array(np.append([-1.0], np.append(roots, [1.0])),
dtype=np.float64)
nodes = (a * (1 - nodes) + b * (1 + nodes)) / 2
return nodes
def quad_custom(fun, n=10):
P_n = [0 for i in range(1, n)] + [1]
P_n_roots = leg.legroots(P_n)
d_P_n = leg.legder(P_n)
d_P_n_vals = leg.legval(P_n_roots, d_P_n)
weights = 2 / ((1 - np.power(P_n_roots, 2)) * np.power(d_P_n_vals, 2))
result = np.sum(weights * fun(P_n_roots))
return result
def gaussian_latitudes(n):
"""Construct latitudes and latitude bounds for a Gaussian grid.
Args:
* n:
The Gaussian grid number (half the number of latitudes in the
grid.
Returns:
A 2-tuple where the first element is a length `n` array of
latitudes (in degrees) and the second element is an `(n, 2)`
array of bounds.
"""
if abs(int(n)) != n:
raise ValueError('n must be a non-negative integer')
nlat = 2 * n
# Create the coefficients of the Legendre polynomial and construct the
# companion matrix:
cs = np.array([0] * nlat + [1], dtype=np.int)
cm = legcompanion(cs)
# Compute the eigenvalues of the companion matrix (the roots of the
# Legendre polynomial) taking advantage of the fact that the matrix is
# symmetric:
roots = la.eigvalsh(cm)
roots.sort()
# Improve the roots by one application of Newton's method, using the
# solved root as the initial guess:
fx = legval(roots, cs)
fpx = legval(roots, legder(cs))
roots -= fx / fpx
# The roots should exhibit symmetry, but with a sign change, so make sure
# this is the case:
roots = (roots - roots[::-1]) / 2.
# Compute the Gaussian weights for each interval:
fm = legval(roots, cs[1:])
fm /= np.abs(fm).max()
fpx /= np.abs(fpx).max()
weights = 1. / (fm * fpx)
# Weights should be symmetric and sum to two (unit weighting over the
# interval [-1, 1]):
weights = (weights + weights[::-1]) / 2.
weights *= 2. / weights.sum()
# Calculate the bounds from the weights, still on the interval [-1, 1]:
bounds1d = np.empty([nlat + 1])
bounds1d[0] = -1
bounds1d[1:-1] = -1 + weights[:-1].cumsum()
bounds1d[-1] = 1
# Convert the bounds to degrees of latitude on [-90, 90]:
bounds1d = np.rad2deg(np.arcsin(bounds1d))
bounds2d = np.empty([nlat, 2])
bounds2d[:, 0] = bounds1d[:-1]
bounds2d[:, 1] = bounds1d[1:]
# Convert the roots from the interval [-1, 1] to latitude values on the
# interval [-90, 90] degrees:
latitudes = np.rad2deg(np.arcsin(roots))
return latitudes, bounds2d
def gaussian_latitudes(n):
"""Construct latitudes and latitude bounds for a Gaussian grid.
Args:
* n:
The Gaussian grid number (half the number of latitudes in the
grid.
Returns:
A 2-tuple where the first element is a length `n` array of
latitudes (in degrees) and the second element is an `(n, 2)`
array of bounds.
"""
if abs(int(n)) != n:
raise ValueError('n must be a non-negative integer')
nlat = 2 * n
# Create the coefficients of the Legendre polynomial and construct the
# companion matrix:
cs = np.array([0] * nlat + [1], dtype=np.int)
cm = legcompanion(cs)
# Compute the eigenvalues of the companion matrix (the roots of the
# Legendre polynomial) taking advantage of the fact that the matrix is
# symmetric:
roots = la.eigvalsh(cm)
roots.sort()
# Improve the roots by one application of Newton's method, using the
# solved root as the initial guess:
fx = legval(roots, cs)
fpx = legval(roots, legder(cs))
roots -= fx / fpx
# The roots should exhibit symmetry, but with a sign change, so make sure
# this is the case:
roots = (roots - roots[::-1]) / 2.
