An extended compact profile iterative method criterion for sparse H-matrices

Abstract

In the theory and applications of Numerical Linear Algebra the class of H -matrices is very important. For example, when one is interested in the solution of linear systems of algebraic equations and their solution is sought by classical iterative methods (e.g., Jacobi, Gauss-Seidel, SOR, SSOR, etc.). In recent years many works have appeared that propose criteria for a matrix A ∈ C n,n to be an H -matrix (see, e.g. 293]). In the present work we propose one more such a criterion together with an algorithm. As far as we know it is the first time the zero pattern of A, which in practical problems is usually large and sparse, is taken into consideration. As a consequence both storage and number of operations required to decide about the H -character of the matrix A are drastically reduced. Finally, we use our method with a set of numerical examples to show its effectiveness.

Diagonal Matrix Scaling and H-Matrices

“…In [8] another choice for a 12 has been made to somehow determine computationally a "small" interval such that both cases of H-and non H-matrix happen for A(a 12 ). So a 12 = 1.416392 was considered for this purpose.…”

Is A∈Cn,n a general H-matrix?

“…2 Obviously, if A is irreducible it is already in its Fnf. In case A is reducible to find a block permutation of the block diagonal of an Fnf of A we use some ideas from the compact profile technique in Kincaid et al [9] and the extended compact profile technique in Hadjidimos [7]. Suppose A is a reducible matrix and C is the matrix obtained and saved by IRR-algorithm: C = C l = spones (I + |A|) n−1 .…”

A parallel characterization of H-matrices

“…In [7], Li et al proposed an efficient iterative algorithm, while in [9] Liu et al presented two efficient algorithmic characterizations which need fewer number of iterations than that of [7]. Other different algorithms are given in [2,5,6,8,10]. However, as all these methods are designed for a sequential computer, they may be not so effective for large scalar matrices which is a common case in many applications.…”