A simple generalization of Geršgorin’s theorem Cvetković, Ljiljana; Kostić, Vladimir; Bru, Rafael ...
Advances in computational mathematics,
11/2011, Letnik:
35, Številka:
2-4
Journal Article
Recenzirano
It is well known that the spectrum of a given matrix
A
belongs to the Geršgorin set Γ(
A
), as well as to the Geršgorin set applied to the transpose of
A
, Γ(
A
T
). So, the spectrum belongs to their ...intersection. But, if we first intersect
i
-th Geršgorin disk Γ
i
(
A
) with the corresponding disk
, and then we make union of such intersections, which are, in fact, the smaller disks of each pair, what we get is not an eigenvalue localization area. The question is what should be added in order to catch all the eigenvalues, while, of course, staying within the set Γ(
A
) ∩ Γ(
A
T
). The answer lies in the appropriate characterization of some subclasses of nonsingular H-matrices. In this paper we give two such characterizations, and then we use them to prove localization areas that answer this question.
A female patient aged 42, started chemotherapy for advanced ovarian carcinoma in June 2016. Considering intraoperative findings, cytoreductive surgery with hyperthermic intraperitoneal chemotherapy ...(HIPEC) were performed, followed by adjuvant chemotherapy. In March 2018, computed tomography (CT) examination showed disease progression in the form of pleural carcinomatosis with increased levels of tumor markers. In April 2018, total parietal pleurectomy, partial visceral pleurectomy, and then hyperthermic intrathoracic chemotherapy (HITHOC) with cisplatin were performed. The procedure was uneventful, as was the postoperative course. The patient was discharged on the 13th postoperative day with no major postoperative complications. Three months after surgery, CT showed no signs of disease relapse. Since this is a relatively new method of treating pleural carcinomatosis, real results are to be expected with larger series of patients and longer postoperative follow-up.
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New criteria for identifying H-matrices CVETKOVIC, Ljiljana; KOSTIC, Vladimir
Journal of computational and applied mathematics,
08/2005, Letnik:
180, Številka:
2
Journal Article
Recenzirano
Odprti dostop
In the recent paper of Gan and Huang (Linear Algebra Appl. 374 (2003) 317), several simple criteria, as well as a necessary condition for nonsingular H-matrices, have been obtained. Inspired by this ...work, we will define several new subclasses of nonsingular H-matrices and give necessary conditions for a matrix to be an H-matrix. Finally, as a result of numerical experiments, we establish relations between defined and some already known subclasses.
Between Geršgorin and minimal Geršgorin sets CVETKOVIE, Ljiljana; KOSTIE, Vladimir
Journal of computational and applied mathematics,
11/2006, Letnik:
196, Številka:
2
Journal Article
Recenzirano
Odprti dostop
The eigenvalues of a given matrix
A can be localized by the well-known Geršgorin theorem: they belong to the Geršgorin set, which is the union of the Geršgorin disks (each of them is a simple ...function of the matrix entries). By applying the same theorem to a similar matrix
X
-
1
AX
, a new inclusion set can be obtained. Taking the intersection over
X, being a (positive) diagonal matrix, will lead us to the minimal Geršgorin set, defined by Varga R.S. Varga, Geršgorin and His Circles, Springer Series in Computational Mathematics, vol. 36, 2004, but this set is not easy to calculate. In this paper we will take the intersection over some special structured matrices
X and show that this intersection can be expressed by the same formula as the eigenvalue inclusion set
C
S
(
A
)
in L.J. Cvetković, V. Kostić, R. Varga, A new Geršgorin-type eigenvalue inclusion set, ETNA 18 (2004) 73–80.
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