# Compute the Gaussian weights for each interval:
fm = legval(roots, cs[1:])
fm /= np.abs(fm).max()
fpx /= np.abs(fpx).max()
weights = 1. / (fm * fpx)
# Weights should be symmetric and sum to two (unit weighting over the
# interval [-1, 1]):
weights = (weights + weights[::-1]) / 2.
weights *= 2. / weights.sum()
# Calculate the bounds from the weights, still on the interval [-1, 1]:
bounds1d = np.empty([nlat + 1])
bounds1d[0] = -1
bounds1d[1:-1] = -1 + weights[:-1].cumsum()
bounds1d[-1] = 1
# Convert the bounds to degrees of latitude on [-90, 90]:
bounds1d = np.rad2deg(np.arcsin(bounds1d))
bounds2d = np.empty([nlat, 2])
bounds2d[:, 0] = bounds1d[:-1]
bounds2d[:, 1] = bounds1d[1:]
# Convert the roots from the interval [-1, 1] to latitude values on the
# interval [-90, 90] degrees:
latitudes = np.rad2deg(np.arcsin(roots))
return latitudes, bounds2d
def apply_H_legendre(self):
'''This applies the Hamiltonian operator, utilizing a builtin capability of the numpy.polynomial.legendre module to get the second derivatives. Note that we have to "pad" the coefficients array with two zeros after taking the second derivative.'''
#taking del^2 has never been easier!
new_coefficients = L.legder(self.coefficients, 2)
new_coefficients = list(new_coefficients)
for i in range(2):
new_coefficients.append(0)
new_coefficients = np.array(new_coefficients) #what a pain!
return (np.array(new_coefficients * (-self.c) +
self.v * self.coefficients * self.period))
def evaluate_basis_derivative(self, x=None, i=0, k=0, output_array=None):
if x is None:
x = self.mesh(False, False)
x = np.atleast_1d(x)
v = self.evaluate_basis(x, i, output_array)
if k > 0:
D = np.zeros((self.N, self.N))
D[:-k, :] = leg.legder(np.eye(self.N), k)
v = np.dot(v, D)
return v
def test_legcubic_path(self):
def asin(x):
A = 2
w = pi / 4
return A * sin(w * x)
def dasin(x):
A = 2
w = pi / 4
return A * w * cos(w * x)
def e(x):
return exp(x) - 1
def de(x):
return exp(x)
p1 = array([asin(0), e(0)])
p2 = array([asin(1), e(1)])
dp1 = array([dasin(0), de(0)])
dp2 = array([dasin(1), de(1)])
c = mathtools.legcubic_path(p1, p2, dp1, dp2)
legsol1 = legendre.legval(0, c)
for i in range(len(p1)):
self.assertAlmostEqual(legsol1[i], p1[i])
legsol2 = legendre.legval(1, c)
for i in range(len(p2)):
self.assertAlmostEqual(legsol2[i], p2[i])
legdsol1 = legendre.legval(0, legendre.legder(c))
for i in range(len(dp1)):
self.assertAlmostEqual(legdsol1[i], dp1[i])
legdsol2 = legendre.legval(1, legendre.legder(c))
for i in range(len(dp2)):
self.assertAlmostEqual(legdsol2[i], dp2[i])
self.assertEquals(legsol1.shape, p1.shape)
self.assertEquals(legsol2.shape, p2.shape)
self.assertEquals(legdsol1.shape, dp1.shape)
self.assertEquals(legdsol2.shape, dp2.shape)
def evaluate_basis_derivative_all(self, x=None, k=0):
if x is None:
x = self.mesh(False, False)
V = self.vandermonde(x)
#assert self.N == V.shape[1]
M = V.shape[1]
if k > 0:
D = np.zeros((M, M))
D[:-k, :] = leg.legder(np.eye(M), k)
V = np.dot(V, D)
return self._composite_basis(V)
def _compute_nodes(self):
"""Computes Gauss-Lobatto integration nodes.
Calculates the Gauss-Lobatto integration nodes via a root calculation of derivatives of the legendre
polynomials.
Note that the precision of float 64 is not guarantied.
