def test_t31(algorithm, const):
def odefun(y, _, k):
return np.sin(y[1]), y[2], -y[3] / k[0], \
((y[0]-1) * np.cos(y[1]) - y[2] / np.cos(y[1]) - k[0] * y[3] * np.tan(y[1])) / k[0]
def odejac(y, _, k):
df_dy = np.array(
[[0, np.cos(y[1]), 0, 0], [0, 0, 1, 0], [0, 0, 0, -1 / k[0]],
[np.cos(y[1]) / k[0], -(np.sin(y[1]) * (y[0] - 1) + k[0] * y[3] * (np.tan(y[1]) ** 2 + 1)
+ (y[2] * np.sin(y[1])) / np.cos(y[1]) ** 2) / k[0], -1 / (k[0] * np.cos(y[1])),
-np.tan(y[1])]])
df_dp = np.empty((4, 0))
return df_dy, df_dp
def bcfun(y0, yf, _, __, ___):
return y0[0], y0[2], yf[0], yf[2]
algo = Shooting(odefun, None, bcfun, algorithm=algorithm, num_arcs=12)
algo.set_derivative_jacobian(odejac)
sol = Trajectory()
sol.t = np.linspace(0, 1, 2)
sol.y = np.array([[0, 0, 0, 0], [0, 0, 0, 0]])
sol.k = np.array([const])
# noinspection PyTypeChecker
cc = np.linspace(const * 10, const, 10)
for c in cc:
sol = copy.deepcopy(sol)
sol.k = np.array([c])
sol = algo.solve(sol)['traj']
assert sol.converged
def test_t33(algorithm, const):
def odefun(y, _, k):
return y[1], (y[0]*y[3] - y[2]*y[1]) / k[0], y[3], y[4], y[5], (-y[2] * y[5] - y[0] * y[1]) / k[0]
def odejac(y, _, k):
df_dy = np.array(
[[0, 1, 0, 0, 0, 0], [y[3] / k[0], -y[2] / k[0], -y[1] / k[0], y[0] / k[0], 0, 0], [0, 0, 0, 1, 0, 0],
[0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1], [-y[1] / k[0], -y[0] / k[0], -y[5] / k[0], 0, 0, -y[2] / k[0]]])
df_dp = np.empty((6, 0))
return df_dy, df_dp
def bcfun(y0, yf, _, __, ___):
return y0[0] + 1, y0[2], y0[3], yf[0] - 1, yf[2], yf[3]
algo = Shooting(odefun, None, bcfun, algorithm=algorithm, num_arcs=12)
algo.set_derivative_jacobian(odejac)
sol = Trajectory()
sol.t = np.linspace(0, 1, 2)
sol.y = np.array([[-1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0]])
sol.k = np.array([const])
# noinspection PyTypeChecker
cc = np.linspace(const * 10, const, 10, dtype=np.float)
for c in cc:
sol = copy.deepcopy(sol)
sol.k = np.array([c])
sol = algo.solve(sol)['traj']
assert sol.converged
def test_t19(const):
def odefun(y, _, k):
return y[1], (-y[1] / k[0]), 1
def odejac(_, __, k):
df_dy = np.array([[0, 1, 0], [0, -1 / k[0], 0], [0, 0, 0]])
df_dp = np.empty((3, 0))
return df_dy, df_dp
def bcfun(y0, yf, _, __, ___):
return y0[0], yf[0], y0[2]
algo = SPBVP(odefun, None, bcfun)
algo.set_derivative_jacobian(odejac)
sol = Trajectory()
sol.t = np.linspace(0, 1, 2)
sol.y = np.array([[0, 0, 0], [0, 0, 1]])
sol.k = np.array([const])
cc = np.linspace(const * 100, const, 10)
for c in cc:
sol = copy.deepcopy(sol)
sol.k = np.array([c])
sol = algo.solve(sol)['traj']
assert sol.converged
def test_t14(algorithm, const):
def odefun(y, _, k):
return 2 * y[1], 2 * ((y[0] - k[0] * np.pi ** 2 * np.cos(np.pi * y[2]) - np.cos(np.pi * y[2])) / k[0]), 2
def odejac(y, _, k):
df_dy = np.array([[0, 2, 0],
[2 / k[0], 0, (2 * (np.pi * np.sin(np.pi * y[2]) + k[0] * np.pi ** 3 * np.sin(np.pi * y[2])))
/ k[0]], [0, 0, 0]])
df_dp = np.empty((3, 0))
return df_dy, df_dp
def bcfun(y0, yf, _, __, ___):
return y0[0], yf[0], y0[2]+1
algo = Shooting(odefun, None, bcfun, algorithm=algorithm, num_arcs=4)
algo.set_derivative_jacobian(odejac)
