NAG Library Routine Document

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))

C Header Interface

#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.

3

Description

f08aff (dorgqr) is intended to be used after a call to f08aef (dgeqrf), f08bef (dgeqpf) or f08bff (dgeqp3). which perform a

Q R

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

Q R

factorization of an

m

p

matrix

A

with

m \geq 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

Q R

factorization routines also yields the

Q R

factorization of the leading

k

columns of

A

, where

k < p

. 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)

4

References

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

5

Arguments

1: $m$ – IntegerInput: On entry: $m$ , the order of the orthogonal matrix $Q$ .

Constraint: $m \geq 0$ .
2: $n$ – IntegerInput: On entry: $n$ , the number of columns of the matrix $Q$ .

Constraint: $m \geq n \geq 0$ .
3: $k$ – IntegerInput: On entry: $k$ , the number of elementary reflectors whose product defines the matrix $Q$ .

Constraint: $n \geq k \geq 0$ .
4: $a (lda *)$ – Real (Kind=nag_wp) arrayInput/Output: Note: the second dimension of the array a must be at least $\max (1, n)$ .

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: $lda$ – IntegerInput: On entry: the first dimension of the array a as declared in the (sub)program from which f08aff (dorgqr) is called.

Constraint: $lda \geq \max (1, m)$ .
6: $tau (*)$ – Real (Kind=nag_wp) arrayInput: Note: the dimension of the array tau must be at least $\max (1, k)$ .

On entry: further details of the elementary reflectors, as returned by f08aef (dgeqrf), f08bef (dgeqpf) or f08bff (dgeqp3).
7: $work (\max (1, lwork))$ – Real (Kind=nag_wp) arrayWorkspace: On exit: if $info = 0$ , $work (1)$ 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 f08aff (dorgqr) is called.
If $lwork = - 1$ , 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, $lwork \geq n \times nb$ , where $nb$ is the optimal block size.

Constraint: $lwork \geq \max (1, n)$ or $lwork = - 1$ .
9: $info$ – IntegerOutput: On exit: $info = 0$ unless the routine detects an error (see Section 6).

6

Error Indicators and Warnings

$info < 0$: If $info = - i$ , 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

{‖E‖}_{2} = O (ε),

where

ε

is the machine precision.

8

Parallelism 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.

9

Further Comments

The total number of floating-point operations is approximately

4 m n k - 2 (m + n) k^{2} + \frac{4}{3} k^{3}

; when

n = k

, the number is approximately

\frac{2}{3} n^{2} (3 m - n)

The complex analogue of this routine is f08atf (zungqr).

10

Example

This example forms the leading

4

columns of the orthogonal matrix

Q

from the

Q R

factorization of the matrix

A

, where

A = (\begin{array}{r} - 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 \end{array}) .

The columns of

Q

form an orthonormal basis for the space spanned by the columns of

A

NAG Library Routine Document

f08aff (dorgqr)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification