NAG FL Interface
f08zbf (dggglm)

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1 Purpose

f08zbf solves a real general Gauss–Markov linear (least squares) model problem.

2 Specification

Fortran Interface
Subroutine f08zbf ( m, n, p, a, lda, b, ldb, d, x, y, work, lwork, info)
Integer, Intent (In) :: m, n, p, lda, ldb, lwork
Integer, Intent (Out) :: info
Real (Kind=nag_wp), Intent (Inout) :: a(lda,*), b(ldb,*), d(m)
Real (Kind=nag_wp), Intent (Out) :: x(n), y(p), work(max(1,lwork))
C Header Interface
#include <nag.h>
void  f08zbf_ (const Integer *m, const Integer *n, const Integer *p, double a[], const Integer *lda, double b[], const Integer *ldb, double d[], double x[], double y[], double work[], const Integer *lwork, Integer *info)
The routine may be called by the names f08zbf, nagf_lapackeig_dggglm or its LAPACK name dggglm.

3 Description

f08zbf solves the real general Gauss–Markov linear model (GLM) problem
minimize x y2  subject to  d=Ax+By  
where A is an m×n matrix, B is an m×p matrix and d is an m element vector. It is assumed that nmn+p, rank(A)=n and rank(E)=m, where E= ( A B ) . Under these assumptions, the problem has a unique solution x and a minimal 2-norm solution y, which is obtained using a generalized QR factorization of the matrices A and B.
In particular, if the matrix B is square and nonsingular, then the GLM problem is equivalent to the weighted linear least squares problem
minimize x B-1(d-Ax)2 .  

4 References

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 QR factorization and its applications Linear Algebra Appl. (Volume 162–164) 243–271

5 Arguments

1: m Integer Input
On entry: m, the number of rows of the matrices A and B.
Constraint: m0.
2: n Integer Input
On entry: n, the number of columns of the matrix A.
Constraint: 0nm.
3: p Integer Input
On entry: p, the number of columns of the matrix B.
Constraint: pm-n.
4: a(lda,*) Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array a must be at least max(1,n).
On entry: the m×n matrix A.
On exit: a is overwritten.
5: lda Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f08zbf is called.
Constraint: ldamax(1,m).
6: b(ldb,*) Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array b must be at least max(1,p).
On entry: the m×p matrix B.
On exit: b is overwritten.
7: ldb Integer Input
On entry: the first dimension of the array b as declared in the (sub)program from which f08zbf is called.
Constraint: ldbmax(1,m).
8: d(m) Real (Kind=nag_wp) array Input/Output
On entry: the left-hand side vector d of the GLM equation.
On exit: d is overwritten.
9: x(n) Real (Kind=nag_wp) array Output
On exit: the solution vector x of the GLM problem.
10: y(p) Real (Kind=nag_wp) array Output
On exit: the solution vector y of the GLM problem.
11: work(max(1,lwork)) Real (Kind=nag_wp) array Workspace
On exit: if info=0, work(1) contains the minimum value of lwork required for optimal performance.
12: lwork Integer Input
On entry: the dimension of the array work as declared in the (sub)program from which f08zbf is called.
If lwork=−1, 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, lworkn+min(m,p)+max(m,p)×nb, where nb is the optimal block size.
Constraint: lwork max(1,m+n+p) or lwork=−1.
13: info Integer Output
On exit: info=0 unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

info<0
If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
info=1
The upper triangular factor R associated with A in the generalized RQ factorization of the pair (A,B) is singular, so that rank(A)<n; the least squares solution could not be computed.
info=2
The bottom (m-n)×(m-n) part of the upper trapezoidal factor T associated with B in the generalized QR factorization of the pair (A,B) is singular, so that rank(AB)<n; the least squares solutions could not be computed.

7 Accuracy

For an error analysis, see Anderson et al. (1992). See also Section 4.6 of Anderson et al. (1999).

8 Parallelism and Performance

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.

9 Further Comments

When p=mn, the total number of floating-point operations is approximately 23(2m3-n3)+4nm2; when p=m=n, the total number of floating-point operations is approximately 143m3.

10 Example

This example solves the weighted least squares problem
minimize x B-1(d-Ax)2 ,  
where
B = ( 0.5 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 2.0 0.0 0.0 0.0 0.0 5.0 ) ,   d= ( 1.32 -4.00 5.52 3.24 )   and   A= ( -0.57 -1.28 -0.39 -1.93 1.08 -0.31 2.30 0.24 -0.40 -0.02 1.03 -1.43 ) .  

10.1 Program Text

Program Text (f08zbfe.f90)

10.2 Program Data

Program Data (f08zbfe.d)

10.3 Program Results

Program Results (f08zbfe.r)