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))
Character (1), Intent (In)	::	vect

C Header Interface

#include <nag.h>

void	f08kff_ (const char vect, const Integer m, const Integer n, const Integer k, double a[], const Integer lda, const double tau[], double work[], const Integer lwork, Integer *info, const Charlen length_vect)

The routine may be called by the names f08kff, nagf_lapackeig_dorgbr or its LAPACK name dorgbr.

3 Description

f08kff is intended to be used after a call to f08kef, which reduces a real rectangular matrix

A

to bidiagonal form

B

by an orthogonal transformation:

A = Q B P^{T}

. f08kef represents the matrices

Q

and

P^{T}

as products of elementary reflectors.

This routine may be used to generate

Q

P^{T}

explicitly as square matrices, or in some cases just the leading columns of

Q

or the leading rows of

P^{T}

The various possibilities are specified by the arguments vect, m, n and k. The appropriate values to cover the most likely cases are as follows (assuming that

A

was an

m \times n

matrix):

1.To form the full $m \times m$ matrix $Q$ :
```
Call dorgbr('Q',m,m,n,...)
```
(note that the array a must have at least $m$ columns).
2.If $m > n$ , to form the $n$ leading columns of $Q$ :
```
Call dorgbr('Q',m,n,n,...)
```
3.To form the full $n \times n$ matrix $P^{T}$ :
```
Call dorgbr('P',n,n,m,...)
```
(note that the array a must have at least $n$ rows).
4.If $m < n$ , to form the $m$ leading rows of $P^{T}$ :
```
Call dorgbr('P',m,n,m,...)
```

4 References

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

5 Arguments

1: $vect$ – Character(1) Input

On entry: indicates whether the orthogonal matrix

Q

P^{T}

is generated.

$vect ='Q'$: $Q$ is generated.
$vect ='P'$: $P^{T}$ is generated.

Constraint:

vect ='Q'

'P'

2: $m$ – Integer Input

On entry:

m

, the number of rows of the orthogonal matrix

Q

P^{T}

to be returned.

Constraint:

m \geq 0

3: $n$ – Integer Input

On entry:

n

, the number of columns of the orthogonal matrix

Q

P^{T}

to be returned.

Constraints:

$n \geq 0$ ;
if $vect ='Q'$ and $m > k$ , $m \geq n \geq k$ ;
if $vect ='Q'$ and $m \leq k$ , $m = n$ ;
if $vect ='P'$ and $n > k$ , $n \geq m \geq k$ ;
if $vect ='P'$ and $n \leq k$ , $n = m$ .

4: $k$ – Integer Input

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

5: $a (lda, *)$ – 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 f08kef.

On exit: the orthogonal matrix

Q

P^{T}

, or the leading rows or columns thereof, as specified by vect, m and n.

6: $lda$ – Integer Input

On entry: the first dimension of the array a as declared in the (sub)program from which f08kff is called.

Constraint:

lda \geq \max (1, m)

7: $tau (*)$ – Real (Kind=nag_wp) array Input

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

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

vect ='Q'

and at least

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

vect ='P'

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

vect ='Q'

, or in its argument taup if

vect ='P'

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

On exit: if

info = 0

work (1)

contains the minimum value of lwork required for optimal performance.

9: $lwork$ – Integer Input

On entry: the dimension of the array work as declared in the (sub)program from which f08kff 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 \min (m, n) \times nb

, where

nb

is the optimal block size.

Constraint:

lwork \geq \max (1, \min (m, n))

lwork = −1

10: $info$ – 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),

where

ε

is the machine precision. A similar statement holds for the computed matrix

P^{T}

8 Parallelism and Performance

f08kff is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08kff 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 for the cases listed in Section 3 are approximately as follows:

1.To form the whole of $Q$ :
- $\frac{4}{3} n (3 m^{2} - 3 m n + n^{2})$ if $m > n$ ,
- $\frac{4}{3} m^{3}$ if $m \leq n$ ;
2.To form the $n$ leading columns of $Q$ when $m > n$ :
- $\frac{2}{3} n^{2} (3 m - n)$ ;
3.To form the whole of $P^{T}$ :
- $\frac{4}{3} n^{3}$ if $m \geq n$ ,
- $\frac{4}{3} m (3 n^{2} - 3 m n + m^{2})$ if $m < n$ ;
4.To form the $m$ leading rows of $P^{T}$ when $m < n$ :
- $\frac{2}{3} m^{2} (3 n - m)$ .

The complex analogue of this routine is f08ktf.

10 Example

For this routine two examples are presented, both of which involve computing the singular value decomposition of a 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})

in the first example 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})

in the second.

A

must first be reduced to bidiagonal form by f08kef. The program then calls f08kff twice to form

Q

and

P^{T}

, and passes these matrices to f08mef, which computes the singular value decomposition of

A

f08kf: FL CL CPP AD

NAG FL Interfacef08kff (dorgbr)

▸▿ 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
f08kff (dorgbr)