# NAG Library Function Document

## 1Purpose

nag_zgglse (f08znc) solves a complex linear equality-constrained least squares problem.

## 2Specification

 #include #include
 void nag_zgglse (Nag_OrderType order, Integer m, Integer n, Integer p, Complex a[], Integer pda, Complex b[], Integer pdb, Complex c[], Complex d[], Complex x[], NagError *fail)

## 3Description

nag_zgglse (f08znc) 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 $A$.

## 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:    $\mathbf{order}$Nag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:    $\mathbf{m}$IntegerInput
On entry: $m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{m}}\ge 0$.
3:    $\mathbf{n}$IntegerInput
On entry: $n$, the number of columns of the matrices $A$ and $B$.
Constraint: ${\mathbf{n}}\ge 0$.
4:    $\mathbf{p}$IntegerInput
On entry: $p$, the number of rows of the matrix $B$.
Constraint: $0\le {\mathbf{p}}\le {\mathbf{n}}\le {\mathbf{m}}+{\mathbf{p}}$.
5:    $\mathbf{a}\left[\mathit{dim}\right]$ComplexInput/Output
Note: the dimension, dim, of the array a must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pda}}×{\mathbf{n}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}×{\mathbf{pda}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
The $\left(i,j\right)$th element of the matrix $A$ is stored in
• ${\mathbf{a}}\left[\left(j-1\right)×{\mathbf{pda}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{a}}\left[\left(i-1\right)×{\mathbf{pda}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the $m$ by $n$ matrix $A$.
On exit: a is overwritten.
6:    $\mathbf{pda}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
7:    $\mathbf{b}\left[\mathit{dim}\right]$ComplexInput/Output
Note: the dimension, dim, of the array b must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdb}}×{\mathbf{n}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{p}}×{\mathbf{pdb}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
The $\left(i,j\right)$th element of the matrix $B$ is stored in
• ${\mathbf{b}}\left[\left(j-1\right)×{\mathbf{pdb}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{b}}\left[\left(i-1\right)×{\mathbf{pdb}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the $p$ by $n$ matrix $B$.
On exit: b is overwritten.
8:    $\mathbf{pdb}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{p}}\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
9:    $\mathbf{c}\left[{\mathbf{m}}\right]$ComplexInput/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 ${\mathbf{c}}\left[{\mathbf{n}}-{\mathbf{p}}\right],{\mathbf{c}}\left[{\mathbf{n}}-{\mathbf{p}}+1\right],\dots ,{\mathbf{c}}\left[{\mathbf{m}}-1\right]$; the remaining elements are overwritten.
10:  $\mathbf{d}\left[{\mathbf{p}}\right]$ComplexInput/Output
On entry: the right-hand side vector $d$ for the equality constraints.
On exit: d is overwritten.
11:  $\mathbf{x}\left[{\mathbf{n}}\right]$ComplexOutput
On exit: the solution vector $x$ of the LSE problem.
12:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\ge 0$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pda}}>0$.
On entry, ${\mathbf{pdb}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdb}}>0$.
NE_INT_2
On entry, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$ and ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
On entry, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry, ${\mathbf{pdb}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry, ${\mathbf{pdb}}=〈\mathit{\text{value}}〉$ and ${\mathbf{p}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{p}}\right)$.
NE_INT_3
On entry, ${\mathbf{p}}=〈\mathit{\text{value}}〉$, ${\mathbf{m}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: $0\le {\mathbf{p}}\le {\mathbf{n}}\le {\mathbf{m}}+{\mathbf{p}}$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_SINGULAR
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.
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.

## 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

nag_zgglse (f08znc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_zgglse (f08znc) 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

## 9Further Comments

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
 $minimize x c-Ax2 subject to Bx=d$
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 ${x}_{2}={x}_{4}$.

### 10.1Program Text

Program Text (f08znce.c)

### 10.2Program Data

Program Data (f08znce.d)

### 10.3Program Results

Program Results (f08znce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017