It is known that the restarted full orthogonalization method (FOM) outperforms the restarted generalized minimum residual (GMRES) method in several circumstances for solving shifted linear systems when the shifts are handled simultaneously. Many variants of them have been proposed to enhance their performance. We show that another restarted method , the restarted Hessenberg method [M. Heyouni, Méthode de Hessenberg Généralisée et Applications (Ph.D. Thesis), Université des Sciences et Technologies de Lille, France, 1996] based on Hessenberg procedure, can effectively be employed, which can provide accelerating convergence rate with respect to the number of restarts. Theoretical analysis shows that the new residual of shifted restarted Hessenberg method is still collinear with each other. In these cases where the proposed algorithm needs less enough elapsed CPU time to converge than the earlier established restarted shifted FOM, the weighted restarted shifted FOM, and some other popular shifted iterative solvers based on the short-term vector recurrence, as shown via extensive numerical experiments involving the recently popular application of handling time fractional differential equations.
Abstract. The contour-integral based eigensolvers are the recent efforts for computing the eigenvalues inside a given region in the complex plane. The best-known members are the Sakurai-Sugiura (SS) method, its stable version CIRR, and the FEAST algorithm. An attractive computational advantage of these methods is that they are easily parallelizable. The FEAST algorithm was developed for the generalized Hermitian eigenvalue problems. It is stable and accurate. However, it may fail when applied to non-Hermitian problems. In this paper, we extend the FEAST algorithm to non-Hermitian problems. The approach can be summarized as follows: (i) to construct a particular contour integral to form a subspace containing the desired eigenspace, and (ii) to use the oblique projection technique to extract desired eigenpairs with appropriately chosen test subspace. The related mathematical framework is established. We also address some implementation issues such as how to choose a suitable starting matrix and design good stopping criteria. Numerical experiments are provided to illustrate that our method is stable and efficient.
A generalized refined Arnoldi method based on the weighted inner product is presented for computing PageRank. The properties of the generalized refined Arnoldi method were studied. To speed up the convergence performance for computing PageRank, we propose to change the weights adaptively where the weights are calculated based on the current residual corresponding to the approximate PageRank vector. Numerical results show that the proposed Arnoldi method converges faster than existing methods, in particular when the damping factor is large.Usually, the matrix A involved is extremely large, so that direct decomposition techniques (such as LU and QR decomposition) cannot be considered for computing PageRank. Iterative methods based on matrix-vector products have been widely studied for this computation.The power method was firstly considered for computing PageRank. However, the eigenvalues of A is the scaling of those of P , except for the dominant eigenvalue, for details, see the proofs in [7,8]. Thus, when the largest eigenvalue of A is not well separated from other eigenvalues and the damping factor˛is close to 1, then the power method converges very slowly. Many researchers proposed several methods to accelerate the convergence of the power method. For instance, the quadratic extrapolation method [9], the adaptive method [10], and the block method [2] are employed. The extrapolation method can improve the convergence performance of the power method, however, it still strongly depends on the convergence speed of the power method and it may not be effective at each step of the extrapolation method.The adaptive method [10] can save the computation of matrix-vector products, while the drawback of the adaptive method is that it cannot improve the convergence performance of the power method. To make the power method more effective, the block structure of the web is taken into use for computing the PageRank. The web structure should be exploited and the matrix should be preprocessed and managed in the right manner so that the effective block method can be employed.On the other hand, other iterative methods are studied for computing PageRank. Iterative methods based on the Arnoldi process are good alternatives for computing the dominant eigenvector. We note that the largest eigenvalue of PageRank matrix A is known to be 1. Golub and Greif [11] proposed an Arnoldi-type method combined with SVD by considering the known largest eigenvalue as a shift so that the computation of the largest Ritz value is avoided. This approach is very efficient as it can avoid the complex arithmetic even if the largest Ritz value is complex. Wu and Wei [12] also proposed an Arnoldi-type method combined with the power method. In their approach, the thick restarted Arnoldi method was used. However, the largest Ritz value is required to be computed in their method. The complex arithmetic may be needed. In addition, the corresponding Ritz vector may be complex-valued, both the real and imaginary parts of the Ritz vector are taken as the restart ...
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