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[{"key": "dc.contributor.advisor", "value": "Juutinen, Petri", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "\u00c5str\u00f6m, Anne", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2022-04-27T04:59:32Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2022-04-27T04:59:32Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2022", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/80733", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4n pro gradu -tutkielman tarkoituksena on syvent\u00e4\u00e4 ja laajentaa lukion MAA12 kurssin numeerisen integroinnin teoriaa. Tutkielmassa k\u00e4sitell\u00e4\u00e4n Newton-Cotesin integrointimenetelm\u00e4 sek\u00e4 Richarsonin ja Rombergin ekstrapolointimenetelm\u00e4. Menetelmi\u00e4 k\u00e4ytet\u00e4\u00e4n yksi- ja kaksiulotteiseen integrointiin.\n\tNumeerinen integroiminen on pinta-alan laskemista. Pinta-ala on funktion kuvaajan ja x-akselin v\u00e4liin j\u00e4\u00e4v\u00e4 alue, joka rajoittuu integrointiv\u00e4liin. Integroinnin m\u00e4\u00e4ritelm\u00e4ksi on valittu koulumatematiikassa yleisesti k\u00e4ytetty Bernhard Riemannin m\u00e4\u00e4ritelm\u00e4, koska siit\u00e4 on luontevaa johtaa numeerisen integroinnin menetelm\u00e4t.\n\tLukion kurssilla esitellyt puolisuunnikass\u00e4\u00e4nt\u00f6 ja Simpsonin 1/3-s\u00e4\u00e4nt\u00f6 ovat Newton-Cotesin numeerisia integrointikaavoja. N\u00e4iden integrointikaavojen johtamisessa on k\u00e4ytetty Lagrangen interpolaatiopolynomeja. Lagrangen interpolaatiokaavalla voidaan m\u00e4\u00e4ritt\u00e4\u00e4 polynomi, joka kulkee valittujen pisteiden kautta. Pisteet voivat olla erillisi\u00e4 tai ne voidaan valita kuvaajalta. Puolisuunnikass\u00e4\u00e4nn\u00f6ss\u00e4 n\u00e4it\u00e4 pisteit\u00e4 on kaksi, ja Simpsonin 1/3-s\u00e4\u00e4nn\u00f6ss\u00e4 on kolme pistett\u00e4. Pisteit\u00e4 lis\u00e4\u00e4m\u00e4ll\u00e4 saadaan johdettua lis\u00e4\u00e4 tarkempia integrointikaavoja, kunhan pisteist\u00f6 on tasav\u00e4linen.\n\tKun pisteit\u00e4 on yli seitsem\u00e4n, tulee kaavoihin negatiivisia kertoimia ja ne eiv\u00e4t ole k\u00e4ytt\u00f6kelpoisia. Yleisimmin integrointiv\u00e4li jaetaankin osav\u00e4leihin, jotka integroidaan erikseen ja osav\u00e4lien integraalit summataan yhteen. N\u00e4it\u00e4 kaavoja kutsutaan yhdistetyiksi kaavoiksi. Tarkkuus paranee, kun osav\u00e4lien m\u00e4\u00e4r\u00e4 lis\u00e4\u00e4ntyy.\n\tKun integraalille ei voida laskea tarkkaa arvoa, on t\u00e4rke\u00e4\u00e4 tiet\u00e4\u00e4 virheen suuruusluokka. Virheen arvioiminen on aina suurimman mahdollisen virheen eli maksimaalisen virheen laskemista. Jokaiselle Newton-Cotesin kaavalle voidaan laskea virhe, ja virheelle voidaan m\u00e4\u00e4ritell\u00e4 asteluku. Asteluku on Oh^n, jossa h on v\u00e4lin pituus. Mit\u00e4 suurempi asteluku on, sit\u00e4 enemm\u00e4n v\u00e4linpituuden muutoksella on vaikutusta virheen suuruusluokkaan. Jos asteluku on Oh^4, niin v\u00e4lin pituuden muutos vaikuttaa virheeseen h^4 kertaisesti. Kun osav\u00e4lej\u00e4 lis\u00e4t\u00e4\u00e4n, tarkkuus paranee t\u00e4m\u00e4n asteluvun rajoissa. Osav\u00e4lien lis\u00e4\u00e4minen ei paranna integraalin tarkkuutta kovinkaan nopeasti.