2014-10-30 23:22:01 +08:00
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*> \brief \b DLASV2 computes the singular value decomposition of a 2-by-2 triangular matrix.
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*
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* =========== DOCUMENTATION ===========
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*
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2018-05-19 05:17:13 +08:00
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* Online html documentation available at
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* http://www.netlib.org/lapack/explore-html/
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2014-10-30 23:22:01 +08:00
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*
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*> \htmlonly
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2018-05-19 05:17:13 +08:00
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*> Download DLASV2 + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasv2.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/dlasv2.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/dlasv2.f">
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2014-10-30 23:22:01 +08:00
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*> [TXT]</a>
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2018-05-19 05:17:13 +08:00
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*> \endhtmlonly
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2014-10-30 23:22:01 +08:00
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*
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* Definition:
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* ===========
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*
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* SUBROUTINE DLASV2( F, G, H, SSMIN, SSMAX, SNR, CSR, SNL, CSL )
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2018-05-19 05:17:13 +08:00
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*
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2014-10-30 23:22:01 +08:00
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* .. Scalar Arguments ..
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* DOUBLE PRECISION CSL, CSR, F, G, H, SNL, SNR, SSMAX, SSMIN
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* ..
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2018-05-19 05:17:13 +08:00
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*
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2014-10-30 23:22:01 +08:00
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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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*> DLASV2 computes the singular value decomposition of a 2-by-2
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*> triangular matrix
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*> [ F G ]
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*> [ 0 H ].
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*> On return, abs(SSMAX) is the larger singular value, abs(SSMIN) is the
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*> smaller singular value, and (CSL,SNL) and (CSR,SNR) are the left and
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*> right singular vectors for abs(SSMAX), giving the decomposition
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*>
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*> [ CSL SNL ] [ F G ] [ CSR -SNR ] = [ SSMAX 0 ]
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*> [-SNL CSL ] [ 0 H ] [ SNR CSR ] [ 0 SSMIN ].
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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] F
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*> \verbatim
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*> F is DOUBLE PRECISION
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*> The (1,1) element of the 2-by-2 matrix.
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*> \endverbatim
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*>
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*> \param[in] G
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*> \verbatim
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*> G is DOUBLE PRECISION
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*> The (1,2) element of the 2-by-2 matrix.
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*> \endverbatim
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*>
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*> \param[in] H
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*> \verbatim
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*> H is DOUBLE PRECISION
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*> The (2,2) element of the 2-by-2 matrix.
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*> \endverbatim
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*>
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*> \param[out] SSMIN
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*> \verbatim
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*> SSMIN is DOUBLE PRECISION
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*> abs(SSMIN) is the smaller singular value.
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*> \endverbatim
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*>
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*> \param[out] SSMAX
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*> \verbatim
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*> SSMAX is DOUBLE PRECISION
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*> abs(SSMAX) is the larger singular value.
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*> \endverbatim
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*>
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*> \param[out] SNL
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*> \verbatim
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*> SNL is DOUBLE PRECISION
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*> \endverbatim
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*>
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*> \param[out] CSL
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*> \verbatim
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*> CSL is DOUBLE PRECISION
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*> The vector (CSL, SNL) is a unit left singular vector for the
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*> singular value abs(SSMAX).
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*> \endverbatim
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*>
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*> \param[out] SNR
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*> \verbatim
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*> SNR is DOUBLE PRECISION
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*> \endverbatim
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*>
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*> \param[out] CSR
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*> \verbatim
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*> CSR is DOUBLE PRECISION
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*> The vector (CSR, SNR) is a unit right singular vector for the
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*> singular value abs(SSMAX).
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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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2018-05-19 05:17:13 +08:00
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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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2014-10-30 23:22:01 +08:00
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*
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2018-05-19 05:17:13 +08:00
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*> \date December 2016
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2014-10-30 23:22:01 +08:00
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*
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2018-05-19 05:17:13 +08:00
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*> \ingroup OTHERauxiliary
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2014-10-30 23:22:01 +08:00
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*
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*> \par Further Details:
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* =====================
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*>
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*> \verbatim
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*>
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*> Any input parameter may be aliased with any output parameter.
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*>
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*> Barring over/underflow and assuming a guard digit in subtraction, all
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*> output quantities are correct to within a few units in the last
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*> place (ulps).
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*>
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*> In IEEE arithmetic, the code works correctly if one matrix element is
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*> infinite.
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*>
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*> Overflow will not occur unless the largest singular value itself
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*> overflows or is within a few ulps of overflow. (On machines with
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*> partial overflow, like the Cray, overflow may occur if the largest
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*> singular value is within a factor of 2 of overflow.)
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*>
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*> Underflow is harmless if underflow is gradual. Otherwise, results
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*> may correspond to a matrix modified by perturbations of size near
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*> the underflow threshold.
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*> \endverbatim
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*>
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* =====================================================================
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SUBROUTINE DLASV2( F, G, H, SSMIN, SSMAX, SNR, CSR, SNL, CSL )
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*
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2018-05-19 05:17:13 +08:00
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* -- LAPACK auxiliary routine (version 3.7.0) --
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2014-10-30 23:22:01 +08:00
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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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2018-05-19 05:17:13 +08:00
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* December 2016
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2014-10-30 23:22:01 +08:00
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*
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* .. Scalar Arguments ..
