NAG Library Routine Document

Integer, Intent (In)	::	m, n, k, lda, ldc, lwork
Integer, Intent (Out)	::	info
Real (Kind=nag_wp), Intent (In)	::	tau(*)
Real (Kind=nag_wp), Intent (Inout)	::	a(lda,), c(ldc,)
Real (Kind=nag_wp), Intent (Out)	::	work(max(1,lwork))
Character (1), Intent (In)	::	side, trans

C Header Interface

#include <nagmk26.h>

void	f08ckf_ (const char side, const char trans, const Integer m, const Integer n, const Integer k, double a[], const Integer lda, const double tau[], double c[], const Integer ldc, double work[], const Integer lwork, Integer *info, const Charlen length_side, const Charlen length_trans)

The routine may be called by its LAPACK name dormrq.

3

Description

f08ckf (dormrq) is intended to be used following a call to f08chf (dgerqf), which performs an

R Q

factorization of a real matrix

A

and represents the orthogonal matrix

Q

as a product of elementary reflectors.

This routine may be used to form one of the matrix products

Q C, Q^{T} C, C Q, C Q^{T},

overwriting the result on

C

, which may be any real rectangular

m

n

matrix.

A common application of this routine is in solving underdetermined linear least squares problems, as described in the F08 Chapter Introduction, and illustrated in Section 10 in f08chf (dgerqf).

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 http://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: $side$ – Character(1)Input

On entry: indicates how

Q

Q^{T}

is to be applied to

C

$side ='L'$: $Q$ or $Q^{T}$ is applied to $C$ from the left.
$side ='R'$: $Q$ or $Q^{T}$ is applied to $C$ from the right.

Constraint:

side ='L'

'R'

2: $trans$ – Character(1)Input

On entry: indicates whether

Q

Q^{T}

is to be applied to

C

$trans ='N'$: $Q$ is applied to $C$ .
$trans ='T'$: $Q^{T}$ is applied to $C$ .

Constraint:

trans ='N'

'T'

3: $m$ – IntegerInput

On entry:

m

, the number of rows of the matrix

C

Constraint:

m \geq 0

4: $n$ – IntegerInput

On entry:

n

, the number of columns of the matrix

C

Constraint:

n \geq 0

5: $k$ – IntegerInput

On entry:

k

, the number of elementary reflectors whose product defines the matrix

Q

Constraints:

if $side ='L'$ , $m \geq k \geq 0$ ;
if $side ='R'$ , $n \geq k \geq 0$ .

6: $a (lda *)$ – Real (Kind=nag_wp) arrayInput/Output

Note: the second dimension of the array a must be at least

\max (1, m)

side ='L'

and at least

\max (1, n)

side ='R'

On entry: the

i

th row of a must contain the vector which defines the elementary reflector

H_{i}

, for

i = 1, 2, \dots, k

, as returned by f08chf (dgerqf).

On exit: is modified by f08ckf (dormrq) but restored on exit.

7: $lda$ – IntegerInput

On entry: the first dimension of the array a as declared in the (sub)program from which f08ckf (dormrq) is called.

Constraint:

lda \geq \max (1, k)

8: $tau (*)$ – Real (Kind=nag_wp) arrayInput

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 (dgerqf).

9: $c (ldc *)$ – Real (Kind=nag_wp) arrayInput/Output

Note: the second dimension of the array c must be at least

\max (1, n)

On entry: the

m

n

matrix

C

On exit: c is overwritten by

Q C

Q^{T} C

C Q

C Q^{T}

as specified by side and trans.

10: $ldc$ – IntegerInput

On entry: the first dimension of the array c as declared in the (sub)program from which f08ckf (dormrq) is called.

Constraint:

ldc \geq \max (1, m)

11: $work (\max (1, lwork))$ – Real (Kind=nag_wp) arrayWorkspace

On exit: if

info = 0

work (1)

contains the minimum value of lwork required for optimal performance.

12: $lwork$ – IntegerInput

On entry: the dimension of the array work as declared in the (sub)program from which f08ckf (dormrq) is called.

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

side ='L'

and at least

m \times nb

side ='R'

, where

nb

is the optimal block size.

Constraints:

if $side ='L'$ , $lwork \geq \max (1, n)$ or $lwork = - 1$ ;
if $side ='R'$ , $lwork \geq \max (1, m)$ or $lwork = - 1$ .

13: $info$ – IntegerOutput

On exit:

info = 0

unless the routine detects an error (see Section 6).

6

Error Indicators and Warnings

$- 999 < info < 0$: If $info = - i$ , argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

$info = - 999$: Dynamic memory allocation failed.

See Section 3.7 in How to Use the NAG Library and its Documentation for further information.
An explanatory message is output, and execution of the program is terminated.

7

Accuracy

The computed result differs from the exact result by a matrix

E

such that

{‖E‖}_{2} = O ε {‖C‖}_{2}

where

ε

is the machine precision.

8

Parallelism and Performance

f08ckf (dormrq) 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

2 n k (2 m - k)

side ='L'

and

2 m k (2 n - k)

side ='R'

The complex analogue of this routine is f08cxf (zunmrq).

10

Example

See Section 10 in f08chf (dgerqf).

NAG Library Manual, Mark 26

NAG AD Library Manual, Mark 26

NAG C Library Manual, Mark 26

F08 (lapackeig) Chapter Contents

F08 (lapackeig) Chapter Introduction

NAG Library Routine Document

f08ckf (dormrq)

▸▿ 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

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