sol = Trajectory()
sol.t = np.linspace(0, 1, 2)
sol.y = np.array([[0, 0, -1], [0, 0, 1]])
sol.k = np.array([const])
# noinspection PyTypeChecker
cc = np.linspace(const * 10, const, 10)
for c in cc:
sol = copy.deepcopy(sol)
sol.k = np.array([c])
sol = algo.solve(sol)['traj']
e1 = np.cos(np.pi * sol.y[:, 2]) + np.exp(-(1 + sol.y[:, 2]) / np.sqrt(sol.k[0])) + np.exp(
-(1 - sol.y[:, 2]) / np.sqrt(sol.k[0]))
e2 = np.exp((sol.y[:, 2] - 1) / np.sqrt(sol.k[0])) / np.sqrt(sol.k[0]) - np.pi * np.sin(
np.pi * sol.y[:, 2]) - 1 / (np.sqrt(sol.k[0]) * np.exp((sol.y[:, 2] + 1) / np.sqrt(sol.k[0])))
assert all(e1 - sol.y[:, 0] < tol)
assert all(e2 - sol.y[:, 1] < tol)
def test_t27(algorithm, const):
def odefun(y, _, k):
return y[1], y[0] * (1 - y[1]) / k[0]
def odejac(y, _, k):
df_dy = np.array([[0, 1], [(1-y[1]) / k[0], -y[0] / k[0]]])
df_dp = np.empty((2, 0))
return df_dy, df_dp
def bcfun(y0, yf, _, __, ___):
return y0[0] - 1, yf[0] - 1 / 3
algo = Shooting(odefun, None, bcfun, algorithm=algorithm, num_arcs=4)
algo.set_derivative_jacobian(odejac)
sol = Trajectory()
sol.t = np.linspace(0, 1, 2)
sol.y = np.array([[1, 0], [1 / 3, 0]])
sol.k = np.array([const])
# noinspection PyTypeChecker
cc = np.linspace(const*10, const, 10)
for c in cc:
sol = copy.deepcopy(sol)
sol.k = np.array([c])
sol = algo.solve(sol)['traj']
assert sol.converged
def test_t30(const):
def odefun(y, _, k):
return y[1], (y[0] - y[0] * y[1]) / k[0]
def odejac(y, _, k):
df_dy = np.array([[0, 1], [(1 - y[1]) / k[0], -y[0] / k[0]]])
df_dp = np.empty((2, 0))
return df_dy, df_dp
def bcfun(y0, yf, _, __, ___):
return y0[0] + 7 / 6, yf[0] - 3 / 2
algo = SPBVP(odefun, None, bcfun)
algo.set_derivative_jacobian(odejac)
sol = Trajectory()
sol.t = np.linspace(0, 1, 2)
sol.y = np.array([[-7 / 6, 0], [3 / 2, 0]])
sol.k = np.array([const])
cc = np.linspace(const * 10, const, 10)
for c in cc:
sol = copy.deepcopy(sol)
sol.k = np.array([c])
sol = algo.solve(sol)['traj']
assert sol.converged
def test_t24(const):
def odefun(x, _, k):
a_mat_x = 1 + x[2]**2
a_mat_xp = 2 * x[2]
y = 1.4
return (x[1],
(((1 + y) / 2 - k[0] * a_mat_xp) * x[0] * x[1] - x[1] / x[0] -
(a_mat_xp / a_mat_x) * (1 - (y - 1) / 2 * x[0]**2)) /
(k[0] * a_mat_x * x[0]), 1)
def odejac(x, _, k):
y = 1.4
df_dy = np.array(
[[0, 1, 0],
[(x[1] * (y / 2 - 2 * k * x[2] + 1 / 2) + x[1] / x[0]**2 +
(4 * x[0] * x[2] * (y / 2 - 1 / 2)) / (x[2]**2 + 1)) /
(k[0] * x[0] * (x[2]**2 + 1)) -
((2 * x[2] * ((y / 2 - 1 / 2) * x[0]**2 - 1)) /
(x[2]**2 + 1) - x[1] / x[0] + x[0] * x[1] *
(y / 2 - 2 * k[0] * x[2] + 1 / 2)) / (k[0] * x[0]**2 *
(x[2]**2 + 1)),
(x[0] * (y / 2 - 2 * k[0] * x[2] + 1 / 2) - 1 / x[0]) /
(k[0] * x[0] * (x[2]**2 + 1)),
-((4 * x[2]**2 * ((y / 2 - 1 / 2) * x[0]**2 - 1)) /
(x[2]**2 + 1)**2 - (2 * ((y / 2 - 1 / 2) * x[0]**2 - 1)) /
(x[2]**2 + 1) + 2 * k[0] * x[0] * x[1]) / (k[0] * x[0] *
(x[2]**2 + 1)) -
(2 * x[2] * ((2 * x[2] * ((y / 2 - 1 / 2) * x[0]**2 - 1)) /
(x[2]**2 + 1) - x[1] / x[0] + x[0] * x[1] *
(y / 2 - 2 * k[0] * x[2] + 1 / 2))) /
(k[0] * x[0] * (x[2]**2 + 1)**2)], [0, 0, 0]])
df_dp = np.empty((3, 0))
return df_dy, df_dp
