Second-Order Krylov Subspace and Arnoldi Procedure

We report our recent work on a second-order Krylov subspace and the corresponding second-order Arnoldi procedure for generating its orthonormal basis. The second-order Krylov subspace is spanned by a sequence of vectors defined via a second-order linear homogeneous recurrence relation with coefficient matrices A and B and an initial vector u. It generalizes the well-known Krylov subspace K n(A;v), which is spanned by a sequence of vectors defined via a first-order linear homogeneous recurrence relation with a single coefficient matrix A and an initial vector v. The applications are shown for the solution of quadratic eigenvalue problems and dimension reduction of second-order dynamical systems. The new approaches preserve essential structures and properties of the quadratic eigenvalue problem and second-order system, and demonstrate superior numerical results over the common approaches based on linearization of these second-order problems.