Landweber relaxation parameter ($0<\tau <2/\Vert A{\Vert }^2$)

The positive parameter $\tau$ is the step in Landweber iteration. For a nonzero bounded linear operator $A$, a standard data-independent condition guaranteeing convergence from zero on data admitting a least-squares solution isIt makes $|1-\tau {\sigma }^2|<1$ on every nonzero singular mode. Thus $\Vert A\Vert <2$ permits $\tau =1/2$, while a unit step is guaranteed by $\Vert A\Vert <\sqrt{2}$.

This is a uniform sufficient range, not a necessary condition for every particular datum. A datum with no component in an unstable mode can still converge for a larger step. At the endpoint $\tau =2/\Vert A{\Vert }^2$, an attained top mode has multiplier $-1$ and can oscillate indefinitely. The strong convergence of relaxed Landweber iteration states the exact-data result without assuming a singular-value-independent contraction rate.