204 lines
5.3 KiB
Fortran
204 lines
5.3 KiB
Fortran
*> \brief \b ZLARFG generates an elementary reflector (Householder matrix).
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*
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* =========== DOCUMENTATION ===========
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*
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* Online html documentation available at
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* http://www.netlib.org/lapack/explore-html/
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*
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*> \htmlonly
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*> Download ZLARFG + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zlarfg.f">
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*> [TGZ]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zlarfg.f">
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*> [ZIP]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlarfg.f">
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*> [TXT]</a>
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*> \endhtmlonly
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*
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* Definition:
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* ===========
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*
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* SUBROUTINE ZLARFG( N, ALPHA, X, INCX, TAU )
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*
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* .. Scalar Arguments ..
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* INTEGER INCX, N
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* COMPLEX*16 ALPHA, TAU
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* ..
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* .. Array Arguments ..
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* COMPLEX*16 X( * )
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* ..
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*
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*
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*> \par Purpose:
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* =============
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*>
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*> \verbatim
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*>
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*> ZLARFG generates a complex elementary reflector H of order n, such
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*> that
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*>
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*> H**H * ( alpha ) = ( beta ), H**H * H = I.
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*> ( x ) ( 0 )
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*>
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*> where alpha and beta are scalars, with beta real, and x is an
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*> (n-1)-element complex vector. H is represented in the form
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*>
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*> H = I - tau * ( 1 ) * ( 1 v**H ) ,
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*> ( v )
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*>
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*> where tau is a complex scalar and v is a complex (n-1)-element
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*> vector. Note that H is not hermitian.
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*>
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*> If the elements of x are all zero and alpha is real, then tau = 0
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*> and H is taken to be the unit matrix.
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*>
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*> Otherwise 1 <= real(tau) <= 2 and abs(tau-1) <= 1 .
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*> \endverbatim
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*
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* Arguments:
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* ==========
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*
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*> \param[in] N
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*> \verbatim
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*> N is INTEGER
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*> The order of the elementary reflector.
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*> \endverbatim
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*>
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*> \param[in,out] ALPHA
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*> \verbatim
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*> ALPHA is COMPLEX*16
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*> On entry, the value alpha.
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*> On exit, it is overwritten with the value beta.
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*> \endverbatim
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*>
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*> \param[in,out] X
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*> \verbatim
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*> X is COMPLEX*16 array, dimension
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*> (1+(N-2)*abs(INCX))
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*> On entry, the vector x.
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*> On exit, it is overwritten with the vector v.
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*> \endverbatim
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*>
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*> \param[in] INCX
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*> \verbatim
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*> INCX is INTEGER
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*> The increment between elements of X. INCX > 0.
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*> \endverbatim
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*>
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*> \param[out] TAU
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*> \verbatim
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*> TAU is COMPLEX*16
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*> The value tau.
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*> \endverbatim
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*
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* Authors:
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* ========
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*
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*> \author Univ. of Tennessee
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*> \author Univ. of California Berkeley
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*> \author Univ. of Colorado Denver
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*> \author NAG Ltd.
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*
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*> \date November 2017
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*
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*> \ingroup complex16OTHERauxiliary
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*
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* =====================================================================
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SUBROUTINE ZLARFG( N, ALPHA, X, INCX, TAU )
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*
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* -- LAPACK auxiliary routine (version 3.8.0) --
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* -- LAPACK is a software package provided by Univ. of Tennessee, --
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* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
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* November 2017
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*
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* .. Scalar Arguments ..
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INTEGER INCX, N
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COMPLEX*16 ALPHA, TAU
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* ..
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* .. Array Arguments ..
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COMPLEX*16 X( * )
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* ..
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*
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* =====================================================================
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*
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* .. Parameters ..
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DOUBLE PRECISION ONE, ZERO
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PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
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* ..
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* .. Local Scalars ..
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INTEGER J, KNT
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DOUBLE PRECISION ALPHI, ALPHR, BETA, RSAFMN, SAFMIN, XNORM
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* ..
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* .. External Functions ..
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DOUBLE PRECISION DLAMCH, DLAPY3, DZNRM2
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COMPLEX*16 ZLADIV
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EXTERNAL DLAMCH, DLAPY3, DZNRM2, ZLADIV
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, DBLE, DCMPLX, DIMAG, SIGN
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* ..
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* .. External Subroutines ..
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EXTERNAL ZDSCAL, ZSCAL
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* ..
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* .. Executable Statements ..
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*
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IF( N.LE.0 ) THEN
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TAU = ZERO
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RETURN
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END IF
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*
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XNORM = DZNRM2( N-1, X, INCX )
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ALPHR = DBLE( ALPHA )
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ALPHI = DIMAG( ALPHA )
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*
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IF( XNORM.EQ.ZERO .AND. ALPHI.EQ.ZERO ) THEN
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*
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* H = I
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*
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TAU = ZERO
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ELSE
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*
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* general case
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*
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BETA = -SIGN( DLAPY3( ALPHR, ALPHI, XNORM ), ALPHR )
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SAFMIN = DLAMCH( 'S' ) / DLAMCH( 'E' )
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RSAFMN = ONE / SAFMIN
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*
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KNT = 0
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IF( ABS( BETA ).LT.SAFMIN ) THEN
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*
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* XNORM, BETA may be inaccurate; scale X and recompute them
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*
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10 CONTINUE
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KNT = KNT + 1
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CALL ZDSCAL( N-1, RSAFMN, X, INCX )
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BETA = BETA*RSAFMN
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ALPHI = ALPHI*RSAFMN
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ALPHR = ALPHR*RSAFMN
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IF( (ABS( BETA ).LT.SAFMIN) .AND. (KNT .LT. 20) )
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$ GO TO 10
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*
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* New BETA is at most 1, at least SAFMIN
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*
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XNORM = DZNRM2( N-1, X, INCX )
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ALPHA = DCMPLX( ALPHR, ALPHI )
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BETA = -SIGN( DLAPY3( ALPHR, ALPHI, XNORM ), ALPHR )
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END IF
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TAU = DCMPLX( ( BETA-ALPHR ) / BETA, -ALPHI / BETA )
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ALPHA = ZLADIV( DCMPLX( ONE ), ALPHA-BETA )
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CALL ZSCAL( N-1, ALPHA, X, INCX )
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*
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* If ALPHA is subnormal, it may lose relative accuracy
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*
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DO 20 J = 1, KNT
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BETA = BETA*SAFMIN
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20 CONTINUE
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ALPHA = BETA
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END IF
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*
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RETURN
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*
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* End of ZLARFG
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*
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END
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