def bcfun(x0, xf, _, __, ___):
return x0[0] - 0.9129, xf[0] - 0.375, x0[2]
algo = SPBVP(odefun, None, bcfun)
algo.set_derivative_jacobian(odejac)
sol = Trajectory()
sol.t = np.linspace(0, 1, 2)
sol.y = np.array([[1, 1, 0], [0.1, 0.1, 1]])
sol.k = np.array([const])
cc = np.linspace(const * 10, const, 10)
for c in cc:
sol = copy.deepcopy(sol)
sol.k = np.array([c])
sol = algo.solve(sol)['traj']
assert sol.converged
def test_t3(algorithm, const):
def odefun(y, _, k):
return (2 * y[1], 2 * (-(2 + np.cos(np.pi * y[2])) * y[1] + y[0] - (1 + k[0] * np.pi * np.pi) * np.cos(
np.pi * y[2]) - (2 + np.cos(np.pi * y[2])) * np.pi * np.sin(np.pi * y[2])) / k[0], 2)
def odejac(y, _, k):
df_dy = np.array([[0, 2, 0],
[2 / k[0], -(2 * np.cos(np.pi * y[2]) + 4)/k[0],
(2*np.pi**2 * np.sin(np.pi * y[2])**2 + 2 * np.pi*np.sin(np.pi*y[2])*(k[0]*np.pi**2 + 1)
- 2*np.pi**2*np.cos(np.pi*y[2])*(np.cos(np.pi*y[2]) + 2)
+ 2*y[1]*np.pi*np.sin(np.pi*y[2]))/k[0]],
[0, 0, 0]], dtype=np.float)
df_dp = np.empty((3, 0))
return df_dy, df_dp
def bcfun(y0, yf, _, __, ___):
return y0[0] + 1, yf[0] + 1, y0[2] + 1
algo = Shooting(odefun, None, bcfun, algorithm=algorithm)
algo.set_derivative_jacobian(odejac)
solinit = Trajectory()
solinit.t = np.linspace(0, 1, 2)
solinit.y = np.array([[-1, 0, -1], [-1, 0, 1]])
solinit.k = np.array([const])
sol = algo.solve(solinit)['traj']
e1 = np.cos(np.pi * sol.y[:, 2])
e2 = -np.pi * np.sin(np.pi * sol.y[:, 2])
assert all(e1 - sol.y[:, 0] < tol)
assert all(e2 - sol.y[:, 1] < tol)
def test_t1(algorithm, const):
def odefun(y, _, k):
return y[1], y[0] / k[0]
def odejac(_, __, k):
df_dy = np.array([[0, 1], [1 / k[0], 0]])
df_dp = np.empty((2, 0))
return df_dy, df_dp
def bcfun(y0, yf, _, __, ___):
return y0[0] - 1, yf[0]
algo = Shooting(odefun, None, bcfun, algorithm=algorithm)
algo.set_derivative_jacobian(odejac)
solinit = Trajectory()
solinit.t = np.linspace(0, 1, 2)
solinit.y = np.array([[0, 1], [0, 1]])
solinit.k = np.array([const])
sol = algo.solve(solinit)['traj']
e1 = (np.exp(-sol.t / np.sqrt(sol.k)) - np.exp((sol.t - 2) / np.sqrt(sol.k))) / (
1 - np.exp(-2.e0 / np.sqrt(sol.k)))
e2 = (1. / (sol.k ** (1 / 2) * np.exp(sol.t / sol.k ** (1 / 2))) + np.exp(
(sol.t - 2) / sol.k ** (1 / 2)) / sol.k ** (1 / 2)) / (1 / np.exp(2 / sol.k ** (1 / 2)) - 1)
assert all(e1 - sol.y[:, 0] < tol)
assert all(e2 - sol.y[:, 1] < tol)
def test_t9(algorithm, const):
def odefun(y, _, k):
return 2 * y[1], 2 * (-(4 * y[2] * y[1] + 2 * y[0]) / (k[0] + y[2] ** 2)), 2
def odejac(y, _, k):
df_dy = np.array([[0, 2, 0],
[-4 / (y[2] ** 2 + k[0]), -(8 * y[2]) / (y[2] ** 2 + k[0]),
(4 * y[2] * (2 * y[0] + 4 * y[1] * y[2])) / (y[2] ** 2 + k[0]) ** 2
- (8 * y[1]) / (y[2] ** 2 + k[0])], [0, 0, 0]])
df_dp = np.empty((3, 0))
return df_dy, df_dp
def bcfun(y0, yf, _, __, k):
return y0[0] - 1 / (1 + k[0]), yf[0] - 1 / (1 + k[0]), y0[2] + 1
algo = Shooting(odefun, None, bcfun, algorithm=algorithm, num_arcs=2)
algo.set_derivative_jacobian(odejac)
solinit = Trajectory()
solinit.t = np.linspace(0., 1., 2)
# noinspection PyTypeChecker
solinit.y = np.array([[1. / (1. + const), 0., -1.], [1. / (1. + const), 1., 1.]])
