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 classification and survey of numerical methods for boundary value problems in ordinary differential equations
Z. Aktas,H. J. Stetter +1 more
TL;DR: A survey and classification of available numerical methods for boundary value problems with an emphasis on two point boundary value problem is presented in this article, where the authors classify numerical methods into five main groups according to their approach in numerical solution.
Journal ArticleDOI
A Tangent Algorithm for Computing the Generalized Singular Value Decomposition
TL;DR: Two new algorithms for floating-point computation of the generalized singular values of a real pair of full column rank matrices and for Floating-point solution of the summarized eigenvalue problem with symmetric, positive definite matrices of H and M are presented.
Journal ArticleDOI
Multi-Fidelity Sensor Selection: Greedy Algorithms to Place Cheap and Expensive Sensors With Cost Constraints
TL;DR: In this article, the authors developed greedy algorithms to approximate the optimal solution to the multi-fidelity sensor selection problem, which is a cost constrained optimization problem prescribing the placement and number of cheap (low signal-to-noise) and expensive (high signal to noise) sensors in an environment or state space.
Journal ArticleDOI
SCDM-k
Anil Damle,Lin Lin,Lexing Ying +2 more
TL;DR: In this article, the authors generalized the selected columns of the SCDM method to KohnSham density functional theory calculations with k-point sampling of the Brillouin zone, which is needed for more general electronic structure calculations for solids.
Some results in estimation and tests of linear hypotheses under the gauss-markoff model
TL;DR: In this article, a number of results, hitherto unrecorded in the literature, on the Gauss-Markoff (G-M) model have been discussed, including explicit expressions for BLU estimators of linear parametric functions, covariance matrix of BLUE's and test criteria for tests of linear hypotheses.
References
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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.