Computations in the neighbourhood of algebraic singularities
It is known that finite precision versus exact computation is a crucial issue only when the computation takes place in the neighbourhood of a singularity. In such a situation, it is essential to know the distance to singularity. Many attention has been dedicated to the relationship between the dista...
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| Veröffentlicht in: | Numerical functional analysis and optimization 1995-01, Vol.16 (3-4), p.287-302 |
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| container_title | Numerical functional analysis and optimization |
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| creator | Chaitin-Chatelin, Françoise Frayssé, Valérie Braconnier, Thierry |
| description | It is known that finite precision versus exact computation is a crucial issue only when the computation takes place in the neighbourhood of a singularity. In such a situation, it is essential to know the distance to singularity. Many attention has been dedicated to the relationship between the distance to singularity δ and the condition number K of the problem under study. The well-known Turing theorem states that, for a linear system Ax = b the distance to singularity, in a normwise measure, is the reciprocal of the normwise condition number ||A
−1
|||A|| In this Paper, we examine the possibility of extending this theorem for nonlinear problems in the neighbourhood of algebraic singularities. After reviewing the literature on that topic ([Demmel 1987, 1990], [Shub and Smale 1992]), we propose and check on the computer a conjecture which makes more explicit Demmel's bounds on the distance to singularity. |
| doi_str_mv | 10.1080/01630569508816619 |
| format | Article |
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−1
|||A|| In this Paper, we examine the possibility of extending this theorem for nonlinear problems in the neighbourhood of algebraic singularities. After reviewing the literature on that topic ([Demmel 1987, 1990], [Shub and Smale 1992]), we propose and check on the computer a conjecture which makes more explicit Demmel's bounds on the distance to singularity.</description><identifier>ISSN: 0163-0563</identifier><identifier>EISSN: 1532-2467</identifier><identifier>DOI: 10.1080/01630569508816619</identifier><identifier>CODEN: NFAODL</identifier><language>eng</language><publisher>Philadelphia, PA: Marcel Dekker, Inc</publisher><subject>Applied sciences ; Automata. Abstract machines. Turing machines ; Computer science; control theory; systems ; Exact sciences and technology ; Mathematics ; Methods of scientific computing (including symbolic computation, algebraic computation) ; Nonlinear algebraic and transcendental equations ; Numerical analysis ; Numerical analysis. Scientific computation ; Sciences and techniques of general use ; Theoretical computing</subject><ispartof>Numerical functional analysis and optimization, 1995-01, Vol.16 (3-4), p.287-302</ispartof><rights>Copyright Taylor & Francis Group, LLC 1995</rights><rights>1995 INIST-CNRS</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c325t-38eff0d3d00c3c0f8a38b80a0ac8178200f85d86837ececaa532fc39e7b897863</citedby><cites>FETCH-LOGICAL-c325t-38eff0d3d00c3c0f8a38b80a0ac8178200f85d86837ececaa532fc39e7b897863</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://www.tandfonline.com/doi/pdf/10.1080/01630569508816619$$EPDF$$P50$$Ginformaworld$$H</linktopdf><linktohtml>$$Uhttps://www.tandfonline.com/doi/full/10.1080/01630569508816619$$EHTML$$P50$$Ginformaworld$$H</linktohtml><link.rule.ids>314,776,780,4010,27900,27901,27902,59620,60409</link.rule.ids><backlink>$$Uhttp://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=3597464$$DView record in Pascal Francis$$Hfree_for_read</backlink></links><search><creatorcontrib>Chaitin-Chatelin, Françoise</creatorcontrib><creatorcontrib>Frayssé, Valérie</creatorcontrib><creatorcontrib>Braconnier, Thierry</creatorcontrib><title>Computations in the neighbourhood of algebraic singularities</title><title>Numerical functional analysis and optimization</title><description>It is known that finite precision versus exact computation is a crucial issue only when the computation takes place in the neighbourhood of a singularity. In such a situation, it is essential to know the distance to singularity. Many attention has been dedicated to the relationship between the distance to singularity δ and the condition number K of the problem under study. The well-known Turing theorem states that, for a linear system Ax = b the distance to singularity, in a normwise measure, is the reciprocal of the normwise condition number ||A
−1
