zgesvj.f

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*> \brief \b ZGESVJ
*
*  =========== DOCUMENTATION ===========
*
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*            http://www.netlib.org/lapack/explore-html/ 
*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE ZGESVJ( JOBA, JOBU, JOBV, M, N, A, LDA, SVA, MV, V,
*                          LDV, CWORK, LWORK, RWORK, LRWORK, INFO )
* 
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, LDV, LWORK, LRWORK, M, MV, N
*       CHARACTER*1        JOBA, JOBU, JOBV
*       ..
*       .. Array Arguments ..
*       COMPLEX*16         A( LDA, * ),  V( LDV, * ), CWORK( LWORK )
*       DOUBLE PRECISION   RWORK( LRWORK ),  SVA( N )
*       ..
*  
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> ZGESVJ computes the singular value decomposition (SVD) of a complex
*> M-by-N matrix A, where M >= N. The SVD of A is written as
*>                                    [++]   [xx]   [x0]   [xx]
*>              A = U * SIGMA * V^*,  [++] = [xx] * [ox] * [xx]
*>                                    [++]   [xx]
*> where SIGMA is an N-by-N diagonal matrix, U is an M-by-N orthonormal
*> matrix, and V is an N-by-N unitary matrix. The diagonal elements
*> of SIGMA are the singular values of A. The columns of U and V are the
*> left and the right singular vectors of A, respectively.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBA
*> \verbatim
*>          JOBA is CHARACTER* 1
*>          Specifies the structure of A.
*>          = 'L': The input matrix A is lower triangular;
*>          = 'U': The input matrix A is upper triangular;
*>          = 'G': The input matrix A is general M-by-N matrix, M >= N.
*> \endverbatim
*>
*> \param[in] JOBU
*> \verbatim
*>          JOBU is CHARACTER*1
*>          Specifies whether to compute the left singular vectors
*>          (columns of U):
*>          = 'U': The left singular vectors corresponding to the nonzero
*>                 singular values are computed and returned in the leading
*>                 columns of A. See more details in the description of A.
*>                 The default numerical orthogonality threshold is set to
*>                 approximately TOL=CTOL*EPS, CTOL=DSQRT(M), EPS=DLAMCH('E').
*>          = 'C': Analogous to JOBU='U', except that user can control the
*>                 level of numerical orthogonality of the computed left
*>                 singular vectors. TOL can be set to TOL = CTOL*EPS, where
*>                 CTOL is given on input in the array WORK.
*>                 No CTOL smaller than ONE is allowed. CTOL greater
*>                 than 1 / EPS is meaningless. The option 'C'
*>                 can be used if M*EPS is satisfactory orthogonality
*>                 of the computed left singular vectors, so CTOL=M could
*>                 save few sweeps of Jacobi rotations.
*>                 See the descriptions of A and WORK(1).
*>          = 'N': The matrix U is not computed. However, see the
*>                 description of A.
*> \endverbatim
*>
*> \param[in] JOBV
*> \verbatim
*>          JOBV is CHARACTER*1
*>          Specifies whether to compute the right singular vectors, that
*>          is, the matrix V:
*>          = 'V' : the matrix V is computed and returned in the array V
*>          = 'A' : the Jacobi rotations are applied to the MV-by-N
*>                  array V. In other words, the right singular vector
*>                  matrix V is not computed explicitly, instead it is
*>                  applied to an MV-by-N matrix initially stored in the
*>                  first MV rows of V.
*>          = 'N' : the matrix V is not computed and the array V is not
*>                  referenced
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the input matrix A. 1/DLAMCH('E') > M >= 0.  
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the input matrix A.
*>          M >= N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is COMPLEX*16 array, dimension (LDA,N)
*>          On entry, the M-by-N matrix A.
*>          On exit,
*>          If JOBU .EQ. 'U' .OR. JOBU .EQ. 'C':
*>                 If INFO .EQ. 0 :
*>                 RANKA orthonormal columns of U are returned in the
*>                 leading RANKA columns of the array A. Here RANKA <= N
*>                 is the number of computed singular values of A that are
*>                 above the underflow threshold DLAMCH('S'). The singular
*>                 vectors corresponding to underflowed or zero singular
*>                 values are not computed. The value of RANKA is returned
*>                 in the array RWORK as RANKA=NINT(RWORK(2)). Also see the
*>                 descriptions of SVA and RWORK. The computed columns of U
*>                 are mutually numerically orthogonal up to approximately
*>                 TOL=SQRT(M)*EPS (default); or TOL=CTOL*EPS (JOBU.EQ.'C'),
*>                 see the description of JOBU.
*>                 If INFO .GT. 0,
*>                 the procedure ZGESVJ did not converge in the given number
*>                 of iterations (sweeps). In that case, the computed
*>                 columns of U may not be orthogonal up to TOL. The output
*>                 U (stored in A), SIGMA (given by the computed singular
*>                 values in SVA(1:N)) and V is still a decomposition of the
*>                 input matrix A in the sense that the residual
*>                 || A - SCALE * U * SIGMA * V^* ||_2 / ||A||_2 is small.
*>          If JOBU .EQ. 'N':
*>                 If INFO .EQ. 0 :
*>                 Note that the left singular vectors are 'for free' in the
*>                 one-sided Jacobi SVD algorithm. However, if only the
*>                 singular values are needed, the level of numerical
*>                 orthogonality of U is not an issue and iterations are
*>                 stopped when the columns of the iterated matrix are
*>                 numerically orthogonal up to approximately M*EPS. Thus,
*>                 on exit, A contains the columns of U scaled with the
*>                 corresponding singular values.
*>                 If INFO .GT. 0 :
*>                 the procedure ZGESVJ did not converge in the given number
*>                 of iterations (sweeps).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] SVA
*> \verbatim
*>          SVA is DOUBLE PRECISION array, dimension (N)
*>          On exit,
*>          If INFO .EQ. 0 :
*>          depending on the value SCALE = RWORK(1), we have:
*>                 If SCALE .EQ. ONE:
*>                 SVA(1:N) contains the computed singular values of A.
*>                 During the computation SVA contains the Euclidean column
*>                 norms of the iterated matrices in the array A.
*>                 If SCALE .NE. ONE:
*>                 The singular values of A are SCALE*SVA(1:N), and this
*>                 factored representation is due to the fact that some of the
*>                 singular values of A might underflow or overflow.
