It is a well-known fact that while reducing a symmetric matrix into a similar tridiagonal one, the already tridiagonal matrix in the partially reduced matrix has as eigenvalues the Lanczos-Ritz values (see e.g. [Golub G. and Van Loan C.] ). This behavior is also shared by the reduction algorithm which transforms symmetric matrices via orthogonal similarity transformations to semiseparable form ...

Abstract. One dimensional stochastic problems on a finite lattice that model the time dependence of epidemics, particle deposition and voter influence can easily be cast into a simple form dV/dt = MV , where V is a vector with components representing the average occupation of the i-th cell and M is a matrix with coefficients drawn from the equations that give rates of evolution of a particular ...

In this paper we consider fast numerical algorithms for solving certain modified matrix eigenvalue problems associated with algebraic equations. The matrices under consideration have the form A = T + uv , where u, v ∈ Rn×n and T = (ti,j) ∈ Rn×n is a tridiagonal matrix such that tj+1,j = ±tj,j+1, 1 ≤ j ≤ n−1. We show that the DQR approach proposed in [Uhlig F., Numer. Math. 76 (1997), no. 4, 515...

In numerical linear algebra much attention has been paid to matrices that are sparse, i.e., containing a lot of zeros. For example, to compute the eigenvalues of a general dense symmetric matrix, this matrix is first reduced to a similar tridiagonal one using an orthogonal similarity transformation. The subsequent QR-algorithm performed on this n×n tridiagonal matrix, takes the sparse structure...

Abstract. A well–known property of an irreducible non–singular M–matrix is that its inverse is non–negative. However, when the matrix is an irreducible and singular M–matrix it is known that it has a generalized inverse which is non–negative, but this is not always true for any generalized inverse. We focus here in characterizing when the Moore–Penrose inverse of a symmetric, singular, irreduci...

A new high-performance general-purpose graphics processing unit (GPGPU) computational fluid dynamics (CFD) library is introduced for use with structured-grid CFD algorithms. A novel set of parallel tridiagonal matrix solvers, implemented in CUDA, is included for use with structured-grid CFD algorithms. The solver library supports both scalar and block-tridiagonal matrices suitable for approxima...

Abstract In this paper a fitted second-order finite difference method is presented for solving singularly perturbed two-point boundary value problems with the boundary layer at both end (left and right) points. We have introduced a fitting factor in second-order tri-diagonal finite difference scheme and it is obtained from the theory of singular perturbations. The efficient Thomas algorithm is ...

The four existing stable factorization methods for symmetric indefinite pivoting (row or column exchanges) maintains a band structure in the reduced matrix and the factors, but destroys symmetry completely once an off-diagonal pivot is used. Two-by-two block pivoting maintains symmetry at all times, but quickly destroys the band structure. Gaussian reduction to tridiagonal also maintains symmet...

A new algorithm is presented, designed to solve tridiagonal matrix problems efficientiy with parallel computers (multiple instruction stream, multiple data stream (MIMD) machines with distributed memory). The algorithm is designed to be extendable to higher order banded diagonal systems.

This paper is concerned with the distance of a symmetric tridiagonal Toeplitz matrix $T$ to manifold similarly structured singular matrices, and determining closest in this manifold. Explicit formulas are presented, exploiting analysis sensitivity spectrum respect structure-preserving perturbations its entries.