1,651 research outputs found
A flexible and adaptive Simpler GMRES with deflated restarting for shifted linear systems
In this paper, two efficient iterative algorithms based on the simpler GMRES
method are proposed for solving shifted linear systems. To make full use of the
shifted structure, the proposed algorithms utilizing the deflated restarting
strategy and flexible preconditioning can significantly reduce the number of
matrix-vector products and the elapsed CPU time. Numerical experiments are
reported to illustrate the performance and effectiveness of the proposed
algorithms.Comment: 17 pages. 9 Tables, 1 figure; Newly update: add some new numerical
results and correct some typos and syntax error
A note on the growth factor in Gaussian elimination for generalized Higham matrices
The Higham matrix is a complex symmetric matrix A=B+iC, where both B and C
are real, symmetric and positive definite and is the
imaginary unit. For any Higham matrix A, Ikramov et al. showed that the growth
factor in Gaussian elimination is less than 3. In this paper, based on the
previous results, a new bound of the growth factor is obtained by using the
maximum of the condition numbers of matrixes B and C for the generalized Higham
matrix A, which strengthens this bound to 2 and proves the Higham's conjecture.Comment: 8 pages, 2 figures; Submitted to MOC on Dec. 22 201
Restarted Hessenberg method for solving shifted nonsymmetric linear systems
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, M\'ethode de Hessenberg G\'en\'eralis\'ee et
Applications, Ph.D. Thesis, Universit\'e 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 CPU time elapsed to converge than the
earlier established restarted shifted FOM, 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 recent
popular applications of handling the time fractional differential equations.Comment: 19 pages, 7 tables. Some corrections for updating the reference
- …