Smale’s Fundamental Theorem of Algebra Reconsidered
In his 1981 Fundamental Theorem of Algebra paper Steve Smale initiated the complexity theory of finding a solution of polynomial equations of one complex variable by a variant of Newton’s method. In this paper we reconsider his algorithm in the light of work done in the intervening years. Smale’s up...
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| Veröffentlicht in: | Foundations of computational mathematics 2014-02, Vol.14 (1), p.85-114 |
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| creator | Armentano, Diego Shub, Michael |
| description | In his 1981 Fundamental Theorem of Algebra paper Steve Smale initiated the complexity theory of finding a solution of polynomial equations of one complex variable by a variant of Newton’s method. In this paper we reconsider his algorithm in the light of work done in the intervening years. Smale’s upper bound estimate was infinite average cost. Ours is polynomial in the Bézout number and the dimension of the input. Hence it is polynomial for any range of dimensions where the Bézout number is polynomial in the input size. In particular it is not just for the case that Smale considered but for a range of dimensions as considered by Bürgisser–Cucker, where the max of the degrees is greater than or equal to
n
1+
ϵ
for some fixed
ϵ
. It is possible that Smale’s algorithm is polynomial cost in all dimensions and our main theorem raises some problems that might lead to a proof of such a theorem. |
| doi_str_mv | 10.1007/s10208-013-9155-y |
| format | Article |
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n
1+
ϵ
for some fixed
ϵ
. It is possible that Smale’s algorithm is polynomial cost in all dimensions and our main theorem raises some problems that might lead to a proof of such a theorem.</description><identifier>ISSN: 1615-3375</identifier><identifier>EISSN: 1615-3383</identifier><identifier>DOI: 10.1007/s10208-013-9155-y</identifier><identifier>CODEN: FCMOA3</identifier><language>eng</language><publisher>Boston: Springer US</publisher><subject>Algebra ; Applications of Mathematics ; Computational mathematics ; Computer Science ; Economics ; Linear and Multilinear Algebras ; Math Applications in Computer Science ; Mathematical research ; Mathematics ; Mathematics and Statistics ; Matrix Theory ; Numerical Analysis ; Polynomials ; Theorems (Mathematics) ; Theory</subject><ispartof>Foundations of computational mathematics, 2014-02, Vol.14 (1), p.85-114</ispartof><rights>SFoCM 2013</rights><rights>COPYRIGHT 2014 Springer</rights><rights>SFoCM 2014</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c455t-1edeb93c010d731d6ab175720079ca4d744a6f05d7354627aca56ebb5f7584ac3</citedby><cites>FETCH-LOGICAL-c455t-1edeb93c010d731d6ab175720079ca4d744a6f05d7354627aca56ebb5f7584ac3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s10208-013-9155-y$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s10208-013-9155-y$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27924,27925,41488,42557,51319</link.rule.ids></links><search><creatorcontrib>Armentano, Diego</creatorcontrib><creatorcontrib>Shub, Michael</creatorcontrib><title>Smale’s Fundamental Theorem of Algebra Reconsidered</title><title>Foundations of computational mathematics</title><addtitle>Found Comput Math</addtitle><description>In his 1981 Fundamental Theorem of Algebra paper Steve Smale initiated the complexity theory of finding a solution of polynomial equations of one complex variable by a variant of Newton’s method. In this paper we reconsider his algorithm in the light of work done in the intervening years. Smale’s upper bound estimate was infinite average cost. Ours is polynomial in the Bézout number and the dimension of the input. Hence it is polynomial for any range of dimensions where the Bézout number is polynomial in the input size. In particular it is not just for the case that Smale considered but for a range of dimensions as considered by Bürgisser–Cucker, where the max of the degrees is greater than or equal to
n
1+
ϵ
for some fixed
ϵ
