Abstract
We present a new algorithm which, given a bidiagonal decomposition of a totally nonnegative matrix, computes all its eigenvalues to high relative accuracy in floating point arithmetic in \(O(n^3)\) time. It also computes exactly the Jordan blocks corresponding to zero eigenvalues in up to \(O(n^4)\) time.
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Notes
This is a situation where we need to “move” only the entry \(d_ix\) to row \(i-1\) as in example (4). We can’t swap the entire zero row \(i-1\) with row i since a nonzero entry \(u_i\) would destroy the tridiagonal structure of the product \(U\cdot J_i(x,y,z)\) on the right hand side of (19). Thus we move just the entry \(d_ix\) one row up, leaving everything else in its place.
The zero eigenvalues are not depicted in the figure since it is log-scale.
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Acknowledgements
I thank James Demmel for suggesting the problem of accurate computations with TN matrices and for conjecturing correctly in [4] that accurate computations with them ought to be possible. I am also thankful for the accommodations during my sabbatical at the University of California, Berkeley, when a part of this research was conducted. I also thank the referees for the careful reading of the manuscript and for their suggestions, which improved the presentation of the material. This work was partially supported by the Woodward Fund for Applied Mathematics at San Jose State University. The Woodward Fund is a gift from the estate of Mrs. Marie Woodward in memory of her son, Henry Teynham Woodward. He was an alumnus of the Mathematics Department at San Jose State University and worked with research groups at NASA Ames.
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Koev, P. Accurate eigenvalues and exact zero Jordan blocks of totally nonnegative matrices. Numer. Math. 141, 693–713 (2019). https://doi.org/10.1007/s00211-019-01022-0
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DOI: https://doi.org/10.1007/s00211-019-01022-0