Lie–Trotter Matrix Splitting
Build a coupled flow from simpler exponentials and measure splitting error.
Free theory articleBuild a coupled flow from simpler exponentials and measure splitting error.
Concept & interpretation
The matrix exponential of a sum A + B describes the coupled flow of two linear vector fields. When matrices do not commute (AB ≠ BA), the standard exponential identityexp(t(A + B)) = exp(tA)exp(tB) fails. Instead, operator splitting methods construct approximate solutions by alternating between the exact flows of A and B over small time steps h = t / n.
Experiment
You can interact with these subflows using the module:
- Open the Lie–Trotter Matrix Splitting module.
- Set Splitting to "Lie Trotter" and Steps to 4. Observe the finite-step error.
- Change Splitting to "Strang" (the symmetric
exp(hA/2)exp(hB)exp(hA/2)) and notice how the Frobenius error decreases significantly, demonstrating its higher global order. - Change Lower generator sign to 1 (making it a hyperbolic flow rather than a rotation) and observe the new trajectories.
- Change Lower generator sign to 0 (where A and B commute) and see that the error drops to 0 immediately.