F08AJF (DORGLQ) (PDF version)
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# NAG Library Routine DocumentF08AJF (DORGLQ)

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

F08AJF (DORGLQ) generates all or part of the real orthogonal matrix $Q$ from an $LQ$ factorization computed by F08AHF (DGELQF).

## 2  Specification

 SUBROUTINE F08AJF ( M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
 INTEGER M, N, K, LDA, LWORK, INFO REAL (KIND=nag_wp) A(LDA,*), TAU(*), WORK(max(1,LWORK))
The routine may be called by its LAPACK name dorglq.

## 3  Description

F08AJF (DORGLQ) is intended to be used after a call to F08AHF (DGELQF), which performs an $LQ$ factorization of a real matrix $A$. The orthogonal matrix $Q$ is represented as a product of elementary reflectors.
This routine 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:
```CALL DORGLQ(N,N,P,A,LDA,TAU,WORK,LWORK,INFO)
```
(note that the array A must have at least $n$ rows) or its leading $p$ rows by:
```CALL DORGLQ(P,N,P,A,LDA,TAU,WORK,LWORK,INFO)
```
The rows of $Q$ returned by the last call form an orthonormal basis for the space spanned by the rows of $A$; thus F08AHF (DGELQF) followed by F08AJF (DORGLQ) can be used to orthogonalize the rows of $A$.
The information returned by the $LQ$ factorization routines 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:
```CALL DORGLQ(N,N,K,A,LDA,TAU,WORK,LWORK,INFO)
```
or its leading $k$ rows by:
```CALL DORGLQ(K,N,K,A,LDA,TAU,WORK,LWORK,INFO)
```

## 4  References

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

## 5  Parameters

1:     M – INTEGERInput
On entry: $m$, the number of rows of the matrix $Q$.
Constraint: ${\mathbf{M}}\ge 0$.
2:     N – INTEGERInput
On entry: $n$, the number of columns of the matrix $Q$.
Constraint: ${\mathbf{N}}\ge {\mathbf{M}}$.
3:     K – INTEGERInput
On entry: $k$, the number of elementary reflectors whose product defines the matrix $Q$.
Constraint: ${\mathbf{M}}\ge {\mathbf{K}}\ge 0$.
4:     A(LDA,$*$) – REAL (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: details of the vectors which define the elementary reflectors, as returned by F08AHF (DGELQF).
On exit: the $m$ by $n$ matrix $Q$.
5:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F08AJF (DORGLQ) is called.
Constraint: ${\mathbf{LDA}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{M}}\right)$.
6:     TAU($*$) – REAL (KIND=nag_wp) arrayInput
Note: the dimension 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 F08AHF (DGELQF).
7:     WORK($\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{LWORK}}\right)$) – REAL (KIND=nag_wp) arrayWorkspace
On exit: if ${\mathbf{INFO}}={\mathbf{0}}$, ${\mathbf{WORK}}\left(1\right)$ contains the minimum value of LWORK required for optimal performance.
8:     LWORK – INTEGERInput
On entry: the dimension of the array WORK as declared in the (sub)program from which F08AJF (DORGLQ) 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, ${\mathbf{LWORK}}\ge {\mathbf{M}}×\mathit{nb}$, where $\mathit{nb}$ is the optimal block size.
Constraint: ${\mathbf{LWORK}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{M}}\right)$ or ${\mathbf{LWORK}}=-1$.
9:     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.

## 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  Further Comments

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 routine is F08AWF (ZUNGLQ).

## 9  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 $A$.

### 9.1  Program Text

Program Text (f08ajfe.f90)

### 9.2  Program Data

Program Data (f08ajfe.d)

### 9.3  Program Results

Program Results (f08ajfe.r)

F08AJF (DORGLQ) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual