The paper suggests a general approach to deriving upper bounds for ‖A−1Q‖∞ from those for ‖A−1‖∞ for matrices A belonging to different subclasses of the class of nonsingular ℌ-matrices. The approach is applied to SDD, S-SDD, OBS, OB, and Nekrasov matrices.
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References
J. H. Ahlberg and E. N. Nilson, “Convergence properties of the spline fit,” J. Soc. Ind. Appl. Math., 11, 95–104 (1963).
L. Cvetković, V. Kostić, and R. Varga, “A new Geršgorin-type eigenvalue inclusion area,” ETNA, 18, 73–80 (2004).
Y. M. Gao and X. H. Wang, “Criteria for generalized diagonal dominant and M-matrices,” Linear Algebra Appl., 169, 257–268 (1992).
L. Yu. Kolotilina, “Pseudoblock conditions of diagonal dominance,” Zap. Nauchn. Semin. POMI, 323, 94–131 (2005); English transl., J. Math. Sci., 137, No. 3, 4815–4834 (2006).
L. Yu. Kolotilina, “Bounds for the determinants and inverses of certain H-matrices,” Zap. Nauchn. Semin. POMI, 346, 81–102 (2007); English transl., J. Math. Sci., 150, 1961–1972 (2008).
L. Yu. Kolotilina, “On bounding inverses to Nekrasov matrices in the infinity norm,” Zap. Nauchn. Semin. POMI, 419, 111–120 (2013); English transl., J. Math. Sci., 199, No. 4, 432–437 (2014).
L. Yu. Kolotilina, “Some bounds for inverses involving matrix sparsity pattern,” Zap. Nauchn. Semin. POMI, 482, 201–219 (2019); English transl., J. Math. Sci., 249, 242–255 (2020).
L. Yu. Kolotilina, “On SDD1 matrices,” Zap. Nauchn. Semin. POMI, 514, 88–112 (2022); English transl., see this issue.
V. R. Kostić, L. Cvetković, and D. I. Cvetković, “Pseudospectra localization and their applications,” Numer. Linear Algebra Appl., 23, 356–372 (2016).
Y. Li and Y. Wang, “Schur complement-based infinity norm bounds for the inverse of GDSDD matrices,” Mathematics, 10, 186 (2022).
J. Liu, Y. Huang, and F. Zhang, “The Schur complements of generalized doubly diagonally dominant matrices,” Linear Algebra Appl., 378, 231–244 (2004).
J. Liu, J. Zhang, and Y. Liu, “The Schur complements of strictly doubly diagonally dominant matrices and its application,” Linear Algebra Appl., 437, 168–183 (2012).
N. Morača, “Upper bounds for the infinity norm of the inverse of SDD and S – SDD matrices,” J. Comput. Appl. Math., 206, 666–678 (2007).
A. M. Ostrowski, “Über die Determinanten mit überwiegender Hauptdiagonale,” Comment. Math. Helv., 10, 69–96 (1937).
S. Z. Pan and S. C. Chen, “An upper bound for ‖A−1‖∞ of strictly doubly diagonally dominant matrices [in Chinese],” J. Fuzhou Univ. Nat. Sci. Ed., 36, 639–642 (2008).
F. Robert, “Blocs-H-matrices et convergence des méthodes itérative,” Linear Algebra Appl., 2, 223–265 (1969).
C. Sang, “Schur complement-based infinity norm bounds for the inverse of DSDD matrices,“ Bull. Iran. Math. Soc., 47, 1379–1398 (2020).
J. M. Varah, “A lower bound for the smallest singular value of a matrix,” Linear Algebra Appl., 11, 3–5 (1975).
R. S. Varga, Geršgorin and His Circles, Springer (2004).
X. R. Yong, “Two properties of diagonally dominant matrices,” Numer. Linear Algebra, 3, 173–177 (1996).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 514, 2022, pp. 77–87.
Translated by L. Yu. Kolotilina.
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Kolotilina, L.Y. Upper Bounds for ‖A−1Q‖∞. J Math Sci 272, 533–540 (2023). https://doi.org/10.1007/s10958-023-06447-5
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DOI: https://doi.org/10.1007/s10958-023-06447-5