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[{"key": "dc.contributor.author", "value": "Kinnunen, Saku", "language": null, "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2025-04-17T05:37:46Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2025-04-17T05:37:46Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2025", "language": null, "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/101493", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "Tutkielmassa k\u00e4sitell\u00e4\u00e4n matriisilaskentaa ja tarkastellaan laskennassa esiintyvi\u00e4 virheit\u00e4. Matriisilaskennassa keskityt\u00e4\u00e4n muutamaan matriisihajotelmaan, vektori- ja matriisinormeihin sek\u00e4 n\u00e4iden kautta siirryt\u00e4\u00e4n matriisilaskennan virheanalyysin pariin. Siisp\u00e4 tutkielmassa t\u00e4hd\u00e4t\u00e4\u00e4n yht\u00e4l\u00f6nratkaisun virheen analysointiin. Siit\u00e4 syyst\u00e4 k\u00e4yd\u00e4\u00e4n tietyt matriisihajotelmat l\u00e4pi sek\u00e4 esitell\u00e4\u00e4n matriisinormien perusteoria. Tutkielmassa k\u00e4sitelt\u00e4v\u00e4t asiat rakentuvat toistensa p\u00e4\u00e4lle, eli esimerkiksi k\u00e4ytyj\u00e4 matriisihajotelmia tullaan tarvitsemaan virheanalyysin parissa. My\u00f6s vektori- ja matriisinormit t\u00e4ytyy k\u00e4yd\u00e4 l\u00e4pi ennen virheanalyysi\u00e4. Johdannon j\u00e4lkeen ensimm\u00e4isess\u00e4 kappaleessa on annettu joitain esi- ja yleistietoja matriiseihin liittyen, mutta tiettyj\u00e4 asioita pidet\u00e4\u00e4n lukijalle selv\u00e4n\u00e4. Ensimm\u00e4inen kappale toimii samalla pienimuotoisena johdatteluna matriisilaskentaan. Tutkielmassa k\u00e4yd\u00e4\u00e4n l\u00e4pi hieman matriisilaskennan historiaa ja sen kehityst\u00e4. Matriisilaskennan historiasta on otettu pintaraapaisu sek\u00e4 matriisihajotelmista Cholesky- ja ominaisarvohajotelmaa k\u00e4sitell\u00e4\u00e4n my\u00f6s hyvin pinnallisesti. LU-hajotelmaan keskityt\u00e4\u00e4n puolestaan perusteellisemmin, koska lopussa paneudutaan LU-hajotelman virheisiin. Ennen matriisinormeja on tarpeen k\u00e4yd\u00e4 l\u00e4pi vektorinormit, sill\u00e4 k\u00e4sittelyss\u00e4 on mm. vektorinormin indusoimia matriisinormeja. N\u00e4in ollen vektorinormeja tarvitaan tietyiss\u00e4 matriisinormeissa. Vektori- ja matriisinormeja k\u00e4yd\u00e4\u00e4n my\u00f6s tutkielmassa perusteellisemmin l\u00e4pi, sill\u00e4 ne ovat t\u00e4rkeit\u00e4 virheanalyysi\u00e4 tarkasteltaessa. Matematiikassa vektorinormin k\u00e4site kuvaa vektorin pituutta tai suuruutta. Vektorinormille on nelj\u00e4 aksioomaa, joiden t\u00e4ytyy olla voimassa. N\u00e4m\u00e4 ovat ei-negatiivisuus, positiivisuus, homogeenisuus ja kolmioep\u00e4yht\u00e4l\u00f6. Matriisinormi on puolestaan matemaattinen k\u00e4site, joka laajentaa vektorinormin idean matriiseihin. Matriisinormi on mitta, joka kertoo kuinka \u201dsuuri\u201d tai \u201dlaaja\u201d matriisi on. Vektorinormin nelj\u00e4 aksioomaa p\u00e4tev\u00e4t my\u00f6s matriisinormille, mutta n\u00e4iden aksioomien lis\u00e4ksi submultiplikatiivisuuden t\u00e4ytyy olla voimassa. Matriisinormien sovelluskohteina ovat esimerkiksi matriisien virheanalyysi ja virheiden arviointi. N\u00e4ihin tutustutaan tarkemmin tutkielman loppupuolella. Matriisille saadaan m\u00e4\u00e4ritelty\u00e4 kohtalaisen helposti h\u00e4iri\u00f6alttiusluku, mik\u00e4 mittaa mm. matriisin alkioissa ilmentyvien py\u00f6ristysvirheiden suuruutta. Matriisin h\u00e4iri\u00f6alttiusluku riippuu valitusta matriisinormista. Mit\u00e4 suurempi h\u00e4iri\u00f6alttiusluku on, sit\u00e4 alttiimpi matriisi on virheille.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.abstract", "value": "", "language": "en", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Miia Hakanen (mihakane@jyu.fi) on 2025-04-17T05:37:46Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2025-04-17T05:37:46Z (GMT). No. of bitstreams: 0\n Previous issue date: 2025", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "54", "language": null, "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": null, "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.subject.other", "value": "", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Matriisilaskennan virheanalyysi\u00e4", "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-202504173315", "language": null, "element": "identifier", "qualifier": "urn", "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": null, "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": null, "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": "dc.type.coar", "value": "http://purl.org/coar/resource_type/c_bdcc", "language": null, "element": "type", "qualifier": "coar", "schema": "dc"}, {"key": "dc.rights.copyright", "value": "\u00a9 The Author(s)", "language": "fi", "element": "rights", "qualifier": "copyright", "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.yso", "value": "matematiikka", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "matriisit", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "matriisilaskenta", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "normit", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}, {"key": "dc.description.accessibilityfeature", "value": "unknown accessibility", "language": "en", "element": "description", "qualifier": "accessibilityfeature", "schema": "dc"}, {"key": "dc.description.accessibilityfeature", "value": "ei tietoa saavutettavuudesta", "language": "fi", "element": "description", "qualifier": "accessibilityfeature", "schema": "dc"}]
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