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Es ist nach Walter Edwin Arnoldi benannt. Im Arnoldi-Verfahren wird zu einer gegebenen Matrix und einem gegebenen Startvektor eine orthonormale Basis des zugeordneten Krylowraumes Der Algorithmus kommt allerdings ohne die vorherige Aufstellung der sogenannten Krylowmatrix aus." , "lang" : "de" } , { "type" : "literal", "value" : "\u0412 \u0447\u0438\u0441\u043B\u0435\u043D\u043D\u043E\u0439 \u043B\u0438\u043D\u0435\u0439\u043D\u043E\u0439 \u0430\u043B\u0433\u0435\u0431\u0440\u0435 \u0438\u0442\u0435\u0440\u0430\u0446\u0438\u044F \u0410\u0440\u043D\u043E\u043B\u044C\u0434\u0438 \u044F\u0432\u043B\u044F\u0435\u0442\u0441\u044F \u0430\u043B\u0433\u043E\u0440\u0438\u0442\u043C\u043E\u043C \u0432\u044B\u0447\u0438\u0441\u043B\u0435\u043D\u0438\u044F \u0441\u043E\u0431\u0441\u0442\u0432\u0435\u043D\u043D\u044B\u0445 \u0437\u043D\u0430\u0447\u0435\u043D\u0438\u0439. \u0410\u0440\u043D\u043E\u043B\u044C\u0434\u0438 \u043D\u0430\u0445\u043E\u0434\u0438\u0442 \u043F\u0440\u0438\u0431\u043B\u0438\u0436\u0435\u043D\u0438\u0435 \u0441\u043E\u0431\u0441\u0442\u0432\u0435\u043D\u043D\u044B\u0445 \u0437\u043D\u0430\u0447\u0435\u043D\u0438\u0439 \u0438 \u0441\u043E\u0431\u0441\u0442\u0432\u0435\u043D\u043D\u044B\u0445 \u0432\u0435\u043A\u0442\u043E\u0440\u043E\u0432 \u043C\u0430\u0442\u0440\u0438\u0446 \u043E\u0431\u0449\u0435\u0433\u043E \u0432\u0438\u0434\u0430(\u0432\u043E\u0437\u043C\u043E\u0436\u043D\u043E \u043D\u0435 \u044D\u0440\u043C\u0438\u0442\u043E\u0432\u043E\u0439) \u0441 \u043F\u043E\u043C\u043E\u0449\u044C\u044E \u043F\u043E\u0441\u0442\u0440\u043E\u0435\u043D\u0438\u044F \u043E\u0440\u0442\u043E\u043D\u043E\u0440\u043C\u0438\u0440\u043E\u0432\u0430\u043D\u043D\u043E\u0433\u043E \u0431\u0430\u0437\u0438\u0441\u0430 \u043F\u043E\u0434\u043F\u0440\u043E\u0441\u0442\u0440\u0430\u043D\u0441\u0442\u0432\u0430 \u041A\u0440\u044B\u043B\u043E\u0432\u0430. \u041C\u0435\u0442\u043E\u0434 \u0410\u0440\u043D\u043E\u043B\u044C\u0434\u0438 \u043E\u0442\u043D\u043E\u0441\u0438\u0442\u0441\u044F \u043A \u0430\u043B\u0433\u043E\u0440\u0438\u0442\u043C\u0430\u043C \u043B\u0438\u043D\u0435\u0439\u043D\u043E\u0439 \u0430\u043B\u0433\u0435\u0431\u0440\u044B, \u043A\u043E\u0442\u043E\u0440\u044B\u0435 \u043F\u043E\u0437\u0432\u043E\u043B\u044F\u044E\u0442 \u043F\u043E\u043B\u0443\u0447\u0438\u0442\u044C \u0447\u0430\u0441\u0442\u0438\u0447\u043D\u043E\u0435 \u0440\u0435\u0448\u0435\u043D\u0438\u0435 \u043F\u043E\u0441\u043B\u0435 \u043D\u0435\u0431\u043E\u043B\u044C\u0448\u043E\u0433\u043E \u043A\u043E\u043B\u0438\u0447\u0435\u0441\u0442\u0432\u0430 \u0438\u0442\u0435\u0440\u0430\u0446\u0438\u0439, \u0432 \u043E\u0442\u043B\u0438\u0447\u0438\u0435 \u043E\u0442 \u0442\u0430\u043A \u043D\u0430\u0437\u044B\u0432\u0430\u0435\u043C\u044B\u0445 \u043F\u0440\u044F\u043C\u044B\u0445 \u043C\u0435\u0442\u043E\u0434\u043E\u0432, \u043A\u043E\u0442\u043E\u0440\u044B\u0435 \u0434\u043E\u043B\u0436\u043D\u044B \u043F\u043E\u043B\u043D\u043E\u0441\u0442\u044C\u044E \u0437\u0430\u0432\u0435\u0440\u0448\u0438\u0442\u044C\u0441\u044F \u0434\u043B\u044F \u043F\u043E\u043B\u0443\u0447\u0435\u043D\u0438\u044F \u043A\u0430\u043A\u0438\u0445-\u043B\u0438\u0431\u043E \u0443\u0434\u043E\u0432\u043B\u0435\u0442\u0432\u043E\u0440\u0438\u0442\u0435\u043B\u044C\u043D\u044B\u0445 \u0440\u0435\u0437\u0443\u043B\u044C\u0442\u0430\u0442\u043E\u0432(\u043D\u0430\u043F\u0440\u0438\u043C\u0435\u0440 \u043E\u0442\u0440\u0430\u0436\u0435\u043D\u0438\u044F \u0425\u0430\u0443\u0441\u0445\u043E\u043B\u0434\u0435\u0440\u0430)." , "lang" : "ru" } , { "type" : "literal", "value" : "In numerical linear algebra, the Arnoldi iteration is an eigenvalue algorithm and an important example of an iterative method. Arnoldi finds an approximation to the eigenvalues and eigenvectors of general (possibly non-Hermitian) matrices by constructing an orthonormal basis of the Krylov subspace, which makes it particularly useful when dealing with large sparse matrices. When applied to Hermitian matrices it reduces to the Lanczos algorithm. The Arnoldi iteration was invented by W. E. Arnoldi in 1951." , "lang" : "en" } ] , "http://xmlns.com/foaf/0.1/depiction" : [ { "type" : "uri", "value" : "http://commons.wikimedia.org/wiki/Special:FilePath/Arnoldi_Iteration.gif" } ] , "http://purl.org/dc/terms/subject" : [ { "type" : "uri", "value" : "http://dbpedia.org/resource/Category:Numerical_linear_algebra" } ] , "http://dbpedia.org/ontology/abstract" : [ { "type" : "literal", "value" : "In der numerischen Mathematik ist das Arnoldi-Verfahren wie das Lanczos-Verfahren ein iteratives Verfahren zur Bestimmung einiger Eigenwerte und zugeh\u00F6riger Eigenvektoren. Es ist nach Walter Edwin Arnoldi benannt. Im Arnoldi-Verfahren wird zu einer gegebenen Matrix und einem gegebenen Startvektor eine orthonormale Basis des zugeordneten Krylowraumes berechnet. Da die Spalten bis auf eine etwaige Skalierung genau den in der Potenzmethode berechneten Vektoren entsprechen, ist es klar, dass der Algorithmus instabil wird, wenn zuerst diese Basis berechnet w\u00FCrde und anschlie\u00DFend, zum Beispiel nach Gram-Schmidt, orthonormalisiert w\u00FCrde. 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