A Useful Form of Unitary Matrix Obtained from Any Sequence of Unit 2-Norm n -Vectors

Charles Sheffield pointed out that the modified Gram-Schmidt (MGS) orthogonalization algorithm for the QR factorization of ... is mathematically equivalent to the QR factorization applied to the matrix B augmented with a ... matrix of zero elements on top. This is true in theory for any method of QR...

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Bibliographic Details
Published inSIAM journal on matrix analysis and applications Vol. 31; no. 2; pp. 565 - 583
Main Author Paige, Christopher C
Format Journal Article
LanguageEnglish
Published Philadelphia Society for Industrial and Applied Mathematics 01.01.2009
Subjects
Algorithms
Augmentation
Equivalence
Error analysis
Factorization
Joints
Linear algebra
Mathematical analysis
Matrix
Rounding
Studies
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Summary:Charles Sheffield pointed out that the modified Gram-Schmidt (MGS) orthogonalization algorithm for the QR factorization of ... is mathematically equivalent to the QR factorization applied to the matrix B augmented with a ... matrix of zero elements on top. This is true in theory for any method of QR factorization, but for Householder's method it is true in the presence of rounding errors as well. This knowledge has been the basis for several successful but difficult rounding error analyses of algorithms which in theory produce orthogonal vectors but significantly fail to do so because of rounding errors. Here the authors show that the same results can be found more directly and easily without recourse to the MGS connection. It is shown that for any sequence of k unit 2-norm n-vectors there is a special (n+k)-square unitary matrix which they call a unitary augmentation of these vectors.(ProQuest: ... denotes formulae/symbols omitted.)
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ISSN:0895-4798
1095-7162
DOI:10.1137/080725167