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A note on inverses of Hessenberg-like matrices. (English) Zbl 0617.65018

Authors’ summary: We generalize some results of Y. Ikebe [ibid. 24, 93-97 (1979; Zbl 0397.15005)] concerning the inverse of an upper Hessenberg matrix. Specifically we prove a theorem to show that if A is a nonsingular block lower s-diagonal matrix (i.e. the blocks \(A_{ij}\) of A satisfy \(A_{ij}=0\) if \(i>j+s)\) and the blocks \(A_{j+s,j}\) are nonsingular, then the inverse of A may be written as \(A^{-1}=XY+Z\). The procedure for the computation of X, Y and Z as developed in the proof of this theorem lays the basis for an efficient implementation of the method on a parallel computer.
Reviewer: F.Szidarovszky

MSC:

65F05 Direct numerical methods for linear systems and matrix inversion
65Y05 Parallel numerical computation

Citations:

Zbl 0397.15005
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References:

[1] Ikebe, Yashuhiko, On inverse of Hessenberg matrices, Linear Algebra Appl., 24, 93-97 (1979) · Zbl 0397.15005
[2] Gantmacher, F. R.; Krein, M. G., (Oszillationsmatrizen (1960), Akademie: Akademie Berlin), 95
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