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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 equality-constrained 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 (Gauss-Markov) 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.

 Operation real/complex solve LSE problem using GRQ LA_GGLSE solve GLM problem using GQR LA_GGGLM     Next: Standard Eigenvalue and Singular Up: Driver Routines Previous: Linear Least Squares (LLS)   Contents   Index
Susan Blackford 2001-08-19