In recent papers Ruhe [10], [12] suggested a rational Krylov method for nonlinear eigenproblems knitting together a secant method for linearizing the nonlinear problem and the Krylov method for the linearized problem. In this note we point out that the method can be understood as an iterative projection method. Similar to the Arnoldi method presented in [13], [14] the search space is expanded b...

The Jacobi-Davidson subspace iteration method ooers possibilities for solving a variety of eigen-problems. In practice one has to apply restarts because of memory limitations, in order to restrict computational overhead, and also if one wants to compute several eigenvalues. In general, restarting has negative eeects on the convergence of subspace methods. We will show how eeective restarts can ...

This paper aims at providing an algorithmic understanding of the “convergence” of Krylov-type methods which relies on asymptotic properties at 0 and ∞. The classical normwise (or analytic) perturbation approach correponds to the limit towards 0. We complement this analysis by the structural perturbation approach provided by the limit to ∞ in the Homotopic Deviation theory. We easily get back th...

Since the rst papers on asymptotic waveform evaluation (AWE), reduced order models have become standard for improving interconnect simulation ee-ciency, and very recent work has demonstrated that bi-orthogonalization algorithms can be used to robustly generate AWE-style macromodels. In this paper we describe using block Arnoldi-based orthogo-nalization methods to generate reduced order models f...

We consider a two-directional Krylov subspace Kk(A[j], b[j]), where besides the dimensionality k of the subspace increases, the matrix A[j] and vector b[j] which induce the subspace may also augment. Specifically, we consider the case where the matrix A[j] and the vector b[j] are augmented by block triangular bordering. We present a two-directional Arnoldi process to efficiently generate a sequ...

The overlap Dirac operator in lattice QCD requires the computation of the sign function of a matrix. While this matrix is usually Hermitian, it becomes non-Hermitian in the presence of a quark chemical potential. We show how the action of the sign function of a non-Hermitian matrix on an arbitrary vector can be computed efficiently on large lattices by an iterative method. A Krylov subspace app...