Linear Least Squares Solutions by Housholder Transformations

  • P. Businger
  • G. H. Golub
Part of the Die Grundlehren der mathematischen Wissenschaften book series (GL, volume 186)


Let A be a given m×n real matrix with mn 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
$$QA = R = {\left( {_{\dddot 0}^{\tilde R}} \right)_{\} (m - n) \times n}}$$
and R is an upper triangular matrix. Clearly,
$$\hat x = {\tilde R^{ - </check_iphenate_word>1}}\tilde c$$
where c denotes the first n components of c.


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  1. [1]
    Householder, A. S.: Unitary Triangularization of a Nonsymmetric Matrix. J. Assoc. Comput. Mach. 5, 339–342 (1958).MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    Martin, R. S., C. Reinsch, and J. H. Wilkinson. Householder’s tridiagonalization of a symmetric matrix. Numer. Math. 11, 181–195 (1968). Cf. II/2.MathSciNetzbMATHCrossRefGoogle Scholar

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© Springer-Verlag Berlin · Heidelberg 1971

Authors and Affiliations

  • P. Businger
  • G. H. Golub

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