SPIKE As A Preconditioner
The SPIKE algorithm can also function as a preconditioner for iterative methods for solving linear systems. To solve a linear system Ax = b using a SPIKE-preconditioned iterative solver, one extracts center bands from A to form a banded preconditioner M and solves linear systems involving M in each iteration with the SPIKE algorithm.
In order for the preconditioner to be effective, row and/or column permutation is usually necessary to move “heavy” elements of A close to the diagonal so that they are covered by the preconditioner. This can be accomplished by computing the weighted spectral reordering of A.
The SPIKE algorithm can be generalized by not restricting the preconditioner to be strictly banded. In particular, the diagonal block in each partition can be a general matrix and thus handled by a direct general linear system solver rather than a banded solver. This enhances the preconditioner, and hence allows better chance of convergence and reduces the number of iterations.
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