*>
*>          If INFO .GT. 0 :
*>          the procedure ZGESVJ did not converge in the given number of
*>          iterations (sweeps) and SCALE*SVA(1:N) may not be accurate.
*> \endverbatim
*>
*> \param[in] MV
*> \verbatim
*>          MV is INTEGER
*>          If JOBV .EQ. 'A', then the product of Jacobi rotations in ZGESVJ
*>          is applied to the first MV rows of V. See the description of JOBV.
*> \endverbatim
*>
*> \param[in,out] V
*> \verbatim
*>          V is COMPLEX*16 array, dimension (LDV,N)
*>          If JOBV = 'V', then V contains on exit the N-by-N matrix of
*>                         the right singular vectors;
*>          If JOBV = 'A', then V contains the product of the computed right
*>                         singular vector matrix and the initial matrix in
*>                         the array V.
*>          If JOBV = 'N', then V is not referenced.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*>          LDV is INTEGER
*>          The leading dimension of the array V, LDV .GE. 1.
*>          If JOBV .EQ. 'V', then LDV .GE. max(1,N).
*>          If JOBV .EQ. 'A', then LDV .GE. max(1,MV) .
*> \endverbatim
*>
*> \param[in,out] CWORK
*> \verbatim
*>          CWORK is COMPLEX*16 array, dimension M+N.
*>          Used as work space.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER.
*>          Length of CWORK, LWORK >= M+N.
*> \endverbatim
*>
*> \param[in,out] RWORK
*> \verbatim
*>          RWORK is DOUBLE PRECISION array, dimension max(6,M+N).
*>          On entry,
*>          If JOBU .EQ. 'C' :
*>          RWORK(1) = CTOL, where CTOL defines the threshold for convergence.
*>                    The process stops if all columns of A are mutually
*>                    orthogonal up to CTOL*EPS, EPS=DLAMCH('E').
*>                    It is required that CTOL >= ONE, i.e. it is not
*>                    allowed to force the routine to obtain orthogonality
*>                    below EPSILON.
*>          On exit,
*>          RWORK(1) = SCALE is the scaling factor such that SCALE*SVA(1:N)
*>                    are the computed singular values of A.
*>                    (See description of SVA().)
*>          RWORK(2) = NINT(RWORK(2)) is the number of the computed nonzero
*>                    singular values.
*>          RWORK(3) = NINT(RWORK(3)) is the number of the computed singular
*>                    values that are larger than the underflow threshold.
*>          RWORK(4) = NINT(RWORK(4)) is the number of sweeps of Jacobi
*>                    rotations needed for numerical convergence.
*>          RWORK(5) = max_{i.NE.j} |COS(A(:,i),A(:,j))| in the last sweep.
*>                    This is useful information in cases when ZGESVJ did
*>                    not converge, as it can be used to estimate whether
*>                    the output is stil useful and for post festum analysis.
*>          RWORK(6) = the largest absolute value over all sines of the
*>                    Jacobi rotation angles in the last sweep. It can be
*>                    useful for a post festum analysis.
*> \endverbatim
*>
*> \param[in] LRWORK
*> \verbatim
*>          LRWORK is INTEGER
*>         Length of RWORK, LRWORK >= MAX(6,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0 : successful exit.
*>          < 0 : if INFO = -i, then the i-th argument had an illegal value
*>          > 0 : ZGESVJ did not converge in the maximal allowed number 
*>                (NSWEEP=30) of sweeps. The output may still be useful. 
*>                See the description of RWORK.
*> \endverbatim
*>
*  Authors:
*  ========
*
*> \author Univ. of Tennessee 
*> \author Univ. of California Berkeley 
*> \author Univ. of Colorado Denver 
*> \author NAG Ltd. 
*
*> \date June 2016
*
*> \ingroup doubleGEcomputational
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*> The orthogonal N-by-N matrix V is obtained as a product of Jacobi plane
*> rotations. In the case of underflow of the tangent of the Jacobi angle, a
*> modified Jacobi transformation of Drmac [3] is used. Pivot strategy uses
*> column interchanges of de Rijk [1]. The relative accuracy of the computed
*> singular values and the accuracy of the computed singular vectors (in
*> angle metric) is as guaranteed by the theory of Demmel and Veselic [2].
*> The condition number that determines the accuracy in the full rank case
*> is essentially min_{D=diag} kappa(A*D), where kappa(.) is the
*> spectral condition number. The best performance of this Jacobi SVD
*> procedure is achieved if used in an  accelerated version of Drmac and
*> Veselic [4,5], and it is the kernel routine in the SIGMA library [6].
*> Some tunning parameters (marked with [TP]) are available for the
*> implementer. 
*> The computational range for the nonzero singular values is the  machine
*> number interval ( UNDERFLOW , OVERFLOW ). In extreme cases, even
*> denormalized singular values can be computed with the corresponding
*> gradual loss of accurate digits.
*> \endverbatim
*
*> \par Contributors:
*  ==================
*>
*> \verbatim
*>
*>  ============
*>
*>  Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)
*> \endverbatim
*
*> \par References:
*  ================
*>
*> [1] P. P. M. De Rijk: A one-sided Jacobi algorithm for computing the
*>    singular value decomposition on a vector computer. 
*>    SIAM J. Sci. Stat. Comp., Vol. 10 (1998), pp. 359-371. 
*> [2] J. Demmel and K. Veselic: Jacobi method is more accurate than QR.
*> [3] Z. Drmac: Implementation of Jacobi rotations for accurate singular
*>    value computation in floating point arithmetic.
*>    SIAM J. Sci. Comp., Vol. 18 (1997), pp. 1200-1222.
*> [4] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm I.
*>    SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1322-1342.
*>    LAPACK Working note 169.
*> [5] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm II.
*>    SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1343-1362.
*>    LAPACK Working note 170.
*> [6] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV,
*>    QSVD, (H,K)-SVD computations.
*>    Department of Mathematics, University of Zagreb, 2008, 2015.
