Calculating Value Function Gradients Across Capital Trajectories
Value function gradients across capital trajectories compute the marginal sensitivity of terminal discounted returns to resource shifts at each state coordinate.
Hamiltonian residual
Meaning
Numerical metrics that measure the discrepancy in the conserved Hamiltonian of a dynamical system over discrete time steps quantify the accumulation of discretization errors. The Hamiltonian residual provides a precise gauge of how closely a simulated path adheres to physical conservation laws. By evaluating this error at each interval, computational scientists can assess the fidelity of symplectic integrators.
Dynamic Error
Differential equations representing physical or economic conservation laws often run into numerical drift during long-term simulation. The Hamiltonian residual measures the difference between the computed total energy and the theoretical constant value. In a perfect simulation, this value remains zero across all steps.
Practical limits of finite precision cause a non-zero remainder to accumulate.
Invariance Drift
Long simulations can suffer from systematic energy gains or losses if the integration method is not energy-preserving. A non-zero Hamiltonian residual signals that the numerical trajectory is drifting away from the true phase space orbit. Symplectic integrators bound this residual over long periods, preventing the catastrophic runaways common in standard schemes.
Such an error metric remains useful for validating the qualitative behavior of simulated orbits.
Conservation Limit
High-dimensional trajectories require adaptive step sizes to maintain the residual within acceptable limits. The Hamiltonian residual is thus used as a feedback trigger to adjust step sizes dynamically, balancing computational speed against energy preservation.