F06QVF : NAG Library, Mark 24

NAG Library Routine Document

F06QVF

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

F06QVF transforms a real upper triangular matrix to an upper Hessenberg matrix by applying a given sequence of plane rotations.

2 Specification

3 Description

F06QVF transforms an

n

n

real upper triangular matrix

U

to an upper Hessenberg matrix

H

, by applying a given sequence of plane rotations from either the left or the right, in planes

k_{1}

k_{2}

;

H

has nonzero subdiagonal elements

h_{k + 1, k}

, for

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

only.

SIDE ='L'

, the rotations are applied from the left:

H = P U,

where

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

SIDE ='R'

, the rotations are applied from the right:

H = U P^{T},

where

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

In either case,

P_{k}

is a rotation in the

(k, k + 1)

plane.

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

1: SIDE – CHARACTER(1)Input

On entry: specifies whether

U

is operated on from the left or the right.

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

Constraint:

SIDE ='L'

'R'

2: N – INTEGERInput

On entry:

n

, the order of the matrices

U

and

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(

*

) – REAL (KIND=nag_wp) arrayInput

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

K2 - K1

On entry:

C (k)

must hold

c_{k}

, the cosine of the rotation

P_{k}

, for

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

6: S(

*

) – REAL (KIND=nag_wp) arrayInput/Output

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

K2 - K1

On entry:

S (k)

must hold

s_{k}

, the sine of the rotation

P_{k}

, for

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

On exit:

S (k)

holds

h_{k + 1, k}

, the subdiagonal element of

H

, 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

n

n

upper triangular matrix

U

On exit: the upper triangular part of the upper Hessenberg matrix

H

8: LDA – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F06QVF is called.

Constraint:

LDA \geq \max (1, N)

NAG Library Routine Document

F06QVF

+− 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 DocumentF06QVF

+− 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

F06QVF