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.const = np.array([const])
sol = algo.solve(solinit)['sol']
e1 = np.cos(np.pi * sol.y[:, 2]) + np.exp(
-(1 + sol.y[:, 2]) / np.sqrt(sol.const[0]))
e2 = -np.exp(-(sol.y[:, 2] + 1) / np.sqrt(sol.const[0])) / np.sqrt(
sol.const[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_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.const = np.array([const])
sol = algo.solve(sol)['sol']
assert sol.converged
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.const = np.array([const])
sol = algo.solve(solinit)['sol']
e1 = (1.e0 - np.exp(
(sol.t - 1.e0) / sol.const)) / (1.e0 - np.exp(-1.e0 / sol.const))
e2 = np.exp((sol.t - 1) / sol.const) / (sol.const *
(1 / np.exp(1 / sol.const) - 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.const = np.array([const])
sol = algo.solve(solinit)['sol']
e1 = np.sin(np.pi * sol.y[:, 2] / (2 * np.sqrt(sol.const[0])))
e2 = (np.pi * np.cos(
(np.pi * sol.y[:, 2]) /
(2 * np.sqrt(sol.const[0])))) / (2 * np.sqrt(sol.const[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.const = np.array([const])
sol = algo.solve(solinit)['sol']
e1 = sol.y[:, 2] / np.sqrt(sol.const[0] + sol.y[:, 2]**2)
e2 = 1 / np.sqrt(sol.y[:, 2]**2 + sol.const[0]) - sol.y[:, 2]**2 / (
sol.y[:, 2]**2 + sol.const[0])**(3 / 2)
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.const = np.array([const])
sol = algo.solve(solinit)['sol']
e1 = 1 / (sol.const[0] + sol.y[:, 2]**2)
e2 = -(2 * sol.y[:, 2]) / (sol.y[:, 2]**2 + sol.const[0])**2
assert all(e1 - sol.y[:, 0] < tol)
assert all(e2 - sol.y[:, 1] < tol)
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.const = np.array([const])
sol = algo.solve(solinit)['sol']
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_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.const = np.array([const])
sol = algo.solve(solinit)['sol']
e1 = 1 + erf(sol.y[:, 2] / np.sqrt(2 * sol.const[0])) / erf(
1 / np.sqrt(2 * sol.const[0]))
e2 = np.sqrt(2) / (np.sqrt(np.pi) * np.sqrt(sol.const[0]) * np.exp(
sol.y[:, 2]**2 /
(2 * sol.const[0])) * erf(np.sqrt(2) / (2 * np.sqrt(sol.const[0]))))
assert all(e1 - sol.y[:, 0] < tol)
assert all(e2 - sol.y[:, 1] < tol)
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.const = np.array([const])
cc = np.linspace(const * 10, const, 10)
for c in cc:
sol = copy.deepcopy(sol)
sol.const = np.array([c])
sol = algo.solve(sol)['sol']
assert sol.converged
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.const = np.array([const])
sol = algo.solve(solinit)['sol']
e1 = (2 - np.exp(-1 / sol.const[0]) - np.exp(
-sol.y[:, 2] / sol.const[0])) / (1 - np.exp(-1 / sol.const[0]))
e2 = -1 / (sol.const[0] * np.exp(sol.y[:, 2] / sol.const[0]) *
(1 / np.exp(1 / sol.const[0]) - 1))
assert all(e1 - sol.y[:, 0] < tol)
assert all(e2 - sol.y[:, 1] < tol)
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.const = np.array([const])
sol = algo.solve(solinit)['sol']
e1 = np.exp(sol.y[:, 2] - 1) + np.exp(-((1 + sol.const[0]) *
(1 + sol.y[:, 2]) / sol.const[0]))
e2 = np.exp(sol.y[:, 2] - 1) - (sol.const[0] + 1) / (sol.const[0] * np.exp(
(sol.y[:, 2] + 1) * (sol.const[0] + 1) / sol.const[0]))
assert all(e1 - sol.y[:, 0] < tol)
assert all(e2 - sol.y[:, 1] < tol)
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.const = np.array([const])
cc = np.linspace(const * 100, const, 10)
for c in cc:
sol = copy.deepcopy(sol)
sol.const = np.array([c])
sol = algo.solve(sol)['sol']
assert sol.converged
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.const = np.array([const])
sol = algo.solve(sol)['sol']
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.const = np.array([const])
cc = np.linspace(const * 10, const, 10)
for c in cc:
sol = copy.deepcopy(sol)
sol.const = np.array([c])
sol = algo.solve(sol)['sol']
assert sol.converged
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.dynamical_parameters = np.array([1])
solinit.const = np.array([])
out = algo.solve(solinit)['sol']
