On Simplex Trapezoidal Rule Families
We consider quadrature rules of a special type for the n-dimensional simplex. These form natural generalizations of the one-dimensional trapezoidal rule, and employ function values at points on a symmetric equally spaced grid, which includes the simplex vertices. The weight assigned to an abscissa i...
Gespeichert in:
| Veröffentlicht in: | SIAM journal on numerical analysis 1980-02, Vol.17 (1), p.126-147 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | eng |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| container_end_page | 147 |
|---|---|
| container_issue | 1 |
| container_start_page | 126 |
| container_title | SIAM journal on numerical analysis |
| container_volume | 17 |
| creator | Lyness, J. N. Genz, A. C. |
| description | We consider quadrature rules of a special type for the n-dimensional simplex. These form natural generalizations of the one-dimensional trapezoidal rule, and employ function values at points on a symmetric equally spaced grid, which includes the simplex vertices. The weight assigned to an abscissa is proportional to m-n, is the same at each interior grid point, and is the same at each grid point lying within the same d-dimensional boundary simplex. Families of such quadrature rules are defined, and some of their properties are derived. The principal result is the derivation of an error expansion for each rule; this expansion is an asymptotic expansion in inverse powers of m for the quadrature discretization error. Conditions under which this expansion is even are obtained together with the conditions under which the rule family is symmetric, or is of polynomial degree zero or one. |
| doi_str_mv | 10.1137/0717014 |
| format | Article |
| fullrecord | <record><control><sourceid>jstor_proqu</sourceid><recordid>TN_cdi_proquest_journals_923610350</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><jstor_id>2156555</jstor_id><sourcerecordid>2156555</sourcerecordid><originalsourceid>FETCH-LOGICAL-c203t-fa0d0c888839b5b94b555b8812711830fc07d24839a9b6a5deb2a472c63b6bb83</originalsourceid><addsrcrecordid>eNo9kE9LxDAQxYMoWFfxC3goIniqziTNnx5lcVVYWND1HJI2hZZ2W5MW1E9vll2cy_CYH28ej5BrhAdEJh9BogTMT0iCUPBsr05JAsBEhjktzslFCC1ErZAl5G6zSz-afuzcd7r1ZnS_Q1OZLn2fO5euTN90jQuX5Kw2XXBXx70gn6vn7fI1W29e3pZP66ykwKasNlBBqeKwwnJb5JZzbpVCKhEVg7oEWdE8Xk1hheGVs9TkkpaCWWGtYgtye_Ad_fA1uzDpdpj9Lr7UBWUCgXGI0P0BKv0Qgne1Hn3TG_-jEfS-AX1sIJI3B7IN0-D_MYpcxGDsDw00U30</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>923610350</pqid></control><display><type>article</type><title>On Simplex Trapezoidal Rule Families</title><source>SIAM Journals Online</source><source>JSTOR Mathematics & Statistics</source><source>JSTOR Archive Collection A-Z Listing</source><creator>Lyness, J. N. ; Genz, A. C.</creator><creatorcontrib>Lyness, J. N. ; Genz, A. C.</creatorcontrib><description>We consider quadrature rules of a special type for the n-dimensional simplex. These form natural generalizations of the one-dimensional trapezoidal rule, and employ function values at points on a symmetric equally spaced grid, which includes the simplex vertices. The weight assigned to an abscissa is proportional to m-n, is the same at each interior grid point, and is the same at each grid point lying within the same d-dimensional boundary simplex. Families of such quadrature rules are defined, and some of their properties are derived. The principal result is the derivation of an error expansion for each rule; this expansion is an asymptotic expansion in inverse powers of m for the quadrature discretization error. Conditions under which this expansion is even are obtained together with the conditions under which the rule family is symmetric, or is of polynomial degree zero or one.