*> \endverbatim
*
*>  \par Bugs, examples and comments:
*   =================================
*>
*> \verbatim
*>  ===========================
*>  Please report all bugs and send interesting test examples and comments to
*>  drmac@math.hr. Thank you.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE zgesvj( JOBA, JOBU, JOBV, M, N, A, LDA, SVA, MV, V, 
     $                   ldv, cwork, lwork, rwork, lrwork, info )
*
*  -- LAPACK computational routine (version 3.6.1) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     June 2016
*
      IMPLICIT NONE 
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, LDV, LWORK, LRWORK, M, MV, N
      CHARACTER*1        JOBA, JOBU, JOBV
*     ..
*     .. Array Arguments ..
      COMPLEX*16         A( lda, * ),  V( ldv, * ), CWORK( lwork )
      DOUBLE PRECISION   RWORK( lrwork ), SVA( n )
*     ..
*
*  =====================================================================
*
*     .. Local Parameters ..
      DOUBLE PRECISION   ZERO,         HALF,         ONE
      parameter( zero = 0.0d0, half = 0.5d0, one = 1.0d0)
      COMPLEX*16      CZERO,                  CONE
      parameter( czero = (0.0d0, 0.0d0), cone = (1.0d0, 0.0d0) )
      INTEGER      NSWEEP
      parameter( nsweep = 30 )
*     ..
*     .. Local Scalars ..
      COMPLEX*16 AAPQ, OMPQ
      DOUBLE PRECISION    AAPP, AAPP0, AAPQ1, AAQQ, APOAQ, AQOAP, BIG, 
     $        bigtheta, cs, ctol, epsln, large, mxaapq, 
     $        mxsinj, rootbig, rooteps, rootsfmin, roottol, 
     $        skl, sfmin, small, sn, t, temp1, theta, thsign, tol
      INTEGER BLSKIP, EMPTSW, i, ibr, IERR, igl, IJBLSK, ir1,
     $        iswrot, jbc, jgl, kbl, lkahead, mvl, n2, n34, 
     $        n4, nbl, notrot, p, pskipped, q, rowskip, swband
      LOGICAL APPLV, GOSCALE, LOWER, LSVEC, NOSCALE, ROTOK, 
     $        rsvec, uctol, upper
*     ..
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC abs, dmax1, dmin1, dconjg, dble, min0, max0, 
     $          dsign, dsqrt
*     ..
*     .. External Functions ..
*     ..
*     from BLAS
      DOUBLE PRECISION   DZNRM2
      COMPLEX*16         ZDOTC
      EXTERNAL           zdotc, dznrm2
      INTEGER            IDAMAX
      EXTERNAL           idamax
*     from LAPACK
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           dlamch
      LOGICAL            LSAME
      EXTERNAL           lsame
*     ..
*     .. External Subroutines ..
*     ..
*     from BLAS
      EXTERNAL           zcopy, zrot, zdscal, zswap
*     from LAPACK
      EXTERNAL           dlascl, zlascl, zlaset, zlassq, xerbla
      EXTERNAL           zgsvj0, zgsvj1
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      lsvec = lsame( jobu, 'U' )
      uctol = lsame( jobu, 'C' )
      rsvec = lsame( jobv, 'V' )
      applv = lsame( jobv, 'A' )
      upper = lsame( joba, 'U' )
      lower = lsame( joba, 'L' )
*
      IF( .NOT.( upper .OR. lower .OR. lsame( joba, 'G' ) ) ) THEN
         info = -1
      ELSE IF( .NOT.( lsvec .OR. uctol .OR. lsame( jobu, 'N' ) ) ) THEN
         info = -2
      ELSE IF( .NOT.( rsvec .OR. applv .OR. lsame( jobv, 'N' ) ) ) THEN
         info = -3
      ELSE IF( m.LT.0 ) THEN
         info = -4
      ELSE IF( ( n.LT.0 ) .OR. ( n.GT.m ) ) THEN
         info = -5
      ELSE IF( lda.LT.m ) THEN
         info = -7
      ELSE IF( mv.LT.0 ) THEN
         info = -9
      ELSE IF( ( rsvec .AND. ( ldv.LT.n ) ) .OR.
     $          ( applv .AND. ( ldv.LT.mv ) ) ) THEN
         info = -11
      ELSE IF( uctol .AND. ( rwork( 1 ).LE.one ) ) THEN
         info = -12
      ELSE IF( lwork.LT.( m+n ) ) THEN
         info = -13
      ELSE IF( lrwork.LT.max0( n, 6 ) ) THEN
         info = -15   
      ELSE
         info = 0
      END IF
*
*     #:(
      IF( info.NE.0 ) THEN
         CALL xerbla( 'ZGESVJ', -info )
         RETURN
      END IF
*
* #:) Quick return for void matrix
*
      IF( ( m.EQ.0 ) .OR. ( n.EQ.0 ) )RETURN
*
*     Set numerical parameters
*     The stopping criterion for Jacobi rotations is
*
*     max_{i<>j}|A(:,i)^* * A(:,j)| / (||A(:,i)||*||A(:,j)||) < CTOL*EPS
*
*     where EPS is the round-off and CTOL is defined as follows:
*
      IF( uctol ) THEN
*        ... user controlled
         ctol = rwork( 1 )
      ELSE
*        ... default
         IF( lsvec .OR. rsvec .OR. applv ) THEN
            ctol = dsqrt( dble( m ) )
         ELSE
            ctol = dble( m )
         END IF
      END IF
*     ... and the machine dependent parameters are
*[!]  (Make sure that DLAMCH() works properly on the target machine.)
*
      epsln = dlamch( 'Epsilon' )
      rooteps = dsqrt( epsln )
      sfmin = dlamch( 'SafeMinimum' )
      rootsfmin = dsqrt( sfmin )
      small = sfmin / epsln
      big = dlamch( 'Overflow' )
*     BIG         = ONE    / SFMIN
      rootbig = one / rootsfmin
      large = big / dsqrt( dble( m*n ) )
      bigtheta = one / rooteps
*
      tol = ctol*epsln
      roottol = dsqrt( tol )
*
      IF( dble( m )*epsln.GE.one ) THEN
         info = -4
         CALL xerbla( 'ZGESVJ', -info )
         RETURN
      END IF
*
*     Initialize the right singular vector matrix.
*
      IF( rsvec ) THEN
         mvl = n
         CALL zlaset( 'A', mvl, n, czero, cone, v, ldv )
      ELSE IF( applv ) THEN
         mvl = mv
      END IF
      rsvec = rsvec .OR. applv
*
*     Initialize SVA( 1:N ) = ( ||A e_i||_2, i = 1:N )
*(!)  If necessary, scale A to protect the largest singular value
*     from overflow. It is possible that saving the largest singular
*     value destroys the information about the small ones.