assert abs(out.dynamical_parameters - 2) < tol
def test_t7(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 = 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.const = np.array([const])
sol = algo.solve(solinit)['sol']
e1 = np.cos(np.pi * sol.y[:, 2]) + sol.y[:, 2] + (
sol.y[:, 2] * erf(sol.y[:, 2] / np.sqrt(2.0e0 * sol.const[0])) +
np.sqrt(2 * sol.const[0] / np.pi) * np.exp(-sol.y[:, 2]**2 /
(2 * sol.const[0]))
) / (erf(1.0e0 / np.sqrt(2 * sol.const[0])) +
np.sqrt(2.0e0 * sol.const[0] / np.pi) * np.exp(-1 /
(2 * sol.const[0])))
e2 = erf((np.sqrt(2) * sol.y[:, 2]) / (2 * np.sqrt(sol.const[0]))) / (
erf(np.sqrt(2) / (2 * np.sqrt(sol.const[0]))) +
(np.sqrt(2) * np.sqrt(sol.const[0])) /
(np.sqrt(np.pi) * np.exp(1 / (2 * sol.const[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_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.const = np.array([const])
sol = algo.solve(solinit)['sol']
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.const = np.array([const])
sol = algo.solve(solinit)['sol']
assert sol.converged
def test_r18(const):
def odefun(y, _, k):
return -y[0] / k[0]
def quadfun(y, _, __):
return y[0]
def bcfun(_, q0, __, qf, ___, ____, k):
return q0[0] - 1, qf[0] - np.exp(-1 / k[0])
algo = SPBVP(odefun, quadfun, bcfun)
solinit = Trajectory()
solinit.t = np.linspace(0, 1, 2)
solinit.y = np.array([[1], [1]])
solinit.q = np.array([[0], [0]])
solinit.const = np.array([const])
sol = algo.solve(solinit)['sol']
e1 = np.exp(-sol.t / sol.const[0])
e2 = -1 / (sol.const[0] * np.exp(sol.t / sol.const[0]))
assert all(e1 - sol.q[:, 0] < tol)
assert all(e2 - sol.y[:, 0] < tol)
def test_t6():
# This is a "special" case not using the difficulty settings above.
def odefun(y, _, k):
return (2 * y[1],
2 * ((-y[2] * y[1] - 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],
[
0, -2 * y[2] / k[0],
-(2 * (y[1] + np.pi * np.sin(np.pi * y[2]) -
k[0] * np.pi**3 * np.sin(np.pi * y[2]) +
np.pi**2 * y[2] * np.cos(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] + 2, yf[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.const = np.array([1])
sol = algo.solve(solinit)['sol']
e1 = np.cos(np.pi * sol.y[:, 2]) + erf(sol.y[:, 2] / np.sqrt(
2 * sol.const[0])) / erf(1 / np.sqrt(2 * sol.const[0]))
e2 = np.sqrt(2) / (
np.sqrt(np.pi) * np.sqrt(sol.const[0]) * np.exp(sol.y[:, 2]**2 /
(2 * sol.const[0])) *
erf(np.sqrt(2) /
(2 * np.sqrt(sol.const[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_t3(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 = 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.const = np.array([const])
sol = algo.solve(solinit)['sol']
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_spbvp_1():
# Full 2PBVP test problem
# This is the simplest BVP
def odefun(y, _, __):
return y[1], -abs(y[0])
def bcfun(y0, yf, _, __, ___):
return y0[0], yf[0] + 2
algo = SPBVP(odefun, None, bcfun)
solinit = Trajectory()
solinit.t = np.linspace(0, 4, 4)
solinit.y = np.array([[0, 1], [0, 1], [0, 1], [0, 1]])
solinit.const = np.array([])
out = algo.solve(solinit)['sol']
assert out.y[0][0] < tol
assert out.y[0][1] - 2.06641646 < tol
assert out.y[-1][0] + 2 < tol
assert out.y[-1][1] + 2.87588998 < tol
assert out.t[-1] - 4 < tol
assert abs(out.y[0][1] - solinit.y[0][1]) > tol
assert abs(out.y[-1][0] - solinit.y[-1][0]) - 2 < tol
def test_r8(const):
def odefun(y, _, k):
return -y[0] / k[0]
def quadfun(y, _, __):
return y[0]
def bcfun(_, q0, __, qf, ___, ____, _____):
return q0[0] - 1, qf[0] - 2
algo = SPBVP(odefun, quadfun, bcfun)
solinit = Trajectory()
solinit.t = np.linspace(0, 1, 2)
solinit.y = np.array([[1], [1]])
solinit.q = np.array([[0], [0]])
solinit.const = np.array([const])
sol = algo.solve(solinit)['sol']
e1 = (1.e0 - np.exp(
(sol.t - 1.e0) / sol.const)) / (1.e0 - np.exp(-1.e0 / sol.const))
e2 = np.exp((sol.t - 1) / sol.const) / (sol.const *
(1 / np.exp(1 / sol.const) - 1))
assert all(e1 - sol.q[:, 0] < tol)
assert all(e2 - sol.y[:, 0] < tol)