NAG Library Function Document

nag_dorgbr (f08kfc)

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

n

matrix):

1.	To form the full $m$ by $m$ matrix $Q$ : nag_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_dorgbr(order,Nag_FormQ,m,n,n,...)
3.	To form the full $n$ by $n$ matrix $P^{T}$ : nag_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_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_OrderTypeInput

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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: vect – Nag_VectTypeInput

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 – IntegerInput

On entry:

m

, the number of rows of the orthogonal matrix

Q

P^{T}

to be returned.

Constraint:

m \geq 0

4: n – IntegerInput

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 – IntegerInput

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[ $\dim$ ] – doubleInput/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 nag_dgebrd (f08kec).

On exit: the orthogonal matrix

Q

P^{T}

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

order = Nag_ColMajor

, the

(i, j)

th element of the matrix is stored in

a [(j - 1) \times pda + i - 1]

order = Nag_RowMajor

, the

(i, j)

th element of the matrix is stored in

a [(i - 1) \times pda + j - 1]

7: pda – IntegerInput

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 doubleInput

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 nag_dgebrd (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 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
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.

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

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

nag_dorgbr (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 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 nag_zungbr (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 tridiagonal form by nag_dgebrd (f08kec). The program then calls nag_dorgbr (f08kfc) twice to form

Q

and

P^{T}

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

A

NAG Library Function Documentnag_dorgbr (f08kfc)

+− 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 Library Function Document

nag_dorgbr (f08kfc)