</description><identifier>ISSN: 0036-1429</identifier><identifier>EISSN: 1095-7170</identifier><identifier>DOI: 10.1137/0717014</identifier><language>eng</language><publisher>Philadelphia: Society for Industrial and Applied Mathematics</publisher><subject>Applied mathematics ; Coefficients ; Degrees of polynomials ; Integers ; Mathematical induction ; Mathematical theorems ; Numerical quadratures ; Polynomials ; Tetrahedrons ; Trapezoidal rule ; Vertices</subject><ispartof>SIAM journal on numerical analysis, 1980-02, Vol.17 (1), p.126-147</ispartof><rights>Copyright 1980 Society for Industrial and Applied Mathematics</rights><rights>[Copyright] © 1980 Society for Industrial and Applied Mathematics</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c203t-fa0d0c888839b5b94b555b8812711830fc07d24839a9b6a5deb2a472c63b6bb83</citedby><cites>FETCH-LOGICAL-c203t-fa0d0c888839b5b94b555b8812711830fc07d24839a9b6a5deb2a472c63b6bb83</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://www.jstor.org/stable/pdf/2156555$$EPDF$$P50$$Gjstor$$H</linktopdf><linktohtml>$$Uhttps://www.jstor.org/stable/2156555$$EHTML$$P50$$Gjstor$$H</linktohtml><link.rule.ids>314,780,784,803,832,3184,27924,27925,58017,58021,58250,58254</link.rule.ids></links><search><creatorcontrib>Lyness, J. N.</creatorcontrib><creatorcontrib>Genz, A. C.</creatorcontrib><title>On Simplex Trapezoidal Rule Families</title><title>SIAM journal on numerical analysis</title><description>We consider quadrature rules of a special type for the n-dimensional simplex. These form natural generalizations of the one-dimensional trapezoidal rule, and employ function values at points on a symmetric equally spaced grid, which includes the simplex vertices. The weight assigned to an abscissa is proportional to m-n, is the same at each interior grid point, and is the same at each grid point lying within the same d-dimensional boundary simplex. Families of such quadrature rules are defined, and some of their properties are derived. The principal result is the derivation of an error expansion for each rule; this expansion is an asymptotic expansion in inverse powers of m for the quadrature discretization error. Conditions under which this expansion is even are obtained together with the conditions under which the rule family is symmetric, or is of polynomial degree zero or one.</description><subject>Applied mathematics</subject><subject>Coefficients</subject><subject>Degrees of polynomials</subject><subject>Integers</subject><subject>Mathematical induction</subject><subject>Mathematical theorems</subject><subject>Numerical quadratures</subject><subject>Polynomials</subject><subject>Tetrahedrons</subject><subject>Trapezoidal rule</subject><subject>Vertices</subject><issn>0036-1429</issn><issn>1095-7170</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>1980</creationdate><recordtype>article</recordtype><sourceid>8G5</sourceid><sourceid>ABUWG</sourceid><sourceid>AFKRA</sourceid><sourceid>AZQEC</sourceid><sourceid>BENPR</sourceid><sourceid>CCPQU</sourceid><sourceid>DWQXO</sourceid><sourceid>GNUQQ</sourceid><sourceid>GUQSH</sourceid><sourceid>M2O</sourceid><recordid>eNo9kE9LxDAQxYMoWFfxC3goIniqziTNnx5lcVVYWND1HJI2hZZ2W5MW1E9vll2cy_CYH28ej5BrhAdEJh9BogTMT0iCUPBsr05JAsBEhjktzslFCC1ErZAl5G6zSz-afuzcd7r1ZnS_Q1OZLn2fO5euTN90jQuX5Kw2XXBXx70gn6vn7fI1W29e3pZP66ykwKasNlBBqeKwwnJb5JZzbpVCKhEVg7oEWdE8Xk1hheGVs9TkkpaCWWGtYgtye_Ad_fA1uzDpdpj9Lr7UBWUCgXGI0P0BKv0Qgne1Hn3TG_-jEfS-AX1sIJI3B7IN0-D_MYpcxGDsDw00U30</recordid><startdate>19800201</startdate><enddate>19800201</enddate><creator>Lyness, J. N.</creator><creator>Genz, A. C.</creator><general>Society for Industrial and Applied Mathematics</general><scope>AAYXX</scope><scope>CITATION</scope><scope>3V.</scope><scope>7WY</scope><scope>7WZ</scope><scope>7X2</scope><scope>7XB</scope><scope>87Z</scope><scope>88A</scope><scope>88F</scope><scope>88I</scope><scope>88K</scope><scope>8AL</scope><scope>8FE</scope><scope>8FG</scope><scope>8FH</scope><scope>8FK</scope><scope>8FL</scope><scope>8G5</scope><scope>ABJCF</scope><scope>ABUWG</scope><scope>AFKRA</scope><scope>ARAPS</scope><scope>ATCPS</scope><scope>AZQEC</scope><scope>BBNVY</scope><scope>BENPR</scope><scope>BEZIV</scope><scope>BGLVJ</scope><scope>BHPHI</scope><scope>CCPQU</scope><scope>D1I</scope><scope>DWQXO</scope><scope>FRNLG</scope><scope>F~G</scope><scope>GNUQQ</scope><scope>GUQSH</scope><scope>HCIFZ</scope><scope>JQ2</scope><scope>K60</scope><scope>K6~</scope><scope>K7-</scope><scope>KB.