H-matrix (iterative method)

In mathematics, an H-matrix is a matrix whose comparison matrix is an M-matrix. It is useful in iterative methods.

Definition: Let $A = (a ij)$ be a $n \times n$ complex matrix. Then comparison matrix M(A) of complex matrix A is defined as $M (A) = α ij$ where $α ij = -| A ij |$ for all $i \neq j, 1 \leq i,j \leq n$ and $α ij = | A ij |$ for all $i = j, 1 \leq i,j \leq n$ . If M(A) is a M-matrix, A is a H-matrix.

Invertible H-matrix guarantees convergence of Gauss–Seidel iterative methods.^[1]

References[edit]

^ Zhang, Cheng-yi; Ye, Dan; Zhong, Cong-Lei; SHUANGHUA, SHUANGHUA (2015). "Convergence on Gauss–Seidel iterative methods for linear systems with general H-matrices". The Electronic Journal of Linear Algebra. 30: 843–870. arXiv:1410.3196. doi:10.13001/1081-3810.1972. Retrieved 21 June 2018.

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[1] Zhang, Cheng-yi; Ye, Dan; Zhong, Cong-Lei; SHUANGHUA, SHUANGHUA (2015). "Convergence on Gauss–Seidel iterative methods for linear systems with general H-matrices". The Electronic Journal of Linear Algebra. 30: 843–870. arXiv:1410.3196. doi:10.13001/1081-3810.1972. Retrieved 21 June 2018.

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