# NAG FL Interfacef08zaf (dgglse)

## 1Purpose

f08zaf solves a real linear equality-constrained least squares problem.

## 2Specification

Fortran Interface
 Subroutine f08zaf ( m, n, p, a, lda, b, ldb, c, d, x, 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,*), c(m), d(p) Real (Kind=nag_wp), Intent (Out) :: x(n), work(max(1,lwork))
#include <nag.h>
 void f08zaf_ (const Integer *m, const Integer *n, const Integer *p, double a[], const Integer *lda, double b[], const Integer *ldb, double c[], double d[], double x[], double work[], const Integer *lwork, Integer *info)
The routine may be called by the names f08zaf, nagf_lapackeig_dgglse or its LAPACK name dgglse.

## 3Description

f08zaf solves the real linear equality-constrained least squares (LSE) problem
 $minimize x c-Ax2 subject to Bx=d$
where $A$ is an $m$ by $n$ matrix, $B$ is a $p$ by $n$ matrix, $c$ is an $m$ element vector and $d$ is a $p$ element vector. It is assumed that $p\le n\le m+p$, $\mathrm{rank}\left(B\right)=p$ and $\mathrm{rank}\left(E\right)=n$, where $E=\left(\begin{array}{c}A\\ B\end{array}\right)$. These conditions ensure that the LSE problem has a unique solution, which is obtained using a generalized $RQ$ factorization of the matrices $B$ and $A$.

## 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
Eldèn L (1980) Perturbation theory for the least squares problem with linear equality constraints SIAM J. Numer. Anal. 17 338–350

## 5Arguments

1: $\mathbf{m}$Integer Input
On entry: $m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{m}}\ge 0$.
2: $\mathbf{n}$Integer Input
On entry: $n$, the number of columns of the matrices $A$ and $B$.
Constraint: ${\mathbf{n}}\ge 0$.
3: $\mathbf{p}$Integer Input
On entry: $p$, the number of rows of the matrix $B$.
Constraint: $0\le {\mathbf{p}}\le {\mathbf{n}}\le {\mathbf{m}}+{\mathbf{p}}$.
4: $\mathbf{a}\left({\mathbf{lda}},*\right)$Real (Kind=nag_wp) array Input/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: $\mathbf{lda}$Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f08zaf 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) array Input/Output
Note: the second dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the $p$ by $n$ matrix $B$.
On exit: b is overwritten.
7: $\mathbf{ldb}$Integer Input
On entry: the first dimension of the array b as declared in the (sub)program from which f08zaf is called.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{p}}\right)$.
8: $\mathbf{c}\left({\mathbf{m}}\right)$Real (Kind=nag_wp) array Input/Output
On entry: the right-hand side vector $c$ for the least squares part of the LSE problem.
On exit: the residual sum of squares for the solution vector $x$ is given by the sum of squares of elements ${\mathbf{c}}\left({\mathbf{n}}-{\mathbf{p}}+1\right),{\mathbf{c}}\left({\mathbf{n}}-{\mathbf{p}}+2\right),\dots ,{\mathbf{c}}\left({\mathbf{m}}\right)$; the remaining elements are overwritten.
9: $\mathbf{d}\left({\mathbf{p}}\right)$Real (Kind=nag_wp) array Input/Output
On entry: the right-hand side vector $d$ for the equality constraints.
On exit: d is overwritten.
10: $\mathbf{x}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Output
On exit: the solution vector $x$ of the LSE problem.
11: $\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$Real (Kind=nag_wp) array Workspace
On exit: if ${\mathbf{info}}={\mathbf{0}}$, ${\mathbf{work}}\left(1\right)$ contains the minimum value of lwork required for optimal performance.
12: $\mathbf{lwork}$Integer Input
On entry: the dimension of the array work as declared in the (sub)program from which f08zaf 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, ${\mathbf{lwork}}\ge {\mathbf{p}}+\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)+\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)×\mathit{nb}$, 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: $\mathbf{info}$Integer Output
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 $B$ in the generalized $RQ$ factorization of the pair $\left(B,A\right)$ is singular, so that $\mathrm{rank}\left(B\right); the least squares solution could not be computed.
${\mathbf{info}}=2$
The $\left(N-P\right)$ by $\left(N-P\right)$ part of the upper trapezoidal factor $T$ associated with $A$ in the generalized $RQ$ factorization of the pair $\left(B,A\right)$ is singular, so that the rank of the matrix ($E$) comprising the rows of $A$ and $B$ is less than $n$; the least squares solutions could not be computed.

## 7Accuracy

For an error analysis, see Anderson et al. (1992) and Eldèn (1980). See also Section 4.6 of Anderson et al. (1999).

## 8Parallelism and Performance

f08zaf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08zaf 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 $m\ge n=p$, the total number of floating-point operations is approximately $\frac{2}{3}{n}^{2}\left(6m+n\right)$; if $p\ll n$, the number reduces to approximately $\frac{2}{3}{n}^{2}\left(3m-n\right)$.
e04ncf/​e04nca may also be used to solve LSE problems. It differs from f08zaf in that it uses an iterative (rather than direct) method, and that it allows general upper and lower bounds to be specified for the variables $x$ and the linear constraints $Bx$.

## 10Example

This example solves the least squares problem
 $minimize x c-Ax2 subject to Bx=d$
where
 $c = -1.50 -2.14 1.23 -0.54 -1.68 0.82 ,$
 $A = -0.57 -1.28 -0.39 0.25 -1.93 1.08 -0.31 -2.14 2.30 0.24 0.40 -0.35 -1.93 0.64 -0.66 0.08 0.15 0.30 0.15 -2.13 -0.02 1.03 -1.43 0.50 ,$
 $B = 1.0 0 -1.0 0 0 1.0 0 -1.0$
and
 $d = 0 0 .$
The constraints $Bx=d$ correspond to ${x}_{1}={x}_{3}$ and ${x}_{2}={x}_{4}$.

### 10.1Program Text

Program Text (f08zafe.f90)

### 10.2Program Data

Program Data (f08zafe.d)

### 10.3Program Results

Program Results (f08zafe.r)