Generalized Linear Least Squares (LSE and GLM) Problems

Driver routines are provided for two types of generalized linear least squares problems.
The first is

$$\min _{x} \Vert c - Ax\Vert _2 \;\;\; \mbox{subject to} \;\;\; B x = d$$ (2.2)

where $A$ is an $m \times n$ matrix and $B$ is a $p \times n$ matrix, $c$ is a given $m$-vector, and $d$ is a given $p$-vector, with $p \leq n \leq m+p$. This is called a linear equality-constrained least squares problem (LSE). The routine LA_GGLSE solves this problem using the generalized $RQ$ (GRQ) factorization, on the assumptions that $B$ has full row rank $p$ and the matrix $\left( \begin{array}{c} A \\ B \end{array} \right)$ has full column rank $n$. Under these assumptions, the problem LSE has a unique solution.
The second generalized linear least squares problem is

$$\min _{x} \Vert y\Vert _2 \;\;\; \mbox{subject to} \;\;\; d = A x + B y$$ (2.3)

where $A$ is an $n \times m$ matrix, $B$ is an $n \times p$ matrix, and $d$ is a given $n$-vector, with $m \leq n \leq m+p$. This is sometimes called a general (Gauss-Markov) linear model problem (GLM). When $B = I$, the identity matrix, the problem reduces to an ordinary linear least squares problem. When $B$ is square and nonsingular, the GLM problem is equivalent to the weighted linear least squares problem:

$$\min_x \Vert B^{-1}(d-Ax) \Vert _2$$

The routine LA_GGGLM solves this problem using the generalized $QR$ (GQR) factorization, on the assumptions that $A$ has full column rank $m$ and the matrix $( A, B )$ has full row rank $n$. Under these assumptions, the problem is always consistent, and there are unique solutions $x$ and $y$. The driver routines for generalized linear least squares problems are listed in Table 2.4.

Table 2.4: Driver routines for generalized linear least squares problems

Operation	real/complex
solve LSE problem using GRQ	LA_GGLSE
solve GLM problem using GQR	LA_GGGLM