"""
roots = leg.legroots(leg.legder(np.array([0] * (self.num_nodes - 1) +
[1], dtype=np.float64)))
self._nodes = np.array(np.append([-1.0], np.append(roots, [1.0])),
dtype=np.float64)
def legReconstructDerivative(freqs, locations, spectrum, numPts):
"""
:param freqs:
:param locations:
:param spectrum:
:param numPts:
:return: derivative of iFFT of the spectrum as an array with shape (Npts, Ncomponent) or (Ncomponent,) if 1-d
"""
deriv=leg.legder(spectrum, axis=0)
return genericLegVal(locations, deriv)
def evaluate_basis_derivative_all(self, x=None, k=0, argument=0):
if x is None:
x = self.mesh(False, False)
V = self.vandermonde(x)
#assert self.N == V.shape[1]
N, M = self.shape(False), self.shape(True)
if k > 0:
D = np.zeros((M, N))
D[:-k] = leg.legder(np.eye(M, N), k)
V = np.dot(V, D)
return self._composite_basis(V, argument=argument)
def director_field_vectorized(self, k, x, jac=False):
""" Returns the value of the vector field for a nonlinear class at a given point.
args:
k (int) : indexes which nonlinear class
x (numpy.array) : (2,N)
kwargs:
jac: If True then returns the Jacobian of the vector-field too.
returns:
out1: ndarray (2,N)
out2: ndarray (2,2,N) if jac==True
"""
from numpy.polynomial.legendre import legval2d, legder
theta = legval2d(
x[0]/self.width,
x[1]/self.height,
self.theta_coeffs[k]
)
out1 = np.array([np.cos(theta), np.sin(theta)])
if jac:
dtheta_dx = legval2d(
x[0]/self.width,
x[1]/self.height,
legder( self.theta_coeffs[k], axis=0)
) / self.width
dtheta_dy = legval2d(
x[0]/self.width,
x[1]/self.height,
legder( self.theta_coeffs[k], axis=1)
) / self.height
N = x.shape[1]
out2 = np.zeros((2,2,N))
out2[0,0,:] = -out1[1]*dtheta_dx
out2[0,1,:] = -out1[1]*dtheta_dy
out2[1,0,:] = out1[0]*dtheta_dx
out2[1,1,:] = out1[0]*dtheta_dy
return out1, out2
return out1
def GaussQuad(f, xi, xf, n, M):
I = 0
c = np.append(np.zeros(M), 1)
x = lg.legroots(c)
w = 0 * x
d = lg.legder(c)
for i in range(len(x)):
w[i] = 2. / ((1 - x[i]**2) * (lg.legval(x[i], d))**2)
for i in range(M):
I += w[i] * f(n, interval(x[i], xf, xi))
I *= 0.5 * (xf - xi)
return I
def director_field(self, k, x, jac=False ):
""" Returns the value of the vector field for a nonlinear class.
args:
x: ndarray (2,)
k: int
kwargs:
jac: If True then returns the Jacobian of the vector-field too.
returns:
out1: ndarray (2,)
out2: ndarray (2,2) if jac==True
"""
from numpy.polynomial.legendre import legval2d, legder
theta = legval2d(
x[0]/self.width,
x[1]/self.height,
self.theta_coeffs[k]
)
out1 = np.array([np.cos(theta), np.sin(theta)])
if jac:
dtheta_dx = legval2d(
x[0]/self.width,
x[1]/self.height,
legder( self.theta_coeffs[k], axis=0)
) / self.width
dtheta_dy = legval2d(
x[0]/self.width,
x[1]/self.height,
legder( self.theta_coeffs[k], axis=1)
) / self.height
out2 = np.zeros( (2,2) )
out2[0,0] = - out1[1]*dtheta_dx
out2[0,1] = - out1[1]*dtheta_dy
out2[1,0] = out1[0]*dtheta_dx
out2[1,1] = out1[0]*dtheta_dy
return out1, out2
return out1
def weights_roots(order, rules='Gauss-Lobatto'):
c_leg = [1] if order == 0 else [i//order for i in xrange(order+1)]
c_dleg = lgd.legder(c_leg)
if rules == 'Gauss-Lobatto':
xs = np.array( [-1] + list( lgd.legroots(c_dleg) ) + [1] )
ws = 2 / ( order * (order + 1) * (lgd.legval(xs, c_leg)**2) )
elif rules == 'Gauss-Legendre':
xs = lgd.legroots(c_leg)
ws = 2 / ( (1 - xs**2) * (lgd.legval(xs, c_dleg)**2) )
return xs, ws
def Hamiltonian_Legendre_polynomial(c, potential, domain, N):
#potential is a constant in this case
x = np.linspace(-domain / 2, domain / 2, N)
delta_x = domain / (N - 1)
#here, the normalized legendre polynomical has been used
# for the nth polynomials, normalization constant is sqrt(2/(2n + 1))
#kinetic term
K = np.zeros((N, N))
for ii in range(N):
legen_left = np.zeros(N)
legen_left[ii] = mt.sqrt((2 * ii + 1) / 2)
for jj in range(N):
deriva_array = np.zeros(N + 2)
deriva_array[jj] = mt.sqrt((2 * jj + 1) / 2)
