Computation of matrix functions with deflated restarting (Q455847)

The problem is to evaluate \(f(A)v\) with \(A\in\mathbb{C}^{n\times n}\), \(v\in\mathbb{C}^{n\times 1}\) and \(f\) a complex function. For large \(n\), this is often approximated by projecting the problem on a lower dimensional Krylov subspace. This converges superlinearly, but that property is lost when restarting is implemented. To mend this shortcomming, an Arnoldi method with deflated restarts is proposed. This means that instead of just keeping the last Krylov vector as a starting vector at restart while all the rest is removed, here only the less important vectors are removed, hence keeping more meaningful information while still reducing the dimension of the subspace, and hence the computational effort. Superlinear convergence is preserved for sufficiently smooth \(f\). This is illustrated with numerical examples.