N.L. Zamarashkin

A theory of pseudoskeleton approximations

Abstract Let an m × n matrix A be approximated by a rank- r matrix with an accuracy e. We prove that it is possible to choose r columns and r rows of A forming a so-called pseudoskeleton component which approximates A with O ( e √ r (√ m + √ n )) accuracy in the sense of the 2-norm. On the way to this estimate we study the interconnection between the volume (i.e., the determinant in the absolute value) and the minimal singular value σ r of r × r submatrices of an n × r matrix with orthogonal columns. We propose a lower bound (better than one given by Chandrasekaran and Ipsen and by Hong and Pan) for the maximum of σ r over all these submatrices and formulate a hypothesis on a tighter bound.

SPECTRA OF MULTILEVEL TOEPLITZ MATRICES : ADVANCED THEORY VIA SIMPLE MATRIX RELATIONSHIPS

We consider the eigenvalue and singular-value distributions for m-level Toeplitz matrices generated by a complex-valued periodic function ƒ of m real variables. We show that familiar formulations for ƒ L∞ (due to Szegő and others) can be preserved so long as f L1, and what is more, with G. Weyls definitions just a bit changed. In contrast to other approaches, the one we follow is based on simple matrix relationships.

Clusters, preconditioners, convergence

Abstract We present a technique that relates the existence of a cluster of singular values to the existence of a cluster of eigenvalues. We also analyze some geometrical properties of eigenvalue clusters and their clusterization speed.

Toeplitz eigenvalues for Radon measures

AbstractIt is well known that for Toeplitz matrices generated by a “sufficiently smooth” real-valuedsymbol, the eigenvalues behave asymptotically as the values of the symbol on uniform mesheswhile the singular values, even for complex-valued functions, do as those values in modulus.These facts are expressed analytically by the Szego and Szego-like formulas, and, as is provedrecently, the “smoothness” assumptions are as mild as those of L 1 . In this paper, it is shownthat the Szego-like formulas hold true even for Toeplitz matrices generated by the so-calledRadon measures.

Thin structure of eigenvalue clusters for non-Hermitian Toeplitz matrices

In contrast to the Hermitian case, the “unfair behavior” of non-Hermitian Toeplitz eigenvalues is still to be unravelled. We propose a general technique for this, which reveals the eigenvalue clusters for symbols from L∞. Moreover, we study a thin structure of those clusters in the terms of properly defined subclusters. In some cases, this leads to as much as the Szego-like formulas.

On the decay of the elements of inverse triangular Toeplitz matrices

We consider half-infinite triangular Toeplitz matrices with slow decay of the elements and prove under a monotonicity condition that elements of the inverse matrix, as well as elements of the fundamental matrix, decay to zero. We also provide a quantitative description of the decay of the fundamental matrix in terms of p-norms. Finally, we prove that for matrices with slow log-convex decay the inverse matrix has fast decay, i.e. is bounded. The results are compared with the classical results of Jaffard and Veccio and illustrated by numerical example.

Fast truncation of mode ranks for bilinear tensor operations

SUMMARY We propose a fast algorithm for mode rank truncation of the result of a bilinear operation on 3-tensors given in the Tucker or canonical form. If the arguments and the result have mode sizes n and mode ranks r, the computation costs (nr3 + r4). The algorithm is based on the cross approximation of Gram matrices, and the accuracy of the resulted Tucker approximation is limited by square root of the machine precision. We apply the proposed algorithms for the evaluation of the Hadamard square of the electron density and demonstrate that it outperforms previously used methods for this purpose. We also check the accuracy of the resulted Tucker approximation and show that one iteration of the Tucker-ALS method improves it almost up to the machine precision. Copyright

Tensor-Train Ranks for Matrices and Their Inverses

Abstract We show that the recent tensor-train (TT) decompositions of matrices come up from its recursive Kronecker-product representations with a systematic use of common bases. The names TTM and QTT used in this case stress the relation with multilevel matrices or quantization that increases artificially the number of levels. Then we investigate how the tensor-train ranks of a matrix can be related to those of its inverse. In the case of a banded Toeplitz matrix, we prove that the tensor-train ranks of its inverse are bounded above by 1+(l+u)^2, where l and u are the bandwidths in the lower and upper parts of the matrix without the main diagonal.

On eigen and singular value clusters

Usually when singular values are clustered, the eigenvalues behave similarly. However, it is not the case if we make no assumptions. Here we present examples when the singular values are clustered whereas the eigenvalues are not, and vice versa. Besides, the necessary and sufficient assumptions are discussed under which the former implies the latter. We also present a new algebraic approach to one-point clusters.

A general equidistribution theorem for the roots of orthogonal polynomials

Abstract It is well-known that the roots of any two orthogonal polynomials are distributed equally if the weights satisfy the Szegő condition. In this paper, we propose a general equidistribution theorem that does not use the Szegő condition and admits an elementary proof.