legen_right_deriva = legen.legder(deriva_array, 2)
#multiply them
legen_multiply = legen.legmul(legen_left, legen_right_deriva)
#integral
legen_integral = legen.legint(legen_multiply)
#calculate the matrix elements
K[ii][jj] = legen.legval(domain / 2, legen_integral) - \
legen.legval(-domain / 2, legen_integral)
#the S matrix, inside the [-1, 1] domain, the legendre ploynomial can be treatedas basis and satisfying <xi|xj> = delta ij, thus S matrix is a identity matrix
S = np.zeros((N, N))
for ii in range(N):
legen_left_S = np.zeros(N)
legen_left_S[ii] = mt.sqrt((2 * ii + 1) / 2)
legen_multiply_S = legen.legmul(legen_left_S, legen_left_S)
legen_integral_S = legen.legint(legen_multiply_S)
S[ii][ii] = legen.legval(domain / 2, legen_integral_S) - \
legen.legval(-domain / 2, legen_integral_S)
K = K * -1 * c
#because the potential is just a constant here, we can calculate the V matrix simply by multiply the matrix S a constant potential value
V = potential * S
##divide the obtained Hamiltonian by the S matrix
H = K + V
return H
def jac_ode_function( xy, t, alpha, width, height, speed=1.0 ):
""" returns velocity feild for input into odeint
args:
xy (np.ndarray) : shape = (2,)
t (float):
alpha: coefficients for angles
returns:
out (np.ndarray) : jacobian of velocity field, shape = (2,2)
"""
x = xy[0]
y = xy[1]
theta = legval2d( x / width, y / height, alpha )
theta_x = legval2d( x / width, y / height, legder( alpha, axis=0) ) / width
theta_y = legval2d( x / width, y / height, legder( alpha, axis=1) ) / height
out = np.zeros( (2,2) )
out[0,0] = - np.sin(theta)*theta_x
out[0,1] = - np.sin(theta)*theta_y
out[1,0] = np.cos(theta)*theta_x
out[1,1] = np.cos(theta)*theta_y
out *= speed
return out
def jac_ode_function(xy, t, alpha, width, height, speed=1.0):
""" returns velocity feild for input into odeint
args:
xy (np.ndarray) : shape = (2,)
t (float):
alpha: coefficients for angles
returns:
out (np.ndarray) : jacobian of velocity field, shape = (2,2)
"""
x = xy[0]
y = xy[1]
theta = legval2d(x / width, y / height, alpha)
theta_x = legval2d(x / width, y / height, legder(alpha, axis=0)) / width
theta_y = legval2d(x / width, y / height, legder(alpha, axis=1)) / height
out = np.zeros((2, 2))
out[0, 0] = -np.sin(theta) * theta_x
out[0, 1] = -np.sin(theta) * theta_y
out[1, 0] = np.cos(theta) * theta_x
out[1, 1] = np.cos(theta) * theta_y
out *= speed
return out
def test_legder(self) :
# check exceptions
assert_raises(ValueError, leg.legder, [0], .5)
assert_raises(ValueError, leg.legder, [0], -1)
# check that zeroth deriviative does nothing
for i in range(5) :
tgt = [0]*i + [1]
res = leg.legder(tgt, m=0)
assert_equal(trim(res), trim(tgt))
# check that derivation is the inverse of integration
for i in range(5) :
for j in range(2, 5) :
tgt = [0]*i + [1]
res = leg.legder(leg.legint(tgt, m=j), m=j)
assert_almost_equal(trim(res), trim(tgt))
# check derivation with scaling
for i in range(5) :
for j in range(2, 5) :
tgt = [0]*i + [1]
res = leg.legder(leg.legint(tgt, m=j, scl=2), m=j, scl=.5)
assert_almost_equal(trim(res), trim(tgt))
def test_legder(self) :
# check exceptions
assert_raises(ValueError, leg.legder, [0], .5)
assert_raises(ValueError, leg.legder, [0], -1)
# check that zeroth deriviative does nothing
for i in range(5) :
tgt = [0]*i + [1]
res = leg.legder(tgt, m=0)
assert_equal(trim(res), trim(tgt))
# check that derivation is the inverse of integration
for i in range(5) :
for j in range(2,5) :
tgt = [0]*i + [1]
res = leg.legder(leg.legint(tgt, m=j), m=j)
assert_almost_equal(trim(res), trim(tgt))
# check derivation with scaling
for i in range(5) :
for j in range(2,5) :
tgt = [0]*i + [1]
res = leg.legder(leg.legint(tgt, m=j, scl=2), m=j, scl=.5)
assert_almost_equal(trim(res), trim(tgt))
def gauss_lobatto_points(start, stop, num_points):
r"""Get the node points for Gauss-Lobatto quadrature.
