Based on Arnoldi's method, a version of generalized Arnoldi algorithm has been developed for the reduction of gyroscopic eigenvalue problems. By utilizing the skew symmetry of system matrix, a very simple recurren...Based on Arnoldi's method, a version of generalized Arnoldi algorithm has been developed for the reduction of gyroscopic eigenvalue problems. By utilizing the skew symmetry of system matrix, a very simple recurrence scheme, named gyroscopic Arnoldi reduction algorithm has been obtained, which is even simpler than the Lanczos algorithm for symmetric eigenvalue problems. The complex number computation is completely avoided. A restart technique is used to enable the reduction algorithm to have iterative characteristics. It has been found that the restart technique is not only effective for the convergence of multiple eigenvalues but it also furnishes the reduction algorithm with a technique to check and compute missed eigenvalues. By combining it with the restart technique, the algorithm is made practical for large-scale gyroscopic eigenvalue problems. Numerical examples are given to demonstrate the effectiveness of the method proposed.展开更多
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 vecto...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.展开更多
基金This research is supported by The National Science FoundationThe Doctoral Training Foundation
文摘Based on Arnoldi's method, a version of generalized Arnoldi algorithm has been developed for the reduction of gyroscopic eigenvalue problems. By utilizing the skew symmetry of system matrix, a very simple recurrence scheme, named gyroscopic Arnoldi reduction algorithm has been obtained, which is even simpler than the Lanczos algorithm for symmetric eigenvalue problems. The complex number computation is completely avoided. A restart technique is used to enable the reduction algorithm to have iterative characteristics. It has been found that the restart technique is not only effective for the convergence of multiple eigenvalues but it also furnishes the reduction algorithm with a technique to check and compute missed eigenvalues. By combining it with the restart technique, the algorithm is made practical for large-scale gyroscopic eigenvalue problems. Numerical examples are given to demonstrate the effectiveness of the method proposed.
文摘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.
