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Estimates for the asymptotic convergence factor of two intervals
Let E be the union of two real intervals not containing zero. Then L n r ( E ) denotes the supremum norm of that polynomial P n of degree less than or equal to n , which is minimal with respect to the supremum norm provided that P n ( 0 ) = 1 . It is well known that the limit κ ( E ) ≔ lim n → ∞ L n...
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| Published in: | Journal of computational and applied mathematics 2011-08, Vol.236 (1), p.28-38 |
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| Language: | English |
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| cites | cdi_FETCH-LOGICAL-c402t-38c512a302bf4685d0b5d586efb6fbe875fe3e0bd9b42592034a66266b725d383 |
| container_end_page | 38 |
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| container_title | Journal of computational and applied mathematics |
| container_volume | 236 |
| creator | Schiefermayr, Klaus |
| description | Let
E
be the union of two real intervals not containing zero. Then
L
n
r
(
E
)
denotes the supremum norm of that polynomial
P
n
of degree less than or equal to
n
, which is minimal with respect to the supremum norm provided that
P
n
(
0
)
=
1
. It is well known that the limit
κ
(
E
)
≔
lim
n
→
∞
L
n
r
(
E
)
n
exists, where
κ
(
E
)
is called the asymptotic convergence factor, since it plays a crucial role for certain iterative methods solving large-scale matrix problems. The factor
κ
(
E
)
can be expressed with the help of Jacobi’s elliptic and theta functions, where this representation is very involved. In this paper, we give precise upper and lower bounds for
κ
(
E
)
in terms of elementary functions of the endpoints of
E
. |
| doi_str_mv | 10.1016/j.cam.2010.06.008 |
| format | article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_miscellaneous_919907111</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><els_id>S0377042710003523</els_id><sourcerecordid>919907111</sourcerecordid><originalsourceid>FETCH-LOGICAL-c402t-38c512a302bf4685d0b5d586efb6fbe875fe3e0bd9b42592034a66266b725d383</originalsourceid><addsrcrecordid>eNp9kE1LAzEQhoMoWKs_wNtexNOuk-xusosXpdQPKHjRc8hmJ5rS3dQkrfTfm9Li0dPLwDMzvA8h1xQKCpTfLQuthoJBmoEXAM0JmdBGtDkVojklEyiFyKFi4pxchLAEAN7SakIe5iHaQUUMmXE-i1-YqbAb1tFFqzPtxi36Txw1ZkbpmAhnsvjjMjtG9Fu1CpfkzKTAq2NOycfT_H32ki_enl9nj4tcV8BiXja6pkyVwDpT8abuoav7uuFoOm46bERtsETo-rarWN0yKCvFOeO8E6zuy6acktvD3bV33xsMUQ42aFyt1IhuE2RL2xYEpTSR9EBq70LwaOTap4p-JynIvSy5lEmW3MuSwGWSlXZujtdV0GplvBq1DX-LrOK0rigk7v7AYaq6tehl0Havp7cedZS9s_98-QXU_X5r</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>919907111</pqid></control><display><type>article</type><title>Estimates for the asymptotic convergence factor of two intervals</title><source>Elsevier:Jisc Collections:Elsevier Read and Publish Agreement 2022-2024:Freedom Collection (Reading list)</source><creator>Schiefermayr, Klaus</creator><creatorcontrib>Schiefermayr, Klaus</creatorcontrib><description>Let
E
be the union of two real intervals not containing zero. Then
L
n
r
(
E
)
denotes the supremum norm of that polynomial
P
n
of degree less than or equal to
n
, which is minimal with respect to the supremum norm provided that
P
n
(
0
)
=
1
. It is well known that the limit
κ
(
E
)
≔
lim
n
→
∞
L
n
r
(
E
)
n
exists, where
κ
(
E
)
is called the asymptotic convergence factor, since it plays a crucial role for certain iterative methods solving large-scale matrix problems. The factor
κ
(
E
)
can be expressed with the help of Jacobi’s elliptic and theta functions, where this representation is very involved. In this paper, we give precise upper and lower bounds for
κ
(
E
)
in terms of elementary functions of the endpoints of
E