solinit.k = np.array([const])
sol = algo.solve(solinit)['traj']
e1 = 1 / (sol.k[0] + sol.y[:, 2] ** 2)
e2 = -(2 * sol.y[:, 2]) / (sol.y[:, 2] ** 2 + sol.k[0]) ** 2
assert all(e1 - sol.y[:, 0] < tol)
assert all(e2 - sol.y[:, 1] < tol)
def test_t7(algorithm, const):
def odefun(y, _, k):
return 2 * y[1], 2 * ((-y[2] * y[1] + y[0] - (1.0e0 + k[0] * np.pi ** 2) * np.cos(np.pi * y[2]) - np.pi *
y[2] * np.sin(np.pi * y[2])) / k[0]), 2
def odejac(y, _, k):
df_dy = np.array([[0, 2, 0],
[2 / k[0], -2 * y[2] / k[0],
-(2 * (y[1] + np.pi * np.sin(np.pi * y[2]) + np.pi ** 2 * y[2] * np.cos(np.pi * y[2])
- np.pi * np.sin(np.pi * y[2]) * (k[0] * np.pi ** 2 + 1))) / k[0]],
[0, 0, 0]])
df_dp = np.empty((3, 0))
return df_dy, df_dp
def bcfun(y0, yf, _, __, ___):
return y0[0] + 1, yf[0] - 1, y0[2] + 1
algo = Shooting(odefun, None, bcfun, algorithm=algorithm)
algo.set_derivative_jacobian(odejac)
solinit = Trajectory()
solinit.t = np.linspace(0, 1, 2)
solinit.y = np.array([[-1, 0, -1], [1, 0, 1]])
solinit.k = np.array([const])
sol = algo.solve(solinit)['traj']
e1 = np.cos(np.pi * sol.y[:, 2]) + sol.y[:, 2] + (
sol.y[:, 2] * erf(sol.y[:, 2] / np.sqrt(2.0e0 * sol.k[0]))
+ np.sqrt(2 * sol.k[0] / np.pi) * np.exp(-sol.y[:, 2] ** 2 / (2 * sol.k[0]))) / (
erf(1.0e0 / np.sqrt(2 * sol.k[0])) + np.sqrt(2.0e0 * sol.k[0] / np.pi)
* np.exp(-1 / (2 * sol.k[0])))
e2 = erf((np.sqrt(2) * sol.y[:, 2]) / (2 * np.sqrt(sol.k[0]))) / (
erf(np.sqrt(2) / (2 * np.sqrt(sol.k[0]))) + (np.sqrt(2) * np.sqrt(sol.k[0])) / (
np.sqrt(np.pi) * np.exp(1 / (2 * sol.k[0])))) - np.pi * np.sin(np.pi * sol.y[:, 2]) + 1
assert all(e1 - sol.y[:, 0] < tol)
assert all(e2 - sol.y[:, 1] < tol)
def test_t33(const):
def odefun(y, _, k):
return y[1], (y[0] * y[3] - y[2] * y[1]) / k[0], y[3], y[4], y[5], (
-y[2] * y[5] - y[0] * y[1]) / k[0]
def odejac(y, _, k):
df_dy = np.array(
[[0, 1, 0, 0, 0, 0],
[y[3] / k[0], -y[2] / k[0], -y[1] / k[0], y[0] / k[0], 0, 0],
[0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1],
[-y[1] / k[0], -y[0] / k[0], -y[5] / k[0], 0, 0, -y[2] / k[0]]])
df_dp = np.empty((6, 0))
return df_dy, df_dp
def bcfun(y0, yf, _, __, ___):
return y0[0] + 1, y0[2], y0[3], yf[0] - 1, yf[2], yf[3]
algo = SPBVP(odefun, None, bcfun)
algo.set_derivative_jacobian(odejac)
sol = Trajectory()
sol.t = np.linspace(0, 1, 2)
sol.y = np.array([[-1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0]])
sol.k = np.array([const])
sol = algo.solve(sol)['traj']
assert sol.converged
def test_t4(const):
def odefun(y, _, k):
return 2 * y[1], 2 * (((1 + k[0]) * y[0] - y[1]) / k[0]), 2
def odejac(_, __, k):
df_dy = np.array([[0, 2,
0], [2 * (1 + k[0]) / k[0], 2 * (-1) / k[0], 0],
[0, 0, 0]])
df_dp = np.empty((3, 0))
return df_dy, df_dp
def bcfun(y0, yf, _, __, k):
return y0[0] - 1 - np.exp(-2), yf[0] - 1 - np.exp(
-2 * (1 + k[0]) / k[0]), y0[2] + 1
algo = SPBVP(odefun, None, bcfun)
