LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ dlarfg()

 subroutine dlarfg ( integer N, double precision ALPHA, double precision, dimension( * ) X, integer INCX, double precision TAU )

DLARFG generates an elementary reflector (Householder matrix).

Purpose:
``` DLARFG generates a real elementary reflector H of order n, such
that

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

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

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

where tau is a real scalar and v is a real (n-1)-element
vector.

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

Otherwise  1 <= tau <= 2.```
Parameters
 [in] N ``` N is INTEGER The order of the elementary reflector.``` [in,out] ALPHA ``` ALPHA is DOUBLE PRECISION On entry, the value alpha. On exit, it is overwritten with the value beta.``` [in,out] X ``` X is DOUBLE PRECISION 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 DOUBLE PRECISION The value tau.```

Definition at line 105 of file dlarfg.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  DOUBLE PRECISION ALPHA, TAU
114 * ..
115 * .. Array Arguments ..
116  DOUBLE PRECISION 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 BETA, RSAFMN, SAFMIN, XNORM
128 * ..
129 * .. External Functions ..
130  DOUBLE PRECISION DLAMCH, DLAPY2, DNRM2
131  EXTERNAL dlamch, dlapy2, dnrm2
132 * ..
133 * .. Intrinsic Functions ..
134  INTRINSIC abs, sign
135 * ..
136 * .. External Subroutines ..
137  EXTERNAL dscal
138 * ..
139 * .. Executable Statements ..
140 *
141  IF( n.LE.1 ) THEN
142  tau = zero
143  RETURN
144  END IF
145 *
146  xnorm = dnrm2( n-1, x, incx )
147 *
148  IF( xnorm.EQ.zero ) THEN
149 *
150 * H = I
151 *
152  tau = zero
153  ELSE
154 *
155 * general case
156 *
157  beta = -sign( dlapy2( alpha, xnorm ), alpha )
158  safmin = dlamch( 'S' ) / dlamch( 'E' )
159  knt = 0
160  IF( abs( beta ).LT.safmin ) THEN
161 *
162 * XNORM, BETA may be inaccurate; scale X and recompute them
163 *
164  rsafmn = one / safmin
165  10 CONTINUE
166  knt = knt + 1
167  CALL dscal( n-1, rsafmn, x, incx )
168  beta = beta*rsafmn
169  alpha = alpha*rsafmn
170  IF( (abs( beta ).LT.safmin) .AND. (knt .LT. 20) )
171  \$ GO TO 10
172 *
173 * New BETA is at most 1, at least SAFMIN
174 *
175  xnorm = dnrm2( n-1, x, incx )
176  beta = -sign( dlapy2( alpha, xnorm ), alpha )
177  END IF
178  tau = ( beta-alpha ) / beta
179  CALL dscal( n-1, one / ( alpha-beta ), x, incx )
180 *
181 * If ALPHA is subnormal, it may lose relative accuracy
182 *
183  DO 20 j = 1, knt
184  beta = beta*safmin
185  20 CONTINUE
186  alpha = beta
187  END IF
188 *
189  RETURN
190 *
191 * End of DLARFG
192 *
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
double precision function dlapy2(X, Y)
DLAPY2 returns sqrt(x2+y2).
Definition: dlapy2.f:63
subroutine dscal(N, DA, DX, INCX)
DSCAL
Definition: dscal.f:79
real(wp) function dnrm2(n, x, incx)
DNRM2
Definition: dnrm2.f90:89
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