F06QRF : NAG Library, Mark 25

NAG Library Routine Document

F06QRF

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

F06QRF performs a

Q R

R Q

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

2 Specification

3 Description

F06QRF transforms an

n

n

real upper Hessenberg matrix

H

to upper triangular form

R

by applying an orthogonal matrix

P

from the left or the right.

H

is assumed to have nonzero subdiagonal elements

h_{k + 1, k}

, for

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

, only.

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

SIDE ='R'

, the rotations are applied from the right:

H P^{T} = R,

where

P = P_{k_{1}} P_{k_{1} + 1} \dots P_{k_{2} - 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} c_{k} & s_{k} \\ - s_{k} & c_{k} \end{array}) .

5 Parameters

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'

N

– INTEGERInput

On entry:

n

, the order of the matrix

H

Constraint:

N \geq 0

K1

– INTEGERInput

K2

– INTEGERInput

On entry: the values

k_{1}

and

k_{2}

K1 < 1

K2 \leq K1

K2 > N

, an immediate return is effected.

C (K2 - 1)

– REAL (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

S (K2 - 1)

– REAL (KIND=nag_wp) arrayInput/Output

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

A (LDA *)

– REAL (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

LDA

– INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F06QRF is called.

Constraint:

LDA \geq \max (1, N)

NAG Library Routine Document

F06QRF

▸▿ 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

NAG Library Routine DocumentF06QRF

▸▿ 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

NAG Library Routine Document

F06QRF