An extension of the criteria for generalized diagonally dominant matrices

“…An H -matrix (usually A ∈ C n,n ) is closely related to an M-matrix and many convergence properties of an M-matrix carry over, almost directly, to convergence properties for an H -matrix. This explains the fact that in the last decade so many works ( [2][3][4][5]7,9,11,12]) have appeared to propose various criteria (algorithms) that decide whether the matrix A of the linear system to be solved is an H -matrix. The majority of the criteria to make this decision are iterative in nature and assume, without explicitly stating it, that the matrix A in question is dense.…”

Section: Introductionmentioning

confidence: 95%

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

Hadjidimos¹

2004

Linear Algebra and its Applications

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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.

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“…In general this is not an easy task. In recent years, there are quite few studies on this subject, and several methods some of which are quite efficient have been proposed for determining if a given A is or isn't a GDDM and possibly a diagonal matrix D in case when it is GDDM (see, e.g., [4,5,6,7,8,9,10,11,12,13,14,15]). These methods can be divided into two categories: direct methods and iterative ones.…”

Section: Introductionmentioning

confidence: 99%

Self-Corrective Iterations (SCI) for generalized diagonally dominant matrices

Guan

et al. 2016

Journal of Computational and Applied Mathematics

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A suggestive indicator is proposed for predicting whether a given (complex or real) square matrix A is or isn't a generalized diagonally dominant matrix (GDDM) by which we mean if A can be brought into a strictly diagonally dominant matrix by post-multiplying some diagonal matrix D. Based on the indicator, three self-corrective iterative algorithms (SCI) are presented for determining if an irreducible A is or is not a GDDM and at the same time delivering the matrix D in case when A is a GDDM. The basic idea is to push A towards being (strictly) diagonally dominant when the indicator suggests that A is likely a GDDM or towards being off-diagonally dominant otherwise. Among the three algorithms, each has their own feature: one takes the fewest number of iterations but the most amount of work per iteration, one takes the most number of iterations but the least amount of work per iteration, and the third one strikes a balance between the two extremes. It is shown that the algorithms will terminate in finite many steps under the assumption that A has no zero entries and the comparison matrix of A is nonsingular. Comparing with existing methods, new algorithms are more efficient, as demonstrated on known difficult examples in the literature, newly designed random matrices, as well as matrices from the University of Florida Sparse Matrix Collection.

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An extension of the criteria for generalized diagonally dominant matrices

Cited by 19 publications

References 2 publications

New criterion for generalized diagonally dominant matrices andM-matrices

New criterion for generalized diagonally dominant matrices andM-matrices

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

Self-Corrective Iterations (SCI) for generalized diagonally dominant matrices

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