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Approximate inverse preconditioners via sparse-sparse iterations. (English) Zbl 0922.65034
The authors consider iteration methods for finding approximate inverse preconditioners \(M\) to the inverse \(A^{-1}\) of a given \((n\times n)\)-matrix \(A\). The iterative methods aim at the minimization of the functional \[ F(M):= \| I- AM\|^2_F= \sum^n_{j=1}\| e_j- Am_j\|^2_2 \] on the space of all \((n\times n)\)-matrices, where \(\|\cdot\|_F\) denotes the Frobenius norm, \(\|\cdot\|_2\) is the Euclidean norm in \(\mathbb{R}^n\), \(e_j\) and \(m_j\) are the \(j\)th columns of the identity matrix \(I\) and of the matrix \(M\), respectively. The authors propose and analyze several iterative methods (Newton, MR, GMRES) with several modifications (numerical dropping in the iterates or in the search directions, self-preconditioning etc.). The different techniques are compared numerically on several examples taken from the well-known Harwell-Boeing collection and from matrices generated by the fluid dynamics analysis package FIDAP.
Reviewer: U.Langer (Linz)

MSC:
65F35 Numerical computation of matrix norms, conditioning, scaling
65F10 Iterative numerical methods for linear systems
65F50 Computational methods for sparse matrices
65Y05 Parallel numerical computation
Software:
FIDAP; ILUS
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