slasv2.f

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*> \brief \b SLASV2 computes the singular value decomposition of a 2-by-2 triangular matrix.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at 
*            http://www.netlib.org/lapack/explore-html/ 
*
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*> \endhtmlonly 
*
*  Definition:
*  ===========
*
*       SUBROUTINE SLASV2( F, G, H, SSMIN, SSMAX, SNR, CSR, SNL, CSL )
* 
*       .. Scalar Arguments ..
*       REAL               CSL, CSR, F, G, H, SNL, SNR, SSMAX, SSMIN
*       ..
*  
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> SLASV2 computes the singular value decomposition of a 2-by-2
*> triangular matrix
*>    [  F   G  ]
*>    [  0   H  ].
*> On return, abs(SSMAX) is the larger singular value, abs(SSMIN) is the
*> smaller singular value, and (CSL,SNL) and (CSR,SNR) are the left and
*> right singular vectors for abs(SSMAX), giving the decomposition
*>
*>    [ CSL  SNL ] [  F   G  ] [ CSR -SNR ]  =  [ SSMAX   0   ]
*>    [-SNL  CSL ] [  0   H  ] [ SNR  CSR ]     [  0    SSMIN ].
*> \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
*>          abs(SSMIN) is the smaller singular value.
*> \endverbatim
*>
*> \param[out] SSMAX
*> \verbatim
*>          SSMAX is REAL
*>          abs(SSMAX) is the larger singular value.
*> \endverbatim
*>
*> \param[out] SNL
*> \verbatim
*>          SNL is REAL
*> \endverbatim
*>
*> \param[out] CSL
*> \verbatim
*>          CSL is REAL
*>          The vector (CSL, SNL) is a unit left singular vector for the
*>          singular value abs(SSMAX).
*> \endverbatim
*>
*> \param[out] SNR
*> \verbatim
*>          SNR is REAL
*> \endverbatim
*>
*> \param[out] CSR
*> \verbatim
*>          CSR is REAL
*>          The vector (CSR, SNR) is a unit right singular vector for the
*>          singular value abs(SSMAX).
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee 
*> \author Univ. of California Berkeley 
*> \author Univ. of Colorado Denver 
*> \author NAG Ltd. 
*
*> \date September 2012
*
*> \ingroup auxOTHERauxiliary
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  Any input parameter may be aliased with any output parameter.
*>
*>  Barring over/underflow and assuming a guard digit in subtraction, all
*>  output quantities are correct to within a few units in the last
*>  place (ulps).
*>
*>  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 slasv2( F, G, H, SSMIN, SSMAX, SNR, CSR, SNL, CSL )
*
*  -- 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               csl, csr, f, g, h, snl, snr, ssmax, ssmin
*     ..
*
* =====================================================================
*
*     .. Parameters ..
      REAL               zero
      parameter( zero = 0.0e0 )
      REAL               half
      parameter( half = 0.5e0 )
      REAL               one
      parameter( one = 1.0e0 )
      REAL               two
      parameter( two = 2.0e0 )
      REAL               four
      parameter( four = 4.0e0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            gasmal, swap
      INTEGER            pmax
      REAL               a, clt, crt, d, fa, ft, ga, gt, ha, ht, l, m,
     $                   mm, r, s, slt, srt, t, temp, tsign, tt
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, sign, sqrt
*     ..
*     .. External Functions ..
      REAL               slamch
      EXTERNAL           slamch
*     ..
*     .. Executable Statements ..
*
      ft = f
      fa = abs( ft )
      ht = h
      ha = abs( h )
*
*     PMAX points to the maximum absolute element of matrix
*       PMAX = 1 if F largest in absolute values
*       PMAX = 2 if G largest in absolute values
*       PMAX = 3 if H largest in absolute values
*
      pmax = 1
      swap = ( ha.GT.fa )
      IF( swap ) THEN
         pmax = 3
         temp = ft
         ft = ht
         ht = temp
         temp = fa
         fa = ha
         ha = temp
*
*        Now FA .ge. HA
*
      END IF
      gt = g
      ga = abs( gt )
      IF( ga.EQ.zero ) THEN
*
*        Diagonal matrix
*
         ssmin = ha
         ssmax = fa
         clt = one
         crt = one
         slt = zero
         srt = zero
      ELSE
         gasmal = .true.
         IF( ga.GT.fa ) THEN
            pmax = 2
            IF( ( fa / ga ).LT.slamch( 'EPS' ) ) THEN
*
*              Case of very large GA
*
               gasmal = .false.
               ssmax = ga
               IF( ha.GT.one ) THEN
                  ssmin = fa / ( ga / ha )
               ELSE
                  ssmin = ( fa / ga )*ha
               END IF
               clt = one
               slt = ht / gt
               srt = one
               crt = ft / gt
            END IF
         END IF
         IF( gasmal ) THEN
*
*           Normal case
*
            d = fa - ha
            IF( d.EQ.fa ) THEN
*
*              Copes with infinite F or H
*
               l = one
            ELSE
               l = d / fa
            END IF
*
*           Note that 0 .le. L .le. 1
*
            m = gt / ft
*
*           Note that abs(M) .le. 1/macheps
*
            t = two - l
*
*           Note that T .ge. 1
*
            mm = m*m
            tt = t*t
            s = sqrt( tt+mm )
*
*           Note that 1 .le. S .le. 1 + 1/macheps
*
            IF( l.EQ.zero ) THEN
               r = abs( m )
            ELSE
               r = sqrt( l*l+mm )
            END IF
*
*           Note that 0 .le. R .le. 1 + 1/macheps
*
            a = half*( s+r )
*
*           Note that 1 .le. A .le. 1 + abs(M)
*
            ssmin = ha / a
            ssmax = fa*a
            IF( mm.EQ.zero ) THEN
*
*              Note that M is very tiny
*
               IF( l.EQ.zero ) THEN
                  t = sign( two, ft )*sign( one, gt )
               ELSE
                  t = gt / sign( d, ft ) + m / t
               END IF
            ELSE
               t = ( m / ( s+t )+m / ( r+l ) )*( one+a )
            END IF
            l = sqrt( t*t+four )
            crt = two / l
            srt = t / l
            clt = ( crt+srt*m ) / a
            slt = ( ht / ft )*srt / a
         END IF
      END IF
      IF( swap ) THEN
         csl = srt
         snl = crt
         csr = slt
         snr = clt
      ELSE
         csl = clt
         snl = slt
         csr = crt
         snr = srt
      END IF
*
*     Correct signs of SSMAX and SSMIN
*
      IF( pmax.EQ.1 )
     $   tsign = sign( one, csr )*sign( one, csl )*sign( one, f )
      IF( pmax.EQ.2 )
     $   tsign = sign( one, snr )*sign( one, csl )*sign( one, g )
      IF( pmax.EQ.3 )
     $   tsign = sign( one, snr )*sign( one, snl )*sign( one, h )
      ssmax = sign( ssmax, tsign )
      ssmin = sign( ssmin, tsign*sign( one, f )*sign( one, h ) )
      return
*
*     End of SLASV2
*
      END