Upper bounds for the condition numbers of the GCD and the reciprocal GCD matrices in spectral norm
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Tarih
2012
Yazarlar
Dergi Başlığı
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Yayıncı
Pergamon-Elsevier Science Ltd
Erişim Hakkı
info:eu-repo/semantics/openAccess
Özet
Let S = {x(1), . . . , x(n)} be a set of n distinct positive integers. The n x n matrix having the greatest common divisor (x(i), x(j)) of x(i) and x(j) as its i, j-entry is called the greatest common divisor (GCD) matrix defined on S, denoted by ((x(i), x(j))), or abbreviated as (S). The n x n matrix (S-1) = (g(ij)), where g(ij) = 1/(x(i),x(j)) is called the reciprocal greatest common divisor (GCD) matrix on S. In this paper, we present upper bounds for the spectral condition numbers of the reciprocal GCD matrix (S-1) and the GCD matrix (S) defined on S = {1, 2, . . . , n}, with n >= 2, as a function of Euler's phi function and n. (C) 2011 Elsevier Ltd. All rights reserved.
Açıklama
Anahtar Kelimeler
GCD matrices, Matrix norms, Euler's phi function
Kaynak
Computers & Mathematics With Applications
WoS Q Değeri
Q1
Scopus Q Değeri
Q1
Cilt
63
Sayı
3
