L.Yu. Kolotilina

A robust AINV-type method for constructing sparse approximate inverse preconditioners in factored form

This paper suggests a new method, called AINV-A, for constructing sparse approximate inverse preconditioners for positive-definite matrices, which can be regarded as a modification of the AINV method proposed by Benzi and Tuma. Numerical results on SPD test matrices coming from different applications demonstrate the robustness of the AINV-A method and its superiority to the original AINV approach. Copyright

Lower bounds for the perron root of a nonnegative matrix

Abstract New lower bounds, frequently better than previous bounds and readily computable, for the Perron root r ( A ) of a nonnegative matrix A are obtained. The theoretical comparison of new bounds with the known ones is supplemented with two numerical examples.

Nonsingularity/singularity criteria for nonstrictly block diagonally dominant matrices

Abstract Let A=(A ij ) N i,j=1 ∈ C n×n be a block irreducible matrix with nonsingular diagonal blocks, v=(v i )∈ C N be a positive vector, and let ∑ N j=1 [−2pt]j≠i ∥A −1 ii A ij ∥v j ⩽v i , i=1,…,N. Under these assumptions, necessary and sufficient conditions for A to be singular are obtained based on a block generalization of Wielandt’s lemma. The pointwise case ( N=n ) of irreducible matrices with nonstrict generalized diagonal dominance is treated separately. For an irreducible matrix A , conditions necessary and sufficient for a boundary point of the union of the Gerschgorin’s circles and of the union of the ovals of Cassini to be an eigenvalue of A are derived.

Generalizations of the Ostrowski-Brauer theorem

Abstract The main theorem of this paper, which generalizes the Ostrowski–Brauer theorem and its previous extensions, provides conditions necessary and sufficient for the singularity of an irreducible matrix A=(a ij )∈ C n×n satisfying the conditions |a ii ||a jj |⩾R i (A)R j (A), where R k (A)=∑ j≠k |a kj |, k=1,…,n, for all i≠j such that |aij|+|aji|≠0 and implies a new description of the location of matrix eigenvalues in terms of ovals of Cassini and Gerschgorin circles.

EIGENVALUE BOUNDS AND INEQUALITIES USING VECTOR AGGREGATION OF MATRICES

Abstract For matrices partitioned into block form, the operation of (block)-vector aggregation, which associates with a given matrix a matrix of smaller size, is introduced. Properties of aggregated matrices are analyzed. In particular, it is shown that in the Hermitian case, the eigenvalues of a block-vector-aggregated matrix interlace those of the original matrix. By using vector aggregation, new eigenvalue bounds and inequalities for normal and Hermitian matrices are derived and put in context with the existing ones. In particular, inequalities interrelating eigenvalues of a block-partitioned Hermitian matrix with those of its diagonal blocks are obtained. Also it is shown that the spectral constants characterizing the block partitioning of a Hermitian matrix are bounded below by the corresponding constants related to associated vector-aggregated matrices.

Two-sided bounds for the inverse of an H-matrix

Abstract The paper presents a simple characterization of a real H -matrix and two-sided componentwise bounds for its inverse in terms of the comparison matrix and the so-called Z- and positive parts. These bounds, improving the well-known Ostrowsky result, are also extended to a larger class of matrices.

An incomplete LU -factorization algorithm based on block bordering.

This paper suggests a new algorithm, called IBBLU, for constructing incomplete LU-factorizations of general non-singular sparse matrices. This algorithm is based on the block bordering idea and uses factorized sparse approximate inverses to approximate the inverses of pivot blocks. The triangular factors L and U are represented in explicit-implicit block form, which enhances the flop performance of the preconditioning. The algorithm suggested is theoretically justified for M- and H-matrices, and its competitiveness with the best available algebraic preconditioning methods in both symmetric and unsymmetric cases is demonstrated numerically. Copyright

Incomplete Block Factorizations as Preconditioners for Sparse SPD Matrices

The paper deals with incomplete block factorization (IBF) preconditionings for block tridiagonal SPD matrices. Two schemes of IBF’s (the standard one and the one in inverse free form) are considered and shown to give rise to SPD preconditioned under certain general assumptions on sparse approximations to pivot blocks. An implementation of CG iterations preconditioned with an IBF in inverse free form on a two-dimensional grid of processors is suggested. Performance of IBF-CG iterations is demonstrated by numerical results for 3D linear elasticity problems.

Optimally conditioned block matrices

This paper provides a description of block 2 × 2 Hermitian positive-definite matrices that are optimally conditioned with respect to block diagonal similarity transformations. Bibliography: 7 titles.

Bounds for the eigenvalues of block 2 × 2 Hermitian positive‐definite matrices

Lower and upper conditional bounds for the eigenvalues of a Hermitian positive-definite block 2 × 2 matrix, describing their closest possible location, are derived. The notions of partially and totally optimal spectra are introduced, and several equivalent characterizations of matrices with partially and totally optimal spectra are presented. It is shown that the block 2 × 2 block Jacobi scaled matrices and the so-called equilibrated matrices have totally optimal spectra. Copyright