\n\tEuler-MacLauren summakaavalla voidaan puolisuunnikass\u00e4\u00e4nn\u00f6n virhe kirjoittaa\nsarjana. T\u00e4st\u00e4 sarjamuodosta saadaan johdettua rekursiokaava, jolla saadaan eliminoitua virhetermej\u00e4. T\u00e4t\u00e4 h^2 virhetermin eliminointia kutsutaan Richardsonin ekstrapolaatioksi. Ekstrapolointimenetelm\u00e4ll\u00e4 saadaan lis\u00e4\u00e4 tarkkuutta nopeammin, koska asteluku paranee kahdella jokaisella ekstrapolointi kerralla. T\u00e4ss\u00e4 menetelm\u00e4ss\u00e4 ensin jaetaan integroitava v\u00e4li osav\u00e4leihin m = 2,4,8,16, . . . ja lasketaan integraalit eri osav\u00e4leille puolisuunnikass\u00e4\u00e4nn\u00f6ll\u00e4. N\u00e4ist\u00e4 arvoista rekursiokaavalla saadaan uudet tarkemmat arvot. Edelleen samalla tavalla voidaan ekstrapoloida n\u00e4ist\u00e4 arvoista tarkempia arvoja rekursiokaavalla. N\u00e4m\u00e4 saadut arvot kirjataan taulukkoon, josta\nvoidaan helposti n\u00e4hd\u00e4 arvojen tarkentuminen. T\u00e4t\u00e4 rekursiomenetelm\u00e4\u00e4 kutsutaan\nRombergin menetelm\u00e4ksi ja taulukkoa Rombergin tauluksi.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Miia Hakanen (mihakane@jyu.fi) on 2022-04-27T04:59:32Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2022-04-27T04:59:32Z (GMT). No. of bitstreams: 0\n Previous issue date: 2022", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "65", "language": "", "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.format.mimetype", "value": "application/pdf", "language": null, "element": "format", "qualifier": "mimetype", "schema": "dc"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.subject.other", "value": "Ronbergin menetelm\u00e4", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "ekstrapolointi", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Lagrangen interpolaatiopolynomi", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Euler-MacLauren summakaava", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "numeerinen integrointi", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Numeerinen integrointi", "language": "", "element": "title", "qualifier": null, "schema": "dc"}, {"key": "dc.type", "value": "master thesis", "language": null, "element": "type", "qualifier": null, "schema": "dc"}, {"key": "dc.identifier.urn", "value": "URN:NBN:fi:jyu-202204272406", "language": "", "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Faculty of Sciences", "language": "en", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Matematiikan ja tilastotieteen laitos", "language": "fi", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Department of Mathematics and Statistics", "language": "en", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikan opettajankoulutus", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Teacher education programme in Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "yvv.contractresearch.funding", "value": "0", "language": "", "element": "contractresearch", "qualifier": "funding", "schema": "yvv"}, {"key": "dc.type.coar", "value": "http://purl.org/coar/resource_type/c_bdcc", "language": null, "element": "type", "qualifier": "coar", "schema": "dc"}, {"key": "dc.rights.accesslevel", "value": "openAccess", "language": null, "element": "rights", "qualifier": "accesslevel", "schema": "dc"}, {"key": "dc.type.publication", "value": "masterThesis", "language": null, "element": "type", "qualifier": "publication", "schema": "dc"}, {"key": "dc.subject.oppiainekoodi", "value": "4041", "language": "", "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.subject.yso", "value": "matematiikka", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "integraalilaskenta", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "numeerinen matematiikka", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.format.content", "value": "fulltext", "language": null, "element": "format", "qualifier": "content", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}, {"key": "dc.type.okm", "value": "G2", "language": null, "element": "type", "qualifier": "okm", "schema": "dc"}]
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