How to compute a special Hessenberg determinant showed by the PDF file? Thank a lot.
Dear Feng,
Use Schur Complement: "Transpose the matrix" and use the following method:
(M/C) = B - AC⁻¹D, (by Brezinski)
and please see the attachment section 2, pp.249-250.
Regards,
Wiwat Wanicharpichat
It depends on what you mean by an explicit or closed form for the determinant. Most likely, you will not find a formula whose computation time is faster than a clever version of Gaussian elimination will be. There are graph-theoretic formulae based on paths in an associated digraph such as the one in the paper
J. Stuart, “The determinant of a Hessenberg L-matrix”, SIAM Journal on Matrix Analysis and Applications 12 (1991) 7-15.
These types of formulae work best if either there are predictable zeros below the subdiagonal, or if there are nice relationships between the values along the paths.
Dear All
The following paper are helpful for understanding this question.
(1) Feng Qi, Jing-Lin Wang, and Bai-Ni Guo, A representation for derangement numbers in terms of a tridiagonal determinant, Kragujevac Journal of Mathematics 42 (2018), no. 1, 7--14.
(2) Feng Qi, A determinantal representation for derangement numbers, Global Journal of Mathematical Analysis 4 (2016), no. 3, 17--17; Available online at http://dx.doi.org/10.14419/gjma.v4i3.6574.
(3) Feng Qi and Bai-Ni Guo, Explicit formulas for derangement numbers and their generating function, Journal of Nonlinear Functional Analysis 2016, Article ID 45, 10 pages.
(4) Feng Qi, Jing-Lin Wang, and Bai-Ni Guo, A recovery of two determinantal representations for derangement numbers, Cogent Mathematics (2016), 3: 1232878, 7 pages; Available online at http://dx.doi.org/10.1080/23311835.2016.1232878.
(5) Feng Qi, Jiao-Lian Zhao, and Bai-Ni Guo, Closed forms for derangement numbers in terms of the Hessenberg determinants, Preprints 2016, 2016100035, 11 pages; Available online at http://dx.doi.org/10.20944/preprints201610.0035.v1.
Feng
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