|||A|| In this Paper, we examine the possibility of extending this theorem for nonlinear problems in the neighbourhood of algebraic singularities. After reviewing the literature on that topic ([Demmel 1987, 1990], [Shub and Smale 1992]), we propose and check on the computer a conjecture which makes more explicit Demmel's bounds on the distance to singularity.</description><subject>Applied sciences</subject><subject>Automata. Abstract machines. Turing machines</subject><subject>Computer science; control theory; systems</subject><subject>Exact sciences and technology</subject><subject>Mathematics</subject><subject>Methods of scientific computing (including symbolic computation, algebraic computation)</subject><subject>Nonlinear algebraic and transcendental equations</subject><subject>Numerical analysis</subject><subject>Numerical analysis. Scientific computation</subject><subject>Sciences and techniques of general use</subject><subject>Theoretical computing</subject><issn>0163-0563</issn><issn>1532-2467</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>1995</creationdate><recordtype>article</recordtype><recordid>eNp1j01Lw0AQhhdRsFZ_gLccvEZnM81mAr1I8QsKXvQcJpvdZCXNlt0U6b83pepFPA3MPM_MvEJcS7iVQHAHUiHkqsyBSColyxMxkzlmabZQxamYHebpBOC5uIjxAwAwK2kmliu_2e5GHp0fYuKGZOxMMhjXdrXfhc77JvE24b41dWCnk-iGdtdzcKMz8VKcWe6jufquc_H--PC2ek7Xr08vq_t1qjHLxxTJWAsNNgAaNVhipJqAgTXJgjKYWnlDirAw2mjm6W-rsTRFTWVBCudCHvfq4GMMxlbb4DYc9pWE6hC_-hN_cm6Ozpaj5t4GHrSLvyLmZbFQiwlbHjE3WB82_OlD31Qj73sffhz8_8oX2aZtCg</recordid><startdate>19950101</startdate><enddate>19950101</enddate><creator>Chaitin-Chatelin, Françoise</creator><creator>Frayssé, Valérie</creator><creator>Braconnier, Thierry</creator><general>Marcel Dekker, Inc</general><general>Taylor & Francis</general><scope>IQODW</scope><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>19950101</creationdate><title>Computations in the neighbourhood of algebraic singularities</title><author>Chaitin-Chatelin, Françoise ; Frayssé, Valérie ; Braconnier, Thierry</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c325t-38eff0d3d00c3c0f8a38b80a0ac8178200f85d86837ececaa532fc39e7b897863</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>1995</creationdate><topic>Applied sciences</topic><topic>Automata. Abstract machines. Turing machines</topic><topic>Computer science; control theory; systems</topic><topic>Exact sciences and technology</topic><topic>Mathematics</topic><topic>Methods of scientific computing (including symbolic computation, algebraic computation)</topic><topic>Nonlinear algebraic and transcendental equations</topic><topic>Numerical analysis</topic><topic>Numerical analysis. Scientific computation</topic><topic>Sciences and techniques of general use</topic><topic>Theoretical computing</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Chaitin-Chatelin, Françoise</creatorcontrib><creatorcontrib>Frayssé, Valérie</creatorcontrib><creatorcontrib>Braconnier, Thierry</creatorcontrib><collection>Pascal-Francis</collection><collection>CrossRef</collection><jtitle>Numerical functional analysis and optimization</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Chaitin-Chatelin, Françoise</au><au>Frayssé, Valérie</au><au>Braconnier, Thierry</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Computations in the neighbourhood of algebraic singularities</atitle><jtitle>Numerical functional analysis and optimization</jtitle><date>1995-01-01</date><risdate>1995</risdate><volume>16</volume><issue>3-4</issue><spage>287</spage><epage>302</epage><pages>287-302</pages><issn>0163-0563</issn><eissn>1532-2467</eissn><coden>NFAODL</coden><abstract>It is known that finite precision versus exact computation is a crucial issue only when the computation takes place in the neighbourhood of a singularity. In such a situation, it is essential to know the distance to singularity. Many attention has been dedicated to the relationship between the distance to singularity δ and the condition number K of the problem under study. The well-known Turing theorem states that, for a linear system Ax = b the distance to singularity, in a normwise measure, is the reciprocal of the normwise condition number ||A
−1
|||A|| In this Paper, we examine the possibility of extending this theorem for nonlinear problems in the neighbourhood of algebraic singularities. After reviewing the literature on that topic ([Demmel 1987, 1990], [Shub and Smale 1992]), we propose and check on the computer a conjecture which makes more explicit Demmel's bounds on the distance to singularity.</abstract><cop>Philadelphia, PA</cop><pub>Marcel Dekker, Inc</pub><doi>10.1080/01630569508816619</doi><tpages>16</tpages></addata></record> |
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| ispartof | Numerical functional analysis and optimization, 1995-01, Vol.16 (3-4), p.287-302 |
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| source | Taylor & Francis |
| subjects | Applied sciences Automata. Abstract machines. Turing machines Computer science control theory systems Exact sciences and technology Mathematics Methods of scientific computing (including symbolic computation, algebraic computation) Nonlinear algebraic and transcendental equations Numerical analysis Numerical analysis. Scientific computation Sciences and techniques of general use Theoretical computing |
| title | Computations in the neighbourhood of algebraic singularities |
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