Contents -- Preface -- Algebra and Matrices -- Operators Preserving Primitivity for Matrix Pairs L. B. Beasley (Utah State University), A. E. Guterman (Moscow State University) -- 1 Introduction -- 2 Preliminaries -- 3 Matrices over the binary Boolean semiring -- 4 Matrices over antinegative semirings without zero divisors -- References -- Decompositions of Quaternions and Their Matrix Equivalents D. Janovska (Institute of Chemical Technology), G. Opfer (University of Hamburg) -- 1 Introduction -- 2 Decompositions of quaternions -- 2.1 Schur decompositions -- 2.2 The polar decomposition -- 2.3 The singular value decomposition (SVD) -- 2.4 The Jordan decomposition -- 2.5 The QR decomposition -- 2.6 The lU decomposition -- References -- Sensitivity Analysis of Hamiltonian and Reversible Systems Prone to Dissipation-Induced Instabilities O. N. Kirillov (Moscow State University) -- 1 Introduction -- 2 A circulatory system with small velocity-dependent forces -- 2.1 Stability of a circulatory system -- 2.2 The influence of small damping and gyroscopic forces on the stability of a circulatory system -- 3 A gyroscopic system with weak damping and circulatory forces -- 3.1 Stability of a gyroscopic system -- 3.2 The influence of small damping and non-conservative positional forces on the stability of a gyroscopic system -- 4 The modified Maxwell-Bloch equations with mechanical applications -- 4.1 Stability of Hauger's gyropendulum -- 4.2 Friction-induced instabilities in rotating elastic bodies of revolution -- 4.2.1 Deformation of the spectral mesh -- Conclusions -- Acknowledgments -- References -- Block Triangular Miniversal Deformations of Matrices and Matrix Pencils L. Klimenko (Information and Computing Centre of Ministry of Labour and Social Policy of Ukraine), V. V. Sergeichuk (Kiev Institute of Mathematics) -- 1 Introduction.

2 Miniversal deformations of matrices -- 3 Miniversal deformations of matrix pencils -- 4 Miniversal deformations of contragredient matrix pencils -- References -- Determining the Schein Rank of Boolean Matrices E. E. Marenich (Murmansk State Pedagogic University) -- 1 Introduction -- 2 The Schein rank of Boolean matrices -- 3 Matrices of Schein rank 2, 3 -- 4 On coding of bipartite graphs by sets -- 5 On canonical bipartite graphs -- 6 On the Schein rank of Boolean matrices and the intersection number of associated graphs -- 7 On coding of bipartite graphs by antichains -- 8 Bipartite intersection graphs k,p,q -- 9 The Schein rank of nxn -- Acknowledgments -- References -- Lattices of Matrix Rows and Matrix Columns. Lattices of Invariant Column Eigenvectors V. Marenich (Murmansk State Pedagogic University) -- 1 Introduction -- 2 Preliminaries -- 2.1 Lattice matrices and column vectors -- 2.2 Brouwerian lattices -- 2.3 Boolean lattices -- 2.4 Systems of linear equations over Brouwerian lattices -- 3 Semilattices and lattices of matrix columns and matrix rows -- 4 Subspaces of A-invariant column eigenvectors -- 5 Lattices of invariant column vectors -- Acknowledgments -- References -- Matrix Algebras and Their Length O. V. Markova (Moscow State University) -- 1 Main definitions and notation -- 2 Introduction -- 3 On the lengths of algebra and its subalgebras -- 3.1 Two block subalgebras in upper triangular matrix algebra -- 3.2 Three block subalgebras in upper triangular matrix algebra -- 3.3 Examples -- References -- On a New Class of Singular Nonsymmetric Matrices with Nonnegative Integer Spectra T. Nahtman (University of Tartu), D. von Rosen (Swedish University of Agricultural Sciences) -- 1 Introduction -- 2 Preparations -- 3 Preliminaries -- 4 Triangular factorization -- 5 Eigenvectors of the matrix B -- References.

Reduction of a Set of Matrices over a Principal Ideal Domain to the Smith Normal Forms by Means of the Same One-Sided Transformations. V. M. Prokip (Pidstryhach Institute for Applied Problems of Mechanics and Mathematics) -- Acknowledgments -- References -- Matrices and Algorithms -- Nonsymmetric Algebraic Riccati Equations Associated with an M-Matrix: Recent Advances and Algorithms D. A . Bini (University of Pisa), B. Iannazzo (University of Insubria), B. Meini (University of Pisa), F. Poloni (Scuola Normale Superiore of Pisa) -- 1 Introduction -- 1.1 Application to fluid queues -- 1.2 Application to transport equation -- 2 Theoretical properties -- 2.1 Some useful facts about nonnegative matrices -- 2.2 The dual equation -- 2.3 Existence of nonnegative solutions -- 2.4 The eigenvalue problem associated with the matrix equation -- 2.5 The eigenvalues of H -- 2.6 The differential of the Riccati operator -- 2.7 The number of positive solutions -- 2.8 Perturbation analysis for the minimal solution -- 3 Numerical methods -- 3.1 Schur method -- 3.2 Functional iterations -- 3.3 Newton's method -- 3.4 Doubling algorithms -- 4 Exploiting the singularity of H -- 4.1 The shift technique -- 4.2 Choosing a new initial value -- 5 Numerical experiments and comparisons -- References -- A Generalized Conjugate Direction Method for Nonsymmetric Large Ill-Conditioned Linear Systems E. R. Boudinov (FORTIS Bank, Brussels), A . I. Manevich (Dnepropetrovsk National University) -- 1 Introduction -- 2 Basic algorithm -- 3 Final algorithm -- 4 Numerical experiments -- 5 Conclusions -- References -- There Exist Normal Hankel ( )-Circulants of Any Order n V. N. Chugunov (INM RAS), Kh. D. Ikramov (Moscow State University) -- References.

