Optimal and Superoptimal Circulant Preconditioners

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Abstract
While applying iterative methods to solve a linear algebraic system with matrix A, one often uses some preconditioner C. Two kinds of preconditioners are investigated: the “optimal” one, which minimizes $\| C - A \|_F$, and the “superoptimal” one, which minimizes $\| I - C^{ - 1} A \|_F $. It is proved that both inherit nonsingularity and positive-definiteness from A. Fast algorithms for finding superoptimal preconditioners are suggested that take $O(n^2 \log _2 n)$ operations in case of arbitrary A of order n, and only $O(n\log _2 n)$ operations in case of Toeplitz or doubly Toeplitz A.

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