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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Analysis of a QR Algorithm for Computing Singular Values
TL;DR: The perturbation result for the smallest singular values of a triangular matrix is stronger than the traditional results because it guarantees high relative accuracy in the smallest plural values after an off-diagonal block of the matrix has been set to zero.
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Subset selection for matrices
F. R. de Hoog,R.M.M. Mattheij +1 more
TL;DR: In this paper, the authors consider the problem of deleting m − k rows from a matrix X ∈ R m × n so that the resulting matrix A ∈ ǫ R k × n is as non-singular as possible.
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Bayesian estimation of hypocenter with origin time eliminated
TL;DR: In this article, a new approach to hypocenter location is formulated from a Bayesian point of view, where the posterior probability density function (pdf) of the hypocenter parameters posterior to observed data is proportional to a product of the likelihood and the prior pdf of the spatial coordinates.
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Fixed points of C2 maps
TL;DR: A numerical implementation of S.N. Chow and J. Yorke's proposed algorithm for computing fixed points of C2 maps that is globally convergent with probability one is presented here, where careful attention has been paid to computational efficiency, accuracy, and robustness.
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.