</scope><scope>L.-</scope><scope>L6V</scope><scope>LK8</scope><scope>M0C</scope><scope>M0K</scope><scope>M0N</scope><scope>M1Q</scope><scope>M2O</scope><scope>M2P</scope><scope>M2T</scope><scope>M7P</scope><scope>M7S</scope><scope>MBDVC</scope><scope>P5Z</scope><scope>P62</scope><scope>PATMY</scope><scope>PDBOC</scope><scope>PQBIZ</scope><scope>PQBZA</scope><scope>PQEST</scope><scope>PQQKQ</scope><scope>PQUKI</scope><scope>PTHSS</scope><scope>PYCSY</scope><scope>Q9U</scope></search><sort><creationdate>19800201</creationdate><title>On Simplex Trapezoidal Rule Families</title><author>Lyness, J. N. ; Genz, A. C.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c203t-fa0d0c888839b5b94b555b8812711830fc07d24839a9b6a5deb2a472c63b6bb83</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>1980</creationdate><topic>Applied mathematics</topic><topic>Coefficients</topic><topic>Degrees of polynomials</topic><topic>Integers</topic><topic>Mathematical induction</topic><topic>Mathematical theorems</topic><topic>Numerical quadratures</topic><topic>Polynomials</topic><topic>Tetrahedrons</topic><topic>Trapezoidal rule</topic><topic>Vertices</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Lyness, J. N.</creatorcontrib><creatorcontrib>Genz, A. C.</creatorcontrib><collection>CrossRef</collection><collection>ProQuest Central (Corporate)</collection><collection>Access via ABI/INFORM (ProQuest)</collection><collection>ABI/INFORM Global (PDF only)</collection><collection>Agricultural Science Collection</collection><collection>ProQuest Central (purchase pre-March 2016)</collection><collection>ABI/INFORM Global (Alumni Edition)</collection><collection>Biology Database (Alumni Edition)</collection><collection>Military Database (Alumni Edition)</collection><collection>Science Database (Alumni Edition)</collection><collection>Telecommunications (Alumni Edition)</collection><collection>Computing Database (Alumni Edition)</collection><collection>ProQuest SciTech Collection</collection><collection>ProQuest Technology Collection</collection><collection>ProQuest Natural Science Collection</collection><collection>ProQuest Central (Alumni) (purchase pre-March 2016)</collection><collection>ABI/INFORM Collection (Alumni Edition)</collection><collection>Research Library (Alumni Edition)</collection><collection>Materials Science & Engineering Collection</collection><collection>ProQuest Central (Alumni Edition)</collection><collection>ProQuest Central UK/Ireland</collection><collection>Advanced Technologies & Aerospace Collection</collection><collection>Agricultural & Environmental Science Collection</collection><collection>ProQuest Central Essentials</collection><collection>Biological Science Collection</collection><collection>ProQuest Central</collection><collection>Business Premium Collection</collection><collection>Technology Collection</collection><collection>Natural Science Collection</collection><collection>ProQuest One Community College</collection><collection>ProQuest Materials Science Collection</collection><collection>ProQuest Central Korea</collection><collection>Business Premium Collection (Alumni)</collection><collection>ABI/INFORM Global (Corporate)</collection><collection>ProQuest Central Student</collection><collection>Research Library Prep</collection><collection>SciTech Premium Collection</collection><collection>ProQuest Computer Science Collection</collection><collection>ProQuest Business Collection (Alumni Edition)</collection><collection>ProQuest Business Collection</collection><collection>Computer Science Database</collection><collection>Materials Science Database</collection><collection>ABI/INFORM Professional Advanced</collection><collection>ProQuest Engineering Collection</collection><collection>ProQuest Biological Science