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Bounds for the inverses of diagonally dominant pentadiagonal matrices. (English) Zbl 1188.15004

One of the most cited works in the study of strictly diagonally-dominated matrices is the contribution of A. M. Ostrowski [Proc. Am. Math. Soc. 3, 26–30 (1952; Zbl 0046.01203)] (though he had written a couple of articles on the same problem, much earlier), where bounds were given for the determinant of a matrix with a dominant principal diagonal. In the study of numerical solutions of partial differential equations and in problems on cubic spline interpolation, tridiagonal matrices arise naturally. In many applications, it is quite useful to have estimates for the entries of the inverse of a tridiagonal matrix concerned.
For a general strictly diagonally-dominated matrix (without any band structure), Ostrowski’s result is one such. Many articles are available in the literature on such estimates for specific matrices arising from particular considerations. More well known among them is the work by P. N. Shivakumar and C. Ji [Linear Algebra Appl. 247, 297–316 (1996; Zbl 0862.65015)] where upper and lower bounds for the inverse elements of finite and infinite diagonally-dominant tridiagonal matrices are presented. These estimates were later improved by R. Nabben [Linear Algebra Appl. 287, 289–305 (1999; Zbl 0951.15005)], among others. For a more recent reference and more general estimates, we point out the work of X.-Q. Liu, T.-Z. Huang and Y.-D. Fu [Appl. Math. Lett. 19, 590–598 (2006; Zbl 1173.15302)].
The authors of the paper under review present four results on upper and lower bounds for inverses of general pentadiagonal matrices. The statements of the results are too technical to be stated here. They also demonstrate by means of a numerical example, that their results are better than the corresponding results of R. Nabben [SIAM J. Matrix Anal. Appl. 20, No. 3, 820–837 (1999; Zbl 0931.15001)].

MSC:

15A09 Theory of matrix inversion and generalized inverses
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