NAG Library Routine Document

F06TRF

▸▿ 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

1 Purpose

F06TRF performs a

Q R

R Q

factorization (as a sequence of plane rotations) of a complex upper Hessenberg matrix.

2 Specification

SUBROUTINE F06TRF (

SIDE, N, K1, K2, C, S, A, LDA)

INTEGER	N, K1, K2, LDA
REAL (KIND=nag_wp)	S(*)
COMPLEX (KIND=nag_wp)	C(K2), A(LDA,*)
CHARACTER(1)	SIDE

3 Description

F06TRF transforms an

n

n

complex upper Hessenberg matrix

H

to upper triangular form

R

by applying a unitary matrix

P

from the left or the right.

H

is assumed to have real nonzero subdiagonal elements

h_{k + 1, k}

, for

k = k_{1}, \dots, k_{2} - 1

, only;

R

has real diagonal elements.

P

is formed as a sequence of plane rotations in planes

k_{1}

k_{2}

SIDE ='L'

, the rotations are applied from the left:

P H = R,

where

P = D P_{k_{2} - 1} \dots P_{k_{1} + 1} P_{k_{1}}

and

D = diag (1, \dots, 1, d_{k_{2}}, 1, \dots, 1)

with

|d_{k_{2}}| = 1

SIDE ='R'

, the rotations are applied from the right:

H P^{H} = R,

where

P = D P_{k_{1}} P_{k_{1} + 1} \dots P_{k_{2} - 1}

and

D = diag (1, \dots, 1, d_{k_{1}}, 1, \dots, 1)

with

|d_{k_{1}}| = 1

In either case,

P_{k}

is a rotation in the

(k, k + 1)

plane, chosen to annihilate

h_{k + 1, k}

The

2

2

plane rotation part of

P_{k}

has the form

(\begin{array}{r} {\bar{c}}_{k} & s_{k} \\ - s_{k} & c_{k} \end{array})

with

s_{k}

real.

4 References

None.

5 Parameters

1: $SIDE$ – CHARACTER(1)Input

On entry: specifies whether

H

is operated on from the left or the right.

$SIDE ='L'$: $H$ is pre-multiplied from the left.
$SIDE ='R'$: $H$ is post-multiplied from the right.

Constraint:

SIDE ='L'

'R'

2: $N$ – INTEGERInput

On entry:

n

, the order of the matrix

H

Constraint:

N \geq 0

3: $K1$ – INTEGERInput

4: $K2$ – INTEGERInput

On entry: the dimension of the array C as declared in the (sub)program from which F06TRF is called. The values

k_{1}

and

k_{2}

K1 < 1

K2 \leq K1

K2 > N

, an immediate return is effected.

5: $C (K2)$ – COMPLEX (KIND=nag_wp) arrayOutput

On exit:

C (k)

holds

c_{k}

, the cosine of the rotation

P_{k}

, for

k = k_{1}, \dots, k_{2} - 1

;

C (k_{2})

holds

d_{k_{2}}

, the

k_{2}

th diagonal element of

D

, if

SIDE ='L'

, or

d_{k_{1}}

, the

k_{1}

th diagonal element of

D

, if

SIDE ='R'

6: $S (*)$ – REAL (KIND=nag_wp) arrayInput/Output

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

K2 - K1

On entry: the nonzero subdiagonal elements of

H

S (k)

must hold

h_{k + 1, k}

, for

k = k_{1}, \dots, k_{2} - 1

On exit:

S (k)

holds

s_{k}

, the sine of the rotation

P_{k}

, for

k = k_{1}, \dots, k_{2} - 1

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

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

N

On entry: the upper triangular part of the

n

n

upper Hessenberg matrix

H

On exit: the upper triangular matrix

R

. The imaginary parts of the diagonal elements are set to zero.

8: $LDA$ – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F06TRF is called.

Constraint:

LDA \geq \max (1, N)

6 Error Indicators and Warnings

None.

7 Accuracy

Not applicable.

8 Parallelism and Performance

F06TRF is not threaded by NAG in any implementation.

F06TRF 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.