Collection</collection><collection>ABI/INFORM Global</collection><collection>Agricultural Science Database</collection><collection>Computing Database</collection><collection>Military Database</collection><collection>Research Library</collection><collection>Science Database</collection><collection>Telecommunications Database</collection><collection>Biological Science Database</collection><collection>Engineering Database</collection><collection>Research Library (Corporate)</collection><collection>Advanced Technologies & Aerospace Database</collection><collection>ProQuest Advanced Technologies & Aerospace Collection</collection><collection>Environmental Science Database</collection><collection>Materials Science Collection</collection><collection>ProQuest One Business</collection><collection>ProQuest One Business (Alumni)</collection><collection>ProQuest One Academic Eastern Edition (DO NOT USE)</collection><collection>ProQuest One Academic</collection><collection>ProQuest One Academic UKI Edition</collection><collection>Engineering Collection</collection><collection>Environmental Science Collection</collection><collection>ProQuest Central Basic</collection><jtitle>SIAM journal on numerical analysis</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Lyness, J. N.</au><au>Genz, A. C.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On Simplex Trapezoidal Rule Families</atitle><jtitle>SIAM journal on numerical analysis</jtitle><date>1980-02-01</date><risdate>1980</risdate><volume>17</volume><issue>1</issue><spage>126</spage><epage>147</epage><pages>126-147</pages><issn>0036-1429</issn><eissn>1095-7170</eissn><abstract>We consider quadrature rules of a special type for the n-dimensional simplex. These form natural generalizations of the one-dimensional trapezoidal rule, and employ function values at points on a symmetric equally spaced grid, which includes the simplex vertices. The weight assigned to an abscissa is proportional to m-n, is the same at each interior grid point, and is the same at each grid point lying within the same d-dimensional boundary simplex. Families of such quadrature rules are defined, and some of their properties are derived. The principal result is the derivation of an error expansion for each rule; this expansion is an asymptotic expansion in inverse powers of m for the quadrature discretization error. Conditions under which this expansion is even are obtained together with the conditions under which the rule family is symmetric, or is of polynomial degree zero or one.</abstract><cop>Philadelphia</cop><pub>Society for Industrial and Applied Mathematics</pub><doi>10.1137/0717014</doi><tpages>22</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0036-1429 |
| ispartof | SIAM journal on numerical analysis, 1980-02, Vol.17 (1), p.126-147 |
| issn | 0036-1429 1095-7170 |
| language | eng |
| recordid | cdi_proquest_journals_923610350 |
| source | SIAM Journals Online; JSTOR Mathematics & Statistics; JSTOR Archive Collection A-Z Listing |
| subjects | Applied mathematics Coefficients Degrees of polynomials Integers Mathematical induction Mathematical theorems Numerical quadratures Polynomials Tetrahedrons Trapezoidal rule Vertices |
| title | On Simplex Trapezoidal Rule Families |
| url | https://sfx.bib-bvb.de/sfx_tum?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2024-12-29T15%3A02%3A11IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-jstor_proqu&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=On%20Simplex%20Trapezoidal%20Rule%20Families&rft.jtitle=SIAM%20journal%20on%20numerical%20analysis&rft.au=Lyness,%20J.%20N.&rft.date=1980-02-01&rft.volume=17&rft.issue=1&rft.spage=126&rft.epage=147&rft.pages=126-147&rft.issn=0036-1429&rft.eissn=1095-7170&rft_id=info:doi/10.1137/0717014&rft_dat=%3Cjstor_proqu%3E2156555%3C/jstor_proqu%3E%3Curl%3E%3C/url%3E&disable_directlink=true&sfx.directlink=off&sfx.report_link=0&rft_id=info:oai/&rft_pqid=923610350&rft_id=info:pmid/&rft_jstor_id=2156555&rfr_iscdi=true |