NAG Library Routine Document

void	f06qrf_ (const char side, const Integer n, const Integer k1, const Integer k2, double c[], double s[], double a[], const Integer *lda, const Charlen length_side)

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}) .

4

References

None.

5

Arguments

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 values

k_{1}

and

k_{2}

k1 < 1

k2 \leq k1

k2 > n

, an immediate return is effected.

5: $c (k2 - 1)$ – 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

6: $s (k2 - 1)$ – 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

7: $a (lda *)$ – 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

8: $lda$ – 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)

6

Error Indicators and Warnings

None.

7

Accuracy

Not applicable.

8

Parallelism and Performance

f06qrf is not threaded in any implementation.

9

Further Comments

None.

10

Example

None.

NAG Library Manual, Mark 26

NAG AD Library Manual, Mark 26

NAG C Library Manual, Mark 26

F06 (blas) Chapter Contents

F06 (blas) Chapter Introduction

NAG Library Routine Document

f06qrf (duhqr)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

6

Error Indicators and Warnings

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example