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Civil-Comp Proceedings
ISSN 1759-3433 CCP: 84
PROCEEDINGS OF THE FIFTH INTERNATIONAL CONFERENCE ON ENGINEERING COMPUTATIONAL TECHNOLOGY Edited by: B.H.V. Topping, G. Montero and R. Montenegro
Paper 93
Algorithms Based on Diagonal Dominance: H-Matrix, Perron Vector and Reducibility C. Corral, I. Giménez and J. Mas
Multidisciplinary Mathematics Institute, Polytechnical University of Valencia, Spain Full Bibliographic Reference for this paper
, "Algorithms Based on Diagonal Dominance: H-Matrix, Perron Vector and Reducibility", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Fifth International Conference on Engineering Computational Technology", Civil-Comp Press, Stirlingshire, UK, Paper 93, 2006. doi:10.4203/ccp.84.93
Keywords: H-matrix, non-negative matrix, irreducible matrix, spectral radius, Perron vector, Jacobi matrix.
Summary
The convergence of several iterative methods for solving linear systems
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Recall that a real square matrix A is an H-matrix if its comparison matrix
Theorem 1 If A is a nonsingular matrix, then A is an H-matrix if, and only if, ![]() ![]() ![]() ![]()
Theorem 2
A is an H-matrix if, and only if, there is a positive diagonal matrix
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Based in the last characterization, in [3,4],
several algorithms are proposed to compute the matrix D to determine if a given matrix is an H-matrix or not. Moreover in [4] a generalization is proposed to compute the spectral radius of nonnegative matrices with constant diagonal entries.
However, the algorithms in [3,4] can fail if the matrix A is reducible. Recall that a matrix A is reducible if a permutation matrix
Moreover, using this decomposition recurrently, one can obtain a permutation matrix In this work, we present several algorithms based on finding the matrix D that ensures the generalized diagonal dominance of the matrix. In the process of finding D, some useful information is obtained that permits the proposed algorithms in order to:
The algorithms require one parameter
When the matrix is reducible, we present some examples to show that it is possible to compute the spectral radius of the Jacobi matrix using the original matrix or its transpose. Moreover, when the algorithm stagnates, the results obtained can be applied to conclude that it is a reducible matrix and to determine the symmetric permutation of the matrix that allows to obtain its canonical form. Later, one can compute the spectral radius of irreducible diagonal blocks. References
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