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)	::	vect, side, trans

C Header Interface

#include <nagmk26.h>

void

f08kgf_ (const char *vect, 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_vect, const Charlen length_side, const Charlen length_trans)

The routine may be called by its LAPACK name dormbr.

3

Description

f08kgf (dormbr) is intended to be used after a call to f08kef (dgebrd), which reduces a real rectangular matrix

A

to bidiagonal form

B

by an orthogonal transformation:

A = Q B P^{T}

. f08kef (dgebrd) represents the matrices

Q

and

P^{T}

as products 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}, P C, P^{T} C, C P ​ or ​ C P^{T},

overwriting the result on

C

(which may be any real rectangular matrix).

4

References

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

5

Arguments

Note: in the descriptions below,

r

denotes the order of

Q

P^{T}

: if

side ='L'

r = m

and if

side ='R'

r = n

1: $vect$ – Character(1)Input

On entry: indicates whether

Q

Q^{T}

P

P^{T}

is to be applied to

C

$vect ='Q'$: $Q$ or $Q^{T}$ is applied to $C$ .
$vect ='P'$: $P$ or $P^{T}$ is applied to $C$ .

Constraint:

vect ='Q'

'P'

2: $side$ – Character(1)Input

On entry: indicates how

Q

Q^{T}

P

P^{T}

is to be applied to

C

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

Constraint:

side ='L'

'R'

3: $trans$ – Character(1)Input

On entry: indicates whether

Q

P

Q^{T}

P^{T}

is to be applied to

C

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

Constraint:

trans ='N'

'T'

4: $m$ – IntegerInput

On entry:

m

, the number of rows of the matrix

C

Constraint:

m \geq 0

5: $n$ – IntegerInput

On entry:

n

, the number of columns of the matrix

C

Constraint:

n \geq 0

6: $k$ – IntegerInput

On entry: if

vect ='Q'

, the number of columns in the original matrix

A

vect ='P'

, the number of rows in the original matrix

A

Constraint:

k \geq 0

7: $a (lda *)$ – Real (Kind=nag_wp) arrayInput

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

\max (1, \min (r, k))

vect ='Q'

and at least

\max (1, r)

vect ='P'

On entry: details of the vectors which define the elementary reflectors, as returned by f08kef (dgebrd).

8: $lda$ – IntegerInput

On entry: the first dimension of the array a as declared in the (sub)program from which f08kgf (dormbr) is called.

Constraints:

if $vect ='Q'$ , $lda \geq \max (1, r)$ ;
if $vect ='P'$ , $lda \geq \max (1, \min (r, k))$ .

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

Note: the dimension of the array tau must be at least

\max (1, \min (r, k))

On entry: further details of the elementary reflectors, as returned by f08kef (dgebrd) in its argument tauq if

vect ='Q'

, or in its argument taup if

vect ='P'

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

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

\max (1, n)

On entry: the matrix

C

On exit: c is overwritten by

Q C

Q^{T} C

C Q

C^{T} Q

P C

P^{T} C

C P

C^{T} P

as specified by vect, side and trans.

11: $ldc$ – IntegerInput

On entry: the first dimension of the array c as declared in the (sub)program from which f08kgf (dormbr) is called.

Constraint:

ldc \geq \max (1, m)

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

On exit: if

info = 0

work (1)

contains the minimum value of lwork required for optimal performance.

13: $lwork$ – IntegerInput

On entry: the dimension of the array work as declared in the (sub)program from which f08kgf (dormbr) 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: $info$ – 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 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

f08kgf (dormbr) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08kgf (dormbr) 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

if $side ='L'$ and $m \geq k$ , $2 n k (2 m - k)$ ;
if $side ='R'$ and $n \geq k$ , $2 m k (2 n - k)$ ;
if $side ='L'$ and $m < k$ , $2 m^{2} n$ ;
if $side ='R'$ and $n < k$ , $2 m n^{2}$ ,

where

k

is the value of the argument k.

The complex analogue of this routine is f08kuf (zunmbr).

10

Example

For this routine two examples are presented. Both illustrate how the reduction to bidiagonal form of a matrix

A

may be preceded by a

Q R

L Q

factorization of

A

In the first example,

m > n

, and

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 routine first performs a

Q R

factorization of

A

A = Q_{a} R

and then reduces the factor

R

to bidiagonal form

B

R = Q_{b} B P^{T}

. Finally it forms

Q_{a}

and calls f08kgf (dormbr) to form

Q = Q_{a} Q_{b}

In the second example,

m < n

, and

A = (\begin{array}{r} - 5.42 & 3.28 & - 3.68 & 0.27 & 2.06 & 0.46 \\ - 1.65 & - 3.40 & - 3.20 & - 1.03 & - 4.06 & - 0.01 \\ - 0.37 & 2.35 & 1.90 & 4.31 & - 1.76 & 1.13 \\ - 3.15 & - 0.11 & 1.99 & - 2.70 & 0.26 & 4.50 \end{array}) .

The routine first performs an

L Q

factorization of

A

A = L P_{a}^{T}

and then reduces the factor

L

to bidiagonal form

B

L = Q B P_{b}^{T}

. Finally it forms

P_{b}^{T}

and calls f08kgf (dormbr) to form

P^{T} = P_{b}^{T} P_{a}^{T}

NAG Library Routine Document

f08kgf (dormbr)

▸▿ 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

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