Using :math:`n` points, this quadrature is accurate to degree
:math:`2n - 3`. The node points are :math:`x_1 = -1`,
:math:`x_n = 1` and the interior are :math:`n - 2` roots of
:math:`P'_{n - 1}(x)`.
Though we don't compute them here, the weights are
:math:`w_1 = w_n = \frac{2}{n(n - 1)}` and for the interior points
.. math::
w_j = \frac{2}{n(n - 1) \left[P_{n - 1}\left(x_j\right)\right]^2}
This is in contrast to the scheme used in Gaussian quadrature, which
use roots of :math:`P_n(x)` as nodes and use the weights
.. math::
w_j = \frac{2}{\left(1 - x_j\right)^2
\left[P'_n\left(x_j\right)\right]^2}
.. note::
This method is **not** generic enough to accommodate non-NumPy
types as it relies on the :mod:`numpy.polynomial.legendre`.
:type start: float
:param start: The beginning of the interval.
:type stop: float
:param stop: The end of the interval.
:type num_points: int
:param num_points: The number of points to use.
:rtype: :class:`numpy.ndarray`
:returns: 1D array, the interior quadrature nodes.
"""
p_n_minus1 = [0] * (num_points - 1) + [1]
inner_nodes = legendre.legroots(legendre.legder(p_n_minus1))
# Utilize symmetry about 0.
inner_nodes = 0.5 * (inner_nodes - inner_nodes[::-1])
if start != -1.0 or stop != 1.0:
# [-1, 1] --> [0, 2] --> [0, stop - start] --> [start, stop]
inner_nodes = start + (inner_nodes + 1.0) * 0.5 * (stop - start)
return np.hstack([[start], inner_nodes, [stop]])
def Legendre_polynomial_basis(c, potential, domain, N, wave_func):
x = np.linspace(-domain / 2, domain / 2, N)
#represent out wave function in the legendre polynomial basis
wave_legen = legen.legfit(x, wave_func, N)
#calculate H |bj>, where H = -c Lap + V
#calculate -c Lap |bj>
Hbj_first = -1 * c * legen.legder(wave_legen, 2)
#calculate V|bj>, here, V is a constant
Hbj_secod = potential * wave_legen
Hbj = Hbj_first + Hbj_secod[0:N - 1]
return Hbj
def get_vandermonde_basis_derivative(self, V, k=0):
"""Return k'th derivatives of basis as a Vandermonde matrix
Parameters
----------
V : array of ndim = 2
Chebyshev Vandermonde matrix
k : int
k'th derivative
"""
assert self.N == V.shape[1]
if k > 0:
D = np.zeros((self.N, self.N))
D[:-k, :] = leg.legder(np.eye(self.N), k)
V = np.dot(V, D)
return self.get_vandermonde_basis(V)
def add_plot(X, lbl, clr, type='o'):
Peq = pos_equilibrium(X)
n = Peq.size
for i in range(n):
plt.plot(Peq[i, 0], 0, type, color=clr)
c = [0] * (n + 2)
c[n + 1] = 1
d = L.legder(c)
P = L.leg2poly(d)
P = mirror(P)
Poly = np.poly1d(P)
x = np.linspace(-1, 1, 100)
y = Poly(x)
plt.plot(x, y, label=lbl, color=clr)
def add_plot(X, lbl, clr, type='o'):
Peq = Newton_Raphson(grad_E, Jacobian_E, X, 100, 1e-8)
n = Peq.size
for i in range(n):
plt.plot(Peq[i, 0], 0, type, color=clr)
c = [0] * (n + 2)
c[n + 1] = 1
d = L.legder(c)
P = L.leg2poly(d)
P = mirror(P)
Poly = np.poly1d(P)
x = np.linspace(-1, 1, 100)
y = Poly(x)
plt.plot(x, y, label=lbl, color=clr)
def _getNodes(self):
"""
Copyright by Dieter Moser, 2014
Computes Gauss-Lobatto integration nodes.
