NAG Library Routine Document

F08KUF (ZUNMBR)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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

1 Purpose

F08KUF (ZUNMBR) multiplies an arbitrary complex

m

n

matrix

C

by one of the complex unitary matrices

Q

P

which were determined by F08KSF (ZGEBRD) when reducing a complex matrix to bidiagonal form.

2 Specification

SUBROUTINE F08KUF (

VECT, SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)

INTEGER	M, N, K, LDA, LDC, LWORK, INFO
COMPLEX (KIND=nag_wp)	A(LDA,), TAU(), C(LDC,*), WORK(max(1,LWORK))
CHARACTER(1)	VECT, SIDE, TRANS

The routine may be called by its LAPACK name zunmbr.

3 Description

F08KUF (ZUNMBR) is intended to be used after a call to F08KSF (ZGEBRD), which reduces a complex rectangular matrix

A

to real bidiagonal form

B

by a unitary transformation:

A = Q B P^{H}

. F08KSF (ZGEBRD) represents the matrices

Q

and

P^{H}

as products of elementary reflectors.

This routine may be used to form one of the matrix products

Q C, Q^{H} C, C Q, C Q^{H}, P C, P^{H} C, C P ​ or ​ C P^{H},

overwriting the result on

C

(which may be any complex rectangular matrix).

4 References

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

5 Parameters

Note: in the descriptions below,

r

denotes the order of

Q

P^{H}

: if

SIDE ='L'

r = M

and if

SIDE ='R'

r = N

1: $VECT$ – CHARACTER(1)Input

On entry: indicates whether

Q

Q^{H}

P

P^{H}

is to be applied to

C

$VECT ='Q'$: $Q$ or $Q^{H}$ is applied to $C$ .
$VECT ='P'$: $P$ or $P^{H}$ is applied to $C$ .

Constraint:

VECT ='Q'

'P'

2: $SIDE$ – CHARACTER(1)Input

On entry: indicates how

Q

Q^{H}

P

P^{H}

is to be applied to

C

$SIDE ='L'$: $Q$ or $Q^{H}$ or $P$ or $P^{H}$ is applied to $C$ from the left.
$SIDE ='R'$: $Q$ or $Q^{H}$ or $P$ or $P^{H}$ is applied to $C$ from the right.

Constraint:

SIDE ='L'

'R'

3: $TRANS$ – CHARACTER(1)Input

On entry: indicates whether

Q

P

Q^{H}

P^{H}

is to be applied to

C

$TRANS ='N'$: $Q$ or $P$ is applied to $C$ .
$TRANS ='C'$: $Q^{H}$ or $P^{H}$ is applied to $C$ .

Constraint:

TRANS ='N'

'C'

4: $M$ – INTEGERInput

On entry:

m

, the number of rows of the matrix

C

Constraint:

M \geq 0

5: $N$ – INTEGERInput

On entry:

n

, the number of columns of the matrix

C

Constraint:

N \geq 0

6: $K$ – 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

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

Note: the second dimension of the array A must be at least

\max (1, \min (r, K))

VECT ='Q'

and at least

\max (1, r)

VECT ='P'

On entry: details of the vectors which define the elementary reflectors, as returned by F08KSF (ZGEBRD).

8: $LDA$ – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F08KUF (ZUNMBR) is called.

Constraints:

if $VECT ='Q'$ , $LDA \geq \max (1, r)$ ;
if $VECT ='P'$ , $LDA \geq \max (1, \min (r, K))$ .

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

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

\max (1, \min (r, K))

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'

10: $C (LDC *)$ – COMPLEX (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array C must be at least

\max (1, N)

On entry: the matrix

C

On exit: C is overwritten by

Q C

Q^{H} C

C Q

C^{H} Q

P C

P^{H} C

C P

C^{H} P

as specified by VECT, SIDE and TRANS.

11: $LDC$ – INTEGERInput

On entry: the first dimension of the array C as declared in the (sub)program from which F08KUF (ZUNMBR) is called.

Constraint:

LDC \geq \max (1, M)

12: $WORK (\max (1, LWORK))$ – 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.

13: $LWORK$ – INTEGERInput

On entry: the dimension of the array WORK as declared in the (sub)program from which F08KUF (ZUNMBR) 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 N \times nb

SIDE ='L'

and at least

M \times nb

SIDE ='R'

, where

nb

is the optimal block size.

Constraints:

if $SIDE ='L'$ , $LWORK \geq \max (1, N)$ or $LWORK = - 1$ ;
if $SIDE ='R'$ , $LWORK \geq \max (1, M)$ or $LWORK = - 1$ .

14: $INFO$ – INTEGEROutput

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 result differs from the exact result by a matrix

E

such that

{‖E‖}_{2} = O (ε) {‖C‖}_{2},

where

ε

is the machine precision.

8 Parallelism and Performance

F08KUF (ZUNMBR) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

F08KUF (ZUNMBR) 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 real floating-point operations is approximately

if $SIDE ='L'$ and $m \geq k$ , $8 n k (2 m - k)$ ;
if $SIDE ='R'$ and $n \geq k$ , $8 m k (2 n - k)$ ;
if $SIDE ='L'$ and $m < k$ , $8 m^{2} n$ ;
if $SIDE ='R'$ and $n < k$ , $8 m n^{2}$ ,

where

k

is the value of the parameter K.

The real analogue of this routine is F08KGF (DORMBR).

10 Example

For this routine two examples are presented. Both illustrate how the reduction to bidiagonal form of a matrix

A

may be preceded by a

Q R

L Q

factorization of

A

In the first example,

m > n

, and

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}) .

The routine first performs a

Q R

factorization of

A

A = Q_{a} R

and then reduces the factor

R

to bidiagonal form

B

R = Q_{b} B P^{H}

. Finally it forms

Q_{a}

and calls F08KUF (ZUNMBR) to form

Q = Q_{a} Q_{b}

In the second example,

m < n

, 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}) .

The routine first performs an

L Q

factorization of

A

A = L P_{a}^{H}

and then reduces the factor

L

to bidiagonal form

B

L = Q B P_{b}^{H}

. Finally it forms

P_{b}^{H}

and calls F08KUF (ZUNMBR) to form

P^{H} = P_{b}^{H} P_{a}^{H}

NAG Library Routine DocumentF08KUF (ZUNMBR)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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 Routine Document

F08KUF (ZUNMBR)