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BIT Numerical Mathematics

, Volume 54, Issue 2, pp 343–356 | Cite as

Accurate and efficient \(\textit{LDU}\) decomposition of almost diagonally dominant Z-matrices

  • A. Barreras
  • J. M. Peña
Article

Abstract

For a class of \(n\times n\) nonsingular almost row diagonally dominant \(Z\)-matrices and given adequate parameters, an efficient method to compute its \(\textit{LDU}\) decomposition with high relative accuracy is provided. It adds an additional cost of \({\fancyscript{O}}(n^2)\) elementary operations over the computational cost of Gaussian elimination. Numerical examples illustrate that the obtained \(\textit{LDU}\) decompositions are rank revealing, and comparisons with alternative procedures are included.

Keywords

\(\textit{LDU}\) decomposition Rank revealing decomposition Accuracy Z-matrices Almost diagonal dominance 

Mathematics Subject Classification (2000)

65F35 65F30 65F05 15A23 

Notes

Acknowledgments

The authors wish to thank the anonymous referees for their valuable suggestions to improve the paper.

References

  1. 1.
    Alfa, A.S., Xue, J., Ye, Q.: Entrywise perturbation theory for diagonally dominant M-matrices with applications. Numer. Math. 90, 401–414 (1999)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Alfa, A.S., Xue, J., Ye, Q.: Accurate computation of the smallest eigenvalue of a diagonally dominant M-matrix. Math. Comp. 71, 217–236 (2001)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Barreras, A., Peña, J.M.: Accurate and efficient LDU decomposition of diagonally dominant \(M\)-matrices. Electr. J. Linear Algebra 24, 153–167 (2012) Google Scholar
  4. 4.
    Demmel, J., Gu, M., Eisenstat, S., Slapnicar, I., Veselic, K., Drmac, Z.: Computing the singular value decomposition with high relative accuracy. Linear Algebra Appl. 299, 21–80 (1999)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Demmel, J., Koev, P.: Accurate SVDs of weakly diagonally dominant M-matrices. Numer. Math. 98, 99–104 (2004)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Demmel, J., Koev, P.: The accurate and efficient solution of a totally positive generalized Vandermonde linear system. SIAM J. Matrix Anal. Appl. 27, 142–152 (2005)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Gasca, M., Peña, J.M.: Total positivity and Neville elimination. Linear Algebra Appl. 165, 25–44 (1992)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Koev, P.: Accurate eigenvalues and SVDs of totally nonnegative matrices. SIAM J. Matrix Anal. Appl. 27, 1–23 (2005)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Peña, J.M.: LDU decompositions with L and U well conditioned. Electr. Trans. Numer. Anal. 18, 198–208 (2004)MATHGoogle Scholar
  10. 10.
    Ye, Q.: Computing singular values of diagonally dominant matrices to high relative accuracy. Math. Comp. 77, 2195–2230 (2008)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department Applied Mathematics/IUMAUniversidad de ZaragozaZaragozaSpain

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