New bounds for roots of polynomials based on Fiedler companion matrices

详细信息    查看全文

• 作者：Fernando De Terá ; n^a ; ^{fteran@math.uc3m.es" class="auth_mail} ; Froilá ; n M. Dopico^b ; ^{dopico@math.uc3m.es" class="auth_mail} ; Javier Pé ; rez^a ; ^{jpalvaro@math.uc3m.es" class="auth_mail}
• 关键词：15A18 ; 15A60 ; 15B99 ; 65F15 ; 65F35 ; 65H04
• 刊名：Linear Algebra and its Applications
• 出版年：15 June 2014
• 年：2014
• 卷：451
• 期：Complete
• 页码：197-230
• 全文大小：478 K

文摘

Several matrix norms of the classical Frobenius companion matrices of a monic polynomial 24379514001578&_mathId=si1.gif&_user=111111111&_pii=S0024379514001578&_rdoc=1&_issn=00243795&md5=2cd2e5205bde1b232fe77a2c8273b17a" title="Click to view the MathML source">p(z)$p(z)$ have been used in the literature to obtain simple lower and upper bounds on the absolute values of the roots 位   of 24379514001578&_mathId=si1.gif&_user=111111111&_pii=S0024379514001578&_rdoc=1&_issn=00243795&md5=2cd2e5205bde1b232fe77a2c8273b17a" title="Click to view the MathML source">p(z)$p(z)$. Recently, M. Fiedler (2003) [9] has introduced a new family of companion matrices of 24379514001578&_mathId=si1.gif&_user=111111111&_pii=S0024379514001578&_rdoc=1&_issn=00243795&md5=2cd2e5205bde1b232fe77a2c8273b17a" title="Click to view the MathML source">p(z)$p(z)$ that has received considerable attention and it is natural to investigate if matrix norms of Fiedler companion matrices may be used to obtain new and sharper lower and upper bounds on 24379514001578&_mathId=si2.gif&_user=111111111&_pii=S0024379514001578&_rdoc=1&_issn=00243795&md5=a5302c277a496d9d787596c17e74d3b7" title="Click to view the MathML source">|位|$|位|$. The development of such bounds requires first to know simple expressions for some relevant matrix norms of Fiedler matrices and we obtain them in the case of the 1- and ∞-matrix norms. With these expressions at hand, we will show that norms of Fiedler matrices produce many new bounds, but that none of them improves significatively the classical bounds obtained from the Frobenius companion matrices. However, we will prove that if the norms of the inverses of Fiedler matrices are used, then another family of new bounds on 24379514001578&_mathId=si2.gif&_user=111111111&_pii=S0024379514001578&_rdoc=1&_issn=00243795&md5=a5302c277a496d9d787596c17e74d3b7" title="Click to view the MathML source">|位|$|位|$ is obtained and some of the bounds in this family improve significatively the bounds coming from the Frobenius companion matrices for certain polynomials.