.</description><identifier>ISSN: 0377-0427</identifier><identifier>EISSN: 1879-1778</identifier><identifier>DOI: 10.1016/j.cam.2010.06.008</identifier><identifier>CODEN: JCAMDI</identifier><language>eng</language><publisher>Kidlington: Elsevier B.V</publisher><subject>Algebra ; Algebraic geometry ; Asymptotic properties ; Convergence ; Estimated asymptotic convergence factor ; Exact sciences and technology ; Inequality ; Intervals ; Jacobian elliptic functions ; Jacobian theta functions ; Mathematical analysis ; Mathematical models ; Mathematics ; Norms ; Numerical analysis ; Numerical analysis. Scientific computation ; Numerical linear algebra ; Representations ; Sciences and techniques of general use ; Sequences, series, summability ; Two intervals ; Unions</subject><ispartof>Journal of computational and applied mathematics, 2011-08, Vol.236 (1), p.28-38</ispartof><rights>2011</rights><rights>2015 INIST-CNRS</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c402t-38c512a302bf4685d0b5d586efb6fbe875fe3e0bd9b42592034a66266b725d383</citedby><cites>FETCH-LOGICAL-c402t-38c512a302bf4685d0b5d586efb6fbe875fe3e0bd9b42592034a66266b725d383</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>309,310,314,780,784,789,790,23929,23930,25139,27923,27924</link.rule.ids><backlink>$$Uhttp://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=24615410$$DView record in Pascal Francis$$Hfree_for_read</backlink></links><search><creatorcontrib>Schiefermayr, Klaus</creatorcontrib><title>Estimates for the asymptotic convergence factor of two intervals</title><title>Journal of computational and applied mathematics</title><description>Let
E
be the union of two real intervals not containing zero. Then
L
n
r
(
E
)
denotes the supremum norm of that polynomial
P
n
of degree less than or equal to
n
, which is minimal with respect to the supremum norm provided that
P
n
(
0
)
=
1
. It is well known that the limit
κ
(
E
)
≔
lim
n
→
∞
L
n
r
(
E
)
n
exists, where
κ
(
E
)
is called the asymptotic convergence factor, since it plays a crucial role for certain iterative methods solving large-scale matrix problems. The factor
κ
(
E
)
can be expressed with the help of Jacobi’s elliptic and theta functions, where this representation is very involved. In this paper, we give precise upper and lower bounds for
κ
(
E
)
in terms of elementary functions of the endpoints of
E
.</description><subject>Algebra</subject><subject>Algebraic geometry</subject><subject>Asymptotic properties</subject><subject>Convergence</subject><subject>Estimated asymptotic convergence factor</subject><subject>Exact sciences and technology</subject><subject>Inequality</subject><subject>Intervals</subject><subject>Jacobian elliptic functions</subject><subject>Jacobian theta functions</subject><subject>Mathematical analysis</subject><subject>Mathematical models</subject><subject>Mathematics</subject><subject>Norms</subject><subject>Numerical analysis</subject><subject>Numerical analysis. Scientific computation</subject><subject>Numerical linear algebra</subject><subject>Representations</subject><subject>Sciences and techniques of general use</subject><subject>Sequences, series, summability</subject><subject>Two intervals</subject><subject>Unions</subject><issn>0377-0427</issn><issn>1879-1778</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2011</creationdate><recordtype>article</recordtype><recordid>eNp9kE1LAzEQhoMoWKs_wNtexNOuk-xusosXpdQPKHjRc8hmJ5rS3dQkrfTfm9Li0dPLwDMzvA8h1xQKCpTfLQuthoJBmoEXAM0JmdBGtDkVojklEyiFyKFi4pxchLAEAN7SakIe5iHaQUUMmXE-i1-YqbAb1tFFqzPtxi36Txw1ZkbpmAhnsvjjMjtG9Fu1CpfkzKTAq2NOycfT_H32ki_enl9nj4tcV8BiXja6pkyVwDpT8abuoav7uuFoOm46bERtsETo-rarWN0yKCvFOeO8E6zuy6acktvD3bV33xsMUQ42aFyt1IhuE2RL2xYEpTSR9EBq70LwaOTap4p-JynIvSy5lEmW3MuSwGWSlXZujtdV0GplvBq1DX-LrOK0rigk7v7AYaq6tehl0Havp7cedZS9s_98-QXU_X5r</recordid><startdate>20110801</startdate><enddate>20110801</enddate><creator>Schiefermayr, Klaus</creator><general>Elsevier B.V</general><general>Elsevier</general><scope>6I.