F06HRF : NAG Library, Mark 24

NAG Library Routine Document

F06HRF

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

F06HRF generates a complex elementary reflection.

2 Specification

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
$P = (\begin{array}{r} θ & 0 \\ 0 & I \end{array}) .$

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

NAG Library Routine Document

F06HRF

+− 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 DocumentF06HRF

+− 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

F06HRF