NAG Library Routine Document

F06HRF

▸▿ 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

1 Purpose

F06HRF generates a complex elementary reflection.

2 Specification

SUBROUTINE F06HRF (

N, ALPHA, X, INCX, TOL, THETA)

INTEGER	N, INCX
REAL (KIND=nag_wp)	TOL
COMPLEX (KIND=nag_wp)	ALPHA, X(*), THETA

3 Description

F06HRF generates details of a complex elementary reflection (Householder matrix),

P

, such that

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

where

P

is unitary,

α

is a complex scalar,

β

is a real scalar, and

x

is an

n

-element complex vector.

P

is given in the form

P = I - γ (\begin{array}{r} ζ \\ z \end{array}) (\begin{array}{r} ζ & z^{H} \end{array}),

where

z

is an

n

-element complex vector,

γ

is a complex scalar such that

Re (γ) = 1

, and

ζ

is a real scalar.

γ

and

ζ

are returned in a single complex value

θ = (ζ, Im (γ))

. Thus

ζ = Re (θ)

and

γ = (1, Im (θ))

x

is such that

\max (|Re (x_{i})|, |Im (x_{i})|) \leq \max (tol, ε \max (|Re (α)|, |Im (α)|)),

where

ε

is the machine precision and

tol

is a user-supplied tolerance, then:

either $θ$ is set to $0$ , in which case $P$ can be taken to be the unit matrix;
or $θ$ is set so that $Re (θ) \leq 0$ and $θ \neq 0$ , in which case
$P = (\begin{array}{r} θ & 0 \\ 0 & I \end{array}) .$

Otherwise

1 \leq Re (θ) \leq \sqrt{2}

4 References

None.

5 Parameters

1: $N$ – INTEGERInput: On entry: $n$ , the number of elements in $x$ and $z$ .
2: $ALPHA$ – COMPLEX (KIND=nag_wp)Input/Output: On entry: the scalar $α$ .

On exit: the scalar $β$ .
3: $X (*)$ – 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 complex elementary reflection.
4: $INCX$ – INTEGERInput: On entry: the increment in the subscripts of X between successive elements of $x$ .

Constraint: $INCX > 0$ .
5: $TOL$ – REAL (KIND=nag_wp)Input: On entry: the value $tol$ .
6: $THETA$ – COMPLEX (KIND=nag_wp)Output: On exit: the scalar $θ$ .

6 Error Indicators and Warnings

None.

7 Accuracy

Not applicable.

8 Parallelism and Performance

F06HRF is not threaded by NAG in any implementation.

F06HRF makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.