We present a randomized algorithm that, on input a weakly diagonally dominant symmetric n-by-n matrix A with m non-zero entries and an n-vector b, produces an x̃ such that ‖x − x̃‖ A ≤ ǫ ‖x‖ A , where Ax = b, in expected time m log n log(1/ǫ). The algorithm applies subgraph preconditioners in a recursive fashion. These preconditioners improve upon the subgraph preconditioners first introduced by ...

1 Flow in deformable porous media I 2 Abstract Many processes in the earth, such as magma migration, can be described by the ow of a low viscosity uid in a viscously deformable, permeable matrix. The purpose of this and a companion paper is to develop a better physical understanding of the equations governing these two-phase ows. This paper presents a series of analytic approximate solutions to...

The purpose of this paper is to show how the problem of finding the zeros of unilateral n-order quaternionic polynomials can be solved by determining the eigenvectors of the corresponding companion matrix. This approach, probably superfluous in the case of quadratic equations for which a closed formula can be given, becomes truly useful for (unilateral) n-order polynomials. To understand the st...

Abstract A number of theoretical and computational problems for matrix polynomials are solved by passing to linearizations. Therefore a perturbation theory, that relates perturbations in the linearization equivalent corresponding polynomial, is needed. In this paper we develop an algorithm finds which coefficients polynomial corresponds given entire pencil. Moreover find transformation matrices...

A fast algorithm for solving systems of linear equations with banded Toeplitz matrices is studied. An important step in the algorithm is a novel method for the spectral factorization of the generating function associated with the Toeplitz matrix. The spectral factorization is extracted from the right deflating subspaces corresponding to the eigenvalues inside and outside the open unit disk of a...

Upper and lower bounds are derived for the absolute values of the eigenvalues of a matrix polynomial (or λ-matrix). The bounds are based on norms of the coefficient matrices and involve the inverses of the leading and trailing coefficient matrices. They generalize various existing bounds for scalar polynomials and single matrices. A variety of tools are used in the derivations, including block ...

The properties of the Volume operator in Loop Quantum Gravity, as constructed by Ashtekar and Lewandowski, are analyzed for the first time at generic vertices of valence greater than four. The present analysis benefits from the general simplified formula for matrix elements of the Volume operator derived in [24], making it feasible to implement it on a computer as a matrix which is then diagona...

This is a companion paper of [Rou05]. It concentrates on asymptotic properties of determinants of some random matrices in the Jacobi ensemble. Let M ∈ Mn1+n2,r(R) (with r ≤ n1 + n2) be a random matrix whose entries are standard i.i.d. Gaussian. We can decompose MT = (MT 1 ,M T 2 ) with M1 ∈ Mn1,r and M2 ∈ Mn2,r. Then, W1 := MT 1 M1 and W2 := MT 2 M2 are independent r× r Wishart matrices with pa...

We present a family of constant–dimension codes for random linear network coding called spread codes. This is a family of optimal codes with maximum minimum distance. A spread code is constructed starting from the algebra defined by the companion matrix of an irreducible polynomial. We give a minimum distance decoding algorithm that is particularly efficient when the dimension of the codewords ...