Journal ArticleDOI
Linear least squares solutions by householder transformations
Peter A. Businger,Gene H. Golub +1 more
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TLDR
In this paper, the euclidean norm is unitarily invariant and a vector x is determined such that x is parallel b-Ax parallel = \parallel c - QAx parallel where c denotes the first n components of c.Abstract:
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.read more
Citations
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A note on subset selection for matrices
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TL;DR: In this paper, the authors gave a constructive proof and moreover a sharper bound for non-singularity of a matrix and showed that it is possible to select subsets of the matrix that are not as singular as possible in a numerical sense.
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TL;DR: In this article , a column selection strategy, named deviation maximization, is introduced to compute rank-revealing QR factorizations as an alternative to the well-known block version of the QR factorization with the column pivoting method, called QP3 and currently implemented in LAPACK.
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Least squares solution of weighted linear systems by G-transformations
TL;DR: Based on permutations and scaling of orthogonal Hessenberg matrices, a family of algorithms (the G- and partial G-algorithms) are presented in this paper which solve the weighted least squares problem (Ax−b)W(Ax −b)=ρ2 for x such that ρ2 is minimal.
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References
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Journal ArticleDOI
Unitary Triangularization of a Nonsymmetric Matrix
TL;DR: This note points out that the same result can be obtained with fewer arithmetic operations, and, in particular, for inverting a square matrix of order N, at most 2(N-1) square roots are required.