Stability of the Peaceman-Rachford approximation
Let \(A\) and \(B\) be selfadjoint bounded from below operators in a Hilbert space. The author investigates stability of approximation (in the operator norm) of \(e^{-t(A+B)}\) by the Peaceman-Rachford formula \[ p(t/2n)^n \to e^{-t(A+B)}, \] where \(p(t) = (1+tA)^{-1}(1-tB)(1+tB)^{-1}(1-tA)\). The approximation is stable if \(\| p(t)\| \leq 1+O(t)\). Sufficient conditions for stability are given; they involve conditions on the commutators of \(\sqrt A\) and \(\sqrt B\).
scientific article; zbMATH DE number 1270901
scientific article