Kronecker-product approximations for some function-related matrices

AbstractA new approximation tool such as sums of Kronecker products is recently found to provide a superb compression property on a series of numerical examples of quite a general nature. The purpose of the paper is explanation of this phenomenon in the form of “existence theorems” for matrix approximations of low Kronecker rank for some classes of function-related matrices including important specimens from potential theory. This lays the grounds for development of new approximation algorithms, for example, in the cases when a matrix is associated with a shift-invariant function on the Cartesian product of nonunform grids, which is of great practical interest in the solution of integral equations on plates or screens

Circulant preconditioners with unbounded inverses

AbstractThe eigenvalue and singular-value distributions for matrices S−1nAn and C−1nAn are examined, where An, Sn, and Cn are Toeplitz matrices, simple circulants, and optimal circulants generated by the Fourier expansion of some function f. Recently it has been proven that a cluster at 1 exists whenever f is from the Wiener class and strictly positive. Both restrictions are now weakened. A proof is given for the case when f may take the zero value, and hence the circulants are to have unbounded inverses. The main requirements on f are that it belong to L2 and be in some sense, sparsely vanishing. Specifically, if f is nonnegative and circulants Sn (or Cn) are positive definite, then the eigenvalues of S−1nAn (or C−1nAn) are clustered at 1. If f is complex-valued and Sn (or Cn) are nonsingular, then the singular values of S−1nAn (or C−1nAn) are clustered at 1 as well. Also proposed and studied are the improved circulants. It is shown that (improved) simple circulants can be much more advantageous than optimal circulants. This depends crucially on the smoothness properties of f. Further, clustering-on theorems are given that pertain to multilevel Toeplitz matrices preconditioned by multilevel simple and optimal circulants

Matrix Bruhat decompositions with a remark on the QR (GR) algorithm

AbstractIn a simple and systematic way we present matrix Bruhat decompositions of two kinds: basic and modified. We show that it is the modified Bruhat decomposition that governs the eigenvalue disorder in the QR (GR) algorithm. This paper can be considered as a commentary on a previous observation about the QR algorithm made by Wilkinson

Procedure of practical exercise with students on the pathogenic effect of accelerations on the organism

The effects of acceleration alone and coupled with administration of either aminazine (chlorpromazine- a sedative) or caffeine (a stimulant) on the development of kinetoses in mice were studied. The problem is presented as a method to teach students and to demonstrate the role of the nervous factor in the development of kinetosis

Influence of matrix operations on the distribution of Eigenvalues and singular values of Toeplitz matrices

AbstractSuppose some Toeplitz matrix families {An(ƒα)} are given, generated by the Fourier expansions for ƒα, and a new family{An} is constructed from An(ƒα) via basic matrix operations. Theorems are proved that describe the singular-value distribution for An in the terms of ƒα, as well as the eigenvalue distribution for H(An)≡(An+ A∗n)2 and K(An)≡(An−A∗n)2i. In particular, if ƒα∈L∞ and only multiplication is used, then we show the singular values of An are distributed as |ƒ(x)|, where ƒ(x)=∏ƒα(x). At the same time, the eigenvalues of H(An) are distributed as Re ƒ(x), while those of K(An) are distributed as Im ƒ(x). The extension to multilevel Toeplitz matrices is also suggested. Finally, an application to circulant preconditioning is discussed

Singular values of cauchy-toeplitz matrices

AbstractThe behavior of singular values of matrices An =[1/(i−j+g)]ni,j=1 with n→∞ is investigated. For any real g which is not integer it is proved that the singular values are clustered at π / ⋎sin π g⋎, which is their upper boundary. The only o(n) singular values are those which lie outside a given ε-neighborhood of the clustering point [o(n)/n→0 as n→∞]; o(n) = O(ln2n) holds if ⋎g⋎ ⩽12. Also proved is that the minimum singular values of An(g) tend to zero provided that ⋎g⋎⩾12

Time stepping free numerical solution of linear differential equations: Krylov subspace versus waveform relaxation

The aim of this paper is two-fold. First, we propose an efficient implementation of the continuous time waveform relaxation method based on block Krylov subspaces. Second, we compare this new implementation against Krylov subspace methods combined with the shift and invert technique

Face vectors of subdivided simplicial complexes

Brenti and Welker have shown that for any simplicial complex X, the face vectors of successive barycentric subdivisions of X have roots which converge to fixed values depending only on the dimension of X. We improve and generalize this result here. We begin with an alternative proof based on geometric intuition. We then prove an interesting symmetry of these roots about the real number -2. This symmetry can be seen via a nice algebraic realization of barycentric subdivision as a simple map on formal power series in two variables. Finally, we use this algebraic machinery with some geometric motivation to generalize the combinatorial statements to arbitrary subdivision methods: any subdivision method will exhibit similar limit behavior and symmetry. Our techniques allow us to compute explicit formulas for the values of the limit roots in the case of barycentric subdivision.Comment: 13 pages, final version, appears in Discrete Mathematics 201

Optimal rank matrix algebras preconditioners

When a linear system Ax = y is solved by means of iterative methods (mainly CG and GMRES) and the convergence rate is slow, one may consider a preconditioner P and move to the preconditioned system P-1 Ax = P(-1)y. The use of such preconditioner changes the spectrum of the matrix defining the system and could result into a great acceleration of the convergence rate. The construction of optimal rank preconditioners is strongly related to the possibility of splitting A as A = P R E. where E is a small perturbation and R is of low rank (Tyrtyshnikov, 1996) [1]. In the present work we extend the black-dot algorithm for the computation of such splitting for P circulant (see Oseledets and Tyrtyshnikov, 2006 [2]), to the case where P is in A, for several known low-complexity matrix algebras A. The algorithm so obtained is particularly efficient when A is Toeplitz plus Hankel like. We finally discuss in detail the existence and the properties of the decomposition A = P+R+E when A is Toeplitz, also extending to the phi-circulant and Hartley-type cases some results previously known for P circulant