The refined Arnoldi method proposed by Jia is used for computing some eigen-pairs of large matrices. In contrast to the Arnoldi method, the fundamental dif-ference is that the refined method seeks certain refined Ritz...The refined Arnoldi method proposed by Jia is used for computing some eigen-pairs of large matrices. In contrast to the Arnoldi method, the fundamental dif-ference is that the refined method seeks certain refined Ritz vectors, which aredifferent from the Ritz vectors obtained by the Arnoldi method, from a projection space with minimal residuals to approximate the desired eigenvectors. In com-parison with the Ritz vectors, the refined Ritz vectors are guaranteed to converge theoretically and can converge much faster numerically. In this paper we propose to replace the Ritz values, obtained by the Arnoldi method with respect to a Krylovsubspace, by the ones obtained with respect to the subspace spanned by the refined Ritz vectors. We discuss how to compute these new approximations cheaply and reliably. Theoretical error bounds between the original Ritz values and the new Ritz values are established. Finally, we present a variant of the refined Arnoldi al-gorithm for an augmented Krylov subspace and discuss restarting issue. Numerical results confirm efficiency of the new algorithm.展开更多
This paper extends the two-grid discretization scheme of the conforming finite elements proposed by Xu and Zhou （Math. Comput., 70 （2001）, pp.17-25） to the nonconforming finite elements for eigenvalue problems. In...This paper extends the two-grid discretization scheme of the conforming finite elements proposed by Xu and Zhou （Math. Comput., 70 （2001）, pp.17-25） to the nonconforming finite elements for eigenvalue problems. In particular, two two-grid discretization schemes based on Rayleigh quotient technique are proposed. By using these new schemes, the solution of an eigenvalue problem on a fine mesh is reduced to that on a much coarser mesh together with the solution of a linear algebraic system on the fine mesh. The resulting solution still maintains an asymptotically optimal accuracy. Comparing with the two-grid discretization scheme of the conforming finite elements, the main advantages of our new schemes are twofold when the mesh size is small enough. First, the lower bounds of the exact eigenvalues in our two-grid discretization schemes can be obtained. Second, the first eigenvalue given by the new schemes has much better accuracy than that obtained by solving the eigenvalue problems on the fine mesh directly.展开更多
In this paper, the theoretical analysis for the Rayleigh quotient matrix is studied, some results of the Rayleigh quotient （matrix） of Hermitian matrices are extended to those for arbitrary matrix on one hand. On th...In this paper, the theoretical analysis for the Rayleigh quotient matrix is studied, some results of the Rayleigh quotient （matrix） of Hermitian matrices are extended to those for arbitrary matrix on one hand. On the other hand, some unitarily invariant norm bounds for singular values are presented for Rayleigh quotient matrices. Our results improve the existing bounds.展开更多
The analogy between eigenvalues and singular values has many faces. The current review brings together several examples of this analogy. One example regards the similarity between Symmetric Rayleigh Quotients and Rect...The analogy between eigenvalues and singular values has many faces. The current review brings together several examples of this analogy. One example regards the similarity between Symmetric Rayleigh Quotients and Rectangular Rayleigh Quotients. Many useful properties of eigenvalues stem are from the Courant-Fischer minimax theorem, from Weyl’s theorem, and their corollaries. Another aspect regards “rectangular” versions of these theorems. Comparing the properties of Rayleigh Quotient matrices with those of Orthogonal Quotient matrices illuminates the subject in a new light. The Orthogonal Quotients Equality is a recent result that converts Eckart-Young’s minimum norm problem into an equivalent maximum norm problem. This exposes a surprising link between the Eckart-Young theorem and Ky Fan’s maximum principle. We see that the two theorems reflect two sides of the same coin: there exists a more general maximum principle from which both theorems are easily derived. Ky Fan has used his extremum principle (on traces of matrices) to derive analog results on determinants of positive definite Rayleigh Quotients matrices. The new extremum principle extends these results to Rectangular Quotients matrices. Bringing all these topics under one roof provides new insight into the fascinating relations between eigenvalues and singular values.展开更多
文摘The refined Arnoldi method proposed by Jia is used for computing some eigen-pairs of large matrices. In contrast to the Arnoldi method, the fundamental dif-ference is that the refined method seeks certain refined Ritz vectors, which aredifferent from the Ritz vectors obtained by the Arnoldi method, from a projection space with minimal residuals to approximate the desired eigenvectors. In com-parison with the Ritz vectors, the refined Ritz vectors are guaranteed to converge theoretically and can converge much faster numerically. In this paper we propose to replace the Ritz values, obtained by the Arnoldi method with respect to a Krylovsubspace, by the ones obtained with respect to the subspace spanned by the refined Ritz vectors. We discuss how to compute these new approximations cheaply and reliably. Theoretical error bounds between the original Ritz values and the new Ritz values are established. Finally, we present a variant of the refined Arnoldi al-gorithm for an augmented Krylov subspace and discuss restarting issue. Numerical results confirm efficiency of the new algorithm.
基金supported by National Natural Science Foundation of China (No. 10761003)by the Foundation of Guizhou Province Scientific Research for Senior Personnel, China
文摘This paper extends the two-grid discretization scheme of the conforming finite elements proposed by Xu and Zhou （Math. Comput., 70 （2001）, pp.17-25） to the nonconforming finite elements for eigenvalue problems. In particular, two two-grid discretization schemes based on Rayleigh quotient technique are proposed. By using these new schemes, the solution of an eigenvalue problem on a fine mesh is reduced to that on a much coarser mesh together with the solution of a linear algebraic system on the fine mesh. The resulting solution still maintains an asymptotically optimal accuracy. Comparing with the two-grid discretization scheme of the conforming finite elements, the main advantages of our new schemes are twofold when the mesh size is small enough. First, the lower bounds of the exact eigenvalues in our two-grid discretization schemes can be obtained. Second, the first eigenvalue given by the new schemes has much better accuracy than that obtained by solving the eigenvalue problems on the fine mesh directly.
文摘In this paper, the theoretical analysis for the Rayleigh quotient matrix is studied, some results of the Rayleigh quotient （matrix） of Hermitian matrices are extended to those for arbitrary matrix on one hand. On the other hand, some unitarily invariant norm bounds for singular values are presented for Rayleigh quotient matrices. Our results improve the existing bounds.
文摘The analogy between eigenvalues and singular values has many faces. The current review brings together several examples of this analogy. One example regards the similarity between Symmetric Rayleigh Quotients and Rectangular Rayleigh Quotients. Many useful properties of eigenvalues stem are from the Courant-Fischer minimax theorem, from Weyl’s theorem, and their corollaries. Another aspect regards “rectangular” versions of these theorems. Comparing the properties of Rayleigh Quotient matrices with those of Orthogonal Quotient matrices illuminates the subject in a new light. The Orthogonal Quotients Equality is a recent result that converts Eckart-Young’s minimum norm problem into an equivalent maximum norm problem. This exposes a surprising link between the Eckart-Young theorem and Ky Fan’s maximum principle. We see that the two theorems reflect two sides of the same coin: there exists a more general maximum principle from which both theorems are easily derived. Ky Fan has used his extremum principle (on traces of matrices) to derive analog results on determinants of positive definite Rayleigh Quotients matrices. The new extremum principle extends these results to Rectangular Quotients matrices. Bringing all these topics under one roof provides new insight into the fascinating relations between eigenvalues and singular values.
