Showing papers on "QR decomposition published in 1982"

Matrix Triangularization By Systolic Arrays

Abstract: Given an n x p matrix X with p < n, matrix triangularization, or triangularization in short, is to determine an n x n nonsingular matrix Al such that MX = [ R 0 where R is p x p upper triangular, and furthermore to compute the entries in R. By triangularization, many matrix problems are reduced to the simpler problem of solving triangular- linear systems (see for example, Stewart). When X is a square matrix, triangularization is the major step in almost all direct methods for solving general linear systems. When M is restricted to be an orthogonal matrix Q, triangularization is also the key step in computing least squares solutions by the QR decomposition, and in computing eigenvalues by the QR algorithm. Triangularization is computationally expensive, however. Algorithms for performing it typically require n3 operations on general n x n matrices. As a result, triangularization has become a bottleneck in some real-time applications.11 This paper sketches unified concepts of using systolic arrays to perform real-time triangularization for both general and band matrices. (Examples and general discussions of systolic architectures can be found in other papers.6.7) Under the same framework systolic triangularization arrays arc derived for the solution of linear systems with pivoting and for least squares computations. More detailed descriptions of the suggested systolic arrays will appear in the final version of the paper.© (1982) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

A Note on the Computation of an Orthonormal Basis for the Null Space of a Matrix

Abstract: A highly regarded method to obtain an orthonormal basis, $Z$, for the null space of a matrix $A^{T}$ is the $QR$ decomposition of $A$, where $Q$ is the product of Householder matrices. In several optimization contexts $A(x)$ varies continuously with $x$ and it is desirable the $Z(x)$ vary continuously also. In this note we demonstrate that the standard implementation of the $QR$ decomposition does not yield an orthonormal basis $Z(x)$ whose elements vary continuously with $x$. We suggest three possible remedies.

A Computational Array for the QR-Method

Abstract: The QR-method is a method for the solution of linear system of equations. The matrix R is upper triangular and Q is a unitary matrix. In equation solving Q is not always computed explicitly. The matrix R can be obtained by applying a sequence of unitary transformations to the matrix defining the system of equations. Householder's method or Given's method can be used to determine unitary transformation matrices. This paper describes a concurrent algorithm and corresponding array for computing the triangular matrix R by Householder transformations. Particular attention is given to issues such as broadcasting and pipelining.

Using the Hessenberg decomposition in control theory

Abstract: Orthogonal matrix techniques are gaining wide acceptance in applied areas by practitioners who appreciate the value of reliable numerical software. Quality programs that can be used to compute the QR Decomposition, the Singular Value Decomposition, and the Schur Decomposition are primarily responsible for this increased appreciation. A fourth orthogonal matrix decomposition, the Hessenberg Decomposition, has recently been put to good use in certain control theory applications. We describe some of these applications and illustrate why this decomposition can frequently replace the much more costly decomposition of Schur.