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

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

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

Approximate iterations for structured matrices

Important matrix-valued functions f (A) are, e.g., the inverse A −1 , the square root √ A and the sign function. Their evaluation for large matrices arising from pdes is not an easy task and needs techniques exploiting appropriate structures of the matrices A and f (A) (often f (A) possesses this structure only approximately). However, intermediate matrices arising during the evaluation may lose the structure of the initial matrix. This would make the computations inefficient and even infeasible. However, the main result of this paper is that an iterative fixed-point like process for the evaluation of f (A) can be transformed, under certain general assumptions, into another process which preserves the convergence rate and benefits from the underlying structure. It is shown how this result applies to matrices in a tensor format with a bounded tensor rank and to the structure of the hierarchical matrix technique. We demonstrate our results by verifying all requirements in the case of the iterative computation of A −1 and √ A

Problems with Jumping Coefficients

We study separability properties of solutions of elliptic equations with piecewise constant coefficients in R d, d ≥ 2. Besides that, we develop efficient tensor-structured preconditioner for the diffusion equation with variable coefficients. It is based only on rank structured decomposition of the tensor of reciprocal coefficient and on the decomposition of the inverse of the Laplacian operator. It can be applied to full vector with linear-logarithmic complexity in the number of unknowns N. It also allows lowrank tensor representation, which has linear complexity in dimension d, hence, it gets rid of the “curse of dimensionality ” and can be used for large values of d. Extensive numerical tests are presented. AMS Subject Classification: 65F30, 65F50, 65N35, 65F10 Key words: structured matrices, elliptic operators, Poisson equation, matrix approximations

An Efficient Block Circulant Preconditioner For Simulating Fracture Using Large Fuse Networks

{\it Critical slowing down} associated with the iterative solvers close to the critical point often hinders large-scale numerical simulation of fracture using discrete lattice networks. This paper presents a block circlant preconditioner for iterative solvers for the simulation of progressive fracture in disordered, quasi-brittle materials using large discrete lattice networks. The average computational cost of the present alorithm per iteration is $O(rs log s) + delops$, where the stiffness matrix ${\bf A}$ is partioned into $r$-by-$r$ blocks such that each block is an $s$-by-$s$ matrix, and $delops$ represents the operational count associated with solving a block-diagonal matrix with $r$-by-$r$ dense matrix blocks. This algorithm using the block circulant preconditioner is faster than the Fourier accelerated preconditioned conjugate gradient (PCG) algorithm, and alleviates the {\it critical slowing down} that is especially severe close to the critical point. Numerical results using random resistor networks substantiate the efficiency of the present algorithm.Comment: 16 pages including 2 figure