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Orthogonal projections of the identity: spectral analysis and applications to approximate inverse preconditioning. (English) Zbl 1095.65042
The construction of optimal preconditioners for linear systems $Ax=b$ is discussed. For the construction of the preconditioner $N$ of the preconditioned system $ANy=b$, $x = Ny$, the minimization problem $\min_{M \in S} \Vert AM-I\Vert _F = \Vert AN-I\Vert _F$ is considered, where $\Vert . \Vert _F$ denotes the Frobenius norm and $S$ is a subspace of the space of all $n \times n$ matrices with real coefficients $M_n(\bbfR)$. In a more general framework, the author analyses the problem $\min_{P \in T} \Vert P-I \Vert _F = \Vert Q-I \Vert _F$ with an arbitrary subspace $T$ of the space $M_n(\bbfR)$. At first some spectral properties of the solution $Q$ are established. Then these results are applied to analyse the effectiveness of the approximate inverse preconditioner $N$. The main result is the following: When the smallest singular value or the smallest eigenvalue’s modulus of the matrix $AN$ increases to 1 the effectiveness of the preconditioner $N$ improves.

65F35Matrix norms, conditioning, scaling (numerical linear algebra)
15A12Conditioning of matrices
15A18Eigenvalues, singular values, and eigenvectors
15A60Applications of functional analysis to matrix theory
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