 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ slarfgp()

 subroutine slarfgp ( integer N, real ALPHA, real, dimension( * ) X, integer INCX, real TAU )

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

Purpose:
``` SLARFGP 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 REAL On entry, the value alpha. On exit, it is overwritten with the value beta.``` [in,out] X ``` X is REAL 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 REAL The value tau.```

Definition at line 103 of file slarfgp.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  REAL ALPHA, TAU
112 * ..
113 * .. Array Arguments ..
114  REAL X( * )
115 * ..
116 *
117 * =====================================================================
118 *
119 * .. Parameters ..
120  REAL TWO, ONE, ZERO
121  parameter( two = 2.0e+0, one = 1.0e+0, zero = 0.0e+0 )
122 * ..
123 * .. Local Scalars ..
124  INTEGER J, KNT
125  REAL BETA, BIGNUM, SAVEALPHA, SMLNUM, XNORM
126 * ..
127 * .. External Functions ..
128  REAL SLAMCH, SLAPY2, SNRM2
129  EXTERNAL slamch, slapy2, snrm2
130 * ..
131 * .. Intrinsic Functions ..
132  INTRINSIC abs, sign
133 * ..
134 * .. External Subroutines ..
135  EXTERNAL sscal
136 * ..
137 * .. Executable Statements ..
138 *
139  IF( n.LE.0 ) THEN
140  tau = zero
141  RETURN
142  END IF
143 *
144  xnorm = snrm2( 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( slapy2( alpha, xnorm ), alpha )
169  smlnum = slamch( 'S' ) / slamch( '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 sscal( 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 = snrm2( n-1, x, incx )
187  beta = sign( slapy2( 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 sscal( 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 SLARFGP
238 *
real function slapy2(X, Y)
SLAPY2 returns sqrt(x2+y2).
Definition: slapy2.f:63
subroutine sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:79
real(wp) function snrm2(n, x, incx)
SNRM2
Definition: snrm2.f90:89
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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