f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_dorglq (f08ajc)

## 1  Purpose

nag_dorglq (f08ajc) generates all or part of the real orthogonal matrix $Q$ from an $LQ$ factorization computed by nag_dgelqf (f08ahc).

## 2  Specification

 #include #include
 void nag_dorglq (Nag_OrderType order, Integer m, Integer n, Integer k, double a[], Integer pda, const double tau[], NagError *fail)

## 3  Description

nag_dorglq (f08ajc) is intended to be used after a call to nag_dgelqf (f08ahc), which performs an $LQ$ factorization of a real matrix $A$. The orthogonal matrix $Q$ is represented as a product of elementary reflectors.
This function may be used to generate $Q$ explicitly as a square matrix, or to form only its leading rows.
Usually $Q$ is determined from the $LQ$ factorization of a $p$ by $n$ matrix $A$ with $p\le n$. The whole of $Q$ may be computed by:
```nag_dorglq(order,n,n,p,a,pda,tau,&fail)
```
(note that the array a must have at least $n$ rows) or its leading $p$ rows by:
```nag_dorglq(order,p,n,p,a,pda,tau,&fail)
```
The rows of $Q$ returned by the last call form an orthonormal basis for the space spanned by the rows of $A$; thus nag_dgelqf (f08ahc) followed by nag_dorglq (f08ajc) can be used to orthogonalize the rows of $A$.
The information returned by the $LQ$ factorization functions also yields the $LQ$ factorization of the leading $k$ rows of $A$, where $k. The orthogonal matrix arising from this factorization can be computed by:
```nag_dorglq(order,n,n,k,a,pda,tau,&fail)
```
or its leading $k$ rows by:
```nag_dorglq(order,k,n,k,a,pda,tau,&fail)
```

## 4  References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 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 matrix $Q$.
Constraint: ${\mathbf{m}}\ge 0$.
3:    $\mathbf{n}$IntegerInput
On entry: $n$, the number of columns of the matrix $Q$.
Constraint: ${\mathbf{n}}\ge {\mathbf{m}}$.
4:    $\mathbf{k}$IntegerInput
On entry: $k$, the number of elementary reflectors whose product defines the matrix $Q$.
Constraint: ${\mathbf{m}}\ge {\mathbf{k}}\ge 0$.
5:    $\mathbf{a}\left[\mathit{dim}\right]$doubleInput/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 ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: details of the vectors which define the elementary reflectors, as returned by nag_dgelqf (f08ahc).
On exit: the $m$ by $n$ matrix $Q$.
If ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, the $\left(i,j\right)$th element of the matrix is stored in ${\mathbf{a}}\left[\left(j-1\right)×{\mathbf{pda}}+i-1\right]$.
If ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, the $\left(i,j\right)$th element of the matrix is stored in ${\mathbf{a}}\left[\left(i-1\right)×{\mathbf{pda}}+j-1\right]$.
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:    $\mathbf{tau}\left[\mathit{dim}\right]$const doubleInput
Note: the dimension, dim, of the array tau must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$.
On entry: further details of the elementary reflectors, as returned by nag_dgelqf (f08ahc).
8:    $\mathbf{fail}$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: ${\mathbf{pda}}>0$.
NE_INT_2
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$ and ${\mathbf{k}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\ge {\mathbf{k}}\ge 0$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$ and ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge {\mathbf{m}}$.
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: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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.

## 7  Accuracy

The computed matrix $Q$ differs from an exactly orthogonal matrix by a matrix $E$ such that
 $E2 = Oε ,$
where $\epsilon$ is the machine precision.

## 8  Parallelism and Performance

nag_dorglq (f08ajc) is not threaded by NAG in any implementation.
nag_dorglq (f08ajc) 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.

The total number of floating-point operations is approximately $4mnk-2\left(m+n\right){k}^{2}+\frac{4}{3}{k}^{3}$; when $m=k$, the number is approximately $\frac{2}{3}{m}^{2}\left(3n-m\right)$.
The complex analogue of this function is nag_zunglq (f08awc).

## 10  Example

This example forms the leading $4$ rows of the orthogonal matrix $Q$ from the $LQ$ factorization of the matrix $A$, where
 $A = -5.42 3.28 -3.68 0.27 2.06 0.46 -1.65 -3.40 -3.20 -1.03 -4.06 -0.01 -0.37 2.35 1.90 4.31 -1.76 1.13 -3.15 -0.11 1.99 -2.70 0.26 4.50 .$
The rows of $Q$ form an orthonormal basis for the space spanned by the rows of $A$.

### 10.1  Program Text

Program Text (f08ajce.c)

### 10.2  Program Data

Program Data (f08ajce.d)

### 10.3  Program Results

Program Results (f08ajce.r)