 LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ clarfg()

 subroutine clarfg ( integer N, complex ALPHA, complex, dimension( * ) X, integer INCX, complex TAU )

CLARFG generates an elementary reflector (Householder matrix).

Purpose:
``` CLARFG generates a complex elementary reflector H of order n, such
that

H**H * ( alpha ) = ( beta ),   H**H * H = I.
(   x   )   (   0  )

where alpha and beta are scalars, with beta real, and x is an
(n-1)-element complex vector. H is represented in the form

H = I - tau * ( 1 ) * ( 1 v**H ) ,
( v )

where tau is a complex scalar and v is a complex (n-1)-element
vector. Note that H is not hermitian.

If the elements of x are all zero and alpha is real, then tau = 0
and H is taken to be the unit matrix.

Otherwise  1 <= real(tau) <= 2  and  abs(tau-1) <= 1 .```
Parameters
 [in] N ``` N is INTEGER The order of the elementary reflector.``` [in,out] ALPHA ``` ALPHA is COMPLEX On entry, the value alpha. On exit, it is overwritten with the value beta.``` [in,out] X ``` X is COMPLEX array, dimension (1+(N-2)*abs(INCX)) On entry, the vector x. On exit, it is overwritten with the vector v.``` [in] INCX ``` INCX is INTEGER The increment between elements of X. INCX > 0.``` [out] TAU ``` TAU is COMPLEX The value tau.```

Definition at line 105 of file clarfg.f.

106 *
107 * -- LAPACK auxiliary routine --
108 * -- LAPACK is a software package provided by Univ. of Tennessee, --
109 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
110 *
111 * .. Scalar Arguments ..
112  INTEGER INCX, N
113  COMPLEX ALPHA, TAU
114 * ..
115 * .. Array Arguments ..
116  COMPLEX X( * )
117 * ..
118 *
119 * =====================================================================
120 *
121 * .. Parameters ..
122  REAL ONE, ZERO
123  parameter( one = 1.0e+0, zero = 0.0e+0 )
124 * ..
125 * .. Local Scalars ..
126  INTEGER J, KNT
127  REAL ALPHI, ALPHR, BETA, RSAFMN, SAFMIN, XNORM
128 * ..
129 * .. External Functions ..
130  REAL SCNRM2, SLAMCH, SLAPY3
132  EXTERNAL scnrm2, slamch, slapy3, cladiv
133 * ..
134 * .. Intrinsic Functions ..
135  INTRINSIC abs, aimag, cmplx, real, sign
136 * ..
137 * .. External Subroutines ..
138  EXTERNAL cscal, csscal
139 * ..
140 * .. Executable Statements ..
141 *
142  IF( n.LE.0 ) THEN
143  tau = zero
144  RETURN
145  END IF
146 *
147  xnorm = scnrm2( n-1, x, incx )
148  alphr = real( alpha )
149  alphi = aimag( alpha )
150 *
151  IF( xnorm.EQ.zero .AND. alphi.EQ.zero ) THEN
152 *
153 * H = I
154 *
155  tau = zero
156  ELSE
157 *
158 * general case
159 *
160  beta = -sign( slapy3( alphr, alphi, xnorm ), alphr )
161  safmin = slamch( 'S' ) / slamch( 'E' )
162  rsafmn = one / safmin
163 *
164  knt = 0
165  IF( abs( beta ).LT.safmin ) THEN
166 *
167 * XNORM, BETA may be inaccurate; scale X and recompute them
168 *
169  10 CONTINUE
170  knt = knt + 1
171  CALL csscal( n-1, rsafmn, x, incx )
172  beta = beta*rsafmn
173  alphi = alphi*rsafmn
174  alphr = alphr*rsafmn
175  IF( (abs( beta ).LT.safmin) .AND. (knt .LT. 20) )
176  \$ GO TO 10
177 *
178 * New BETA is at most 1, at least SAFMIN
179 *
180  xnorm = scnrm2( n-1, x, incx )
181  alpha = cmplx( alphr, alphi )
182  beta = -sign( slapy3( alphr, alphi, xnorm ), alphr )
183  END IF
184  tau = cmplx( ( beta-alphr ) / beta, -alphi / beta )
185  alpha = cladiv( cmplx( one ), alpha-beta )
186  CALL cscal( n-1, alpha, x, incx )
187 *
188 * If ALPHA is subnormal, it may lose relative accuracy
189 *
190  DO 20 j = 1, knt
191  beta = beta*safmin
192  20 CONTINUE
193  alpha = beta
194  END IF
195 *
196  RETURN
197 *
198 * End of CLARFG
199 *
real function slapy3(X, Y, Z)
SLAPY3 returns sqrt(x2+y2+z2).
Definition: slapy3.f:68
subroutine csscal(N, SA, CX, INCX)
CSSCAL
Definition: csscal.f:78
subroutine cscal(N, CA, CX, INCX)
CSCAL
Definition: cscal.f:78