 LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ dlarfgp()

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

DLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.

Purpose:
``` DLARFGP 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, beta is non-negative, 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.```
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 103 of file dlarfgp.f.

104*
105* -- LAPACK auxiliary routine --
106* -- LAPACK is a software package provided by Univ. of Tennessee, --
107* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
108*
109* .. Scalar Arguments ..
110 INTEGER INCX, N
111 DOUBLE PRECISION ALPHA, TAU
112* ..
113* .. Array Arguments ..
114 DOUBLE PRECISION X( * )
115* ..
116*
117* =====================================================================
118*
119* .. Parameters ..
120 DOUBLE PRECISION TWO, ONE, ZERO
121 parameter( two = 2.0d+0, one = 1.0d+0, zero = 0.0d+0 )
122* ..
123* .. Local Scalars ..
124 INTEGER J, KNT
125 DOUBLE PRECISION BETA, BIGNUM, SAVEALPHA, SMLNUM, XNORM
126* ..
127* .. External Functions ..
128 DOUBLE PRECISION DLAMCH, DLAPY2, DNRM2
129 EXTERNAL dlamch, dlapy2, dnrm2
130* ..
131* .. Intrinsic Functions ..
132 INTRINSIC abs, sign
133* ..
134* .. External Subroutines ..
135 EXTERNAL dscal
136* ..
137* .. Executable Statements ..
138*
139 IF( n.LE.0 ) THEN
140 tau = zero
141 RETURN
142 END IF
143*
144 xnorm = dnrm2( n-1, x, incx )
145*
146 IF( xnorm.EQ.zero ) THEN
147*
148* H = [+/-1, 0; I], sign chosen so ALPHA >= 0
149*
150 IF( alpha.GE.zero ) THEN
151* When TAU.eq.ZERO, the vector is special-cased to be
152* all zeros in the application routines. We do not need
153* to clear it.
154 tau = zero
155 ELSE
156* However, the application routines rely on explicit
157* zero checks when TAU.ne.ZERO, and we must clear X.
158 tau = two
159 DO j = 1, n-1
160 x( 1 + (j-1)*incx ) = 0
161 END DO
162 alpha = -alpha
163 END IF
164 ELSE
165*
166* general case
167*
168 beta = sign( dlapy2( alpha, xnorm ), alpha )
169 smlnum = dlamch( 'S' ) / dlamch( 'E' )
170 knt = 0
171 IF( abs( beta ).LT.smlnum ) THEN
172*
173* XNORM, BETA may be inaccurate; scale X and recompute them
174*
175 bignum = one / smlnum
176 10 CONTINUE
177 knt = knt + 1
178 CALL dscal( n-1, bignum, x, incx )
179 beta = beta*bignum
180 alpha = alpha*bignum
181 IF( (abs( beta ).LT.smlnum) .AND. (knt .LT. 20) )
182 \$ GO TO 10
183*
184* New BETA is at most 1, at least SMLNUM
185*
186 xnorm = dnrm2( n-1, x, incx )
187 beta = sign( dlapy2( alpha, xnorm ), alpha )
188 END IF
189 savealpha = alpha
190 alpha = alpha + beta
191 IF( beta.LT.zero ) THEN
192 beta = -beta
193 tau = -alpha / beta
194 ELSE
195 alpha = xnorm * (xnorm/alpha)
196 tau = alpha / beta
197 alpha = -alpha
198 END IF
199*
200 IF ( abs(tau).LE.smlnum ) THEN
201*
202* In the case where the computed TAU ends up being a denormalized number,
203* it loses relative accuracy. This is a BIG problem. Solution: flush TAU
204* to ZERO. This explains the next IF statement.
205*
206* (Bug report provided by Pat Quillen from MathWorks on Jul 29, 2009.)
207* (Thanks Pat. Thanks MathWorks.)
208*
209 IF( savealpha.GE.zero ) THEN
210 tau = zero
211 ELSE
212 tau = two
213 DO j = 1, n-1
214 x( 1 + (j-1)*incx ) = 0
215 END DO
216 beta = -savealpha
217 END IF
218*
219 ELSE
220*
221* This is the general case.
222*
223 CALL dscal( n-1, one / alpha, x, incx )
224*
225 END IF
226*
227* If BETA is subnormal, it may lose relative accuracy
228*
229 DO 20 j = 1, knt
230 beta = beta*smlnum
231 20 CONTINUE
232 alpha = beta
233 END IF
234*
235 RETURN
236*
237* End of DLARFGP
238*
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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