NAG Library Routine Document

1Purpose

f08aff (dorgqr) generates all or part of the real orthogonal matrix $Q$ from a $QR$ factorization computed by f08aef (dgeqrf), f08bef (dgeqpf) or f08bff (dgeqp3).

2Specification

Fortran Interface
 Subroutine f08aff ( m, n, k, a, lda, tau, work, info)
 Integer, Intent (In) :: m, n, k, lda, lwork Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (In) :: tau(*) Real (Kind=nag_wp), Intent (Inout) :: a(lda,*) Real (Kind=nag_wp), Intent (Out) :: work(max(1,lwork))
#include nagmk26.h
 void f08aff_ (const Integer *m, const Integer *n, const Integer *k, double a[], const Integer *lda, const double tau[], double work[], const Integer *lwork, Integer *info)
The routine may be called by its LAPACK name dorgqr.

3Description

f08aff (dorgqr) is intended to be used after a call to f08aef (dgeqrf), f08bef (dgeqpf) or f08bff (dgeqp3). which perform a $QR$ 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 columns.
Usually $Q$ is determined from the $QR$ factorization of an $m$ by $p$ matrix $A$ with $m\ge p$. The whole of $Q$ may be computed by:
```Call dorgqr(m,m,p,a,lda,tau,work,lwork,info)
```
(note that the array a must have at least $m$ columns) or its leading $p$ columns by:
```Call dorgqr(m,p,p,a,lda,tau,work,lwork,info)
```
The columns of $Q$ returned by the last call form an orthonormal basis for the space spanned by the columns of $A$; thus f08aef (dgeqrf) followed by f08aff (dorgqr) can be used to orthogonalize the columns of $A$.
The information returned by the $QR$ factorization routines also yields the $QR$ factorization of the leading $k$ columns of $A$, where $k. The orthogonal matrix arising from this factorization can be computed by:
```Call dorgqr(m,m,k,a,lda,tau,work,lwork,info)
```
or its leading $k$ columns by:
```Call dorgqr(m,k,k,a,lda,tau,work,lwork,info)
```

4References

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

5Arguments

1:     $\mathbf{m}$ – IntegerInput
On entry: $m$, the order of the orthogonal matrix $Q$.
Constraint: ${\mathbf{m}}\ge 0$.
2:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of columns of the matrix $Q$.
Constraint: ${\mathbf{m}}\ge {\mathbf{n}}\ge 0$.
3:     $\mathbf{k}$ – IntegerInput
On entry: $k$, the number of elementary reflectors whose product defines the matrix $Q$.
Constraint: ${\mathbf{n}}\ge {\mathbf{k}}\ge 0$.
4:     $\mathbf{a}\left({\mathbf{lda}},*\right)$ – 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 f08aef (dgeqrf), f08bef (dgeqpf) or f08bff (dgeqp3).
On exit: the $m$ by $n$ matrix $Q$.
5:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08aff (dorgqr) is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
6:     $\mathbf{tau}\left(*\right)$ – 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 f08aef (dgeqrf), f08bef (dgeqpf) or f08bff (dgeqp3).
7:     $\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$ – 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:     $\mathbf{lwork}$ – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08aff (dorgqr) 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, ${\mathbf{lwork}}\ge {\mathbf{n}}×\mathit{nb}$, where $\mathit{nb}$ is the optimal block size.
Constraint: ${\mathbf{lwork}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ or ${\mathbf{lwork}}=-1$.
9:     $\mathbf{info}$ – IntegerOutput
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

6Error Indicators and Warnings

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

7Accuracy

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

8Parallelism and Performance

f08aff (dorgqr) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08aff (dorgqr) 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 routine. 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 $n=k$, the number is approximately $\frac{2}{3}{n}^{2}\left(3m-n\right)$.
The complex analogue of this routine is f08atf (zungqr).

10Example

This example forms the leading $4$ columns of the orthogonal matrix $Q$ from the $QR$ factorization of the matrix $A$, where
 $A = -0.57 -1.28 -0.39 0.25 -1.93 1.08 -0.31 -2.14 2.30 0.24 0.40 -0.35 -1.93 0.64 -0.66 0.08 0.15 0.30 0.15 -2.13 -0.02 1.03 -1.43 0.50 .$
The columns of $Q$ form an orthonormal basis for the space spanned by the columns of $A$.

10.1Program Text

Program Text (f08affe.f90)

10.2Program Data

Program Data (f08affe.d)

10.3Program Results

Program Results (f08affe.r)

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