Shifted-update rotation: simple integration of the many-level Schrِdinger equation to long times

Shifted-update direct propagation algorithms are used to calculate the survival probabilities and spectra of a matrix-scaling model for vibrational energy redistribution with 70 000 active states and nine degrees of freedom. A stabilized first-order method, shifted-update rotation has a simple geometrical interpretation, yet requires no storage overhead. Of the higher-order methods, only a third-order method executes faster in many cases, at the cost of stability and memory overhead. We compare shifted-update methods to others applicable to sparse systems.