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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 A is an m-by-n matrix and B is a p-by-n matrix, c is a given m-vector, and d is a given p-vector, with . 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

| (2.3)

where A is an n-by-m matrix, B is an n-by-p matrix, and d is a given n-vector, with . 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.

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

Next: Standard Eigenvalue and Singular Up: Driver Routines Previous: Linear Least Squares (LLS) Contents Index