A new version of the alternating directions implicit (ADI) iteration for the solution of large-scale Lyapunov equations is introduced. It generalizes the hitherto existing iteration, by incorporating tangential directions in the way they are already available for rational Krylov subspaces. Additionally, first strategies to adaptively select shifts and tangential directions in each iteration are...

Abstract Mechanical systems are often modeled with the multibody system method or finite element and numerically described of differential equations. Increasing demands on detail resulting high complexity these make use model order reduction inevitable. Frequently, moment matching based Krylov subspaces is used for reduction. There, transfer functions full reduced matched at distinct frequency ...

Matrix functions are a central topic of linear algebra, and problems of their numerical approximation appear increasingly often in scientific computing. We review various rational Krylov methods for the computation of large-scale matrix functions. Emphasis is put on the rational Arnoldi method and variants thereof, namely, the extended Krylov subspace method and the shift-and-invert Arnoldi met...

We present a one-dimensional multicomponent model, known to be partially integrable when restricted the subspaces made of only two components. By constructing fully antisymmetrized bases, we find excited eigenstates corresponding totally antisymmetric irreducible representation permutation operator in otherwise nonintegrable subspaces. establish rigorously breakdown integrability those by showi...

In this talk I will discuss necessary and sufficient conditions on a nonsingular matrix A, such that for any initial vector r0, an orthogonal basis of the Krylov subspaces Kn(A, r0) is generated by a short recurrence. Orthogonality here is meant with respect to some unspecified positive definite inner product. This question is closely related to the question of existence of optimal Krylov subsp...

1. Introduction. We consider the resolution of large-scale nonlinear problems arising from the finite-element discretization of geometrically non-linear structural analysis problems. We use a classical Newton Raphson algorithm to handle the non-linearity which leads to the resolution of a sequence of linear systems with non-invariant matrices and right hand sides. The linear systems are solved ...

Newton-Krylov methods, combination of Newton-like methods and Krylov subspace methods for solving the Newton equations, often need adequate preconditioning in order to be successful. Approximations of the Jacobian matrices are required to form preconditioners and this step is very often the dominant cost of Newton-Krylov methods. Therefore, working with preconditioners destroys in principle the...

In recent years, order-reduction techniques based on Krylov subspaces have become the methods of choice for generating macromodels of large multi-port RLC circuits. Despite the success of these techniques and the extensive research efforts in this area, for general RCL circuits, the existing Krylov subspace-based reduction algorithms do not fully preserve all essential structures of the given l...