NAG Library Routine Document

F06FSF

▸▿ 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

F06FSF generates a real elementary reflection in the LINPACK (as opposed to NAG) style.

2 Specification

SUBROUTINE F06FSF (

N, ALPHA, X, INCX, TOL, Z1)

INTEGER	N, INCX
REAL (KIND=nag_wp)	ALPHA, X(*), TOL, Z1

3 Description

F06FSF generates details of a real 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 orthogonal,

α

and

β

are real scalars, and

x

is an

n

-element real vector.

P

is given in the form

P = I - \frac{1}{ζ} (\begin{array}{r} ζ \\ z \end{array}) (\begin{array}{r} ζ & z^{T} \end{array}),

where

z

is an

n

-element real vector and

ζ

is a real scalar. (This form is compatible with that used by LINPACK.)

If the elements of

x

are all zero, or if the elements of

x

are all less than

tol \times |α|

in absolute value, then

ζ

is set to

0

and

P

can be taken to be the unit matrix. Otherwise

ζ

always lies in the range

(1, 2)

4 References

None.

5 Parameters

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

On exit: the scalar $β$ .
3: $X (*)$ – REAL (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 real 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 $tol$ .
If TOL is not in the range $(0, 1)$ , then the value $0$ is used for $tol$ .
6: $Z1$ – REAL (KIND=nag_wp)Output: On exit: the scalar $ζ$ .

6 Error Indicators and Warnings

None.

7 Accuracy

Not applicable.

8 Parallelism and Performance

F06FSF is not threaded by NAG in any implementation.

F06FSF 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.