NAG Library Routine Document

F08KTF (ZUNGBR)

The various possibilities are specified by the parameters 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:
```
CALL ZUNGBR('Q',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:
```
CALL ZUNGBR('Q',m,n,n,...)
```
To form the full n by n matrix PH:
```
CALL ZUNGBR('P',n,n,m,...)
```
(note that the array A must have at least n rows).
If m<n, to form the m leading rows of PH:
```
CALL ZUNGBR('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 Parameters

1: VECT – CHARACTER(1)Input

On entry: indicates whether the unitary matrix

Q

P^{H}

is generated.

$VECT ='Q'$: $Q$ is generated.
$VECT ='P'$: $P^{H}$ is generated.

Constraint:

VECT ='Q'

'P'

2: M – INTEGERInput

On entry:

m

, the number of rows of the unitary matrix

Q

P^{H}

to be returned.

Constraint:

M \geq 0

3: N – INTEGERInput

On entry:

n

, the number of columns of the unitary matrix

Q

P^{H}

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 – 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

5: A(LDA, $*$ ) – COMPLEX (KIND=nag_wp) arrayInput/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 F08KSF (ZGEBRD).

On exit: the unitary matrix

Q

P^{H}

, or the leading rows or columns thereof, as specified by VECT, M and N.

6: LDA – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F08KTF (ZUNGBR) is called.

Constraint:

LDA \geq \max (1, M)

7: TAU( $*$ ) – COMPLEX (KIND=nag_wp) arrayInput

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 F08KSF (ZGEBRD) in its parameter TAUQ if

VECT ='Q'

, or in its parameter TAUP if

VECT ='P'

8: WORK( $\max (1, LWORK)$ ) – COMPLEX (KIND=nag_wp) arrayWorkspace

On exit: if

INFO = 0

, the real part of

WORK (1)

contains the minimum value of LWORK required for optimal performance.

9: LWORK – INTEGERInput

On entry: the dimension of the array WORK as declared in the (sub)program from which F08KTF (ZUNGBR) 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 – INTEGEROutput

On exit:

INFO = 0

unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

Errors or warnings detected by the routine:

$INFO < 0$: 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 unitary 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^{H}

8 Further Comments

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

To form the whole of Q:
- $\frac{16}{3} n (3 m^{2} - 3 m n + n^{2})$ if $m > n$ ,
- $\frac{16}{3} m^{3}$ if $m \leq n$ ;
To form the n leading columns of Q when m>n:
- $\frac{8}{3} n^{2} (3 m - n)$ ;
To form the whole of PH:
- $\frac{16}{3} n^{3}$ if $m \geq n$ ,
- $\frac{16}{3} m^{3} (3 n^{2} - 3 m n + m^{2})$ if $m < n$ ;
To form the m leading rows of PH when m<n:
- $\frac{8}{3} m^{2} (3 n - m)$ .

The real analogue of this routine is F08KFF (DORGBR).

9 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} 0.96 - 0.81 i & - 0.03 + 0.96 i & - 0.91 + 2.06 i & - 0.05 + 0.41 i \\ - 0.98 + 1.98 i & - 1.20 + 0.19 i & - 0.66 + 0.42 i & - 0.81 + 0.56 i \\ 0.62 - 0.46 i & 1.01 + 0.02 i & 0.63 - 0.17 i & - 1.11 + 0.60 i \\ - 0.37 + 0.38 i & 0.19 - 0.54 i & - 0.98 - 0.36 i & 0.22 - 0.20 i \\ 0.83 + 0.51 i & 0.20 + 0.01 i & - 0.17 - 0.46 i & 1.47 + 1.59 i \\ 1.08 - 0.28 i & 0.20 - 0.12 i & - 0.07 + 1.23 i & 0.26 + 0.26 i \end{array})

in the first example and

A = (\begin{array}{r} 0.28 - 0.36 i & 0.50 - 0.86 i & - 0.77 - 0.48 i & 1.58 + 0.66 i \\ - 0.50 - 1.10 i & - 1.21 + 0.76 i & - 0.32 - 0.24 i & - 0.27 - 1.15 i \\ 0.36 - 0.51 i & - 0.07 + 1.33 i & - 0.75 + 0.47 i & - 0.08 + 1.01 i \end{array})

in the second.

A

must first be reduced to tridiagonal form by F08KSF (ZGEBRD). The program then calls F08KTF (ZUNGBR) twice to form

Q

and

P^{H}

, and passes these matrices to F08MSF (ZBDSQR), which computes the singular value decomposition of

A

NAG Library Routine DocumentF08KTF (ZUNGBR)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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 Routine Document

F08KTF (ZUNGBR)