13 Computation using the QR decomposition

Abstract

The QR decomposition is one of the most basic tools for statistical computation, yet it is also one of the most versatile and powerful. The chapter discusses the problem of perfect or nearly perfect linear dependencies, the complex QR decomposition, the QR decomposition in regression, the essential properties of the QR decomposition, and the use of Householder transformations to compute the QR decomposition. A classical alternative to the Householder QR algorithm is the Gram–Schmidt method. Least-squares regression estimates can also be found using either the Cholesky factorization or the singular value decomposition. The QR decomposition approach has been observed to offer excellent numerical properties at reasonable computational cost, while providing factors, Q and R , which are quite generally useful. Although the QR decomposition is a long established technique, it is not in stasis. The QR decomposition is a key to stable and efficient solutions to many least-squares problems. A diverse collection of examples in statistics is given in the chapter and it is certain that there are many more problems for which orthogonalization algorithms can be exploited. In addition, the QR decomposition provides ready insight into the statistical properties of regression estimates.