*     This initial scaling is almost minimal in the sense that the
*     goal is to make sure that no column norm overflows, and that
*     SQRT(N)*max_i SVA(i) does not overflow. If INFinite entries
*     in A are detected, the procedure returns with INFO=-6.
*
      skl = one / dsqrt( dble( m )*dble( n ) )
      noscale = .true.
      goscale = .true.
*
      IF( lower ) THEN
*        the input matrix is M-by-N lower triangular (trapezoidal)
         DO 1874 p = 1, n
            aapp = zero
            aaqq = one
            CALL zlassq( m-p+1, a( p, p ), 1, aapp, aaqq )
            IF( aapp.GT.big ) THEN
               info = -6
               CALL xerbla( 'ZGESVJ', -info )
               RETURN
            END IF
            aaqq = dsqrt( aaqq )
            IF( ( aapp.LT.( big / aaqq ) ) .AND. noscale ) THEN
               sva( p ) = aapp*aaqq
            ELSE
               noscale = .false.
               sva( p ) = aapp*( aaqq*skl )
               IF( goscale ) THEN
                  goscale = .false.
                  DO 1873 q = 1, p - 1
                     sva( q ) = sva( q )*skl
 1873             CONTINUE
               END IF
            END IF
 1874    CONTINUE
      ELSE IF( upper ) THEN
*        the input matrix is M-by-N upper triangular (trapezoidal)
         DO 2874 p = 1, n
            aapp = zero
            aaqq = one
            CALL zlassq( p, a( 1, p ), 1, aapp, aaqq )
            IF( aapp.GT.big ) THEN
               info = -6
               CALL xerbla( 'ZGESVJ', -info )
               RETURN
            END IF
            aaqq = dsqrt( aaqq )
            IF( ( aapp.LT.( big / aaqq ) ) .AND. noscale ) THEN
               sva( p ) = aapp*aaqq
            ELSE
               noscale = .false.
               sva( p ) = aapp*( aaqq*skl )
               IF( goscale ) THEN
                  goscale = .false.
                  DO 2873 q = 1, p - 1
                     sva( q ) = sva( q )*skl
 2873             CONTINUE
               END IF
            END IF
 2874    CONTINUE
      ELSE
*        the input matrix is M-by-N general dense
         DO 3874 p = 1, n
            aapp = zero
            aaqq = one
            CALL zlassq( m, a( 1, p ), 1, aapp, aaqq )
            IF( aapp.GT.big ) THEN
               info = -6
               CALL xerbla( 'ZGESVJ', -info )
               RETURN
            END IF
            aaqq = dsqrt( aaqq )
            IF( ( aapp.LT.( big / aaqq ) ) .AND. noscale ) THEN
               sva( p ) = aapp*aaqq
            ELSE
               noscale = .false.
               sva( p ) = aapp*( aaqq*skl )
               IF( goscale ) THEN
                  goscale = .false.
                  DO 3873 q = 1, p - 1
                     sva( q ) = sva( q )*skl
 3873             CONTINUE
               END IF
            END IF
 3874    CONTINUE
      END IF
*
      IF( noscale )skl = one
*
*     Move the smaller part of the spectrum from the underflow threshold
*(!)  Start by determining the position of the nonzero entries of the
*     array SVA() relative to ( SFMIN, BIG ).
*
      aapp = zero
      aaqq = big
      DO 4781 p = 1, n
         IF( sva( p ).NE.zero )aaqq = dmin1( aaqq, sva( p ) )
         aapp = dmax1( aapp, sva( p ) )
 4781 CONTINUE
*
* #:) Quick return for zero matrix
*
      IF( aapp.EQ.zero ) THEN
         IF( lsvec )CALL zlaset( 'G', m, n, czero, cone, a, lda )
         rwork( 1 ) = one
         rwork( 2 ) = zero
         rwork( 3 ) = zero
         rwork( 4 ) = zero
         rwork( 5 ) = zero
         rwork( 6 ) = zero
         RETURN
      END IF
*
* #:) Quick return for one-column matrix
*
      IF( n.EQ.1 ) THEN
         IF( lsvec )CALL zlascl( 'G', 0, 0, sva( 1 ), skl, m, 1,
     $                           a( 1, 1 ), lda, ierr )
         rwork( 1 ) = one / skl
         IF( sva( 1 ).GE.sfmin ) THEN
            rwork( 2 ) = one
         ELSE
            rwork( 2 ) = zero
         END IF
         rwork( 3 ) = zero
         rwork( 4 ) = zero
         rwork( 5 ) = zero
         rwork( 6 ) = zero
         RETURN
      END IF
*
*     Protect small singular values from underflow, and try to
*     avoid underflows/overflows in computing Jacobi rotations.
*
      sn = dsqrt( sfmin / epsln )
      temp1 = dsqrt( big / dble( n ) )
      IF( ( aapp.LE.sn ) .OR. ( aaqq.GE.temp1 ) .OR.    
     $    ( ( sn.LE.aaqq ) .AND. ( aapp.LE.temp1 ) ) ) THEN
         temp1 = dmin1( big, temp1 / aapp )
*         AAQQ  = AAQQ*TEMP1
*         AAPP  = AAPP*TEMP1
      ELSE IF( ( aaqq.LE.sn ) .AND. ( aapp.LE.temp1 ) ) THEN
         temp1 = dmin1( sn / aaqq, big / (aapp*dsqrt( dble(n)) ) )
*         AAQQ  = AAQQ*TEMP1
*         AAPP  = AAPP*TEMP1
      ELSE IF( ( aaqq.GE.sn ) .AND. ( aapp.GE.temp1 ) ) THEN
         temp1 = dmax1( sn / aaqq, temp1 / aapp )
*         AAQQ  = AAQQ*TEMP1
*         AAPP  = AAPP*TEMP1
      ELSE IF( ( aaqq.LE.sn ) .AND. ( aapp.GE.temp1 ) ) THEN
         temp1 = dmin1( sn / aaqq, big / ( dsqrt( dble( n ) )*aapp ) )
*         AAQQ  = AAQQ*TEMP1
*         AAPP  = AAPP*TEMP1
      ELSE
         temp1 = one
      END IF
*
*     Scale, if necessary
*
      IF( temp1.NE.one ) THEN
         CALL dlascl( 'G', 0, 0, one, temp1, n, 1, sva, n, ierr )
      END IF
      skl = temp1*skl
      IF( skl.NE.one ) THEN
         CALL zlascl( joba, 0, 0, one, skl, m, n, a, lda, ierr )
         skl = one / skl
      END IF
*
*     Row-cyclic Jacobi SVD algorithm with column pivoting
*
      emptsw = ( n*( n-1 ) ) / 2
      notrot = 0
       
      DO 1868 q = 1, n
         cwork( q ) = cone
 1868 CONTINUE     
*
*
*
      swband = 3
*[TP] SWBAND is a tuning parameter [TP]. It is meaningful and effective
*     if ZGESVJ is used as a computational routine in the preconditioned
*     Jacobi SVD algorithm ZGEJSV. For sweeps i=1:SWBAND the procedure
*     works on pivots inside a band-like region around the diagonal.
