Journal ArticleDOI
Linear least squares solutions by householder transformations
Peter A. Businger,Gene H. Golub +1 more
Reads0
Chats0
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
More filters
Journal ArticleDOI
Fundamental solutions for the collation method in planar elastostatics
TL;DR: In this article, a method for solving planar elastostatic problems referred to as the fundamental collocation method is described, which features techniques employed in the boundary integral equation method and the boundary-point-least-squares collocations method.
Journal ArticleDOI
Solving the minimal least squares problem subject to bounds on the variables
TL;DR: The systems of linear equations satisfied by the descent direction and the Lagrange multipliers in the minimization algorithm are solved by direct methods based on QR decompositions or iterative preconditioned conjugate gradient methods.
Book ChapterDOI
Annotated Bibliography on Generalized Inverses and Applications
M.Z. Nashed,L.B. Rall +1 more
TL;DR: Well, Mr. Jacobi, here it is: all the generalized inversion of two generations of invertors who, knowingly or unknowingly, subscribed to (and extended) your dictum.
Journal ArticleDOI
A parallel QR factorization algorithm with controlled local pivoting
TL;DR: A new version of the Householder algorithm with column pivoting for computing a QR factorization that identifies rank and range space of a given matrix that is well suited for implementation on a parallel machine, in particular, a MIMD machine with distributed memory.
Journal ArticleDOI
Scaled givens rotations for the solution of linear least squares problems on systolic arrays
Jesse L. Barlow,Ilse C. F. Ipsen +1 more
TL;DR: A class of Scaled Givens rotations, to be applied to the solution of weighted multiple linear least squares problems on systolic arrays, is discussed, indicating slightly less accumulation of round off error than known sequential methods.
References
More filters
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.