F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF08ZNF (ZGGLSE)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

F08ZNF (ZGGLSE) solves a complex linear equality-constrained least squares problem.

## 2  Specification

 SUBROUTINE F08ZNF ( M, N, P, A, LDA, B, LDB, C, D, X, WORK, LWORK, INFO)
 INTEGER M, N, P, LDA, LDB, LWORK, INFO COMPLEX (KIND=nag_wp) A(LDA,*), B(LDB,*), C(M), D(P), X(N), WORK(max(1,LWORK))
The routine may be called by its LAPACK name zgglse.

## 3  Description

F08ZNF (ZGGLSE) solves the complex 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 $\mathrm{rank}\left(\mathrm{E}\right)=n$, 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 $A$.

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

## 5  Parameters

1:     M – INTEGERInput
On entry: $m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{M}}\ge 0$.
2:     N – INTEGERInput
On entry: $n$, the number of columns of the matrices $A$ and $B$.
Constraint: ${\mathbf{N}}\ge 0$.
3:     P – INTEGERInput
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:     A(LDA,$*$) – COMPLEX (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:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F08ZNF (ZGGLSE) is called.
Constraint: ${\mathbf{LDA}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{M}}\right)$.
6:     B(LDB,$*$) – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array B must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On entry: the $p$ by $n$ matrix $B$.
On exit: B is overwritten.
7:     LDB – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F08ZNF (ZGGLSE) is called.
Constraint: ${\mathbf{LDB}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{P}}\right)$.
8:     C(M) – COMPLEX (KIND=nag_wp) arrayInput/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:     D(P) – COMPLEX (KIND=nag_wp) arrayInput/Output
On entry: the right-hand side vector $d$ for the equality constraints.
On exit: D is overwritten.
10:   X(N) – COMPLEX (KIND=nag_wp) arrayOutput
On exit: the solution vector $x$ of the LSE problem.
11:   WORK($\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{LWORK}}\right)$) – COMPLEX (KIND=nag_wp) arrayWorkspace
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:   LWORK – INTEGERInput
On entry: the dimension of the array WORK as declared in the (sub)program from which F08ZNF (ZGGLSE) 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:   INFO – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

## 6  Error Indicators and Warnings

Errors or warnings detected by the routine:
${\mathbf{INFO}}<0$
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.

## 7  Accuracy

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

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 $\frac{8}{3}{n}^{2}\left(3m-n\right)$.

## 9  Example

This example solves the least squares problem
 $minimize x c-Ax2 subject to Bx=d$
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.

### 9.1  Program Text

Program Text (f08znfe.f90)

### 9.2  Program Data

Program Data (f08znfe.d)

### 9.3  Program Results

Program Results (f08znfe.r)