NAG FL Interface
f06hrf (zhousg)

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1 Purpose

f06hrf generates a complex elementary reflection.

2 Specification

Fortran Interface
Subroutine f06hrf ( n, alpha, x, incx, tol, theta)
Integer, Intent (In) :: n, incx
Real (Kind=nag_wp), Intent (In) :: tol
Complex (Kind=nag_wp), Intent (Inout) :: alpha, x(*)
Complex (Kind=nag_wp), Intent (Out) :: theta
C Header Interface
#include <nag.h>
void  f06hrf_ (const Integer *n, Complex *alpha, Complex x[], const Integer *incx, const double *tol, Complex *theta)
The routine may be called by the names f06hrf or nagf_blas_zhousg.

3 Description

f06hrf generates details of a complex elementary reflection (Householder matrix), P, such that
P ( α x )=( β 0 )  
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 ( ζ z ) ( ζ zH ) ,  
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)).
If x is such that
where ε is the machine precision and tol is a user-supplied tolerance, then:
Otherwise 1Re(θ)2.

4 References


5 Arguments

1: n Integer Input
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) array Input/Output
Note: the dimension of the array x must be at least max(1, 1+(n-1) ×incx ) .
On entry: the n-element vector x. xi must be stored in x(1+(i-1)×incx), for i=1,2,,n.
Intermediate elements of x are not referenced.
On exit: the referenced elements are overwritten by details of the complex elementary reflection.
4: incx Integer Input
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


7 Accuracy

Not applicable.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
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.

9 Further Comments


10 Example