On the maximum computing time of the bisection method for real root isolation

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• 作者：George E. Collins ^{gcollins8@charter.net" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Polynomial roots ; Real roots ; Root isolation ; Computing time ; Algorithm analysis ; Dominance
• 刊名：Journal of Symbolic Computation
• 出版年：2017
• 出版时间：March-April 2017
• 年：2017
• 卷：79
• 期：part_P2
• 页码：444-456
• 全文大小：359 K

文摘

The bisection method for polynomial real root isolation was introduced by Collins and Akritas in 1976. In 1981 Mignotte introduced the polynomials 1600033X&_mathId=si1.gif&_user=111111111&_pii=S074771711600033X&_rdoc=1&_issn=07477171&md5=96f96cf24a97bf95b34fb6bf353ec878" title="Click to view the MathML source">A_a,n(x)=xⁿ−2(ax−1)²$A_{a,n}(x)=x^n-2{(ax-1)}^2$, a   an integer, 1600033X&_mathId=si2.gif&_user=111111111&_pii=S074771711600033X&_rdoc=1&_issn=07477171&md5=6458292c70dff58fc0b60b4f911b475d" title="Click to view the MathML source">a≥2$a\ge 2$ and 1600033X&_mathId=si3.gif&_user=111111111&_pii=S074771711600033X&_rdoc=1&_issn=07477171&md5=244321b719d2355609e6a8c5d4574604" title="Click to view the MathML source">n≥3$n\ge 3$. First we prove that if a   is odd then the computing time of the bisection method when applied to 1600033X&_mathId=si12.gif&_user=111111111&_pii=S074771711600033X&_rdoc=1&_issn=07477171&md5=2aeadb845361eeb0481963c8e4a1fc56" title="Click to view the MathML source">A_a,n$A_{a,n}$ dominates 1600033X&_mathId=si11.gif&_user=111111111&_pii=S074771711600033X&_rdoc=1&_issn=07477171&md5=592b2422ae0a5cff8e366abab9709e79">1600033X-si11.gif">$n^5{(\log d)}^2$ where d   is the maximum norm of 1600033X&_mathId=si12.gif&_user=111111111&_pii=S074771711600033X&_rdoc=1&_issn=07477171&md5=2aeadb845361eeb0481963c8e4a1fc56" title="Click to view the MathML source">A_a,n$A_{a,n}$. Then we prove that if A is any polynomial of degree n with maximum norm d   then the computing time of the bisection method, with a minor improvement regarding homothetic transformations, is dominated by 1600033X&_mathId=si11.gif&_user=111111111&_pii=S074771711600033X&_rdoc=1&_issn=07477171&md5=592b2422ae0a5cff8e366abab9709e79">1600033X-si11.gif">$n^5{(\log d)}^2$. It follows that the maximum computing time of the bisection method is codominant with 1600033X&_mathId=si11.gif&_user=111111111&_pii=S074771711600033X&_rdoc=1&_issn=07477171&md5=592b2422ae0a5cff8e366abab9709e79">1600033X-si11.gif">$n^5{(\log d)}^2$.