algo.set_derivative_jacobian(odejac)
solinit = Trajectory()
solinit.t = np.linspace(0, 1, 2)
solinit.y = np.array([[-1, 0, -1], [-1, 0, 1]])
solinit.k = np.array([const])
sol = algo.solve(solinit)['traj']
e1 = np.exp(sol.y[:, 2] - 1) + np.exp(-((1 + sol.k[0]) *
(1 + sol.y[:, 2]) / sol.k[0]))
e2 = np.exp(sol.y[:, 2] - 1) - (sol.k[0] + 1) / (sol.k[0] * np.exp(
(sol.y[:, 2] + 1) * (sol.k[0] + 1) / sol.k[0]))
assert all(e1 - sol.y[:, 0] < tol)
assert all(e2 - sol.y[:, 1] < tol)
def test_t31(const):
def odefun(y, _, k):
return np.sin(y[1]), y[2], -y[3] / k[0], \
((y[0]-1) * np.cos(y[1]) - y[2] / np.cos(y[1]) - k[0] * y[3] * np.tan(y[1])) / k[0]
def odejac(y, _, k):
df_dy = np.array([[0, np.cos(y[1]), 0, 0], [0, 0, 1, 0],
[0, 0, 0, -1 / k[0]],
[
np.cos(y[1]) / k[0],
-(np.sin(y[1]) * (y[0] - 1) + k[0] * y[3] *
(np.tan(y[1])**2 + 1) +
(y[2] * np.sin(y[1])) / np.cos(y[1])**2) /
k[0], -1 / (k[0] * np.cos(y[1])), -np.tan(y[1])
]])
df_dp = np.empty((4, 0))
return df_dy, df_dp
def bcfun(y0, yf, _, __, ___):
return y0[0], y0[2], yf[0], yf[2]
algo = SPBVP(odefun, None, bcfun)
algo.set_derivative_jacobian(odejac)
sol = Trajectory()
sol.t = np.linspace(0, 1, 2)
sol.y = np.array([[0, 0, 0, 0], [0, 0, 0, 0]])
sol.k = np.array([const])
sol = algo.solve(sol)['traj']
assert sol.converged
def test_t21(const):
def odefun(y, _, k):
return y[1], (y[0] *
(1 + y[0]) - np.exp(-2 * y[2] / np.sqrt(k[0]))) / k[0], 1
def odejac(y, _, k):
df_dy = np.array([[0, 1, 0],
[(2 * y[0] + 1) / k[0], 0,
(2 * np.exp(-(2 * y[2]) / np.sqrt(k[0]))) /
k[0]**(3 / 2)], [0, 0, 0]])
df_dp = np.empty((3, 0))
return df_dy, df_dp
def bcfun(y0, yf, _, __, k):
return y0[0] - 1, yf[0] - np.exp(-1 / np.sqrt(k[0])), y0[2]
algo = SPBVP(odefun, None, bcfun)
algo.set_derivative_jacobian(odejac)
solinit = Trajectory()
solinit.t = np.linspace(0, 1, 2)
solinit.y = np.array([[0, 0, 0], [0, 0, 1]])
solinit.k = np.array([const])
sol = algo.solve(solinit)['traj']
e1 = np.exp(-sol.y[:, 2] / np.sqrt(const))
e2 = -np.exp(-sol.y[:, 2] / np.sqrt(const)) / np.sqrt(const)
assert all(e1 - sol.y[:, 0] < tol)
assert all(e2 - sol.y[:, 1] < tol)
def test_t2(const):
def odefun(y, _, k):
return y[1], y[1] / k[0]
def odejac(_, __, k):
df_dy = np.array([[0, 1], [0, 1 / k[0]]])
df_dp = np.empty((2, 0))
return df_dy, df_dp
def bcfun(y0, yf, _, __, ___):
return y0[0] - 1, yf[0]
algo = SPBVP(odefun, None, bcfun)
algo.set_derivative_jacobian(odejac)
solinit = Trajectory()
solinit.t = np.linspace(0, 1, 2)
solinit.y = np.array([[0, 1], [0, 1]])
solinit.k = np.array([const])
sol = algo.solve(solinit)['traj']
e1 = (1.e0 - np.exp(
(sol.t - 1.e0) / sol.k)) / (1.e0 - np.exp(-1.e0 / sol.k))
e2 = np.exp((sol.t - 1) / sol.k) / (sol.k * (1 / np.exp(1 / sol.k) - 1))
assert all(e1 - sol.y[:, 0] < tol)
assert all(e2 - sol.y[:, 1] < tol)
def test_t16(const):
def odefun(y, _, k):
return 1 * y[1], 1 * (-y[0] * np.pi**2 / (4 * k[0])), 1
