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 <nag.h>

void	f08cjf_ (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 the names f08cjf, nagf_lapackeig_dorgrq or its LAPACK name dorgrq.

3 Description

f08cjf is intended to be used following a call to f08chf, which performs an

R Q

factorization of a real matrix

A

and represents the orthogonal matrix

Q

as a product of

k

elementary reflectors of order

n

This routine may be used to generate

Q

explicitly as a square matrix, or to form only its trailing rows.

Usually

Q

is determined from the

R Q

factorization of a

p \times n

matrix

A

with

p \leq n

. The whole of

Q

may be computed by :

Call dorgrq(n,n,p,a,lda,tau,work,lwork,info)

(note that the matrix

A

must have at least

n

rows) or its trailing

p

rows by :

Call dorgrq(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 f08chf followed by f08cjf can be used to orthogonalize the rows of

A

The information returned by f08chf also yields the

R Q

factorization of the trailing

k

rows of

A

, where

k < p

. The orthogonal matrix arising from this factorization can be computed by :

Call dorgrq(n,n,k,a,lda,tau,work,lwork,info)

or its leading

k

columns by :

Call dorgrq(k,n,k,a,lda,tau,work,lwork,info)

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 https://www.netlib.org/lapack/lug

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

5 Arguments

1: $m$ – Integer Input: On entry: $m$ , the number of rows of the matrix $Q$ .

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

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

Constraint: $m \geq k \geq 0$ .
4: $a (lda, *)$ – Real (Kind=nag_wp) array Input/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 f08chf.

On exit: the $m \times n$ matrix $Q$ .
5: $lda$ – Integer Input: On entry: the first dimension of the array a as declared in the (sub)program from which f08cjf is called.

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

On entry: $tau (i)$ must contain the scalar factor of the elementary reflector $H_{i}$ , as returned by f08chf.
7: $work (\max (1, lwork))$ – Real (Kind=nag_wp) array Workspace: On exit: if $info = 0$ , $work (1)$ contains the minimum value of lwork required for optimal performance.
8: $lwork$ – Integer Input: On entry: the dimension of the array work as declared in the (sub)program from which f08cjf 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: $lwork \geq \max (1, m)$ or $lwork = −1$ .
9: $info$ – Integer Output: 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

and

ε

is the machine precision.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

f08cjf 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

m = k

this becomes

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

The complex analogue of this routine is f08cwf.

10 Example

This example generates the first four rows of the matrix

Q

of the

R Q

factorization of

A

as returned by f08chf, where

A = (\begin{array}{r} - 0.57 & - 1.93 & 2.30 & - 1.93 & 0.15 & - 0.02 \\ - 1.28 & 1.08 & 0.24 & 0.64 & 0.30 & 1.03 \\ - 0.39 & - 0.31 & 0.40 & - 0.66 & 0.15 & - 1.43 \\ 0.25 & - 2.14 & - 0.35 & 0.08 & - 2.13 & 0.50 \end{array}) .

Note that the block size (NB) of

64

assumed in this example is not realistic for such a small problem, but should be suitable for large problems.

f08cj: FL CL CPP AD PY MB

NAG FL Interfacef08cjf (dorgrq)

▸▿ 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

NAG FL Interface
f08cjf (dorgrq)