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A new efficient approach to the characterization of D ‐stable matrices
Author(s) -
Pavani Raffaella
Publication year - 2018
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.4902
Subject(s) - mathematics , stability (learning theory) , context (archaeology) , characterization (materials science) , matrix (chemical analysis) , numerical analysis , algebra over a field , linear algebra , ordinary differential equation , order (exchange) , numerical linear algebra , numerical stability , matrix analysis , symbolic computation , differential equation , pure mathematics , mathematical analysis , eigenvalues and eigenvectors , computer science , geometry , paleontology , materials science , physics , finance , quantum mechanics , machine learning , economics , composite material , biology , nanotechnology
The concept of D ‐stability is relevant for stable square matrices of any order, especially when they appear in ordinary differential systems modeling physical problems. Indeed, D ‐stability was treated from different points of view in the last 50 years, but the problem of characterization of a general D ‐stable matrix was solved for low‐order matrices only (ie, up to order 4). Here, a new approach is proposed within the context of numerical linear algebra. Starting from a known necessary and sufficient condition, other simpler equivalent necessary and sufficient conditions for D ‐stability are proved. Such conditions turn out to be computationally more appealing for symbolic software, as discussed in the reported examples. Therefore, a new symbolic method is proposed to characterize matrices of order greater than 4, and then it is used in some numerical examples, given in details.