# NAG FL Interfacef08znf (zgglse)

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## 1Purpose

f08znf solves a complex linear equality-constrained least squares problem.

## 2Specification

Fortran Interface
 Subroutine f08znf ( 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 Complex (Kind=nag_wp), Intent (Inout) :: a(lda,*), b(ldb,*), c(m), d(p) Complex (Kind=nag_wp), Intent (Out) :: x(n), work(max(1,lwork))
#include <nag.h>
 void f08znf_ (const Integer *m, const Integer *n, const Integer *p, Complex a[], const Integer *lda, Complex b[], const Integer *ldb, Complex c[], Complex d[], Complex x[], Complex work[], const Integer *lwork, Integer *info)
The routine may be called by the names f08znf, nagf_lapackeig_zgglse or its LAPACK name zgglse.

## 3Description

f08znf solves the complex linear equality-constrained least squares (LSE) problem
 $minimize x ‖c-Ax‖2 subject to Bx=d$
where $A$ is an $m×n$ matrix, $B$ is a $p×n$ 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 $\mathrm{rank}\left(\mathrm{E}\right)=n$, 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: $\mathbf{a}\left({\mathbf{lda}},*\right)$Complex (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 $m×n$ 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 f08znf is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
6: $\mathbf{b}\left({\mathbf{ldb}},*\right)$Complex (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 $p×n$ 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 f08znf is called.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{p}}\right)$.
8: $\mathbf{c}\left({\mathbf{m}}\right)$Complex (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: $\mathbf{d}\left({\mathbf{p}}\right)$Complex (Kind=nag_wp) array Input/Output
On entry: the right-hand side vector $d$ for the equality constraints.
On exit: d is overwritten.
10: $\mathbf{x}\left({\mathbf{n}}\right)$Complex (Kind=nag_wp) array Output
On exit: the solution vector $x$ of the LSE problem.
11: $\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$Complex (Kind=nag_wp) array Workspace
On exit: if ${\mathbf{info}}={\mathbf{0}}$, the real part of ${\mathbf{work}}\left(1\right)$ 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 f08znf 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 $\left(N-P\right)×\left(N-P\right)$ 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

f08znf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08znf 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 real floating-point operations is approximately $\frac{8}{3}{n}^{2}\left(6m+n\right)$; if $p\ll n$, the number reduces to approximately $\frac{8}{3}{n}^{2}\left(3m-n\right)$.

## 10Example

This example solves the least squares problem
 $minimize x ‖c-Ax‖2 subject to Bx=d$
where
 $c = ( -2.54+0.09i 1.65-2.26i -2.11-3.96i 1.82+3.30i -6.41+3.77i 2.07+0.66i ) ,$
and
 $A = ( 0.96-0.81i -0.03+0.96i -0.91+2.06i -0.05+0.41i -0.98+1.98i -1.20+0.19i -0.66+0.42i -0.81+0.56i 0.62-0.46i 1.01+0.02i 0.63-0.17i -1.11+0.60i 0.37+0.38i 0.19-0.54i -0.98-0.36i 0.22-0.20i 0.83+0.51i 0.20+0.01i -0.17-0.46i 1.47+1.59i 1.08-0.28i 0.20-0.12i -0.07+1.23i 0.26+0.26i ) ,$
 $B = ( 1.0+0.0i 0.0i+0.0 -1.0+0.0i 0.0i+0.0 0.0i+0.0 1.0+0.0i 0.0i+0.0 -1.0+0.0i )$
and
 $d = ( 0 0 ) .$
The constraints $Bx=d$ correspond to ${x}_{1}={x}_{3}$ and ${x}_{2}={x}_{4}$.
Note that the block size (NB) of $64$ assumed in this example is not realistic for such a small problem, but should be suitable for large problems.

### 10.1Program Text

Program Text (f08znfe.f90)

### 10.2Program Data

Program Data (f08znfe.d)

### 10.3Program Results

Program Results (f08znfe.r)