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[{"key": "dc.contributor.advisor", "value": "Lehrb\u00e4ck, Juha", "language": null, "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Arvio, Ville", "language": null, "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2018-01-23T16:12:49Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2018-01-23T16:12:49Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2017", "language": null, "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.other", "value": "oai:jykdok.linneanet.fi:1815569", "language": null, "element": "identifier", "qualifier": "other", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/56870", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4n tutkielman tavoitteena on k\u00e4sitell\u00e4 numeerisia yht\u00e4l\u00f6nratkaisumenetelmi\u00e4 matematiikan aineenopettajan n\u00e4k\u00f6kulmasta ja toimia lukion numeerisen matematiikan kurssin opettajan taustamateriaalina. Keskeinen sis\u00e4lt\u00f6 k\u00e4sittelee Lipschitz-jatkuvuutta, iteraatiota sek\u00e4 Newton-Raphsonin menetelm\u00e4\u00e4.\n\nYht\u00e4l\u00f6nratkaisu ja kahden lausekkeen yht\u00e4suuruuksien vertailu palautuu aina matematiikan klassiseen ongelmaan funktion nollakohdan etsimisest\u00e4. Keskeiset numeerisen yht\u00e4l\u00f6nratkaisun metodit ovat rekursio ja iteraatio. Rekursio tarkoittaa oleellisesti toistoa. Iteraatiossa edellinen likiratkaisu ohjaa tarkentavasti seuraavan likiratkaisun laskentaa. T\u00e4ll\u00f6in muodostuu tarkentuvien likiratkaisuiden lukujono, joka suotuisassa tapauksessa suppenee kohti alkuper\u00e4isen funktion nollakohtaa. Iteraatiossa yhteys alkuper\u00e4isen funktion nollakohtayht\u00e4l\u00f6n f(x) = 0 sek\u00e4 iteraattifunktion iterointiyht\u00e4l\u00f6n x = phi(x) v\u00e4lill\u00e4 on oleellinen, sill\u00e4 iterointiyht\u00e4l\u00f6n kiintopiste on my\u00f6s alkuper\u00e4isen funktion nollakohta. Siten iterointi toimii ty\u00f6v\u00e4lineen\u00e4 nollakohdan likiarvon m\u00e4\u00e4ritt\u00e4misess\u00e4.\n\nIteraattifunktio ei ole yksik\u00e4sitteinen. Jotkin iteraatiomenetelm\u00e4t toimivat paremmin kuin toiset. Oleellista suppenemiselle on riitt\u00e4v\u00e4n l\u00e4helt\u00e4 nollakohtaa valittu alkuarvaus. Huonosti valittu alkuarvaus ja sopimaton iteraatiomenetelm\u00e4 johtavat suppenemisen sijaan likiratkaisulukujonon hajaantumiseen. Iteraation suppenemisesta voidaan kuitenkin varmistua, jos pystyt\u00e4\u00e4n osoittamaan, ett\u00e4 iteraattifunktio on ratkaisun l\u00e4hiymp\u00e4rist\u00f6ss\u00e4 kutistavasti Lipschitz-jatkuva. T\u00e4ll\u00f6in Lipschitz-ehto takaa, ett\u00e4 funktio k\u00e4ytt\u00e4ytyy nollakohdan l\u00e4hiymp\u00e4rist\u00f6ss\u00e4 maltillisesti ja \"pomppimatta\". K\u00e4yt\u00e4nn\u00f6llisen\u00e4 apuv\u00e4lineen\u00e4 Lipschitz-jatkuvuuden tutkimiseen hy\u00f6dynnet\u00e4\u00e4n usein Lipschitz-jatkuvuuden derivoituvuusehtoa.\n\nYksinkertaisimmillaan yht\u00e4l\u00f6nratkaisumenetelm\u00e4 on rekursiivinen puolitushaku, joka tarkentuu tarkasteluv\u00e4li\u00e4 puolittaen ja samalla nollakohdan v\u00e4lin sis\u00e4puolella s\u00e4ilytt\u00e4en. Yksinkertaisin iteraatio on kiintopistemenetelm\u00e4, joka suppenee lineaarisesti. Tutkittaessa suppenemisnopeuden kiihdytt\u00e4mist\u00e4 p\u00e4\u00e4dyt\u00e4\u00e4n Aitkenin kaavaan, jolla saadaan likiratkaisuiden poikkeaman eli liki- ja tosiratkaisun v\u00e4lisen et\u00e4isyyden suppenemiseen huomattava parannus. Kysymykseen voidaanko yleisesti m\u00e4\u00e4ritell\u00e4 iteraattifunktio, jonka suppenemisnopeus on v\u00e4hint\u00e4\u00e4n neli\u00f6ityv\u00e4, l\u00f6ydet\u00e4\u00e4n ehto, joka johtaa Newton-Raphsonin menetelm\u00e4n palautuskaavaan ja samalla menetelm\u00e4n konstruktiiviseen perusteluun. Oleellisesti Newtonin menetelm\u00e4 perustuu derivaatan k\u00e4ytt\u00f6\u00f6n, jolloin graafisesti seuraava nollakohdan likiarvo l\u00f6ydet\u00e4\u00e4n funktion kuvaajalle piirretyn tangentin ja x-akselin leikkauskohdasta. Ylilineaarisesti suppenevat j\u00e4nne- ja sekanttimenetelm\u00e4 esitell\u00e4\u00e4n lyhyesti, mutta itsen\u00e4isin\u00e4 menetelmin\u00e4 l\u00e4pik\u00e4ymisen sijasta l\u00e4hinn\u00e4 Newtonin menetelm\u00e4n k\u00e4yt\u00e4nn\u00f6llisin\u00e4 sovelluksina, kun funktion derivaatan arvoa on hankala laskea tai ylip\u00e4\u00e4ns\u00e4 derivaatta on vaikea muodostaa.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted using Plone Publishing form by Ville Arvio (vijuarvi) on 2018-01-23 16:12:48.504250. Form: Pro gradu -lomake (https://kirjasto.jyu.fi/julkaisut/julkaisulomakkeet/pro-gradu-lomake). JyX data: [jyx_publishing-allowed (fi) =True]", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by jyx lomake-julkaisija (jyx-julkaisija.group@korppi.jyu.fi) on 2018-01-23T16:12:49Z\nNo. of bitstreams: 2\nURN:NBN:fi:jyu-201801231313.pdf: 2965135 bytes, checksum: d53bb502c1d76dc94b3d2fdfcab937ef (MD5)\nlicense.html: 4791 bytes, checksum: d0a29abd8750324d684ea69d59b12a01 (MD5)", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2018-01-23T16:12:49Z (GMT). No. of bitstreams: 2\nURN:NBN:fi:jyu-201801231313.pdf: 2965135 bytes, checksum: d53bb502c1d76dc94b3d2fdfcab937ef (MD5)\nlicense.html: 4791 bytes, checksum: d0a29abd8750324d684ea69d59b12a01 (MD5)\n Previous issue date: 2017", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "1 verkkoaineisto (133 sivua)", "language": null, "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": "Lipschitz-jatkuvuus", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Newton-Raphsonin menetelm\u00e4", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Numeeriset yht\u00e4l\u00f6nratkaisumenetelm\u00e4t", "language": null, "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-201801231313", "language": null, "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", 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