Volume 41, pp. 478-496, 2014.

A deflated block flexible GMRES-DR method for linear systems with multiple right-hand sides

Jing Meng, Pei-Yong Zhu, Hou-Biao Li, and Xian-Ming Gu


This study is mainly focused on the iterative solution of multiple linear systems with several right-hand sides. To solve such systems efficiently, we first present a flexible version of block GMRES with deflation of eigenvalues according to [R. B. Morgan, Restarted block-GMRES with deflation of eigenvalues, Appl. Numer. Math., 54 (2005), pp. 222–236] and then apply a modified block Arnoldi vector deflation technique to accelerate the convergence of this new flexible version. Incorporating this deflation technique, the new algorithm can address the possible linear dependence at each iteration during the block Arnoldi procedure and reduce computational expense. Moreover, by analyzing its main mathematical properties, we show that the vector deflation procedure arises from the non-increasing behavior of the singular values of the block residual. In addition, the new approach also inherits the property of deflating small eigenvalues to mitigate convergence slowdown. Finally, the effectiveness of the proposed method is illustrated by some numerical experiments.

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Key words

deflated BFGMRES-DR, block Krylov subspace, modified block Arnoldi vector deflation, harmonic Ritz vectors, deflated block flexible Arnoldi procedure, multiple right-hand sides

AMS subject classifications


Links to the cited ETNA articles

[11]Vol. 16 (2003), pp. 129-142 A. El Guennouni, K. Jbilou, and H. Sadok: A block version of BiCGSTAB for linear systems with multiple right-hand sides
[20]Vol. 20 (2005), pp. 75-85 Na Li and Yousef Saad: Crout versions of ILU factorization with pivoting for sparse symmetric matrices
[22]Vol. 30 (2008), pp. 1-9 Hualei Liu and Baojiang Zhong: Simpler Block GMRES for nonsymmetric systems with multiple right-hand sides

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