def extra():
l_bc, r_bc = 0., 1.
lmbda = 12
N = 50
D, x = cheb_vectorized(N)
M = np.dot(D, D)
guess = (.5 * (x + 1)) #**lmbda
N2 = 500
def pseudospectral_ode(y):
out = np.zeros(y.shape)
ypp = M.dot(y)
out = 4 * ypp - lmbda * np.sinh(lmbda * y)
out[0], out[-1] = y[0] - r_bc, y[-1] - l_bc
return out
u = root(pseudospectral_ode, guess, method='lm', tol=1e-9)
print u.success
num_sol = BarycentricInterpolator(x, u.x)
xx = np.linspace(-1, 1, N2)
uu = num_sol.__call__(xx)
plt.plot(x, guess, '*b')
plt.plot(xx, uu, '-r') # Numerical solution via
# the pseudospectral method
plt.axis([-1., 1., 0 - .1, 1.1])
plt.show()
plt.clf()
def extra(): l_bc, r_bc = 0., 1. lmbda = 12 N = 50 D, x = cheb_vectorized(N) M = np.dot(D, D) guess = (.5*(x+1))#**lmbda N2 = 500 def pseudospectral_ode(y): out = np.zeros(y.shape) ypp = M.dot(y) out = 4*ypp - lmbda*np.sinh(lmbda*y) out[0], out[-1] = y[0] - r_bc, y[-1] - l_bc return out u = root(pseudospectral_ode,guess,method='lm',tol=1e-9) print u.success num_sol = BarycentricInterpolator(x,u.x) xx = np.linspace(-1, 1, N2) uu = num_sol.__call__(xx) plt.plot(x,guess,'*b') plt.plot(xx, uu, '-r') # Numerical solution via # the pseudospectral method plt.axis([-1.,1.,0-.1,1.1]) plt.show() plt.clf()
def trajectory():
N = 200
D, x = cheb(N)
eps = 0.
def L_yp(x,yp_):
a_ = a(x)
return a_**3.*yp_*(1 + a_**2.*yp_**2.)**(-1./2) - a_**2.*r(x)
def g(z):
out = D.dot(L_yp( x,D.dot(z) ))
out[0], out[-1] = z[0] - z1, z[-1] - 0
return out
# Use the straight line trajectory as an initial guess.
# Another option would be to use the shortest-time trajectory found
# in heuristic_tractory() as an initial guess.
eps, time = time_functional()
sol = root(g,y(x,eps))
# sol = root(g,z1/2.*(x-(-1.)))
# print sol.success
z = sol.x
poly = BarycentricInterpolator(x,D.dot(z))
num_func = lambda inpt: poly.__call__(inpt)
def L(x):
a_, yp_ = a(x), num_func(x)
return a_*(1 + a_**2.*yp_**2.)**(1./2) - a_**2.*r(x)*yp_
print "The shortest time is approximately "
print integrate.quad(L,-1,1,epsabs=1e-10,epsrel=1e-10)[0]
# x0 = np.linspace(-1,1,200)
# y0 = y(x0,eps)
# plt.plot(x0,y0,'-g',linewidth=2.,label='Initial guess')
# plt.plot(x,z,'-b',linewidth=2.,label="Numerical solution")
# ax = np.linspace(0-.05,z1+.05,200)
# plt.plot(-np.ones(ax.shape),ax,'-k',linewidth=2.5)
# plt.plot(np.ones(ax.shape),ax,'-k',linewidth=2.5)
# plt.plot(-1,0,'*b',markersize=10.)
# plt.plot(1,z1,'*b',markersize=10.)
# plt.xlabel('$x$',fontsize=18)
# plt.ylabel('$y$',fontsize=18)
# plt.legend(loc='best')
# plt.axis([-1.1,1.1,0. -.05,z1 +.05])
# # plt.savefig("minimum_time_rivercrossing.pdf")
# plt.show()
# plt.clf()
return x, z, N
plt.xlabel("$x$")
plt.savefig('pseudospectral_blog1_fig0.jpg')
plt.show()
N = 5
x = np.cos((np.pi / (N - 1)) * np.linspace(0, N - 1, N))
plt.plot(X, f(X), '-k')
plt.plot(x, f(x), '*k')
plt.savefig('pseudospectral_blog1_fig1.jpg')
plt.show()
N = 5
x1 = np.cos((np.pi / N) * np.linspace(0, N, N + 1))
interp = BarycentricInterpolator(x1, f(x1))
u1 = interp.__call__(X)
N = 10
x2 = np.cos((np.pi / N) * np.linspace(0, N, N + 1))
interp = BarycentricInterpolator(x2, f(x2))
u2 = interp.__call__(X)
N = 15
x3 = np.cos((np.pi / N) * np.linspace(0, N, N + 1))
interp = BarycentricInterpolator(x3, f(x3))
u3 = interp.__call__(X)
plt.plot(X, f(X), '-k', label="$f$")
plt.plot(x1, f(x1), '*g')
plt.plot(X, u1, '-g', label="$p_5$")
def nonlinear_minimal_area_surface_of_revolution():
l_bc, r_bc = 1., 7.
N = 80
D, x = cheb_vectorized(N)
M = np.dot(D, D)
guess = 1. + (x - -1.) * ((r_bc - l_bc) / 2.)