def odejac(_, __, k):
df_dy = np.array([[0, 1, 0], [-np.pi**2 / (4 * k[0]), 0, 0], [0, 0,
0]])
df_dp = np.empty((3, 0))
return df_dy, df_dp
def bcfun(y0, yf, _, __, k):
return y0[0], yf[0] - np.sin(np.pi / (2 * np.sqrt(k[0]))), y0[2]
algo = SPBVP(odefun, None, bcfun)
algo.set_derivative_jacobian(odejac)
solinit = Trajectory()
solinit.t = np.linspace(0, 1, 2)
solinit.y = np.array([[0, 0, 0], [0, 0, 1]])
solinit.k = np.array([const])
sol = algo.solve(solinit)['traj']
e1 = np.sin(np.pi * sol.y[:, 2] / (2 * np.sqrt(sol.k[0])))
e2 = (np.pi * np.cos((np.pi * sol.y[:, 2]) /
(2 * np.sqrt(sol.k[0])))) / (2 * np.sqrt(sol.k[0]))
assert all(e1 - sol.y[:, 0] < tol)
assert all(e2 - sol.y[:, 1] < tol)
def test_t17(const):
def odefun(y, _, k):
return 0.2 * y[1], 0.2 * (-3 * k[0] * y[0] / (k[0] + y[2]**2)**2), 0.2
def odejac(y, _, k):
df_dy = np.array([[0, 0.2, 0],
[
-(3 * k[0]) / (5 * (y[2]**2 + k[0])**2), 0,
(12 * k[0] * y[0] * y[2]) / (5 *
(y[2]**2 + k[0])**3)
], [0, 0, 0]])
df_dp = np.empty((3, 0))
return df_dy, df_dp
def bcfun(y0, yf, _, __, k):
return y0[0] + 0.1 / np.sqrt(k[0] + 0.01), yf[0] - 0.1 / np.sqrt(
k[0] + 0.01), y0[2] + 0.1
algo = SPBVP(odefun, None, bcfun)
algo.set_derivative_jacobian(odejac)
solinit = Trajectory()
solinit.t = np.linspace(0, 1, 2)
solinit.y = np.array([[0, 0, 0], [0, 0, 1]])
solinit.k = np.array([const])
sol = algo.solve(solinit)['traj']
e1 = sol.y[:, 2] / np.sqrt(sol.k[0] + sol.y[:, 2]**2)
e2 = 1 / np.sqrt(sol.y[:, 2]**2 + sol.k[0]) - sol.y[:, 2]**2 / (
sol.y[:, 2]**2 + sol.k[0])**(3 / 2)
assert all(e1 - sol.y[:, 0] < tol)
assert all(e2 - sol.y[:, 1] < tol)
def test_t13(const):
def odefun(y, _, k):
return 2 * y[1], 2 * ((y[0] - k[0] * np.pi**2 * np.cos(np.pi * y[2]) -
np.cos(np.pi * y[2])) / k[0]), 2
def odejac(y, _, k):
df_dy = np.array([[0, 2, 0],
[
2 / k[0], 0,
(2 * (np.pi * np.sin(np.pi * y[2]) +
k[0] * np.pi**3 * np.sin(np.pi * y[2]))) /
k[0]
], [0, 0, 0]])
df_dp = np.empty((3, 0))
return df_dy, df_dp
def bcfun(y0, yf, _, __, ___):
return y0[0], yf[0] + 1, y0[2] + 1
algo = SPBVP(odefun, None, bcfun)
algo.set_derivative_jacobian(odejac)
solinit = Trajectory()
solinit.t = np.linspace(0, 1, 2)
solinit.y = np.array([[-1, 0, -1], [0, 0, 1]])
solinit.k = np.array([const])
sol = algo.solve(solinit)['traj']
e1 = np.cos(np.pi * sol.y[:, 2]) + np.exp(
-(1 + sol.y[:, 2]) / np.sqrt(sol.k[0]))
e2 = -np.exp(-(sol.y[:, 2] + 1) / np.sqrt(sol.k[0])) / np.sqrt(
sol.k[0]) - np.pi * np.sin(np.pi * sol.y[:, 2])
assert all(e1 - sol.y[:, 0] < tol)
assert all(e2 - sol.y[:, 1] < tol)
def test_t10(const):
def odefun(y, _, k):
return 2 * y[1], 2 * (-y[2] * y[1] / k[0]), 2
def odejac(y, _, k):
df_dy = np.array([[0, 2,
0], [0, 2 * (-y[2]) / k[0], 2 * (-y[1] / k[0])],
[0, 0, 0]])
df_dp = np.empty((3, 0))
return df_dy, df_dp
def bcfun(y0, yf, _, __, ___):
return y0[0], yf[0] - 2, y0[2] + 1
algo = SPBVP(odefun, None, bcfun)
