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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Numerical linear algebra aspects of globally convergent homotopy methods
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A globally convergent algorithm for computing fixed points of C2 maps
TL;DR: Chow, Mallet-Paret, and Yorke have recently proposed an algorithm for computing Brouwer fixed points of C^2 maps as discussed by the authors, and a numerical implementation of that algorithm is presented here.
Book ChapterDOI
13 Computation using the QR decomposition
TL;DR: The chapter discusses the problem of perfect or nearly perfect linear dependencies, the complex QR decompositions, the QR decomposition in regression, the essential properties of the QR decay, and the use of Householder transformations to compute the QR decompposition.
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New Fast and Accurate Jacobi SVD Algorithm. I
Zlatko Drmač,Krešimir Veselić +1 more
TL;DR: The quest for a highly accurate and efficient SVD algorithm has led to a new, superior variant of the Jacobi algorithm, which has inherited all good high accuracy properties and can outperform the QR algorithm.
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On Rank-Revealing Factorisations
TL;DR: A systematic treatment of algorithms for determining RRQR factorisations and presents "hybrid" algorithms that solve the optimisation problems almost exactly (up to a factor proportional to the size of the matrix).
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.