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):

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).
If m>n, to form the n leading columns of Q:
```
nag_dorgbr(order,Nag_FormQ,m,n,n,...)
```
To form the full n by n matrix PT:
```
nag_dorgbr(order,Nag_FormP,n,n,m,...)
```
(note that the array a must have at least n rows).
If m<n, to form the m leading rows of PT:
```
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

or 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 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 Further Comments

The total number of floating point operations for the cases listed in Section 3 are approximately as follows:

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

The complex analogue of this function is nag_zungbr (f08ktc).

9 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 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Function Document

nag_dorgbr (f08kfc)