algo.set_derivative_jacobian(odejac)
solinit = Trajectory()
solinit.t = np.linspace(0, 1, 2)
solinit.y = np.array([[0, 0, -1], [2, 0, 1]])
solinit.k = np.array([const])
sol = algo.solve(solinit)['traj']
e1 = 1 + erf(sol.y[:, 2] / np.sqrt(2 * sol.k[0])) / erf(
1 / np.sqrt(2 * sol.k[0]))
e2 = np.sqrt(2) / (np.sqrt(np.pi) * np.sqrt(sol.k[0]) * np.exp(
sol.y[:, 2]**2 /
(2 * sol.k[0])) * erf(np.sqrt(2) / (2 * np.sqrt(sol.k[0]))))
assert all(e1 - sol.y[:, 0] < tol)
assert all(e2 - sol.y[:, 1] < tol)
def test_t9(const):
def odefun(y, _, k):
return 2 * y[1], 2 * (-(4 * y[2] * y[1] + 2 * y[0]) /
(k[0] + y[2]**2)), 2
def odejac(y, _, k):
df_dy = np.array(
[[0, 2, 0],
[
-4 / (y[2]**2 + k[0]), -(8 * y[2]) / (y[2]**2 + k[0]),
(4 * y[2] * (2 * y[0] + 4 * y[1] * y[2])) /
(y[2]**2 + k[0])**2 - (8 * y[1]) / (y[2]**2 + k[0])
], [0, 0, 0]])
df_dp = np.empty((3, 0))
return df_dy, df_dp
def bcfun(y0, yf, _, __, k):
return y0[0] - 1 / (1 + k[0]), yf[0] - 1 / (1 + k[0]), y0[2] + 1
algo = SPBVP(odefun, None, bcfun)
algo.set_derivative_jacobian(odejac)
solinit = Trajectory()
solinit.t = np.linspace(0, 1, 2)
solinit.y = np.array([[1 / (1 + const), 0, -1], [1 / (1 + const), 1, 1]])
solinit.k = np.array([const])
sol = algo.solve(solinit)['traj']
e1 = 1 / (sol.k[0] + sol.y[:, 2]**2)
e2 = -(2 * sol.y[:, 2]) / (sol.y[:, 2]**2 + sol.k[0])**2
assert all(e1 - sol.y[:, 0] < tol)
assert all(e2 - sol.y[:, 1] < tol)
def test_t8(const):
def odefun(y, _, k):
return y[1], (-y[1] / k[0]), 1
def odejac(_, __, k):
df_dy = np.array([[0, 1, 0], [0, -1 / k[0], 0], [0, 0, 0]])
df_dp = np.empty((3, 0))
return df_dy, df_dp
def bcfun(y0, yf, _, __, ___):
return y0[0] - 1, yf[0] - 2, y0[2]
algo = SPBVP(odefun, None, bcfun)
algo.set_derivative_jacobian(odejac)
solinit = Trajectory()
solinit.t = np.linspace(0, 1, 2)
solinit.y = np.array([[1, 0, -1], [2, 0, 1]])
solinit.k = np.array([const])
sol = algo.solve(solinit)['traj']
e1 = (2 - np.exp(-1 / sol.k[0]) -
np.exp(-sol.y[:, 2] / sol.k[0])) / (1 - np.exp(-1 / sol.k[0]))
e2 = -1 / (sol.k[0] * np.exp(sol.y[:, 2] / sol.k[0]) *
(1 / np.exp(1 / sol.k[0]) - 1))
assert all(e1 - sol.y[:, 0] < tol)
assert all(e2 - sol.y[:, 1] < tol)
def test_shooting_4():
# This problem contains a quad and tests if the prob solver correctly
# integrates the quadfun. Also tests multiple shooting.
def odefun(x, _, __):
return -x[1], x[0]
def quadfun(x, _, __):
return x[0]
def bcfun(y0, _, __, qf, ___, ____, _____):
return y0[0], y0[1] - 1, qf[0] - 1.0
algo = Shooting(odefun, quadfun, bcfun, num_arcs=4)
solinit = Trajectory()
solinit.t = np.linspace(0, np.pi / 2, 2)
solinit.y = np.array([[1, 0], [1, 0]])
solinit.q = np.array([[0], [0]])
solinit.k = np.array([])
out = algo.solve(solinit)['traj']
assert (out.y[0, 0] - 0) < tol
assert (out.y[0, 1] - 1) < tol
assert (out.q[0, 0] - 2) < tol
assert (out.q[-1, 0] - 1) < tol
def test_spbvp_3():
