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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Application of the Finite Element/Boundary Element Approach to the Analysis of Radiation and Scattering from Fluid-Loaded Elastic and Piezoelectric Structures.
TL;DR: In this paper, a finite element model is combined with a combined Helmholtz integral equation formulation (CHIEF) boundary element model of the wetted surface to compute the in fluid eigenfrequency of a specific mode of the structure.
Journal ArticleDOI
Numerical methods of closed-loop multibody systems with singular configurations based on the geometrical structure of constraints
TL;DR: In this paper, a numerical method of closed-loop multibody systems with singular configurations is presented, in which the traditional hypothesis on independence of constraints is abandoned, and instead of correcting constraint violations at the end of each integration step, positions and velocities are modified to satisfy their constraint equations before they are used to form equations of motion.
Journal ArticleDOI
Further three-dimensional stress analysis of an intersection of a cylindrical shell with a plate
D. Redekop,John Schroeder +1 more
TL;DR: In this paper, the theory presented previously for the three-dimensional state of stress in a plate-cylindrical shell intersection under axisymmetric loading is generalized to include non-axisymmetric loading cases by expansion of loads and stresses in a Fourier series in the circumferential direction.
Book ChapterDOI
Least-Squares Methods
TL;DR: The method can be used to include information from different sources; experimental and theoretical, and is applied to curve fitting problems and to more general situations.
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.