Integer, Intent (In)	::	m, n, k, l, 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 <nag.h>

void

f08bkf_ (const char *side, const char *trans, const Integer *m, const Integer *n, const Integer *k, const Integer *l, 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 the names f08bkf, nagf_lapackeig_dormrz or its LAPACK name dormrz.

3 Description

f08bkf is intended to be used following a call to f08bhf, which performs an

R Z

factorization of a real upper trapezoidal matrix

A

and represents the orthogonal matrix

Z

as a product of elementary reflectors.

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

Z C, Z^{T} C, C Z, C Z^{T},

overwriting the result on

C

, which may be any real rectangular

m \times n

matrix.

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

5 Arguments

1: $side$ – Character(1) Input

On entry: indicates how

Z

Z^{T}

is to be applied to

C

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

Constraint:

side ='L'

'R'

2: $trans$ – Character(1) Input

On entry: indicates whether

Z

Z^{T}

is to be applied to

C

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

Constraint:

trans ='N'

'T'

3: $m$ – Integer Input

On entry:

m

, the number of rows of the matrix

C

Constraint:

m \geq 0

4: $n$ – Integer Input

On entry:

n

, the number of columns of the matrix

C

Constraint:

n \geq 0

5: $k$ – Integer Input

On entry:

k

, the number of elementary reflectors whose product defines the matrix

Z

Constraints:

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

6: $l$ – Integer Input

On entry:

l

, the number of columns of the matrix

A

containing the meaningful part of the Householder reflectors.

Constraints:

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

7: $a (lda, *)$ – Real (Kind=nag_wp) array Input

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 f08bhf.

8: $lda$ – Integer Input

On entry: the first dimension of the array a as declared in the (sub)program from which f08bkf is called.

Constraint:

lda \geq \max (1, k)

9: $tau (*)$ – 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 f08bhf.

10: $c (ldc, *)$ – Real (Kind=nag_wp) array Input/Output

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

\max (1, n)

On entry: the

m \times n

matrix

C

On exit: c is overwritten by

Z C

Z^{T} C

C Z

Z^{T} C

as specified by side and trans.

11: $ldc$ – Integer Input

On entry: the first dimension of the array c as declared in the (sub)program from which f08bkf is called.

Constraint:

ldc \geq \max (1, m)

12: $work (\max (1, lwork))$ – Real (Kind=nag_wp) array Workspace

On exit: if

info = 0

work (1)

contains the minimum value of lwork required for optimal performance.

13: $lwork$ – Integer Input

On entry: the dimension of the array work as declared in the (sub)program from which f08bkf 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 $lwork = −1$ .

14: $info$ – Integer Output

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 9 in the Introduction to the NAG Library FL Interface 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

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

f08bkf 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.