*     The boundaries are determined dynamically, based on the number of
*     pivots above a threshold.
*
      kbl = min0( 8, n )
*[TP] KBL is a tuning parameter that defines the tile size in the
*     tiling of the p-q loops of pivot pairs. In general, an optimal
*     value of KBL depends on the matrix dimensions and on the
*     parameters of the computer's memory.
*
      nbl = n / kbl
      IF( ( nbl*kbl ).NE.n )nbl = nbl + 1
*
      blskip = kbl**2
*[TP] BLKSKIP is a tuning parameter that depends on SWBAND and KBL.
*
      rowskip = min0( 5, kbl )
*[TP] ROWSKIP is a tuning parameter.
*
      lkahead = 1
*[TP] LKAHEAD is a tuning parameter.
*
*     Quasi block transformations, using the lower (upper) triangular
*     structure of the input matrix. The quasi-block-cycling usually
*     invokes cubic convergence. Big part of this cycle is done inside
*     canonical subspaces of dimensions less than M.
*
      IF( ( lower .OR. upper ) .AND. ( n.GT.max0( 64, 4*kbl ) ) ) THEN
*[TP] The number of partition levels and the actual partition are
*     tuning parameters.
         n4 = n / 4
         n2 = n / 2
         n34 = 3*n4
         IF( applv ) THEN
            q = 0
         ELSE
            q = 1
         END IF
*
         IF( lower ) THEN
*
*     This works very well on lower triangular matrices, in particular
*     in the framework of the preconditioned Jacobi SVD (xGEJSV).
*     The idea is simple:
*     [+ 0 0 0]   Note that Jacobi transformations of [0 0]
*     [+ + 0 0]                                       [0 0]
*     [+ + x 0]   actually work on [x 0]              [x 0]
*     [+ + x x]                    [x x].             [x x]
*
            CALL zgsvj0( jobv, m-n34, n-n34, a( n34+1, n34+1 ), lda,
     $                   cwork( n34+1 ), sva( n34+1 ), mvl,
     $                   v( n34*q+1, n34+1 ), ldv, epsln, sfmin, tol,
     $                   2, cwork( n+1 ), lwork-n, ierr )

            CALL zgsvj0( jobv, m-n2, n34-n2, a( n2+1, n2+1 ), lda,
     $                   cwork( n2+1 ), sva( n2+1 ), mvl,
     $                   v( n2*q+1, n2+1 ), ldv, epsln, sfmin, tol, 2,
     $                   cwork( n+1 ), lwork-n, ierr )

            CALL zgsvj1( jobv, m-n2, n-n2, n4, a( n2+1, n2+1 ), lda,
     $                   cwork( n2+1 ), sva( n2+1 ), mvl,
     $                   v( n2*q+1, n2+1 ), ldv, epsln, sfmin, tol, 1,
     $                   cwork( n+1 ), lwork-n, ierr )

            CALL zgsvj0( jobv, m-n4, n2-n4, a( n4+1, n4+1 ), lda,
     $                   cwork( n4+1 ), sva( n4+1 ), mvl,
     $                   v( n4*q+1, n4+1 ), ldv, epsln, sfmin, tol, 1,
     $                   cwork( n+1 ), lwork-n, ierr )
*
            CALL zgsvj0( jobv, m, n4, a, lda, cwork, sva, mvl, v, ldv,
     $                   epsln, sfmin, tol, 1, cwork( n+1 ), lwork-n,
     $                   ierr )
*
            CALL zgsvj1( jobv, m, n2, n4, a, lda, cwork, sva, mvl, v,
     $                   ldv, epsln, sfmin, tol, 1, cwork( n+1 ),
     $                   lwork-n, ierr )
*
*
         ELSE IF( upper ) THEN
*
*
            CALL zgsvj0( jobv, n4, n4, a, lda, cwork, sva, mvl, v, ldv,
     $                   epsln, sfmin, tol, 2, cwork( n+1 ), lwork-n,
     $                   ierr )
*
            CALL zgsvj0( jobv, n2, n4, a( 1, n4+1 ), lda, cwork( n4+1 ),
     $                   sva( n4+1 ), mvl, v( n4*q+1, n4+1 ), ldv,
     $                   epsln, sfmin, tol, 1, cwork( n+1 ), lwork-n,
     $                   ierr )
*
            CALL zgsvj1( jobv, n2, n2, n4, a, lda, cwork, sva, mvl, v,
     $                   ldv, epsln, sfmin, tol, 1, cwork( n+1 ),
     $                   lwork-n, ierr )
*
            CALL zgsvj0( jobv, n2+n4, n4, a( 1, n2+1 ), lda,
     $                   cwork( n2+1 ), sva( n2+1 ), mvl,
     $                   v( n2*q+1, n2+1 ), ldv, epsln, sfmin, tol, 1,
     $                   cwork( n+1 ), lwork-n, ierr )

         END IF
*
      END IF
*
*     .. Row-cyclic pivot strategy with de Rijk's pivoting ..
*
      DO 1993 i = 1, nsweep
*
*     .. go go go ...
*
         mxaapq = zero
         mxsinj = zero
         iswrot = 0
*
         notrot = 0
         pskipped = 0
*
*     Each sweep is unrolled using KBL-by-KBL tiles over the pivot pairs
*     1 <= p < q <= N. This is the first step toward a blocked implementation
*     of the rotations. New implementation, based on block transformations,
*     is under development.
