sqrt12.f

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*> \brief \b SQRT12
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at 
*            http://www.netlib.org/lapack/explore-html/ 
*
*  Definition:
*  ===========
*
*       REAL             FUNCTION SQRT12( M, N, A, LDA, S, WORK, LWORK )
* 
*       .. Scalar Arguments ..
*       INTEGER            LDA, LWORK, M, N
*       ..
*       .. Array Arguments ..
*       REAL               A( LDA, * ), S( * ), WORK( LWORK )
*       ..
*  
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> SQRT12 computes the singular values `svlues' of the upper trapezoid
*> of A(1:M,1:N) and returns the ratio
*>
*>      || s - svlues||/(||svlues||*eps*max(M,N))
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is REAL array, dimension (LDA,N)
*>          The M-by-N matrix A. Only the upper trapezoid is referenced.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.
*> \endverbatim
*>
*> \param[in] S
*> \verbatim
*>          S is REAL array, dimension (min(M,N))
*>          The singular values of the matrix A.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is REAL array, dimension (LWORK)
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The length of the array WORK. LWORK >= max(M*N + 4*min(M,N) +
*>          max(M,N), M*N+2*MIN( M, N )+4*N).
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee 
*> \author Univ. of California Berkeley 
*> \author Univ. of Colorado Denver 
*> \author NAG Ltd. 
*
*> \date November 2011
*
*> \ingroup single_lin
*
*  =====================================================================
      REAL             FUNCTION sqrt12( M, N, A, LDA, S, WORK, LWORK )
*
*  -- LAPACK test routine (version 3.4.0) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     November 2011
*
*     .. Scalar Arguments ..
      INTEGER            LDA, LWORK, M, N
*     ..
*     .. Array Arguments ..
      REAL               A( lda, * ), S( * ), WORK( lwork )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ZERO, ONE
      parameter                ( zero = 0.0e0, one = 1.0e0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, INFO, ISCL, J, MN
      REAL               ANRM, BIGNUM, NRMSVL, SMLNUM
*     ..
*     .. External Functions ..
      REAL               SASUM, SLAMCH, SLANGE, SNRM2
      EXTERNAL           sasum, slamch, slange, snrm2
*     ..
*     .. External Subroutines ..
      EXTERNAL           saxpy, sbdsqr, sgebd2, slabad, slascl, slaset,
     $                   xerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          max, min, real
*     ..
*     .. Local Arrays ..
      REAL               DUMMY( 1 )
*     ..
*     .. Executable Statements ..
*
      sqrt12 = zero
*
*     Test that enough workspace is supplied
*
      IF( lwork.LT.max( m*n+4*min( m, n )+max( m, n ),
     $                  m*n+2*min( m, n )+4*n) ) THEN
         CALL xerbla( 'SQRT12', 7 )
         RETURN
      END IF
*
*     Quick return if possible
*
      mn = min( m, n )
      IF( mn.LE.zero )
     $   RETURN
*
      nrmsvl = snrm2( mn, s, 1 )
*
*     Copy upper triangle of A into work
*
      CALL slaset( 'Full', m, n, zero, zero, work, m )
      DO 20 j = 1, n
         DO 10 i = 1, min( j, m )
            work( ( j-1 )*m+i ) = a( i, j )
   10    CONTINUE
   20 CONTINUE
*
*     Get machine parameters
*
      smlnum = slamch( 'S' ) / slamch( 'P' )
      bignum = one / smlnum
      CALL slabad( smlnum, bignum )
*
*     Scale work if max entry outside range [SMLNUM,BIGNUM]
*
      anrm = slange( 'M', m, n, work, m, dummy )
      iscl = 0
      IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
*
*        Scale matrix norm up to SMLNUM
*
         CALL slascl( 'G', 0, 0, anrm, smlnum, m, n, work, m, info )
         iscl = 1
      ELSE IF( anrm.GT.bignum ) THEN
*
*        Scale matrix norm down to BIGNUM
*
         CALL slascl( 'G', 0, 0, anrm, bignum, m, n, work, m, info )
         iscl = 1
      END IF
*
      IF( anrm.NE.zero ) THEN
*
*        Compute SVD of work
*
         CALL sgebd2( m, n, work, m, work( m*n+1 ), work( m*n+mn+1 ),
     $                work( m*n+2*mn+1 ), work( m*n+3*mn+1 ),
     $                work( m*n+4*mn+1 ), info )
         CALL sbdsqr( 'Upper', mn, 0, 0, 0, work( m*n+1 ),
     $                work( m*n+mn+1 ), dummy, mn, dummy, 1, dummy, mn,
     $                work( m*n+2*mn+1 ), info )
*
         IF( iscl.EQ.1 ) THEN
            IF( anrm.GT.bignum ) THEN
               CALL slascl( 'G', 0, 0, bignum, anrm, mn, 1,
     $                      work( m*n+1 ), mn, info )
            END IF
            IF( anrm.LT.smlnum ) THEN
               CALL slascl( 'G', 0, 0, smlnum, anrm, mn, 1,
     $                      work( m*n+1 ), mn, info )
            END IF
         END IF
*
      ELSE
*
         DO 30 i = 1, mn
            work( m*n+i ) = zero
   30    CONTINUE
      END IF
*
*     Compare s and singular values of work
*
      CALL saxpy( mn, -one, s, 1, work( m*n+1 ), 1 )
      sqrt12 = sasum( mn, work( m*n+1 ), 1 ) /
     $         ( slamch( 'Epsilon' )*REAL( MAX( M, N ) ) )
      IF( nrmsvl.NE.zero )
     $   sqrt12 = sqrt12 / nrmsvl
*
      RETURN
*
*     End of SQRT12
*
      END