On the Treatment of Boundary Artifacts in Image Restoration by Reflection and/or Anti-Reflection M. Donatelli (University of Insubria), S. Serra-Capizzano (University of Insubria) -- 1 Introduction -- 2 Reflection for image extension and BCs -- 3 Image extension by anti-reflection -- 4 Numerical experiments -- Acknowledgment -- References -- Zeros of Determinants of -Matrices W. Gander (ETH, Zurich) -- 1 Introduction -- 2 Matlab reverses computing -- 3 Evaluating the characteristic polynomial -- 4 Suppression instead of deflation -- 5 Example -- 6 Generalization to -matrices -- 7 Conclusion -- References -- How to Find a Good Submatrix S . A . Goreinov (INM RAS), I . V. Oseledets (INM RAS), D. V. Savostyanov (INM RAS), E. E. Tyrtyshnikov (INM RAS), N. L. Zamarashkin (INM RAS) -- 1 Introduction -- 1.1 Notation -- 1.2 Definitions and basic lemmas -- 2 Algorithm maxvol -- 3 maxvol-based maximization methods -- 3.1 Theoretical estimates -- 3.2 Search for the maximum element in random low-rank matrices -- 3.3 Maximization of bivariate functions -- 4 Conclusion and future work -- References -- Conjugate and Semi-Conjugate Direction Methods with Preconditioning Projectors V. P. 'in (Novosibirsk Institute of Compo Math. and Math. Geophys) -- 1 Introduction -- 2 Multiplicative projector methods -- 3 Additive projective methods -- 4 Iterations in Krylov subspaces with dynamic preconditioning -- References -- Some Relationships between Optimal Preconditioner and Superoptimal Preconditioner J.-B. Chen (Shanghai Maritime University), X.-Q . Jin (University of Macau), Y .-M. Wei (Fudan University), Zh.-L. Xu (Shanghai Maritime University) -- 1 Introduction -- 2 Relationships between cu and tu -- References -- Scaling, Preconditioning, and Superlinear Convergence in GMRES-Type Iterations I. Kaporin (Computing Center of Russian Academy of Sciences).

1 Introduction -- 2 Problem setting -- 3 Scaling techniques -- 4 How to estimate GMRES convergence -- 5 Superlinear GMRES convergence via scaled error matrix -- 5.1 Main result -- 5.2 The corresponding GMRES iteration number bound -- 5.3 Relating the new estimate to scaling -- 5.4 Relating the new estimate to ILU preconditioning -- 6 Numerical experiments -- 7 Conclusion -- Acknowledgments -- References -- Toeplitz and Toeplitz-Block-Toeplitz Matrices and Their Correlation with Syzygies of Polynomials H. Khalil (Institute Camille Jordan), B. Mourrain (INRIA), M. Schatzman (Institute Camille Jordan) -- 1 Introduction -- 2 Univariate case -- 2.1 Moving lines and Toeplitz matrices -- 2.2 Euclidean division -- 2.3 Construction of the generators -- 3 Bivariate case -- 3.1 Moving hyperplanes -- 3.2 Generators and reduction -- 3.3 Interpolation -- 4 Conclusions -- References -- Concepts of Data-Sparse Tensor-Product Approximation in Many-Particle Modelling H.-J. Flad (TU Berlin), W. Hackbusch (Max-Planck-Institute, Leipzig), R . Schneider (TU Berlin) -- 1 Introduction -- 2 Basic principles of electronic structure calculations -- 2.1 Density matrix formulation of Hartree-Fock and OFT models -- 3 Hyperbolic cross approximation in wavelet bases -- 4 Toolkit for tensor-product approximations -- 4.1 Tensor-product representations in higher dimension -- 4.2 Separable approximation of functions -- 4.2.1 Sine interpolation methods -- 4.2.2 Integral representation methods -- 4.2.3 On the best approximation by exponential sums -- 5 Data sparse formats for univariate components -- 5.1 Hierarchical matrix techniques -- 5.2 Hierarchical Kronecker tensor-product approximations -- 5.3 Wavelet Kronecker tensor-product approximations -- 6 Linear scaling methods for Hartree-Fock and Kohn-Sham equations -- 6.1 Matrix-valued functions approach for density matrices.

6.2 Computation of Hartree potentials in tensor-product formats.