</scope><scope>AAFTH</scope><scope>IQODW</scope><scope>AAYXX</scope><scope>CITATION</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>20110801</creationdate><title>Estimates for the asymptotic convergence factor of two intervals</title><author>Schiefermayr, Klaus</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c402t-38c512a302bf4685d0b5d586efb6fbe875fe3e0bd9b42592034a66266b725d383</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2011</creationdate><topic>Algebra</topic><topic>Algebraic geometry</topic><topic>Asymptotic properties</topic><topic>Convergence</topic><topic>Estimated asymptotic convergence factor</topic><topic>Exact sciences and technology</topic><topic>Inequality</topic><topic>Intervals</topic><topic>Jacobian elliptic functions</topic><topic>Jacobian theta functions</topic><topic>Mathematical analysis</topic><topic>Mathematical models</topic><topic>Mathematics</topic><topic>Norms</topic><topic>Numerical analysis</topic><topic>Numerical analysis. Scientific computation</topic><topic>Numerical linear algebra</topic><topic>Representations</topic><topic>Sciences and techniques of general use</topic><topic>Sequences, series, summability</topic><topic>Two intervals</topic><topic>Unions</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Schiefermayr, Klaus</creatorcontrib><collection>ScienceDirect Open Access Titles</collection><collection>Elsevier:ScienceDirect:Open Access</collection><collection>Pascal-Francis</collection><collection>CrossRef</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>Journal of computational and applied mathematics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Schiefermayr, Klaus</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Estimates for the asymptotic convergence factor of two intervals</atitle><jtitle>Journal of computational and applied mathematics</jtitle><date>2011-08-01</date><risdate>2011</risdate><volume>236</volume><issue>1</issue><spage>28</spage><epage>38</epage><pages>28-38</pages><issn>0377-0427</issn><eissn>1879-1778</eissn><coden>JCAMDI</coden><abstract>Let
E
be the union of two real intervals not containing zero. Then
L
n
r
(
E
)
denotes the supremum norm of that polynomial
P
n
of degree less than or equal to
n
, which is minimal with respect to the supremum norm provided that
P
n
(
0
)
=
1
. It is well known that the limit
κ
(
E
)
≔
lim
n
→
∞
L
n
r
(
E
)
n
exists, where
κ
(
E
)
is called the asymptotic convergence factor, since it plays a crucial role for certain iterative methods solving large-scale matrix problems. The factor
κ
(
E
)
can be expressed with the help of Jacobi’s elliptic and theta functions, where this representation is very involved. In this paper, we give precise upper and lower bounds for
κ
(
E
)
in terms of elementary functions of the endpoints of
E
.</abstract><cop>Kidlington</cop><pub>Elsevier B.V</pub><doi>10.1016/j.cam.2010.06.008</doi><tpages>11</tpages><oa>free_for_read</oa></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0377-0427 |
| ispartof | Journal of computational and applied mathematics, 2011-08, Vol.236 (1), p.28-38 |
| issn | 0377-0427 1879-1778 |
| language | eng |
| recordid | cdi_proquest_miscellaneous_919907111 |
| source | Elsevier:Jisc Collections:Elsevier Read and Publish Agreement 2022-2024:Freedom Collection (Reading list) |
| subjects | Algebra Algebraic geometry Asymptotic properties Convergence Estimated asymptotic convergence factor Exact sciences and technology Inequality Intervals Jacobian elliptic functions Jacobian theta functions Mathematical analysis Mathematical models Mathematics Norms Numerical analysis Numerical analysis. Scientific computation Numerical linear algebra Representations Sciences and techniques of general use Sequences, series, summability Two intervals Unions |
| title | Estimates for the asymptotic convergence factor of two intervals |
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