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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 MS AM-I F =AN-I F is considered, where · F denotes the Frobenius norm and S is a subspace of the space of all n×n matrices with real coefficients M n ().

In a more general framework, the author analyses the problem min PT P-I F =Q-I F with an arbitrary subspace T of the space M n (). 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.

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