NAG Library Routine Document

F06FRF

+− Contents

1 Purpose

9 Example

F06FRF generates a real elementary reflection in the NAG (as opposed to LINPACK) style.

SUBROUTINE F06FRF (

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

F06FRF 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 - (\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.

x

is such that

\max |x_{i}| \leq \max (tol, ε |α|)

where

ε

is the machine precision and

tol

is a user-supplied tolerance, then

ζ

is set to

0

, and

P

can be taken to be the unit matrix. Otherwise

1 \leq ζ \leq \sqrt{2}

None.

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 $tol$ .
6: ZETA – REAL (KIND=nag_wp)Output: On exit: the scalar $ζ$ .