. It is possible that Smale’s algorithm is polynomial cost in all dimensions and our main theorem raises some problems that might lead to a proof of such a theorem.</description><subject>Algebra</subject><subject>Applications of Mathematics</subject><subject>Computational mathematics</subject><subject>Computer Science</subject><subject>Economics</subject><subject>Linear and Multilinear Algebras</subject><subject>Math Applications in Computer Science</subject><subject>Mathematical research</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Matrix Theory</subject><subject>Numerical Analysis</subject><subject>Polynomials</subject><subject>Theorems (Mathematics)</subject><subject>Theory</subject><issn>1615-3375</issn><issn>1615-3383</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2014</creationdate><recordtype>article</recordtype><recordid>eNp1kd9KwzAUh4soqNMH8K7glRedOU1O016O4XQgCDqvQ5qe1kr_aNKBu_M1fD2fxIyJOpjkIuHk-5KT_ILgDNgYGJOXDljM0ogBjzJAjFZ7wREkgBHnKd__WUs8DI6de2YMMANxFOBDqxv6fP9w4WzZFbqlbtBNuHii3lIb9mU4aSrKrQ7vyfSdqwuyVJwEB6VuHJ1-z6PgcXa1mN5Et3fX8-nkNjICcYiACsozbhiwQnIoEp2DRBn7hjOjRSGF0EnJ0G-iSGKpjcaE8hxLianQho-C8825L7Z_XZIb1HO_tJ2_UoHIODAQCfxSlX-KqruyH6w2be2MmnCZIso0Rk9FO6iKOrK66Tsqa1_e4sc7eD8KamuzU7jYEjwz0NtQ6aVzav5wv83ChjW2d85SqV5s3Wq7UsDUOlC1CVT5QNU6ULXyTrxxnGe7iuyfz_hX-gK9_6Am</recordid><startdate>20140201</startdate><enddate>20140201</enddate><creator>Armentano, Diego</creator><creator>Shub, Michael</creator><general>Springer US</general><general>Springer</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><scope>ISR</scope><scope>7SC</scope><scope>7TB</scope><scope>8FD</scope><scope>FR3</scope><scope>JQ2</scope><scope>KR7</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope></search><sort><creationdate>20140201</creationdate><title>Smale’s Fundamental Theorem of Algebra Reconsidered</title><author>Armentano, Diego ; Shub, Michael</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c455t-1edeb93c010d731d6ab175720079ca4d744a6f05d7354627aca56ebb5f7584ac3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2014</creationdate><topic>Algebra</topic><topic>Applications of Mathematics</topic><topic>Computational mathematics</topic><topic>Computer Science</topic><topic>Economics</topic><topic>Linear and Multilinear Algebras</topic><topic>Math Applications in Computer Science</topic><topic>Mathematical research</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Matrix Theory</topic><topic>Numerical Analysis</topic><topic>Polynomials</topic><topic>Theorems (Mathematics)</topic><topic>Theory</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Armentano, Diego</creatorcontrib><creatorcontrib>Shub, Michael</creatorcontrib><collection>CrossRef</collection><collection>Gale In Context: Science</collection><collection>Computer and Information Systems Abstracts</collection><collection>Mechanical & Transportation Engineering Abstracts</collection><collection>Technology Research Database</collection><collection>Engineering Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Civil Engineering Abstracts</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><jtitle>Foundations of computational mathematics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Armentano, Diego</au><au>Shub, Michael</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Smale’s Fundamental Theorem of Algebra Reconsidered</atitle><jtitle>Foundations of computational mathematics</jtitle><stitle>Found Comput Math</stitle><date>2014-02-01</date><risdate>2014</risdate><volume>14</volume><issue>1</issue><spage>85</spage><epage>114</epage><pages>85-114</pages><issn>1615-3375</issn><eissn>1615-3383</eissn><coden>FCMOA3</coden><abstract>In his 1981 Fundamental Theorem of Algebra paper Steve Smale initiated the complexity theory of finding a solution of polynomial equations of one complex variable by a variant of Newton’s method. In this paper we reconsider his algorithm in the light of work done in the intervening years. Smale’s upper bound estimate was infinite average cost. Ours is polynomial in the Bézout number and the dimension of the input. Hence it is polynomial for any range of dimensions where the Bézout number is polynomial in the input size. In particular it is not just for the case that Smale considered but for a range of dimensions as considered by Bürgisser–Cucker, where the max of the degrees is greater than or equal to
n
1+
ϵ
for some fixed
ϵ
. It is possible that Smale’s algorithm is polynomial cost in all dimensions and our main theorem raises some problems that might lead to a proof of such a theorem.</abstract><cop>Boston</cop><pub>Springer US</pub><doi>10.1007/s10208-013-9155-y</doi><tpages>30</tpages></addata></record> |
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| subjects | Algebra Applications of Mathematics Computational mathematics Computer Science Economics Linear and Multilinear Algebras Math Applications in Computer Science Mathematical research Mathematics Mathematics and Statistics Matrix Theory Numerical Analysis Polynomials Theorems (Mathematics) Theory |
| title | Smale’s Fundamental Theorem of Algebra Reconsidered |
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