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Generalized Linear Least Squares (LSE and GLM) Problems
Driver routines are provided for two types of generalized linear least squares
problems.
The first is

(2.2) 
where is an matrix and is a matrix,
is a given vector, and is a given vector,
with
.
This is
called a linear equalityconstrained least squares problem (LSE).
The routine LA_GGLSE
solves this problem using the generalized
(GRQ) factorization,
on the
assumptions that has full row rank and
the matrix
has full column rank .
Under these assumptions, the problem LSE has a unique solution.
The second generalized linear least squares problem is

(2.3) 
where is an matrix, is an matrix,
and is a given vector,
with
.
This is sometimes called a general (GaussMarkov) linear model problem (GLM).
When , the identity matrix, the problem reduces to an ordinary linear least squares problem.
When is square and nonsingular, the GLM problem is equivalent to the
weighted linear least squares problem:
The routine LA_GGGLM
solves this problem using the generalized (GQR)
factorization,
on the
assumptions that has full column rank and the
matrix has full row rank . Under these assumptions, the
problem is always consistent, and there are unique solutions and .
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 
Next: Standard Eigenvalue and Singular
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Susan Blackford
20010819