LAPACK  3.9.1
LAPACK: Linear Algebra PACKage

◆ zlarfg()

subroutine zlarfg ( integer  N,
complex*16  ALPHA,
complex*16, dimension( * )  X,
integer  INCX,
complex*16  TAU 
)

ZLARFG generates an elementary reflector (Householder matrix).

Download ZLARFG + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 ZLARFG 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*16
          On entry, the value alpha.
          On exit, it is overwritten with the value beta.
[in,out]X
          X is COMPLEX*16 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*16
          The value tau.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Definition at line 105 of file zlarfg.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*16 ALPHA, TAU
114 * ..
115 * .. Array Arguments ..
116  COMPLEX*16 X( * )
117 * ..
118 *
119 * =====================================================================
120 *
121 * .. Parameters ..
122  DOUBLE PRECISION ONE, ZERO
123  parameter( one = 1.0d+0, zero = 0.0d+0 )
124 * ..
125 * .. Local Scalars ..
126  INTEGER J, KNT
127  DOUBLE PRECISION ALPHI, ALPHR, BETA, RSAFMN, SAFMIN, XNORM
128 * ..
129 * .. External Functions ..
130  DOUBLE PRECISION DLAMCH, DLAPY3, DZNRM2
131  COMPLEX*16 ZLADIV
132  EXTERNAL dlamch, dlapy3, dznrm2, zladiv
133 * ..
134 * .. Intrinsic Functions ..
135  INTRINSIC abs, dble, dcmplx, dimag, sign
136 * ..
137 * .. External Subroutines ..
138  EXTERNAL zdscal, zscal
139 * ..
140 * .. Executable Statements ..
141 *
142  IF( n.LE.0 ) THEN
143  tau = zero
144  RETURN
145  END IF
146 *
147  xnorm = dznrm2( n-1, x, incx )
148  alphr = dble( alpha )
149  alphi = dimag( 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( dlapy3( alphr, alphi, xnorm ), alphr )
161  safmin = dlamch( 'S' ) / dlamch( '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 zdscal( 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 = dznrm2( n-1, x, incx )
181  alpha = dcmplx( alphr, alphi )
182  beta = -sign( dlapy3( alphr, alphi, xnorm ), alphr )
183  END IF
184  tau = dcmplx( ( beta-alphr ) / beta, -alphi / beta )
185  alpha = zladiv( dcmplx( one ), alpha-beta )
186  CALL zscal( 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 ZLARFG
199 *
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
double precision function dlapy3(X, Y, Z)
DLAPY3 returns sqrt(x2+y2+z2).
Definition: dlapy3.f:68
subroutine zdscal(N, DA, ZX, INCX)
ZDSCAL
Definition: zdscal.f:78
subroutine zscal(N, ZA, ZX, INCX)
ZSCAL
Definition: zscal.f:78
complex *16 function zladiv(X, Y)
ZLADIV performs complex division in real arithmetic, avoiding unnecessary overflow.
Definition: zladiv.f:64
double precision function dznrm2(N, X, INCX)
DZNRM2
Definition: dznrm2.f:75
Here is the call graph for this function:
Here is the caller graph for this function: