F06HQF : NAG Library, Mark 24

NAG Library Routine Document

F06HQF

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

F06HQF generates a sequence of complex plane rotations.

2 Specification

3 Description

F06HQF generates the parameters of a complex unitary matrix

P

, of order

n + 1

, chosen so as to set to zero the elements of a supplied

n

-element complex vector

x

PIVOT ='F'

and

DIRECT ='F'

, or if

PIVOT ='V'

and

DIRECT ='B'

P (\begin{array}{r} α \\ x \end{array}) = (\begin{array}{r} β \\ 0 \end{array});

PIVOT ='F'

and

DIRECT ='B'

, or if

PIVOT ='V'

and

DIRECT ='F'

P (\begin{array}{r} x \\ α \end{array}) = (\begin{array}{r} 0 \\ β \end{array}) .

Here

α

and

β

are complex scalars.

P

is represented as a sequence of

n

plane rotations

P_{k}

, as specified by PIVOT and DIRECT;

P_{k}

is chosen to annihilate

x_{k}

, and its

2

2

plane rotation part has the form

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

with

c_{k}

real. The tangent of the rotation,

t_{k}

, is overwritten on

x_{k}

5 Parameters

1: PIVOT – CHARACTER(1)Input

On entry: specifies the plane rotated by

P_{k}

$PIVOT ='V'$ (variable pivot): $P_{k}$ rotates the $(k, k + 1)$ plane.
$PIVOT ='F'$ (fixed pivot): $P_{k}$ rotates the $(1, k + 1)$ plane if $DIRECT ='F'$ , or the $(k, n + 1)$ plane if $DIRECT ='B'$ .

Constraint:

PIVOT ='V'

'F'

2: DIRECT – CHARACTER(1)Input

On entry: specifies the sequence direction.

$DIRECT ='F'$ (forward sequence): $P = P_{n} \dots P_{2} P_{1}$ .
$DIRECT ='B'$ (backward sequence): $P = P_{1} P_{2} \dots P_{n}$ .

Constraint:

DIRECT ='F'

'B'

3: N – INTEGERInput

On entry:

n

, the number of elements in

x

4: ALPHA – COMPLEX (KIND=nag_wp)Input/Output

On entry: the scalar

α

On exit: the scalar

β

5: X(

*

) – COMPLEX (KIND=nag_wp) arrayInput/Output

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

\max (1, 1 + (N - 1) \times INCX)

On entry: the

n

-element vector

x

x_{i}

must be stored in

X (1 + (i - 1) \times INCX)

, for

i = 1, 2, \dots, N

Intermediate elements of X are not referenced.

On exit: the referenced elements are overwritten by details of the plane rotations.

6: INCX – INTEGERInput

On entry: the increment in the subscripts of X between successive elements of

x

Constraint:

INCX > 0

7: C(N) – REAL (KIND=nag_wp) arrayOutput

On exit: the values

c_{k}

, the cosines of the rotations.

8: S(N) – COMPLEX (KIND=nag_wp) arrayOutput

On exit: the values

s_{k}

, the sines of the rotations.

NAG Library Routine Document

F06HQF

+− 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 DocumentF06HQF

+− 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

F06HQF