Chapter 17 - Implementing the QR Decomposition

Abstract

After reviewing the reduced QR decomposition done using Gram-Schmidt, this chapter develops two efficient methods for computing the QR decomposition, using Givens rotations and Householder reflections. Givens rotations are defined, and the use of a rotation to zero out a particular entry in a vector is developed. This is followed by showing how to use Givens rotations to zero out multiple entries in a vector. If J(i,j,c,s) is a Givens rotation and A is a matrix, the product J(i,j,c,s)*A can be performed by modifying only two rows of A. These ideas are then applied to zeroing out entries in a column of a matrix. Due to possibility of overflow, the Givens parameters c and s must be computed carefully. This material is put together to develop algorithms for computing the QR and reduced QR decompositions of a general m × n matrix using Givens rotations, and the flop count is determined. Next, Householder reflections are defined and many of their properties developed. It is shown how to find a vector u such that Hu*A zeros out all the elements below a diagonal element of A. A careful analysis is done to assure accurate computation of u. As with Givens rotations, Householder reflections can be applied implicitly rather than actually building a Householder matrix. The chapter ends with the development of the QR decomposition using Householder reflections, which is the standard algorithm. Studying Givens rotations was useful because they are needed when computing eigenvalues and the SVD.