In this paper, the modified Gauss-Seidel method with the new preconditioner for solving the linear system Ax = b, where A is a nonsingular M -matrix with unit diagonal, is considered. The convergence property and the comparison theorems of the proposed method are established. Two examples are given to show the efficiency and effectiveness of the modified Gauss-Seidel method with the presented n...

A Schur-type algorithm is presented for computing recursively the triangular factorization R = LU of a strongly nonsingular n n matrix satisfying a displacement equation RY V R = GH with Hessenberg matrices Y and V and n matrices G, H. If is small compared with n and the matrices Y and V admit fast matrix-vector multiplication, the new algorithm is fast in the sense that it will require less th...

A complex matrix A is ray-nonsingular if det(X 0 A) f 0 for every matrix X with positive entries. A sufficient condition for ray nonsingularity is that the origin is not in the relative interior of the convex hull of the signed transversal products of A. The concept of an isolated set of transversals is defined and used to obtain a neces*Work supported by an NSERC research grant. LINEAR ALGEBRA...

Let A be a nonsingular matrix with positive inverse and B a nonnegative matrix. Let the inverse of A + vB be positive for 0 < v < v < +00 and at least one of its entries be equal to zero for v = v* ; an algorithm to compute v* is described in this paper. Furthermore, it is shown that if A + A is positive definite, then the inverse of A + v (BB ) is positive for 0 < v < v* .

An M-matrix is a matrix that can be expressed as αI − P, where P is entry wise nonnegative and α ≥ ρ(P ). It is well known that the inverse of a nonsingular irreducible M-matrix is positive. In this paper, matrices of the form αI − P, where P is an irreducible eventually nonnegative matrix and α > ρ(P ), are studied. It is shown that if index0(P ) ≤ 1, then there exists a positive number λ such...

It is well known that Newton’s method for a nonlinear system has quadratic convergence when the Jacobian is a nonsingular matrix in a neighborhood of the solution. Here we present a modification of this method for nonlinear systems whose Jacobian matrix is singular. We prove, under certain conditions, that this modified Newton’s method has quadratic convergence. Moreover, different numerical te...

Let F = C or R and A ∈ GLn(F). Let s(A) ∈ R+ be the singular values of A, λ(A) ∈ C the unordered n-tuple of eigenvalues of A, a(A) := diag R ∈ R+, where A = QR is the QR decomposition of A, u(A) := diag U ∈ C, where A = LωU is any Gelfand-Naimark decomposition. We obtain complete relations between (1) u(A) and a(A), (2) u(A) and s(A), (3) u(A) and λ(A), and (4) a(A) and λ(A). We also study the ...

Let F = C or R and A ∈ GLn(F). Let s(A) ∈ R+ be the singular values of A, λ(A) ∈ C the unordered n-tuple of eigenvalues of A, a(A) := diagR ∈ R+, where A = QR is the QR decomposition of A, u(A) := diagU ∈ C, where A = LωU is any Gelfand-Naimark decomposition. We obtain complete relations between (1) u(A) and a(A), (2) u(A) and s(A), (3) u(A) and λ(A), and (4) a(A) and λ(A). We also study the re...

In this paper, we investigate condition numbers of eigenvalue problems of matrix polynomials with nonsingular leading coefficients, generalizing classical results of matrix perturbation theory. We provide a relation between the condition numbers of eigenvalues and the pseudospectral growth rate. We obtain that if a simple eigenvalue of a matrix polynomial is ill-conditioned in some respects, th...

In this paper we discuss the numerical solution of projected generalized Lyapunov equations using the matrix sign function method. Such equations arise in stability analysis and control problems for descriptor systems including model reduction based on balanced truncation. It is known that the matrix sign function method applied to a matrix pencil λE−A converges if and only if λE−A is of index ...