*
         DO 2000 ibr = 1, nbl
*
            igl = ( ibr-1 )*kbl + 1
*
            DO 1002 ir1 = 0, min0( lkahead, nbl-ibr )
*
               igl = igl + ir1*kbl
*
               DO 2001 p = igl, min0( igl+kbl-1, n-1 )
*
*     .. de Rijk's pivoting
*
                  q = idamax( n-p+1, sva( p ), 1 ) + p - 1
                  IF( p.NE.q ) THEN
                     CALL zswap( m, a( 1, p ), 1, a( 1, q ), 1 )
                     IF( rsvec )CALL zswap( mvl, v( 1, p ), 1,  
     $                                           v( 1, q ), 1 )
                     temp1 = sva( p )
                     sva( p ) = sva( q )
                     sva( q ) = temp1
                     aapq = cwork(p)
                     cwork(p) = cwork(q)
                     cwork(q) = aapq
                  END IF
*
                  IF( ir1.EQ.0 ) THEN
*
*        Column norms are periodically updated by explicit
*        norm computation.
*[!]     Caveat:
*        Unfortunately, some BLAS implementations compute DZNRM2(M,A(1,p),1)
*        as SQRT(S=CDOTC(M,A(1,p),1,A(1,p),1)), which may cause the result to
*        overflow for ||A(:,p)||_2 > SQRT(overflow_threshold), and to
*        underflow for ||A(:,p)||_2 < SQRT(underflow_threshold).
*        Hence, DZNRM2 cannot be trusted, not even in the case when
*        the true norm is far from the under(over)flow boundaries.
*        If properly implemented SCNRM2 is available, the IF-THEN-ELSE-END IF
*        below should be replaced with "AAPP = DZNRM2( M, A(1,p), 1 )".
*
                     IF( ( sva( p ).LT.rootbig ) .AND.     
     $                    ( sva( p ).GT.rootsfmin ) ) THEN
                        sva( p ) = dznrm2( m, a( 1, p ), 1 )
                     ELSE
                        temp1 = zero
                        aapp = one
                        CALL zlassq( m, a( 1, p ), 1, temp1, aapp )
                        sva( p ) = temp1*dsqrt( aapp )
                     END IF
                     aapp = sva( p )
                  ELSE
                     aapp = sva( p )
                  END IF
*
                  IF( aapp.GT.zero ) THEN
*
                     pskipped = 0
*
                     DO 2002 q = p + 1, min0( igl+kbl-1, n )
*
                        aaqq = sva( q )
*
                        IF( aaqq.GT.zero ) THEN
*
                           aapp0 = aapp
                           IF( aaqq.GE.one ) THEN
                              rotok = ( small*aapp ).LE.aaqq
                              IF( aapp.LT.( big / aaqq ) ) THEN
                                 aapq = ( zdotc( m, a( 1, p ), 1, 
     $                                   a( 1, q ), 1 ) / aaqq ) / aapp
                              ELSE
                                 CALL zcopy( m, a( 1, p ), 1,   
     $                                        cwork(n+1), 1 )
                                 CALL zlascl( 'G', 0, 0, aapp, one, 
     $                                m, 1, cwork(n+1), lda, ierr )
                                 aapq = zdotc( m, cwork(n+1), 1,
     $                                   a( 1, q ), 1 ) / aaqq
                              END IF
                           ELSE
                              rotok = aapp.LE.( aaqq / small )
                              IF( aapp.GT.( small / aaqq ) ) THEN
                                 aapq = ( zdotc( m, a( 1, p ), 1, 
     $                                    a( 1, q ), 1 ) / aaqq ) / aapp
                              ELSE
                                 CALL zcopy( m, a( 1, q ), 1,   
     $                                        cwork(n+1), 1 )
                                 CALL zlascl( 'G', 0, 0, aaqq,
     $                                         one, m, 1,
     $                                         cwork(n+1), lda, ierr )
                                 aapq = zdotc( m, a(1, p ), 1,
     $                                   cwork(n+1), 1 ) / aapp
                              END IF
                           END IF
*
*                           AAPQ = AAPQ * DCONJG( CWORK(p) ) * CWORK(q) 
                           aapq1  = -abs(aapq) 
                           mxaapq = dmax1( mxaapq, -aapq1 )
*
*        TO rotate or NOT to rotate, THAT is the question ...
*
                           IF( abs( aapq1 ).GT.tol ) THEN
*
*           .. rotate
*[RTD]      ROTATED = ROTATED + ONE
*
                              IF( ir1.EQ.0 ) THEN
                                 notrot = 0
                                 pskipped = 0
                                 iswrot = iswrot + 1
                              END IF
*
                              IF( rotok ) THEN
*
                                ompq = aapq / abs(aapq) 
                                aqoap = aaqq / aapp
                                 apoaq = aapp / aaqq
                                 theta = -half*abs( aqoap-apoaq )/aapq1
*
                                 IF( abs( theta ).GT.bigtheta ) THEN
* 
                                    t  = half / theta
                                    cs = one

                                    CALL zrot( m, a(1,p), 1, a(1,q), 1,
     $                                          cs, dconjg(ompq)*t )
                                    IF ( rsvec ) THEN
                                        CALL zrot( mvl, v(1,p), 1, 
     $                                  v(1,q), 1, cs, dconjg(ompq)*t )
                                    END IF
                                    
                                    sva( q ) = aaqq*dsqrt( dmax1( zero, 
     $                                          one+t*apoaq*aapq1 ) )
                                    aapp = aapp*dsqrt( dmax1( zero,
     $                                          one-t*aqoap*aapq1 ) )
                                    mxsinj = dmax1( mxsinj, abs( t ) )
*
                                 ELSE
*
*                 .. choose correct signum for THETA and rotate
*
                                    thsign = -dsign( one, aapq1 )
                                    t = one / ( theta+thsign*       
     $                                   dsqrt( one+theta*theta ) )
                                    cs = dsqrt( one / ( one+t*t ) )
                                    sn = t*cs
*
                                    mxsinj = dmax1( mxsinj, abs( sn ) )
                                    sva( q ) = aaqq*dsqrt( dmax1( zero,
     $                                          one+t*apoaq*aapq1 ) )
                                    aapp = aapp*dsqrt( dmax1( zero,  
     $                                      one-t*aqoap*aapq1 ) )
*
                                    CALL zrot( m, a(1,p), 1, a(1,q), 1,
     $                                          cs, dconjg(ompq)*sn )
                                    IF ( rsvec ) THEN
                                        CALL zrot( mvl, v(1,p), 1, 
     $                                  v(1,q), 1, cs, dconjg(ompq)*sn )
                                    END IF 
                                 END IF 
                                 cwork(p) = -cwork(q) * ompq 
*
                                 ELSE
*              .. have to use modified Gram-Schmidt like transformation
                                 CALL zcopy( m, a( 1, p ), 1,
     $                                       cwork(n+1), 1 )
                                 CALL zlascl( 'G', 0, 0, aapp, one, m,
     $                                        1, cwork(n+1), lda,
     $                                        ierr )
                                 CALL zlascl( 'G', 0, 0, aaqq, one, m,
     $                                        1, a( 1, q ), lda, ierr )
                                 CALL zaxpy( m, -aapq, cwork(n+1), 1,
     $                                       a( 1, q ), 1 )
                                 CALL zlascl( 'G', 0, 0, one, aaqq, m,
     $                                        1, a( 1, q ), lda, ierr )
                                 sva( q ) = aaqq*dsqrt( dmax1( zero,
     $                                      one-aapq1*aapq1 ) )
                                 mxsinj = dmax1( mxsinj, sfmin )
                              END IF
*           END IF ROTOK THEN ... ELSE
*
*           In the case of cancellation in updating SVA(q), SVA(p)
*           recompute SVA(q), SVA(p).