Calculates the Gauss-Lobatto integration nodes via a root calculation of derivatives of the legendre
polynomials. Note that the precision of float 64 is not guarantied.
"""
M = self.num_nodes
a = self.tleft
b = self.tright
roots = leg.legroots(leg.legder(np.array([0] * (M - 1) + [1], dtype=np.float64)))
nodes = np.array(np.append([-1.0], np.append(roots, [1.0])), dtype=np.float64)
nodes = (a * (1 - nodes) + b * (1 + nodes)) / 2
return nodes
def __accel__(t, vi, deg=[0]):
"""Equation for acceleration due to gravity.
:t: current time
:vi: current position and velocity vector
:n: degrees of perturbation terms to use ([0] = 2-body)
"""
from numpy.polynomial.legendre import legval, legder
# initialize solution vector
v = np.zeros(len(vi))
# dr/dt = v
v[:3] = vi[3:]
# magnitude of position vector
r = norm(vi[:3])
# sin(phi) = z/r
sinphi = vi[2] / r
xi = re / r
# Common coefficients for Legendre polynomials and derivatives
mult = [J[n] * xi**n if n in deg else 0 for n in xrange(max(deg) + 1)]
# compute coefficients for Legendre polynomial evaluation
cPn = (np.arange(max(deg) + 1) + 1) * mult
# Legendre summation
Pn = legval(sinphi, cPn)
# Legendre derivative summation
Pn_prime = legval(sinphi, legder(mult))
# common acceleration
v[3:] = (mu / r**3) * (Pn + sinphi * Pn_prime) * vi[:3]
# additive term for z-acceleration
v[-1] -= (mu / r**3) * (r * Pn_prime)
return v
def calculate_roots(deg):
rez = []
for i in range(1, deg // 2 + deg % 2 + 1):
x_0 = math.cos(math.pi * (4 * i - 1) / (4 * deg + 2))
x_k = x_0
coefs = np.zeros(i)
coefs[i - 1] = 1
der_coefs = legndr.legder(coefs)
x_k_1 = x_k - legndr.legval([x_k], coefs)[0] / legndr.legval(
[x_k], der_coefs)[0]
e = 2 / (deg * 10)
while x_k_1 - x_k > e:
temp = x_k_1
x_k_1 = x_k - legndr.legval([x_k], coefs)[0] / legndr.legval(
[x_k], der_coefs)[0]
x_k = temp
rez.append(x_k)
return rez
def __init__(self, n, k, a, m):
self.k2 = 2 * k
# Uniform grid points
x = np.linspace(-1, 1, n)
# Legendre Polynomials on grid
self.V = legvander(x, m - 1)
# Do QR factorization of Vandermonde for least squares
self.Q, self.R = qr(self.V, mode='economic')
I = np.eye(m)
D = np.zeros((m, m))
D[:-self.k2, :] = legder(I, self.k2)
# Legendre modal approximation of differential operator
self.A = I - a * D
# Store LU factors for repeated solves
self.PLU = lu_factor(self.A[:-self.k2, self.k2:])
def weights(order):
x = roots(order)
w = np.empty(order)
coef = []
for n in range(0, order+1):
if(n == order):
coef.append(1)
else:
coef.append(0)
derivative_coefficients = leg.legder(coef)
del coef[-1]
del coef[-1]
coef.append(1)
for i in range(0, order):
tmp = leg.Legendre(derivative_coefficients, domain=[-1, 1])(2*x[i] - 1)
tmp2 = leg.Legendre(coef, domain=[-1, 1])(2.0*x[i] - 1)
w[i] = 1/(order*tmp*tmp2)
# Possibly some bug here, get a factor 2 different using mathematica
return w
def get_basis(deg, dofset=None):
# Legendre polynomials are used as the basis of polynomials. In the basis of
# Legendre polynomials is row of eye
polyset = np.eye(deg+1)
