Linear least squares solutions by householder transformations
In: Numerische Mathematik, Jg. 7 (1965-06-01), S. 269-276
Online
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Zugriff:
Let A be a given m×n real matrix with m≧n and of rank n and b a given vector. We wish to determine a vector x such that $$\parallel b - A\hat x\parallel = \min .$$ where ∥ … ∥ indicates the euclidean norm. Since the euclidean norm is unitarily invariant $$\parallel b - Ax\parallel = \parallel c - QAx\parallel $$ where c=Q b and Q T Q = I. We choose Q so that $$QA = R = {\left( {_{\dddot 0}^{\tilde R}} \right)_{\} (m - n) \times n}}$$ (1) and R is an upper triangular matrix. Clearly, $$\hat x = {\tilde R^{ - 1}}\tilde c$$ where c denotes the first n components of c.
Titel: |
Linear least squares solutions by householder transformations
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Autor/in / Beteiligte Person: | Businger, Peter A. ; Golub, Gene H. |
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Zeitschrift: | Numerische Mathematik, Jg. 7 (1965-06-01), S. 269-276 |
Veröffentlichung: | Springer Science and Business Media LLC, 1965 |
Medientyp: | unknown |
ISSN: | 0945-3245 (print) ; 0029-599X (print) |
DOI: | 10.1007/bf01436084 |
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