Abstract
In this paper, we consider how to factor symmetric totally nonpositive matrices and their inverses by taking advantage of the symmetric property. It is well-known that the Bunch-Kaufman algorithm is the most commonly used pivoting strategy which can, however, produce arbitrarily large entries in the lower triangular factor for such matrices as illustrated by our example. Therefore, it is interesting to show that when the Bunch-Parlett algorithm is simplified for these matrices, it only requires O(n 2) comparisons with the growth factor being nicely bounded by 4. These facts, together with a nicely bounded lower triangular factor and a pleasantly small relative backward error, show that the Bunch-Parlett algorithm is more preferable than the Bunch-Kaufman algorithm when dealing with these matrices.
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Notes
In fact, \(a^{(2)}_{ij}\) should be \(a^{(2)}_{i-1,j-1}\) or \(a^{(2)}_{ij}\). For the sake of the convenience, we perform a minor abuse of notation throughout the paper.
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The author is very grateful to the referees and Prof. D. Kressner for their very careful reading of the paper and valuable suggestions.
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Communicated by Daniel Kressner.
This work was supported by the National Natural Science Foundation for Youths of China (Grant No. 11001233), the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20094301120002), the Hunan Provincial Natural Science Foundation of China (Grant No. 11JJ4004).
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Huang, R. Factoring symmetric totally nonpositive matrices and inverses with a diagonal pivoting method. Bit Numer Math 53, 443–458 (2013). https://doi.org/10.1007/s10543-012-0404-2
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DOI: https://doi.org/10.1007/s10543-012-0404-2