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DOUBLE PRECISION CSL, CSR, F, G, H, SNL, SNR, SSMAX, SSMIN
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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 ZERO
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PARAMETER ( ZERO = 0.0D0 )
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DOUBLE PRECISION HALF
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PARAMETER ( HALF = 0.5D0 )
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DOUBLE PRECISION ONE
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PARAMETER ( ONE = 1.0D0 )
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DOUBLE PRECISION TWO
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PARAMETER ( TWO = 2.0D0 )
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DOUBLE PRECISION FOUR
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PARAMETER ( FOUR = 4.0D0 )
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* ..
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* .. Local Scalars ..
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LOGICAL GASMAL, SWAP
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INTEGER PMAX
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DOUBLE PRECISION A, CLT, CRT, D, FA, FT, GA, GT, HA, HT, L, M,
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$ MM, R, S, SLT, SRT, T, TEMP, TSIGN, TT
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, SIGN, SQRT
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* ..
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* .. External Functions ..
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DOUBLE PRECISION DLAMCH
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EXTERNAL DLAMCH
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* ..
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* .. Executable Statements ..
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*
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FT = F
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FA = ABS( FT )
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HT = H
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HA = ABS( H )
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*
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* PMAX points to the maximum absolute element of matrix
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* PMAX = 1 if F largest in absolute values
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* PMAX = 2 if G largest in absolute values
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* PMAX = 3 if H largest in absolute values
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*
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PMAX = 1
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SWAP = ( HA.GT.FA )
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IF( SWAP ) THEN
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PMAX = 3
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TEMP = FT
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FT = HT
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HT = TEMP
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TEMP = FA
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FA = HA
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HA = TEMP
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*
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* Now FA .ge. HA
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*
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END IF
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GT = G
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GA = ABS( GT )
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IF( GA.EQ.ZERO ) THEN
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*
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* Diagonal matrix
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*
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SSMIN = HA
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SSMAX = FA
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CLT = ONE
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CRT = ONE
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SLT = ZERO
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SRT = ZERO
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ELSE
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GASMAL = .TRUE.
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IF( GA.GT.FA ) THEN
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PMAX = 2
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IF( ( FA / GA ).LT.DLAMCH( 'EPS' ) ) THEN
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*
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* Case of very large GA
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*
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GASMAL = .FALSE.
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SSMAX = GA
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IF( HA.GT.ONE ) THEN
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SSMIN = FA / ( GA / HA )
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ELSE
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SSMIN = ( FA / GA )*HA
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END IF
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CLT = ONE
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SLT = HT / GT
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SRT = ONE
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CRT = FT / GT
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END IF
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END IF
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IF( GASMAL ) THEN
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*
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* Normal case
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*
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D = FA - HA
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IF( D.EQ.FA ) THEN
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*
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* Copes with infinite F or H
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*
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L = ONE
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ELSE
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L = D / FA
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END IF
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*
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* Note that 0 .le. L .le. 1
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*
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M = GT / FT
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*
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* Note that abs(M) .le. 1/macheps
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*
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T = TWO - L
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*
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* Note that T .ge. 1
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*
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MM = M*M
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TT = T*T
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S = SQRT( TT+MM )
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*
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* Note that 1 .le. S .le. 1 + 1/macheps
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*
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IF( L.EQ.ZERO ) THEN
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R = ABS( M )
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ELSE
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R = SQRT( L*L+MM )
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END IF
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*
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* Note that 0 .le. R .le. 1 + 1/macheps
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*
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A = HALF*( S+R )
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*
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* Note that 1 .le. A .le. 1 + abs(M)
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*
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SSMIN = HA / A
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SSMAX = FA*A
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IF( MM.EQ.ZERO ) THEN
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*
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* Note that M is very tiny
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*
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IF( L.EQ.ZERO ) THEN
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T = SIGN( TWO, FT )*SIGN( ONE, GT )
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ELSE
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T = GT / SIGN( D, FT ) + M / T
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END IF
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ELSE
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T = ( M / ( S+T )+M / ( R+L ) )*( ONE+A )
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END IF
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L = SQRT( T*T+FOUR )
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CRT = TWO / L
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SRT = T / L
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CLT = ( CRT+SRT*M ) / A
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SLT = ( HT / FT )*SRT / A
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END IF
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END IF
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IF( SWAP ) THEN
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CSL = SRT
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SNL = CRT
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CSR = SLT
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SNR = CLT
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ELSE
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CSL = CLT
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SNL = SLT
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CSR = CRT
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SNR = SRT
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END IF
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*
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* Correct signs of SSMAX and SSMIN
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*
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IF( PMAX.EQ.1 )
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$ TSIGN = SIGN( ONE, CSR )*SIGN( ONE, CSL )*SIGN( ONE, F )
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IF( PMAX.EQ.2 )
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$ TSIGN = SIGN( ONE, SNR )*SIGN( ONE, CSL )*SIGN( ONE, G )
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IF( PMAX.EQ.3 )
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$ TSIGN = SIGN( ONE, SNR )*SIGN( ONE, SNL )*SIGN( ONE, H )
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SSMAX = SIGN( SSMAX, TSIGN )
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SSMIN = SIGN( SSMIN, TSIGN*SIGN( ONE, F )*SIGN( ONE, H ) )
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RETURN
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*
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* End of DLASV2
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*
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END
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