F06TRF : NAG Library, Mark 24

NAG Library Routine Document

F06TRF

Note: before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

1 Purpose

F06TRF performs a

Q R

R Q

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

2 Specification

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.

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)

NAG Library Routine Document

F06TRF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

NAG Library Routine DocumentF06TRF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

NAG Library Routine Document

F06TRF