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| # An example of calculating a geodesic | |
| # model takes the form y(t,x) = e^(-x[0]*t) + e^(-x[1]*t) for time points t = 0.5, 2.0 | |
| # We enforce x[i] > 0 by going to log parameters: | |
| # y(t,x) = e^(-exp(x[0])*t) + e^(-exp(x[1])*t) | |
| from geodesic import geodesic, InitialVelocity | |
| import numpy as np | |
| exp = np.exp | |
| # Model predictions | |
| def r(x): | |
| return np.array([ exp(-exp(x[0])*0.5) + exp(-exp(x[1])*0.5), exp(-exp(x[0])*2) + exp(-exp(x[1])*2)]) | |
| # Jacobian | |
| def j(x): | |
| return np.array([ [ -0.5*exp(x[0])*exp(-x[0]*0.5), -2.0*exp(x[0])*exp(-x[0]*2)], | |
| [ -0.5*exp(x[1])*exp(-x[1]*0.5), -2.0*exp(x[1])*exp(-x[1]*2)] ] ) | |
| # Directional second derivative | |
| def Avv(x,v): | |
| h = 1e-4 | |
| return (r(x + h*v) + r(x - h*v) - 2*r(x))/h/h | |
| # Choose starting parameters | |
| x = np.log([1.0, 2.0]) | |
| v = InitialVelocity(x, j, Avv) | |
| # Callback function used to monitor the geodesic after each step | |
| def callback(geo): | |
| # Integrate until the norm of the velocity has grown by a factor of 10 | |
| # and print out some diagnotistic along the way | |
| print "Iteration: %i, tau: %f, |v| = %f" %(len(geo.vs), geo.ts[-1], np.linalg.norm(geo.vs[-1])) | |
| return np.linalg.norm(geo.vs[-1]) < 10.0 | |
| # Construct the geodesic | |
| # It is usually not necessary to be very accurate here, so we set small tolerances | |
| geo = geodesic(r, j, Avv, 2, 2, x, v, atol = 1e-2, rtol = 1e-2, callback = callback) | |
| # Integrate | |
| geo.integrate(25.0) | |
| import pylab | |
| # plot the geodesic path to find the limit | |
| # This should show the singularity at the "fold line" x[0] = x[1] | |
| pylab.plot(geo.ts, geo.xs) | |
| pylab.xlabel("tau") | |
| pylab.ylabel("Parameter Values") | |
| pylab.show() |