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 an id="bbr0020">1976a>an>. In an id="bbr0090">1981a>an> Mignotte introduced the polynomials an id="mmlsi1" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S074771711600033X&_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)²an>an class="mathContainer hidden">an class="mathCode">ath altimg="si1.gif" overflow="scroll">Aa,nalse">(xalse">)=xn−2alse">(ax−1alse">)2ath>an>an>an>, a   an integer, an id="mmlsi2" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S074771711600033X&_mathId=si2.gif&_user=111111111&_pii=S074771711600033X&_rdoc=1&_issn=07477171&md5=6458292c70dff58fc0b60b4f911b475d" title="Click to view the MathML source">a≥2an>an class="mathContainer hidden">an class="mathCode">ath altimg="si2.gif" overflow="scroll">a≥2ath>an>an>an> and an id="mmlsi3" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S074771711600033X&_mathId=si3.gif&_user=111111111&_pii=S074771711600033X&_rdoc=1&_issn=07477171&md5=244321b719d2355609e6a8c5d4574604" title="Click to view the MathML source">n≥3an>an class="mathContainer hidden">an class="mathCode">ath altimg="si3.gif" overflow="scroll">n≥3ath>an>an>an>. First we prove that if a   is odd then the computing time of the bisection method when applied to an id="mmlsi12" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S074771711600033X&_mathId=si12.gif&_user=111111111&_pii=S074771711600033X&_rdoc=1&_issn=07477171&md5=2aeadb845361eeb0481963c8e4a1fc56" title="Click to view the MathML source">A_a,nan>an class="mathContainer hidden">an class="mathCode">ath altimg="si12.gif" overflow="scroll">Aa,nath>an>an>an> dominates an id="mmlsi11" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S074771711600033X&_mathId=si11.gif&_user=111111111&_pii=S074771711600033X&_rdoc=1&_issn=07477171&md5=592b2422ae0a5cff8e366abab9709e79">ass="imgLazyJSB inlineImage" height="16" width="62" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S074771711600033X-si11.gif">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si11.gif" overflow="scroll">n5alse">(athvariant="normal">log⁡ace width="0.2em">ace>dalse">)2ath>an>an>an> where d   is the maximum norm of an id="mmlsi12" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S074771711600033X&_mathId=si12.gif&_user=111111111&_pii=S074771711600033X&_rdoc=1&_issn=07477171&md5=2aeadb845361eeb0481963c8e4a1fc56" title="Click to view the MathML source">A_a,nan>an class="mathContainer hidden">an class="mathCode">ath altimg="si12.gif" overflow="scroll">Aa,nath>an>an>an>. 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 an id="mmlsi11" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S074771711600033X&_mathId=si11.gif&_user=111111111&_pii=S074771711600033X&_rdoc=1&_issn=07477171&md5=592b2422ae0a5cff8e366abab9709e79">ass="imgLazyJSB inlineImage" height="16" width="62" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S074771711600033X-si11.gif">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si11.gif" overflow="scroll">n5alse">(athvariant="normal">log⁡ace width="0.2em">ace>dalse">)2ath>an>an>an>. It follows that the maximum computing time of the bisection method is codominant with an id="mmlsi11" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S074771711600033X&_mathId=si11.gif&_user=111111111&_pii=S074771711600033X&_rdoc=1&_issn=07477171&md5=592b2422ae0a5cff8e366abab9709e79">ass="imgLazyJSB inlineImage" height="16" width="62" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S074771711600033X-si11.gif">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si11.gif" overflow="scroll">n5alse">(athvariant="normal">log⁡ace width="0.2em">ace>dalse">)2ath>an>an>an>.