if dofset is None: dofset = np.linspace(-1, 1, deg+1)
# Reatange dofs to have external first
dofset = np.r_[dofset[0], dofset[-1], dofset[1:-1]]
# Compute the nodal matrix
A = np.array([legval(dofset, base) for base in polyset])
# New coefficients
B = np.linalg.inv(A)
# Combine the basis according to new weights
basis = [lambda x, c=c: legval(x, c) for c in B]
# Check that the basis is nodal
assert np.allclose(np.array([f(dofset) for f in basis]), polyset)
# First deriv
dbasis = [lambda x, c=legder(c): legval(x, c) for c in B]
return basis, dbasis
def __init__(self,n,k,a,m):
self.k2 = 2*k
# Uniform grid points
x = np.linspace(-1,1,n)
# Legendre Polynomials on grid
self.V = legvander(x,m-1)
# Do QR factorization of Vandermonde for least squares
self.Q,self.R = qr(self.V,mode='economic')
I = np.eye(m)
D = np.zeros((m,m))
D[:-self.k2,:] = legder(I,self.k2)
# Legendre modal approximation of differential operator
self.A = I-a*D
# Store LU factors for repeated solves
self.PLU = lu_factor(self.A[:-self.k2,self.k2:])
def legder(cs, m=1, scl=1) :
from numpy.polynomial.legendre import legder
return legder(cs, m, scl)
fl = lambdify(x, f, "numpy") ul = lambdify(x, u, "numpy") n = 32 domain = sys.argv[-1] if len(sys.argv) == 2 else "C1" # Chebyshev-Gauss nodes and weights points, w = leg.leggauss(n+1) # Chebyshev Vandermonde matrix V = leg.legvander(points, n) scl=1.0 # First derivative matrix zero padded to (n+1)x(n+1) D1 = np.zeros((n+1,n+1)) D1[:-1,:] = leg.legder(np.eye(n+1), 1, scl=scl) Vx = np.dot(V, D1) # Matrix of trial functions P = np.zeros((n+1,n+1)) P[:,0] = (V[:,0] - V[:,1])/2 P[:,1:-1] = V[:,:-2] - V[:,2:] P[:,-1] = (V[:,0] + V[:,1])/2 # Matrix of first derivatives of trial functions Px = np.zeros((n+1,n+1)) Px[:,0] = (Vx[:,0] - Vx[:,1])/2 Px[:,1:-1] = Vx[:,:-2] - Vx[:,2:] Px[:,-1] = (Vx[:,0] + Vx[:,1])/2
from scipy.interpolate import interp1d
Nin = 10 # Number of data points in
Nout = 50 # Number of points we want to project onto
Nfine = Nin**2 # Number of points for plotting pretty
Xin = np.linspace(-1,1,Nin) # The space of points
Fin = np.cos(20*Xin)+np.sin(7*Xin) # Evaluate some fake data
Xfine = np.linspace(-1,1,Nfine) # Linspace for plotting pretty
Fspline = interp1d(Xin,Fin,kind='cubic') # A spline interpolation of the data
Xout = np.linspace(-1,1,Nout-1) # Probably don't need this since we want to plot at the GLL nodes
I = np.eye(Nout) # A hack for choosing the right polynomial orders
zero = np.zeros(Nout-1)
# Build our set of GLL nodes
pts = legendre.legroots(legendre.legder(I[-1])) # GLL points
Xi = [-1, list(pts), 1]
Xi = np.hstack(Xi)
#print "GLL Points:", Xi
# Build the weight set
weights = np.zeros([len(Xi)])
for i in range(len(Xi)-2):
weights[i+1] = 2/((Nout)*(Nout-1)*(legendre.legval(Xi[i+1],I[-1]))**2)
weights[0]=2.0/((Nout)*(Nout-1))
weights[-1]=weights[0]
#print "GLL Weights:", weights
# Calculate the norms
norms = np.zeros(Nout)