Least Squares Arnoldi for Large Nonsymmetric Eigenproblems

Akira Nishida

University of Tokyo, 113, JAPAN


In this paper, we propose a highly efficient accelerating method for the restarted Arnoldi iteration to compute the eigenvalues of a large nonsymmetric matrix. Its effectiveness is proved by various numerical experiments and comparisons with other approaches. Several new results on the characteristics of the polynomial acceleration are also reported.
The Arnoldi iteration has been the most popular method for nonsymmetric large eigenproblems. Although the defect of increasing computational complexity per iteration step can be improved with the explicitly restarting technique, by which the dimensions of the Krylov subspaces are kept modest, the dimension of the subspace becomes large, in particular when the required eigenvalues are clustered. In our study, an accelerating polynomial is chosen to minimize an $L_2$ norm of the polynomial on the boundary of the convex hull with respect to some suitable weight function. A new simple algorithm is proposed for the efficient computation of the mini-max polynomial to accelerate the convergence of the Arnoldi iteration.
>From the numerical results, we can derive the strong dependency of the polynomial acceleration on the distribution of spectrum, which proves the better performance of our algorithm than the Chebyshev acceleration, in the cases where the moduli of the wanted eigenvalues are considerably larger than those of the unwanted eigenvalues, and the faster convergence than those of all the other approaches, especially when the shape of the convex hull of the unwanted eigenvalues bears little resemblance with an ellipse.
Finally, we propose a new parallelization technique for the nonsymmetric double shifted QR algorithm with perfect load balance and uninterrupted pipelining on distributed memory parallel architectures, which is strongly required from the viewpoint of complexity of the Arnoldi iteration. Its parallel efficiency is much higher than those reported in other papers.