A tridiagonal matrix with entries given by square matrices is a block tridiagonal matrix; the matrix is banded if off-diagonal blocks are upper or lower triangular. Such matrices are of great importance in numerical analysis and physics, and to obtain general properties is of great utility. The blocks of the inverse matrix of a block tridiagonal matrix can be factored in terms of two sets of ma...

Abstract. This paper generalizes to the nonlinear case a standard way to solve general sparse systems of linear equations. In particular, Duff [J. Inst. Math. Appl., 19 (1977), pp. 339-342] has suggested that row and column interchanges be applied to permute the coefficient matrix of a linear system into block lower triangular form., The linear system then can be solved by using the associated ...

This paper generalizes to the nonlinear case a standard way to solve general sparse systems of linear equations. In particular, Duu 1977] has suggested that row and column interchanges be applied to permute the coeecient matrix of a linear system into block lower triangular form. The linear system then can be solved by using the associated block Gauss-Seidel or forward block substitution scheme...

This paper provides new versions of necessary and sufficient conditions for compatibility of finite discrete conditional distributions, and of the uniqueness for those compatible conditional distributions. We note that the ratio matrix (the matrix C in Arnold and Press (1989)), after interchanging its rows and/or columns, can be rearranged to be an irreducible block diagonal matrix. We find tha...

We consider saddle-point problems that typically arise from the mixed finite element discretization of second order elliptic problems. By proper equivalent algebraic operations the considered saddle-point problem is transformed to another saddle-point problem. The resulting problem can then be efficiently preconditioned by a block-diagonal matrix or by a factored block-matrix (the blocks corres...

This research investigates the implementation mechanism of block-wise ILU(k) preconditioner on GPU. The block-wise ILU(k) algorithm requires both the level k and the block size to be designed as variables. A decoupled ILU(k) algorithm consists of a symbolic phase and a factorization phase. In the symbolic phase, a ILU(k) nonzero pattern is established from the point-wise structure extracted fro...

In this paper we consider numerical solution of 3D linear elasticity equations described by a coupled system of second order elliptic partial differential equations. This system is discretized by trilinear parallelepipedal finite elements. Preconditioned Conjugate Gradient iterative method is used for solving large-scale linear algebraic systems arising after the Finite Element Method (FEM) dis...

Subspace clustering is a useful technique for many computer vision applications in which the intrinsic dimension of high-dimensional data is often smaller than the ambient dimension. Spectral clustering, as one of the main approaches to subspace clustering, often takes on a sparse representation or a low-rank representation to learn a block diagonal self-representation matrix for subspace gener...