Purpose

LA_GGGLM solves the general (Gauss-Markov) linear model (GLM) problem:

$$\min_x \Vert y \Vert _2 \;\;\;\;\mbox{ subject} \;\mbox{ to}\;\;\;\; d = A\,x + B\,y$$

where $A$ and $B$ are real or complex rectangular matrices and $d$ is a real or complex vector. Further, $A$ is $n \times m$, $B$ is $n \times p$, and $d$ is $n\times 1$, and it is assumed that

$$m \leq n \leq m+p, \;\;\;\; \mbox{rank}(A) = m, \;\;\;\; \mbox{rank}(A, B) = n.$$

These conditions ensure that the GLM problem has unique solution vectors $x$ and $y$. The problem is solved using the generalized $QR$ factorization of $A$ and $B$.
If matrix $B$ is square and nonsingular, then the GLM problem is equivalent to the weighted linear least squares problem

$$\min_x \Vert B^{-1}\:(d-A\,x) \Vert _2.$$