f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_zggglm (f08zpc)

## 1  Purpose

nag_zggglm (f08zpc) solves a complex general Gauss–Markov linear (least squares) model problem.

## 2  Specification

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

## 3  Description

nag_zggglm (f08zpc) solves the complex 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
 $minimize x B-1 d-Ax 2 .$

## 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

## 5  Arguments

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 3.2.1.3 in the Essential Introduction 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 matrices $A$ and $B$.
Constraint: ${\mathbf{m}}\ge 0$.
3:    $\mathbf{n}$IntegerInput
On entry: $n$, the number of columns of the matrix $A$.
Constraint: $0\le {\mathbf{n}}\le {\mathbf{m}}$.
4:    $\mathbf{p}$IntegerInput
On entry: $p$, the number of columns of the matrix $B$.
Constraint: ${\mathbf{p}}\ge {\mathbf{m}}-{\mathbf{n}}$.
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
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdb}}×{\mathbf{p}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}×{\mathbf{pdb}}\right)$ 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 $m$ by $p$ 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 ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{p}}\right)$.
9:    $\mathbf{d}\left[{\mathbf{m}}\right]$ComplexInput/Output
On entry: the left-hand side vector $d$ of the GLM equation.
On exit: d is overwritten.
10:  $\mathbf{x}\left[{\mathbf{n}}\right]$ComplexOutput
On exit: the solution vector $x$ of the GLM problem.
11:  $\mathbf{y}\left[{\mathbf{p}}\right]$ComplexOutput
On exit: the solution vector $y$ of the GLM problem.
12:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.2.1.2 in the Essential Introduction for further information.
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, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pda}}>0$.
On entry, ${\mathbf{pdb}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdb}}>0$.
NE_INT_2
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: $0\le {\mathbf{n}}\le {\mathbf{m}}$.
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 ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
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: ${\mathbf{p}}\ge {\mathbf{m}}-{\mathbf{n}}$.
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 3.6.6 in the Essential Introduction for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.
NE_SINGULAR
The bottom $\left(N-M\right)$ by $\left(N-M\right)$ 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.
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 $\mathrm{rank}\left(\begin{array}{cc}B& A\end{array}\right); the least squares solutions could not be computed.

## 7  Accuracy

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

## 8  Parallelism and Performance

nag_zggglm (f08zpc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_zggglm (f08zpc) 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.

When $p=m\ge n$, the total number of real floating-point operations is approximately $\frac{8}{3}\left(2{m}^{3}-{n}^{3}\right)+16n{m}^{2}$; when $p=m=n$, the total number of real floating-point operations is approximately $\frac{56}{3}{m}^{3}$.

## 10  Example

This example solves the weighted least squares problem
 $minimize x B-1 d-Ax 2 ,$
where
 $B = 0.5-1.0i .0i+0.0 .0i+0.0 .0i+0.0 .0i+0.0 1.0-2.0i .0i+0.0 .0i+0.0 .0i+0.0 .0i+0.0 2.0-3.0i .0i+0.0 .0i+0.0 .0i+0.0 .0i+0.0 5.0-4.0i ,$
 $d = 6.00-0.40i -5.27+0.90i 2.72-2.13i -1.30-2.80i$
and
 $A = 0.96-0.81i -0.03+0.96i -0.91+2.06i -0.98+1.98i -1.20+0.19i -0.66+0.42i 0.62-0.46i 1.01+0.02i 0.63-0.17i 1.08-0.28i 0.20-0.12i -0.07+1.23i .$

### 10.1  Program Text

Program Text (f08zpce.c)

### 10.2  Program Data

Program Data (f08zpce.d)

### 10.3  Program Results

Program Results (f08zpce.r)