sbdt03.f

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*> \brief \b SBDT03
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at 
*            http://www.netlib.org/lapack/explore-html/ 
*
*  Definition:
*  ===========
*
*       SUBROUTINE SBDT03( UPLO, N, KD, D, E, U, LDU, S, VT, LDVT, WORK,
*                          RESID )
* 
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            KD, LDU, LDVT, N
*       REAL               RESID
*       ..
*       .. Array Arguments ..
*       REAL               D( * ), E( * ), S( * ), U( LDU, * ),
*      $                   VT( LDVT, * ), WORK( * )
*       ..
*  
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> SBDT03 reconstructs a bidiagonal matrix B from its SVD:
*>    S = U' * B * V
*> where U and V are orthogonal matrices and S is diagonal.
*>
*> The test ratio to test the singular value decomposition is
*>    RESID = norm( B - U * S * VT ) / ( n * norm(B) * EPS )
*> where VT = V' and EPS is the machine precision.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          Specifies whether the matrix B is upper or lower bidiagonal.
*>          = 'U':  Upper bidiagonal
*>          = 'L':  Lower bidiagonal
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix B.
*> \endverbatim
*>
*> \param[in] KD
*> \verbatim
*>          KD is INTEGER
*>          The bandwidth of the bidiagonal matrix B.  If KD = 1, the
*>          matrix B is bidiagonal, and if KD = 0, B is diagonal and E is
*>          not referenced.  If KD is greater than 1, it is assumed to be
*>          1, and if KD is less than 0, it is assumed to be 0.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is REAL array, dimension (N)
*>          The n diagonal elements of the bidiagonal matrix B.
*> \endverbatim
*>
*> \param[in] E
*> \verbatim
*>          E is REAL array, dimension (N-1)
*>          The (n-1) superdiagonal elements of the bidiagonal matrix B
*>          if UPLO = 'U', or the (n-1) subdiagonal elements of B if
*>          UPLO = 'L'.
*> \endverbatim
*>
*> \param[in] U
*> \verbatim
*>          U is REAL array, dimension (LDU,N)
*>          The n by n orthogonal matrix U in the reduction B = U'*A*P.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*>          LDU is INTEGER
*>          The leading dimension of the array U.  LDU >= max(1,N)
*> \endverbatim
*>
*> \param[in] S
*> \verbatim
*>          S is REAL array, dimension (N)
*>          The singular values from the SVD of B, sorted in decreasing
*>          order.
*> \endverbatim
*>
*> \param[in] VT
*> \verbatim
*>          VT is REAL array, dimension (LDVT,N)
*>          The n by n orthogonal matrix V' in the reduction
*>          B = U * S * V'.
*> \endverbatim
*>
*> \param[in] LDVT
*> \verbatim
*>          LDVT is INTEGER
*>          The leading dimension of the array VT.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is REAL array, dimension (2*N)
*> \endverbatim
*>
*> \param[out] RESID
*> \verbatim
*>          RESID is REAL
*>          The test ratio:  norm(B - U * S * V') / ( n * norm(A) * EPS )
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee 
*> \author Univ. of California Berkeley 
*> \author Univ. of Colorado Denver 
*> \author NAG Ltd. 
*
*> \date November 2011
*
*> \ingroup single_eig
*
*  =====================================================================
      SUBROUTINE sbdt03( UPLO, N, KD, D, E, U, LDU, S, VT, LDVT, WORK,
     $                   resid )
*
*  -- 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 ..
      CHARACTER          UPLO
      INTEGER            KD, LDU, LDVT, N
      REAL               RESID
*     ..
*     .. Array Arguments ..
      REAL               D( * ), E( * ), S( * ), U( ldu, * ),
     $                   vt( ldvt, * ), work( * )
*     ..
*
* ======================================================================
*
*     .. Parameters ..
      REAL               ZERO, ONE
      parameter                ( zero = 0.0e+0, one = 1.0e+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, J
      REAL               BNORM, EPS
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ISAMAX
      REAL               SASUM, SLAMCH
      EXTERNAL           lsame, isamax, sasum, slamch
*     ..
*     .. External Subroutines ..
      EXTERNAL           sgemv
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, max, min, real
*     ..
*     .. Executable Statements ..
*
*     Quick return if possible
*
      resid = zero
      IF( n.LE.0 )
     $   RETURN
*
*     Compute B - U * S * V' one column at a time.
*
      bnorm = zero
      IF( kd.GE.1 ) THEN
*
*        B is bidiagonal.
*
         IF( lsame( uplo, 'U' ) ) THEN
*
*           B is upper bidiagonal.
*
            DO 20 j = 1, n
               DO 10 i = 1, n
                  work( n+i ) = s( i )*vt( i, j )
   10          CONTINUE
               CALL sgemv( 'No transpose', n, n, -one, u, ldu,
     $                     work( n+1 ), 1, zero, work, 1 )
               work( j ) = work( j ) + d( j )
               IF( j.GT.1 ) THEN
                  work( j-1 ) = work( j-1 ) + e( j-1 )
                  bnorm = max( bnorm, abs( d( j ) )+abs( e( j-1 ) ) )
               ELSE
                  bnorm = max( bnorm, abs( d( j ) ) )
               END IF
               resid = max( resid, sasum( n, work, 1 ) )
   20       CONTINUE
         ELSE
*
*           B is lower bidiagonal.
*
            DO 40 j = 1, n
               DO 30 i = 1, n
                  work( n+i ) = s( i )*vt( i, j )
   30          CONTINUE
               CALL sgemv( 'No transpose', n, n, -one, u, ldu,
     $                     work( n+1 ), 1, zero, work, 1 )
               work( j ) = work( j ) + d( j )
               IF( j.LT.n ) THEN
                  work( j+1 ) = work( j+1 ) + e( j )
                  bnorm = max( bnorm, abs( d( j ) )+abs( e( j ) ) )
               ELSE
                  bnorm = max( bnorm, abs( d( j ) ) )
               END IF
               resid = max( resid, sasum( n, work, 1 ) )
   40       CONTINUE
         END IF
      ELSE
*
*        B is diagonal.
*
         DO 60 j = 1, n
            DO 50 i = 1, n
               work( n+i ) = s( i )*vt( i, j )
   50       CONTINUE
            CALL sgemv( 'No transpose', n, n, -one, u, ldu, work( n+1 ),
     $                  1, zero, work, 1 )
            work( j ) = work( j ) + d( j )
            resid = max( resid, sasum( n, work, 1 ) )
   60    CONTINUE
         j = isamax( n, d, 1 )
         bnorm = abs( d( j ) )
      END IF
*
*     Compute norm(B - U * S * V') / ( n * norm(B) * EPS )
*
      eps = slamch( 'Precision' )
*
      IF( bnorm.LE.zero ) THEN
         IF( resid.NE.zero )
     $      resid = one / eps
      ELSE
         IF( bnorm.GE.resid ) THEN
            resid = ( resid / bnorm ) / ( REAL( n )*EPS )
         ELSE
            IF( bnorm.LT.one ) THEN
               resid = ( min( resid, REAL( n )*BNORM ) / bnorm ) /
     $                 ( REAL( n )*EPS )
            ELSE
               resid = min( resid / bnorm, REAL( N ) ) /
     $                 ( REAL( n )*EPS )
            END IF
         END IF
      END IF
*
      RETURN
*
*     End of SBDT03
*
      END