Entrywise relative perturbation bounds for exponentials of essentially non-negative matrices
The authors consider the perturbation behavior of essentially non-negative matrix exponentials. A matrix is said to be essentially non-negative if \(a_{ij} \geq 0\) for all \(i \neq j\). The generator matrix \(Q\) of a continuous-time Markov chain is used as an example with the mentioned property. The \((i,j)\) entry of the matrix exponential exp\((Qt)\) is the probability that the chain has at state \(j\) at time \(t\) if it is at state \(i\) initially. Especially, taking a sensitivity analysis one is interested in entries of the transition matrix exp\((Qt)\) which represent small probability events. Because normwise perturbation bounds do not reflect these small entries sharp entrywise relative perturbation bounds for the exponential matrices are established that are determined entrywise by the absolute accuracy of the diagonal entries and by the relative accuracy of the off-diagonal elements.
scientific article; zbMATH DE number 5377099