New subclasses of block H-matrices with applications to parallel decomposition-type relaxation methods

The parallel decomposition-type relaxation methods for solving large sparse systems of linear equations on SIMD multiprocessor systems have been proposed in [3] and [2]. In case when the coefficient matrix of the linear system is a block $H$-matrix, sufficient conditions for the convergence of methods given in [2], [3] have been further improved in [5] and [4]. From the practical point of view, the convergence area obtained there is not always suitable for computation, so we propose new, easily computable ones, for some special subclasses of block $H$-matrices. Furthermore, this approach improves the already known convergence area for the class of block strictly diagonally dominant (SDD) matrices.

[1]  Zhong-Zhi Bai,et al.  A class of parallel decomposition-type relaxation methods for large sparse systems of linear equations , 1998 .

[2]  Ljiljana Cvetković.  Some convergence conditions for a class of parallel decomposition-type linear relaxation methods , 2002 .

[3]  Yangfeng Su,et al.  On the convergence of a class of parallel decomposition-type relaxation methods , 1997 .

[4]  J. Gillis,et al.  Matrix Iterative Analysis , 1961 .

[5]  Zhong-Zhi Bai,et al.  A unified framework for the construction of various matrix multisplitting iterative methods for large sparse system of linear equations , 1996 .

[6]  Shu-huang Xiang,et al.  Weak block diagonally dominant matrices, weak block H-matrix and their applications , 1998 .

[7]  Louis A. Hageman,et al.  Iterative Solution of Large Linear Systems. , 1971 .

[8]  R. Varga,et al.  Block diagonally dominant matrices and generalizations of the Gerschgorin circle theorem , 1962 .

[9]  Apostolos Hadjidimos,et al.  Accelerated overrelaxation method , 1978 .

[10]  Ljiljana Cvetković,et al.  Some convergence results of PD relaxation methods , 2000, Appl. Math. Comput..

[11]  L. Jackson,et al.  A generalized solution of the boundary value problem for $y^{\prime\prime}=f(x,\,y,\,y^{\prime})$. , 1962 .

[12]  R. Varga,et al.  A NEW GERŠGORIN-TYPE EIGENVALUE INCLUSION SET , 2022 .

[13]  Zi-Cai Li,et al.  Penalty combinations of the Ritz-Galerkin and finite difference methods for singularity problems , 1997 .