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*> \brief \b SLAS2 computes singular values of a 2-by-2 triangular matrix.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download SLAS2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/slas2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/slas2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slas2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE SLAS2( F, G, H, SSMIN, SSMAX )
*
* .. Scalar Arguments ..
* REAL F, G, H, SSMAX, SSMIN
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> SLAS2 computes the singular values of the 2-by-2 matrix
*> [ F G ]
*> [ 0 H ].
*> On return, SSMIN is the smaller singular value and SSMAX is the
*> larger singular value.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] F
*> \verbatim
*> F is REAL
*> The (1,1) element of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[in] G
*> \verbatim
*> G is REAL
*> The (1,2) element of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[in] H
*> \verbatim
*> H is REAL
*> The (2,2) element of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[out] SSMIN
*> \verbatim
*> SSMIN is REAL
*> The smaller singular value.
*> \endverbatim
*>
*> \param[out] SSMAX
*> \verbatim
*> SSMAX is REAL
*> The larger singular value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup OTHERauxiliary
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> Barring over/underflow, all output quantities are correct to within
*> a few units in the last place (ulps), even in the absence of a guard
*> digit in addition/subtraction.
*>
*> In IEEE arithmetic, the code works correctly if one matrix element is
*> infinite.
*>
*> Overflow will not occur unless the largest singular value itself
*> overflows, or is within a few ulps of overflow. (On machines with
*> partial overflow, like the Cray, overflow may occur if the largest
*> singular value is within a factor of 2 of overflow.)
*>
*> Underflow is harmless if underflow is gradual. Otherwise, results
*> may correspond to a matrix modified by perturbations of size near
*> the underflow threshold.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE SLAS2( F, G, H, SSMIN, SSMAX )
*
* -- LAPACK auxiliary routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
REAL F, G, H, SSMAX, SSMIN
* ..
*
* ====================================================================
*
* .. Parameters ..
REAL ZERO
PARAMETER ( ZERO = 0.0E0 )
REAL ONE
PARAMETER ( ONE = 1.0E0 )
REAL TWO
PARAMETER ( TWO = 2.0E0 )
* ..
* .. Local Scalars ..
REAL AS, AT, AU, C, FA, FHMN, FHMX, GA, HA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
FA = ABS( F )
GA = ABS( G )
HA = ABS( H )
FHMN = MIN( FA, HA )
FHMX = MAX( FA, HA )
IF( FHMN.EQ.ZERO ) THEN
SSMIN = ZERO
IF( FHMX.EQ.ZERO ) THEN
SSMAX = GA
ELSE
SSMAX = MAX( FHMX, GA )*SQRT( ONE+
$ ( MIN( FHMX, GA ) / MAX( FHMX, GA ) )**2 )
END IF
ELSE
IF( GA.LT.FHMX ) THEN
AS = ONE + FHMN / FHMX
AT = ( FHMX-FHMN ) / FHMX
AU = ( GA / FHMX )**2
C = TWO / ( SQRT( AS*AS+AU )+SQRT( AT*AT+AU ) )
SSMIN = FHMN*C
SSMAX = FHMX / C
ELSE
AU = FHMX / GA
IF( AU.EQ.ZERO ) THEN
*
* Avoid possible harmful underflow if exponent range
* asymmetric (true SSMIN may not underflow even if
* AU underflows)
*
SSMIN = ( FHMN*FHMX ) / GA
SSMAX = GA
ELSE
AS = ONE + FHMN / FHMX
AT = ( FHMX-FHMN ) / FHMX
C = ONE / ( SQRT( ONE+( AS*AU )**2 )+
$ SQRT( ONE+( AT*AU )**2 ) )
SSMIN = ( FHMN*C )*AU
SSMIN = SSMIN + SSMIN
SSMAX = GA / ( C+C )
END IF
END IF
END IF
RETURN
*
* End of SLAS2
*
END
|
