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 $Q$ :
```
nag_lapackeig_dorgbr(order,Nag_FormQ,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

nag_lapackeig_dorgbr(order,Nag_FormQ,m,n,n,...)

3.To form the full $n \times n$ matrix $P^{T}$ :
```
nag_lapackeig_dorgbr(order,Nag_FormP,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}

nag_lapackeig_dorgbr(order,Nag_FormP,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: $order$ – Nag_OrderType Input

On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by

order = Nag_RowMajor

. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $vect$ – Nag_VectType Input

On entry: indicates whether the orthogonal matrix

Q

P^{T}

is generated.

$vect = Nag_FormQ$: $Q$ is generated.
$vect = Nag_FormP$: $P^{T}$ is generated.

Constraint:

vect = Nag_FormQ

Nag_FormP

3: $m$ – Integer Input

On entry:

m

, the number of rows of the orthogonal matrix

Q

P^{T}

to be returned.

Constraint:

m \geq 0

4: $n$ – Integer Input

On entry:

n

, the number of columns of the orthogonal matrix

Q

P^{T}

to be returned.

Constraints:

$n \geq 0$ ;
if $vect = Nag_FormQ$ and $m > k$ , $m \geq n \geq k$ ;
if $vect = Nag_FormQ$ and $m \leq k$ , $m = n$ ;
if $vect = Nag_FormP$ and $n > k$ , $n \geq m \geq k$ ;
if $vect = Nag_FormP$ and $n \leq k$ , $n = m$ .

5: $k$ – Integer Input

On entry: if

vect = Nag_FormQ

, the number of columns in the original matrix

A

vect = Nag_FormP

, the number of rows in the original matrix

A

Constraint:

k \geq 0

6: $a [\dim]$ – double Input/Output

Note: the dimension, dim, of the array a must be at least

$\max (1, pda \times n)$ when $order = Nag_ColMajor$ ;
$\max (1, m \times pda)$ when $order = Nag_RowMajor$ .

On entry: details of the vectors which define the elementary reflectors, as returned by f08kec.

On exit: the orthogonal matrix

Q

P^{T}

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

7: $pda$ – Integer Input

On entry: the stride separating row or column elements (depending on the value of order) in the array a.

Constraints:

if $order = Nag_ColMajor$ , $pda \geq \max (1, m)$ ;
if $order = Nag_RowMajor$ , $pda \geq \max (1, n)$ .

8: $tau [\dim]$ – const double Input

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

$\max (1, \min (m, k))$ when $vect = Nag_FormQ$ ;
$\max (1, \min (n, k))$ when $vect = Nag_FormP$ .

On entry: further details of the elementary reflectors, as returned by f08kec in its argument tauq if

vect = Nag_FormQ

, or in its argument taup if

vect = Nag_FormP

9: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_ENUM_INT_3: On entry, $vect = ⟨ value ⟩$ , $m = ⟨ value ⟩$ , $n = ⟨ value ⟩$ and $k = ⟨ value ⟩$ .
Constraint: $n \geq 0$ and
if $vect = Nag_FormQ$ and $m > k$ , $m \geq n \geq k$ ;
if $vect = Nag_FormQ$ and $m \leq k$ , $m = n$ ;
if $vect = Nag_FormP$ and $n > k$ , $n \geq m \geq k$ ;
if $vect = Nag_FormP$ and $n \leq k$ , $n = m$ .
NE_INT: On entry, $k = ⟨ value ⟩$ .
Constraint: $k \geq 0$ .

On entry, $m = ⟨ value ⟩$ .
Constraint: $m \geq 0$ .

On entry, $pda = ⟨ value ⟩$ .
Constraint: $pda > 0$ .
NE_INT_2: On entry, $pda = ⟨ value ⟩$ and $m = ⟨ value ⟩$ .
Constraint: $pda \geq \max (1, m)$ .

On entry, $pda = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pda \geq \max (1, n)$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

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

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

f08kfc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08kfc 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 function. 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 function is f08ktc.

10 Example

For this function 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 f08kec. The program then calls f08kfc twice to form

Q

and

P^{T}

, and passes these matrices to f08mec, which computes the singular value decomposition of

A

f08kf: FL CL CPP AD PY MB

NAG CL Interfacef08kfc (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 CL Interface
f08kfc (dorgbr)