Abstract
This short note proves that if \(A\) is accretive-dissipative, then the growth factor for such \(A\) in Gaussian elimination is less than \(4\). If \(A\) is a Higham matrix, i.e., the accretive-dissipative matrix \(A\) is complex symmetric, then the growth factor is less than \(2\sqrt{2}\). The result obtained improves those of George et al. in [Numer. Linear Algebra Appl. 9, 107–114 (2002)] and is one step closer to the final solution of Higham’s conjecture.
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Notes
In [2], accretive-dissipative matrix is called generalized Higham matrix.
References
George, A., Ikramov, Kh.D.: On the growth factor in Gaussian elimination for matrices with sharp angular field of values. Calcolo 41, 27–36 (2004)
George, A., Ikramov, Kh.D., Kucherov, A.B.: On the growth factor in Gaussian elimination for generalized Higham matrices. Numer. Linear Algebra Appl. 9, 107–114 (2002)
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Acknowledgments
The author thanks Prof. Henry Wolkowicz for valuable discussion on a draft of this paper, and both referees for their comments on the presentation of the submitted version.
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Lin, M. A note on the growth factor in Gaussian elimination for accretive-dissipative matrices. Calcolo 51, 363–366 (2014). https://doi.org/10.1007/s10092-013-0089-1
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DOI: https://doi.org/10.1007/s10092-013-0089-1