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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TL;DR: In this article, the authors developed an algorithm for adaptively estimating the noise subspace of a data matrix, as is required in signal processing applications employing the signal subspace approach, using a rank-revealing QR factorization instead of the more expensive singular value or eigenvalue decompositions.
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A Report on the Accuracy of Some Widely Used Least Squares Computer Programs
TL;DR: In this paper, a linear least square test based on fifth degree polynomials has been run on more than twenty different computer programs in order to assess their numerical accuracy, and it was found that those programs using orthogonal Householder transformations, classical Gram-Schmidt orthonormalization or modified GramSchmidt Orthogonalization were generally much more accurate than those using elimination algorithms.
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Computing approximate Fekete points by QR factorizations of Vandermonde matrices
Alvise Sommariva,Marco Vianello +1 more
TL;DR: Numerical tests are presented for the interval and the square, which show that approximate Fekete points are well suited for polynomial interpolation and cubature.
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Note on the iterative refinement of least squares solution
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Data-Driven Sparse Sensor Placement for Reconstruction
TL;DR: In this paper, the singular value decomposition and QR pivoting are used to find sparse point sensors for signal reconstruction in high-dimensional high-bandwidth systems, and a tailored library of features extracted from training data is used.
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.