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In [McDonald, Pestana and Wathen, SIAM J. Sci. Comput., 40 (2018), pp. A1012-A1033], a block circulant preconditioner is proposed for all-at-once linear systems arising from evolutionary partial differential equations, in which the preconditioned matrix is proven to be diagonalizable and to have identity-plus-low-rank decomposition under the setting of heat equation. In this paper, we generalize the block circulant preconditioner by introducing a small parameter ǫ > 0 into the top-right block of the block circulant preconditioner. The implementation of the generalized preconditioner requires the same computational complexity as that of the block circulant one. Theoretically, we prove that (i) the generalization preserves the diagonalizability and the identityplus-low-rank decomposition; (ii) all eigenvalues of the new preconditioned matrix are clustered at 1 with the clustering radius positively related to ǫ; (iii) GMRES method for the preconditioned system has a linear convergence rate independent of size of the linear system when ǫ is taken to be smaller or comparable with square root of time-step size. Numerical results are reported to confirm the efficiency of the proposed preconditioner and to show that the generalization improves the performance of block circulant preconditioner.

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