2df1e3372d
Inf values in the input vector will survive rescaling, causing an infinite loop. The value of 1000 is arbitrarily chosen as a large but finite value with the intention to never interfere with regular calculations.
273 lines
7.7 KiB
Fortran
273 lines
7.7 KiB
Fortran
*> \brief \b CLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.
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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 CLARFGP + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/clarfgp.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/clarfgp.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/clarfgp.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 CLARFGP( 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 ALPHA, TAU
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* ..
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* .. Array Arguments ..
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* COMPLEX 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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*> CLARFGP 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, beta is real and non-negative, and
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*> x is an (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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*> \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
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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 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
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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 December 2016
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*
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*> \ingroup complexOTHERauxiliary
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*
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* =====================================================================
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SUBROUTINE CLARFGP( N, ALPHA, X, INCX, TAU )
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*
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* -- LAPACK auxiliary routine (version 3.7.0) --
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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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* December 2016
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*
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* .. Scalar Arguments ..
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INTEGER INCX, N
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COMPLEX ALPHA, TAU
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* ..
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* .. Array Arguments ..
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COMPLEX 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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REAL TWO, ONE, ZERO
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PARAMETER ( TWO = 2.0E+0, ONE = 1.0E+0, ZERO = 0.0E+0 )
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* ..
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* .. Local Scalars ..
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INTEGER J, KNT
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REAL ALPHI, ALPHR, BETA, BIGNUM, SMLNUM, XNORM
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COMPLEX SAVEALPHA
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* ..
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* .. External Functions ..
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REAL SCNRM2, SLAMCH, SLAPY3, SLAPY2
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COMPLEX CLADIV
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EXTERNAL SCNRM2, SLAMCH, SLAPY3, SLAPY2, CLADIV
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, AIMAG, CMPLX, REAL, SIGN
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* ..
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* .. External Subroutines ..
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EXTERNAL CSCAL, CSSCAL
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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 = SCNRM2( N-1, X, INCX )
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ALPHR = REAL( ALPHA )
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ALPHI = AIMAG( ALPHA )
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*
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IF( XNORM.EQ.ZERO ) THEN
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*
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* H = [1-alpha/abs(alpha) 0; 0 I], sign chosen so ALPHA >= 0.
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*
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IF( ALPHI.EQ.ZERO ) THEN
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IF( ALPHR.GE.ZERO ) THEN
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* When TAU.eq.ZERO, the vector is special-cased to be
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* all zeros in the application routines. We do not need
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* to clear it.
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TAU = ZERO
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ELSE
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* However, the application routines rely on explicit
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* zero checks when TAU.ne.ZERO, and we must clear X.
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TAU = TWO
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DO J = 1, N-1
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X( 1 + (J-1)*INCX ) = ZERO
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END DO
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ALPHA = -ALPHA
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END IF
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ELSE
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* Only "reflecting" the diagonal entry to be real and non-negative.
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XNORM = SLAPY2( ALPHR, ALPHI )
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TAU = CMPLX( ONE - ALPHR / XNORM, -ALPHI / XNORM )
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DO J = 1, N-1
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X( 1 + (J-1)*INCX ) = ZERO
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END DO
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ALPHA = XNORM
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END IF
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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( SLAPY3( ALPHR, ALPHI, XNORM ), ALPHR )
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SMLNUM = SLAMCH( 'S' ) / SLAMCH( 'E' )
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BIGNUM = ONE / SMLNUM
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*
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KNT = 0
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IF( ABS( BETA ).LT.SMLNUM ) 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 CSSCAL( N-1, BIGNUM, X, INCX )
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BETA = BETA*BIGNUM
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ALPHI = ALPHI*BIGNUM
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ALPHR = ALPHR*BIGNUM
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IF( ABS( BETA ).LT.SMLNUM .AND. KNT .LT. 1000 )
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$ GO TO 10
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*
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* New BETA is at most 1, at least SMLNUM
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*
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XNORM = SCNRM2( N-1, X, INCX )
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ALPHA = CMPLX( ALPHR, ALPHI )
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BETA = SIGN( SLAPY3( ALPHR, ALPHI, XNORM ), ALPHR )
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END IF
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SAVEALPHA = ALPHA
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ALPHA = ALPHA + BETA
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IF( BETA.LT.ZERO ) THEN
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BETA = -BETA
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TAU = -ALPHA / BETA
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ELSE
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ALPHR = ALPHI * (ALPHI/REAL( ALPHA ))
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ALPHR = ALPHR + XNORM * (XNORM/REAL( ALPHA ))
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TAU = CMPLX( ALPHR/BETA, -ALPHI/BETA )
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ALPHA = CMPLX( -ALPHR, ALPHI )
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END IF
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ALPHA = CLADIV( CMPLX( ONE ), ALPHA )
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*
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IF ( ABS(TAU).LE.SMLNUM ) THEN
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*
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* In the case where the computed TAU ends up being a denormalized number,
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* it loses relative accuracy. This is a BIG problem. Solution: flush TAU
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* to ZERO (or TWO or whatever makes a nonnegative real number for BETA).
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*
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* (Bug report provided by Pat Quillen from MathWorks on Jul 29, 2009.)
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* (Thanks Pat. Thanks MathWorks.)
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*
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ALPHR = REAL( SAVEALPHA )
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ALPHI = AIMAG( SAVEALPHA )
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IF( ALPHI.EQ.ZERO ) THEN
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IF( ALPHR.GE.ZERO ) THEN
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TAU = ZERO
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ELSE
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TAU = TWO
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DO J = 1, N-1
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X( 1 + (J-1)*INCX ) = ZERO
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END DO
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BETA = -SAVEALPHA
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END IF
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ELSE
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XNORM = SLAPY2( ALPHR, ALPHI )
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TAU = CMPLX( ONE - ALPHR / XNORM, -ALPHI / XNORM )
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DO J = 1, N-1
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X( 1 + (J-1)*INCX ) = ZERO
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END DO
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BETA = XNORM
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END IF
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*
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ELSE
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*
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* This is the general case.
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*
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CALL CSCAL( N-1, ALPHA, X, INCX )
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*
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END IF
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*
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* If BETA 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*SMLNUM
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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 CLARFGP
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*
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END
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