Modifying the QR-Decomposition to Constrained and Weighted Linear Least Squares

A new way of looking at a class of methods for the weighted linear least squares problem $\min _x \| M^{ - ( 1/2 )} ( b - Ax ) \|_2 $ where $M = {\operatorname{diag}}(\mu _i )$ is presented by introducing a modified QR-decomposition with QM-invariant, i.e., $QMQ^T = M$. One of the main advantages with this approach is that linear constraints are easily incorporated by letting the corresponding diagonal elements in M become zero. Householder reflections are generalized to M-invariant reflections, and an algorithm for solving the constrained and weighted linear least squares problem is described. The system equations (or the augmented system equations) are used to derive condition numbers, and the connection between these condition numbers and the rounding error in the solution is investigated.