The routine may be called by the names f08zbf, nagf_lapackeig_dggglm or its LAPACK name dggglm.
3Description
f08zbf solves the real general Gauss–Markov linear model (GLM) problem
where is an matrix, is an matrix and is an element vector. It is assumed that , and , where . Under these assumptions, the problem has a unique solution and a minimal -norm solution , which is obtained using a generalized factorization of the matrices and .
In particular, if the matrix is square and nonsingular, then the GLM problem is equivalent to the weighted linear least squares problem
4References
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia
Anderson E, Bai Z and Dongarra J (1992) Generalized factorization and its applications Linear Algebra Appl. (Volume 162–164) 243–271
5Arguments
1: – IntegerInput
On entry: , the number of rows of the matrices and .
Constraint:
.
2: – IntegerInput
On entry: , the number of columns of the matrix .
Constraint:
.
3: – IntegerInput
On entry: , the number of columns of the matrix .
Constraint:
.
4: – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a
must be at least
.
On exit: if , contains the minimum value of lwork required for optimal performance.
12: – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08zbf is called.
If , a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued.
Suggested value:
for optimal performance, , where is the optimal block size.
Constraint:
or .
13: – IntegerOutput
On exit: unless the routine detects an error (see Section 6).
6Error Indicators and Warnings
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
The upper triangular factor associated with in the generalized factorization of the pair is singular, so that ; the least squares solution could not be computed.
The bottom part of the upper trapezoidal factor associated with in the generalized factorization of the pair is singular, so that ; the least squares solutions could not be computed.
Background information to multithreading can be found in the Multithreading documentation.
f08zbf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08zbf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
9Further Comments
When , the total number of floating-point operations is approximately ; when , the total number of floating-point operations is approximately .
10Example
This example solves the weighted least squares problem