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

Random Matrix Theory and Entanglement in Quantum Spin Chains

We compute the entropy of entanglement in the ground states of a general class of quantum spin-chain Hamiltonians - those that are related to quadratic forms of Fermi operators - between the first N spins and the rest of the system in the limit of infinite total chain length. We show that the entropy can be expressed in terms of averages over the classical compact groups and establish an explicit correspondence between the symmetries of a given Hamiltonian and those characterizing the Haar measure of the associated group. These averages are either Toeplitz determinants or determinants of combinations of Toeplitz and Hankel matrices. Recent generalizations of the Fisher-Hartwig conjecture are used to compute the leading order asymptotics of the entropy as N --> infinity . This is shown to grow logarithmically with N. The constant of proportionality is determined explicitly, as is the next (constant) term in the asymptotic expansion. The logarithmic growth of the entropy was previously predicted on the basis of numerical computations and conformal-field-theoretic calculations. In these calculations the constant of proportionality was determined in terms of the central charge of the Virasoro algebra. Our results therefore lead to an explicit formula for this charge. We also show that the entropy is related to solutions of ordinary differential equations of Painlev\'e type. In some cases these solutions can be evaluated to all orders using recurrence relations.Comment: 39 pages, 1 table, no figures. Revised version: minor correction

Distribution Functions for Random Variables for Ensembles of positive Hermitian Matrices

Distribution functions for random variables that depend on a parameter are computed asymptotically for ensembles of positive Hermitian matrices. The inverse Fourier transform of the distribution is shown to be a Fredholm determinant of a certain operator that is an analogue of a Wiener-Hopf operator. The asymptotic formula shows that up to the terms of order $o(1)$, the distributions are Gaussian

Determinants of Hankel Matrices

The purpose of this paper is to compute asymptotically Hankel determinants for weights that are supported in a semi-infinite interval.The main idea is to reduce the problem to determinants of other operators whose determinant asymptotics are well known.Comment: 18 pages, LaTeX fil

Convolution operations arising from Vandermonde matrices

Different types of convolution operations involving large Vandermonde matrices are considered. The convolutions parallel those of large Gaussian matrices and additive and multiplicative free convolution. First additive and multiplicative convolution of Vandermonde matrices and deterministic diagonal matrices are considered. After this, several cases of additive and multiplicative convolution of two independent Vandermonde matrices are considered. It is also shown that the convergence of any combination of Vandermonde matrices is almost sure. We will divide the considered convolutions into two types: those which depend on the phase distribution of the Vandermonde matrices, and those which depend only on the spectra of the matrices. A general criterion is presented to find which type applies for any given convolution. A simulation is presented, verifying the results. Implementations of all considered convolutions are provided and discussed, together with the challenges in making these implementations efficient. The implementation is based on the technique of Fourier-Motzkin elimination, and is quite general as it can be applied to virtually any combination of Vandermonde matrices. Generalizations to related random matrices, such as Toeplitz and Hankel matrices, are also discussed.Comment: Submitted to IEEE Transactions on Information Theory. 16 pages, 1 figur