*
                              IF( ( sva( q ) / aaqq )**2.LE.rooteps )
     $                            THEN
                                 IF( ( aaqq.LT.rootbig ) .AND.
     $                               ( aaqq.GT.rootsfmin ) ) THEN
                                    sva( q ) = dznrm2( m, a( 1, q ), 1 )
                                 ELSE
                                    t = zero
                                    aaqq = one
                                    CALL zlassq( m, a( 1, q ), 1, t,
     $                                           aaqq )
                                    sva( q ) = t*dsqrt( aaqq )
                                 END IF
                              END IF
                              IF( ( aapp / aapp0 ).LE.rooteps ) THEN
                                 IF( ( aapp.LT.rootbig ) .AND.
     $                               ( aapp.GT.rootsfmin ) ) THEN
                                    aapp = dznrm2( m, a( 1, p ), 1 )
                                 ELSE
                                    t = zero
                                    aapp = one
                                    CALL zlassq( m, a( 1, p ), 1, t,
     $                                           aapp )
                                    aapp = t*dsqrt( aapp )
                                 END IF
                                 sva( p ) = aapp
                              END IF
*
                           ELSE
*                             A(:,p) and A(:,q) already numerically orthogonal
                              IF( ir1.EQ.0 )notrot = notrot + 1
*[RTD]      SKIPPED  = SKIPPED + 1
                              pskipped = pskipped + 1
                           END IF
                        ELSE
*                          A(:,q) is zero column
                           IF( ir1.EQ.0 )notrot = notrot + 1
                           pskipped = pskipped + 1
                        END IF
*
                        IF( ( i.LE.swband ) .AND.
     $                      ( pskipped.GT.rowskip ) ) THEN
                           IF( ir1.EQ.0 )aapp = -aapp
                           notrot = 0
                           GO TO 2103
                        END IF
*
 2002                CONTINUE
*     END q-LOOP
*
 2103                CONTINUE
*     bailed out of q-loop
*
                     sva( p ) = aapp
*
                  ELSE
                     sva( p ) = aapp
                     IF( ( ir1.EQ.0 ) .AND. ( aapp.EQ.zero ) )
     $                   notrot = notrot + min0( igl+kbl-1, n ) - p
                  END IF
*
 2001          CONTINUE
*     end of the p-loop
*     end of doing the block ( ibr, ibr )
 1002       CONTINUE
*     end of ir1-loop
*
* ... go to the off diagonal blocks
*
            igl = ( ibr-1 )*kbl + 1
*
            DO 2010 jbc = ibr + 1, nbl
*
               jgl = ( jbc-1 )*kbl + 1
*
*        doing the block at ( ibr, jbc )
*
               ijblsk = 0
               DO 2100 p = igl, min0( igl+kbl-1, n )
*
                  aapp = sva( p )
                  IF( aapp.GT.zero ) THEN
*
                     pskipped = 0
*
                     DO 2200 q = jgl, min0( jgl+kbl-1, n )
*
                        aaqq = sva( q )
                        IF( aaqq.GT.zero ) THEN
                           aapp0 = aapp
*
*     .. M x 2 Jacobi SVD ..
*
*        Safe Gram matrix computation
*
                           IF( aaqq.GE.one ) THEN
                              IF( aapp.GE.aaqq ) THEN
                                 rotok = ( small*aapp ).LE.aaqq
                              ELSE
                                 rotok = ( small*aaqq ).LE.aapp
                              END IF
                              IF( aapp.LT.( big / aaqq ) ) THEN
                                 aapq = ( zdotc( m, a( 1, p ), 1, 
     $                                  a( 1, q ), 1 ) / aaqq ) / aapp
                              ELSE
                                 CALL zcopy( m, a( 1, p ), 1,
     $                                       cwork(n+1), 1 )
                                 CALL zlascl( 'G', 0, 0, aapp,
     $                                        one, m, 1,
     $                                        cwork(n+1), lda, ierr )
                                 aapq = zdotc( m, cwork(n+1), 1,
     $                                  a( 1, q ), 1 ) / aaqq
                              END IF
                           ELSE
                              IF( aapp.GE.aaqq ) THEN
                                 rotok = aapp.LE.( aaqq / small )
                              ELSE
                                 rotok = aaqq.LE.( aapp / small )
                              END IF
                              IF( aapp.GT.( small / aaqq ) ) THEN
                                 aapq = ( zdotc( m, a( 1, p ), 1, 
     $                                   a( 1, q ), 1 ) / aaqq ) / aapp
                              ELSE
                                 CALL zcopy( m, a( 1, q ), 1,
     $                                       cwork(n+1), 1 )
                                 CALL zlascl( 'G', 0, 0, aaqq,
     $                                        one, m, 1,
     $                                        cwork(n+1), lda, ierr )
                                 aapq = zdotc( m, a( 1, p ), 1,
     $                                  cwork(n+1),  1 ) / aapp
                              END IF
                           END IF
*
*                           AAPQ = AAPQ * DCONJG(CWORK(p))*CWORK(q)   
                           aapq1  = -abs(aapq)
                           mxaapq = dmax1( mxaapq, -aapq1 )
*
*        TO rotate or NOT to rotate, THAT is the question ...