# This problem contains a parameter, but it is not explicit in the BCs.
# Since time is buried in the ODEs, this tests if the BVP solver calculates
# sensitivities with respect to parameters.
def odefun(_, p, __):
return 1 * p[0]
def bcfun(y0, yf, _, __, ___):
return y0[0] - 0, yf[0] - 2
algo = SPBVP(odefun, None, bcfun)
solinit = Trajectory()
solinit.t = np.linspace(0, 1, 4)
solinit.y = np.array([[0], [0], [0], [0]])
solinit.p = np.array([1])
solinit.k = np.array([])
out = algo.solve(solinit)['traj']
assert abs(out.p - 2) < tol
def scale_sol(sol: Trajectory, scale_factors, inv=False):
sol = copy.deepcopy(sol)
if inv:
op = np.multiply
else:
op = np.divide
sol.t = op(sol.t, scale_factors[0])
sol.y = op(sol.y, scale_factors[1])
sol.q = op(sol.q, scale_factors[2])
if sol.u.size > 0:
sol.u = op(sol.u, scale_factors[3])
sol.p = op(sol.p, scale_factors[4])
sol.nu = op(sol.nu, scale_factors[5])
sol.k = op(sol.k, scale_factors[6])
return sol
def test_t19(algorithm, const):
def odefun(y, _, k):
return y[1], (-y[1] / k[0]), 1
def odejac(_, __, k):
df_dy = np.array([[0, 1, 0], [0, -1 / k[0], 0], [0, 0, 0]])
df_dp = np.empty((3, 0))
return df_dy, df_dp
def bcfun(y0, yf, _, __, ___):
return y0[0], yf[0], y0[2]
algo = Shooting(odefun, None, bcfun, algorithm=algorithm)
algo.set_derivative_jacobian(odejac)
solinit = Trajectory()
solinit.t = np.linspace(0, 1, 2)
solinit.y = np.array([[0, 0, 0], [0, 0, 1]])
solinit.k = np.array([const])
sol = algo.solve(solinit)['traj']
assert sol.converged
def test_t23(algorithm, const):
def odefun(y, _, k):
return y[1], 1 / k[0] * np.sinh(y[0] / k[0])
def odejac(y, _, k):
df_dy = np.array([[0, 1], [np.cosh(y[0] / k[0]) / k[0] ** 2, 0]])
df_dp = np.empty((2, 0))
return df_dy, df_dp
def bcfun(y0, yf, _, __, ___):
return y0[0], yf[0] - 1
algo = Shooting(odefun, None, bcfun, algorithm=algorithm)
algo.set_derivative_jacobian(odejac)
solinit = Trajectory()
solinit.t = np.linspace(0, 1, 2)
solinit.y = np.array([[0, 0], [1, 0]])
solinit.k = np.array([const])
sol = algo.solve(solinit)['traj']
assert sol.converged
def test_t22(const):
def odefun(y, _, k):
return y[1], -(y[1] + y[0] * y[0]) / k[0]
def odejac(y, _, k):
df_dy = np.array([[0, 1], [-(2 * y[0]) / k[0], -1 / k[0]]])
df_dp = np.empty((2, 0))
return df_dy, df_dp
def bcfun(y0, yf, _, __, ___):
return y0[0], yf[0] - 1 / 2
algo = SPBVP(odefun, None, bcfun)
algo.set_derivative_jacobian(odejac)
solinit = Trajectory()
solinit.t = np.linspace(0, 1, 2)
solinit.y = np.array([[0, 0], [0, 0]])
solinit.k = np.array([const])
sol = algo.solve(solinit)['traj']
assert sol.converged
def test_t29(algorithm, const):
def odefun(y, _, k):
return y[1], (y[0] - y[0]*y[1]) / k[0]
def odejac(y, _, k):
df_dy = np.array([[0, 1], [(1-y[1]) / k[0], -y[0] / k[0]]])
df_dp = np.empty((2, 0))
return df_dy, df_dp
def bcfun(y0, yf, _, __, ___):
return y0[0], yf[0] - 3/2
algo = Shooting(odefun, None, bcfun, algorithm=algorithm, num_arcs=1)
algo.set_derivative_jacobian(odejac)
sol = Trajectory()
sol.t = np.linspace(0, 1, 2)
sol.y = np.array([[0, 0], [3/2, 0]])
sol.k = np.array([const])
sol = algo.solve(sol)['traj']
assert sol.converged
def test_t15(const):
def odefun(y, _, k):
return 2 * y[1], 2 * (y[2] * y[0] / k[0]), 2
def odejac(y, _, k):
df_dy = np.array([[0, 2, 0], [2 * (y[2] / k[0]), 0, 2 * (y[0] / k[0])],
[0, 0, 0]])
df_dp = np.empty((3, 0))
return df_dy, df_dp
def bcfun(y0, yf, _, __, ___):
return y0[0] - 1, yf[0] - 1, y0[2] + 1
algo = SPBVP(odefun, None, bcfun)
algo.set_derivative_jacobian(odejac)
solinit = Trajectory()
solinit.t = np.linspace(0, 1, 2)
solinit.y = np.array([[1, 0, -1], [0, 0, 1]])
solinit.k = np.array([const])
sol = algo.solve(solinit)['traj']
assert sol.converged
