Generalized Linear Least Squares (LSE and GLM) Problems

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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

where A is an m-by-m matrix and B is a p-by-n matrix, c is a given m-vector, and d is a given p-vector, with p < = n < = m + p. This is called a linear equality-constrained least squares problem (LSE). The routine xGGLSE    solves this problem using the generalized RQ (GRQ) factorization,   on the assumptions that B has full row rank p and the matrix has full column rank n. Under these assumptions, the problem LSE  has a unique solution.

The second generalized linear least squares problem is

where A is an n-by-m matrix, B is an n-by-p matrix, and d is a given n-vector, with m < = n < = m + p. This is sometimes called a general (Gauss-Markov) linear model problem (GLM).      When B = I, 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:

The routine xGGGLM     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.

------------------------------------------------------------------
Single precision   Double precision
Operation                      real     complex   real     complex
------------------------------------------------------------------
solve LSE problem using GQR    SGGLSE   CGGLSE    DGGLSE   ZGGLSE
solve GLM problem using GQR    SGGGLM   CGGGLM    DGGGLM   ZGGGLM
------------------------------------------------------------------
Table 2.4: Driver routines for generalized linear least squares problems

Tue Nov 29 14:03:33 EST 1994