# NAG Library Routine Document

## 1Purpose

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

## 2Specification

Fortran Interface
 Subroutine f08zbf ( m, n, p, a, lda, b, ldb, d, x, y, work, 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))
#include <nagmk26.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 its LAPACK name dggglm.

## 3Description

f08zbf (dggglm) solves the real general Gauss–Markov linear model (GLM) problem
 $minimize x y2 subject to d=Ax+By$
where $A$ is an $m$ by $n$ matrix, $B$ is an $m$ by $p$ matrix and $d$ is an $m$ element vector. It is assumed that $n\le m\le n+p$, $\mathrm{rank}\left(A\right)=n$ and $\mathrm{rank}\left(E\right)=m$, where $E=\left(\begin{array}{cc}A& B\end{array}\right)$. 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
 $minimize x B-1 d-Ax 2 .$

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

## 5Arguments

1:     $\mathbf{m}$ – IntegerInput
On entry: $m$, the number of rows of the matrices $A$ and $B$.
Constraint: ${\mathbf{m}}\ge 0$.
2:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of columns of the matrix $A$.
Constraint: $0\le {\mathbf{n}}\le {\mathbf{m}}$.
3:     $\mathbf{p}$ – IntegerInput
On entry: $p$, the number of columns of the matrix $B$.
Constraint: ${\mathbf{p}}\ge {\mathbf{m}}-{\mathbf{n}}$.
4:     $\mathbf{a}\left({\mathbf{lda}},*\right)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the $m$ by $n$ matrix $A$.
On exit: a is overwritten.
5:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08zbf (dggglm) is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
6:     $\mathbf{b}\left({\mathbf{ldb}},*\right)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{p}}\right)$.
On entry: the $m$ by $p$ matrix $B$.
On exit: b is overwritten.
7:     $\mathbf{ldb}$ – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f08zbf (dggglm) is called.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
8:     $\mathbf{d}\left({\mathbf{m}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: the left-hand side vector $d$ of the GLM equation.
On exit: d is overwritten.
9:     $\mathbf{x}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the solution vector $x$ of the GLM problem.
10:   $\mathbf{y}\left({\mathbf{p}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the solution vector $y$ of the GLM problem.
11:   $\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$ – Real (Kind=nag_wp) arrayWorkspace
On exit: if ${\mathbf{info}}={\mathbf{0}}$, ${\mathbf{work}}\left(1\right)$ contains the minimum value of lwork required for optimal performance.
12:   $\mathbf{lwork}$ – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08zbf (dggglm) is called.
If ${\mathbf{lwork}}=-1$, 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, ${\mathbf{lwork}}\ge {\mathbf{n}}+\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{p}}\right)+\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{p}}\right)×\mathit{nb}$, where $\mathit{nb}$ is the optimal block size.
Constraint: ${\mathbf{lwork}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}+{\mathbf{n}}+{\mathbf{p}}\right)$ or ${\mathbf{lwork}}=-1$.
13:   $\mathbf{info}$ – IntegerOutput
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

## 6Error Indicators and Warnings

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{info}}=1$
The upper triangular factor $R$ associated with $A$ in the generalized $RQ$ factorization of the pair $\left(A,B\right)$ is singular, so that $\mathrm{rank}\left(A\right); the least squares solution could not be computed.
${\mathbf{info}}=2$
The bottom $\left(m-n\right)$ by $\left(m-n\right)$ part of the upper trapezoidal factor $T$ associated with $B$ in the generalized $QR$ factorization of the pair $\left(A,B\right)$ is singular, so that $\mathrm{rank}\left(\begin{array}{cc}A& B\end{array}\right); the least squares solutions could not be computed.

## 7Accuracy

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

## 8Parallelism and Performance

f08zbf (dggglm) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08zbf (dggglm) 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.

When $p=m\ge n$, the total number of floating-point operations is approximately $\frac{2}{3}\left(2{m}^{3}-{n}^{3}\right)+4n{m}^{2}$; when $p=m=n$, the total number of floating-point operations is approximately $\frac{14}{3}{m}^{3}$.

## 10Example

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

### 10.1Program Text

Program Text (f08zbfe.f90)

### 10.2Program Data

Program Data (f08zbfe.d)

### 10.3Program Results

Program Results (f08zbfe.r)