N2 = 50
def pseudospectral_ode(y):
out = np.zeros(y.shape)
yp, ypp = D.dot(y), M.dot(y)
out = y * ypp - 1. - yp**2.
out[0], out[-1] = y[0] - r_bc, y[-1] - l_bc
return out
u = root(pseudospectral_ode, guess, method='lm', tol=1e-9)
num_sol = BarycentricInterpolator(x, u.x)
# Up to this point we have found the numerical solution
# using the pseudospectral method. In the code that follows
# we check that solution with the analytic solution,
# and graph the results
def f(x):
return np.array([
x[1] * np.cosh((-1. + x[0]) / x[1]) - l_bc, x[1] * np.cosh(
(1. + x[0]) / x[1]) - r_bc
])
parameters = root(f, np.array([1., 1.]), method='lm', tol=1e-9)
A, B = parameters.x[0], parameters.x[1]
def analytic_solution(x):
out = B * np.cosh((x + A) / B)
return out
xx = np.linspace(-1, 1, N2)
uu = num_sol.__call__(xx)
# print "Max error is ", np.max(np.abs(uu - analytic_solution(xx)))
plt.plot(x, guess, '-b')
plt.plot(xx, uu, '-r') # Numerical solution via
# the pseudospectral method
plt.plot(xx, analytic_solution(xx), '*k') # Analytic solution
plt.axis([-1., 1., l_bc - 1., r_bc + 1.])
# plt.show()
plt.clf()
theta = np.linspace(0, 2 * np.pi, N2)
X, Theta = np.meshgrid(xx, theta, indexing='ij')
print "\nxx = \n", xx
print "\nuu = \n", uu
F = uu[:, np.newaxis] + np.zeros(uu.shape)
print "\nX = \n", X
print "\nTheta = \n", Theta
print "\nF = \n", F
Y = F * np.cos(Theta)
Z = F * np.sin(Theta)
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
# X, Y, Z = axes3d.get_test_data(0.05)
ax.plot_wireframe(X, Y, Z, rstride=1, cstride=1)
print ax.azim, ax.elev
ax.azim = -65
ax.elev = 0
# ax.view_init(elev=-60, azim=30)
# plt.savefig('minimal_surface.pdf')
plt.show()
npts = 7
noiseAmp = 0.03
xdata = np.linspace(0.2, 10., npts) + 0.3*np.random.randn(npts)
xdata = np.sort(xdata)
ydata = f(xdata) * (1.0 + noiseAmp * np.random.randn(npts))
np.savetxt('logCurveData9.txt', zip(xdata, ydata), fmt='%10.2f')
# Create x array spanning data set plus 5%
xmin, xmax = frangeAdd(xdata, 0.05)
x = np.linspace(xmin, xmax, 200)
# Plot data and "fit"
plt.plot(xdata, ydata, 'or', label='data')
plt.plot(x, f(x), 'k-', label='fitting function')
# Create y array from cubic spline
f_cubic = interp1d(xdata, ydata, kind='cubic', bounds_error=False)
plt.plot(x, f_cubic(x), 'k-', label='cubic spline')
# Create y array from univariate spline
f_univar = UnivariateSpline(xdata, ydata, w=None, bbox=[xmin, xmax], k=3)
plt.plot(x, f_univar(x), label='univariate spline')
# Create y array from barycentric interpolation
f_bary = BarycentricInterpolator(xdata, ydata)
plt.plot(x, f_bary.__call__(x), label='barycentric interp')
plt.legend(loc='best')
plt.show()
def nonlinear_minimal_area_surface_of_revolution():
l_bc, r_bc = 1., 7.
N = 80
D, x = cheb_vectorized(N)
M = np.dot(D, D)
guess = 1. + (x--1.)*((r_bc - l_bc)/2.)
N2 = 50
def pseudospectral_ode(y):
out = np.zeros(y.shape)
yp, ypp = D.dot(y), M.dot(y)
out = y*ypp - 1. - yp**2.
out[0], out[-1] = y[0] - r_bc, y[-1] - l_bc
return out
u = root(pseudospectral_ode,guess,method='lm',tol=1e-9)
num_sol = BarycentricInterpolator(x,u.x)
# Up to this point we have found the numerical solution
# using the pseudospectral method. In the code that follows
# we check that solution with the analytic solution,
# and graph the results
def f(x):
return np.array([ x[1]*np.cosh((-1.+x[0])/x[1])-l_bc,
x[1]*np.cosh((1.+x[0])/x[1])-r_bc])
parameters = root(f,np.array([1.,1.]),method='lm',tol=1e-9)
A, B = parameters.x[0], parameters.x[1]
def analytic_solution(x):
out = B*np.cosh((x + A)/B)
return out
xx = np.linspace(-1, 1, N2)
uu = num_sol.__call__(xx)
# print "Max error is ", np.max(np.abs(uu - analytic_solution(xx)))
plt.plot(x,guess,'-b')
plt.plot(xx, uu, '-r') # Numerical solution via
# the pseudospectral method
plt.plot(xx, analytic_solution(xx), '*k') # Analytic solution
plt.axis([-1.,1.,l_bc-1.,r_bc +1.])
# plt.show()
plt.clf()
theta = np.linspace(0,2*np.pi,N2)
X,Theta = np.meshgrid(xx,theta,indexing='ij')
print "\nxx = \n", xx
print "\nuu = \n", uu
F = uu[:,np.newaxis] +np.zeros(uu.shape)
print "\nX = \n", X
print "\nTheta = \n", Theta
print "\nF = \n", F
Y = F*np.cos(Theta)
Z = F*np.sin(Theta)
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
# X, Y, Z = axes3d.get_test_data(0.05)
ax.plot_wireframe(X, Y, Z, rstride=1, cstride=1)
print ax.azim, ax.elev
ax.azim=-65; ax.elev = 0
# ax.view_init(elev=-60, azim=30)
# plt.savefig('minimal_surface.pdf')
plt.show()
