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  • This paper provides two results on the numerical behavior of the classical Gram-Schmidt algorithm. The first result states that, provided the normal equations associated with the initial vectors are numerically nonsingular, the loss of orthogonality of the vectors computed by the classical Gram-Schmidt algorithm depends quadratically on the condition number of the initial vectors. The second result states that, provided the initial set of vectors has numerical full rank, two iterations of the classical Gram-Schmidt algorithm are enough for ensuring the orthogonality of the computed vectors to be close to the unit roundoff level.
  • This paper provides two results on the numerical behavior of the classical Gram-Schmidt algorithm. The first result states that, provided the normal equations associated with the initial vectors are numerically nonsingular, the loss of orthogonality of the vectors computed by the classical Gram-Schmidt algorithm depends quadratically on the condition number of the initial vectors. The second result states that, provided the initial set of vectors has numerical full rank, two iterations of the classical Gram-Schmidt algorithm are enough for ensuring the orthogonality of the computed vectors to be close to the unit roundoff level. (en)
  • Článek obsahuje dva fundamentální výsledky týkající se numerické stability klasického Gram-Schmidtova ortogonalizačního procesu. První výsledek dává do souvislosti ztrátu ortogonality mezi vektory vypočtenými klasickou variantou Gram-Schmidtova procesu a číslem podmíněnosti matice vektorů vstupujících do ortogonalizace. Ukazuje, že za předpokladu numerické nesingularity soustavy normálních rovnic je tato závislost kvadratická. Druhý výsledek ukazuje, že za předpokladu numerické plné hodnosti této matice, stačí dvě iterace klasické Gram-Schmidtova procesu k tomu, aby byla výsledná ortogonalita vypočtených vektorů na hladině strojové přesnosti dané aritmetiky. (cs)
Title
  • Rounding Error Analysis of the Classical Gram-Schmidt Orthogonalization Process
  • Analýza zaokrouhlovacích chyb klasického Gram-Schmidtova ortogonalizačního procesu (cs)
  • Rounding Error Analysis of the Classical Gram-Schmidt Orthogonalization Process (en)
skos:prefLabel
  • Rounding Error Analysis of the Classical Gram-Schmidt Orthogonalization Process
  • Analýza zaokrouhlovacích chyb klasického Gram-Schmidtova ortogonalizačního procesu (cs)
  • Rounding Error Analysis of the Classical Gram-Schmidt Orthogonalization Process (en)
skos:notation
  • RIV/67985807:_____/05:00405259!RIV06-AV0-67985807
http://linked.open.../vavai/riv/strany
  • 87;100
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  • P(1ET400300415), P(IAA1030405), Z(AV0Z10300504)
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  • -
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  • 541529
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  • RIV/67985807:_____/05:00405259
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  • rounding error analysis; orthogonalization; classical Gram-Schmidt process (en)
http://linked.open.../riv/klicoveSlovo
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  • DE - Spolková republika Německo
http://linked.open...ontrolniKodProRIV
  • [65325AF5FF99]
http://linked.open...i/riv/nazevZdroje
  • Numerische Mathematik
http://linked.open...in/vavai/riv/obor
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  • 101
http://linked.open...iv/tvurceVysledku
  • Rozložník, Miroslav
  • Giraud, L.
  • Langou, J.
  • van den Eshof, J.
http://linked.open...n/vavai/riv/zamer
issn
  • 0029-599X
number of pages
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