*
                           IF( abs( aapq1 ).GT.tol ) THEN
                              notrot = 0
*[RTD]      ROTATED  = ROTATED + 1
                              pskipped = 0
                              iswrot = iswrot + 1
*
                              IF( rotok ) THEN
*
                                         ompq = aapq / abs(aapq) 
                                 aqoap = aaqq / aapp
                                 apoaq = aapp / aaqq
                                 theta = -half*abs( aqoap-apoaq )/ aapq1
                                 IF( aaqq.GT.aapp0 )theta = -theta
*
                                 IF( abs( theta ).GT.bigtheta ) THEN
                                    t  = half / theta
                                    cs = one 
                                    CALL zrot( m, a(1,p), 1, a(1,q), 1,
     $                                          cs, dconjg(ompq)*t )
                                    IF( rsvec ) THEN
                                        CALL zrot( mvl, v(1,p), 1, 
     $                                  v(1,q), 1, cs, dconjg(ompq)*t )
                                    END IF
                                    sva( q ) = aaqq*dsqrt( dmax1( zero,
     $                                         one+t*apoaq*aapq1 ) )
                                    aapp = aapp*dsqrt( dmax1( zero,
     $                                     one-t*aqoap*aapq1 ) )
                                    mxsinj = dmax1( mxsinj, abs( t ) )
                                 ELSE
*
*                 .. choose correct signum for THETA and rotate
*
                                    thsign = -dsign( one, aapq1 )
                                    IF( aaqq.GT.aapp0 )thsign = -thsign
                                    t = one / ( theta+thsign*
     $                                  dsqrt( one+theta*theta ) )
                                    cs = dsqrt( one / ( one+t*t ) )
                                    sn = t*cs
                                    mxsinj = dmax1( mxsinj, abs( sn ) )
                                    sva( q ) = aaqq*dsqrt( dmax1( zero,
     $                                         one+t*apoaq*aapq1 ) )
                                    aapp = aapp*dsqrt( dmax1( zero,  
     $                                         one-t*aqoap*aapq1 ) )
*
                                    CALL zrot( m, a(1,p), 1, a(1,q), 1,
     $                                          cs, dconjg(ompq)*sn ) 
                                    IF( rsvec ) THEN
                                        CALL zrot( mvl, v(1,p), 1, 
     $                                  v(1,q), 1, cs, dconjg(ompq)*sn )
                                    END IF
                                 END IF
                                 cwork(p) = -cwork(q) * ompq 
*
                              ELSE
*              .. have to use modified Gram-Schmidt like transformation
                               IF( aapp.GT.aaqq ) THEN
                                    CALL zcopy( m, a( 1, p ), 1,
     $                                          cwork(n+1), 1 )
                                    CALL zlascl( 'G', 0, 0, aapp, one,
     $                                           m, 1, cwork(n+1),lda,
     $                                           ierr )
                                    CALL zlascl( 'G', 0, 0, aaqq, one,
     $                                           m, 1, a( 1, q ), lda,
     $                                           ierr )
                                    CALL zaxpy( m, -aapq, cwork(n+1),
     $                                          1, a( 1, q ), 1 )
                                    CALL zlascl( 'G', 0, 0, one, aaqq,
     $                                           m, 1, a( 1, q ), lda,
     $                                           ierr )
                                    sva( q ) = aaqq*dsqrt( dmax1( zero,
     $                                         one-aapq1*aapq1 ) )
                                    mxsinj = dmax1( mxsinj, sfmin )
                               ELSE
                                   CALL zcopy( m, a( 1, q ), 1,
     $                                          cwork(n+1), 1 )
                                    CALL zlascl( 'G', 0, 0, aaqq, one,
     $                                           m, 1, cwork(n+1),lda,
     $                                           ierr )
                                    CALL zlascl( 'G', 0, 0, aapp, one,
     $                                           m, 1, a( 1, p ), lda,
     $                                           ierr )
                                    CALL zaxpy( m, -dconjg(aapq), 
     $                                   cwork(n+1), 1, a( 1, p ), 1 )
                                    CALL zlascl( 'G', 0, 0, one, aapp,
     $                                           m, 1, a( 1, p ), lda,
     $                                           ierr )
                                    sva( p ) = aapp*dsqrt( dmax1( zero,
     $                                         one-aapq1*aapq1 ) )
                                    mxsinj = dmax1( mxsinj, sfmin )
                               END IF
                              END IF
*           END IF ROTOK THEN ... ELSE
*
*           In the case of cancellation in updating SVA(q), SVA(p)
*           .. recompute SVA(q), SVA(p)
                              IF( ( sva( q ) / aaqq )**2.LE.rooteps )
     $                            THEN
                                 IF( ( aaqq.LT.rootbig ) .AND.
     $                               ( aaqq.GT.rootsfmin ) ) THEN
                                    sva( q ) = dznrm2( m, a( 1, q ), 1)
                                  ELSE
                                    t = zero
                                    aaqq = one
                                    CALL zlassq( m, a( 1, q ), 1, t,
     $                                           aaqq )
                                    sva( q ) = t*dsqrt( aaqq )
                                 END IF
                              END IF
                              IF( ( aapp / aapp0 )**2.LE.rooteps ) THEN
                                 IF( ( aapp.LT.rootbig ) .AND.
     $                               ( aapp.GT.rootsfmin ) ) THEN
                                    aapp = dznrm2( m, a( 1, p ), 1 )
                                 ELSE
                                    t = zero
                                    aapp = one
                                    CALL zlassq( m, a( 1, p ), 1, t,
     $                                           aapp )
                                    aapp = t*dsqrt( aapp )
                                 END IF
                                 sva( p ) = aapp
                              END IF
*              end of OK rotation
                           ELSE
                              notrot = notrot + 1
*[RTD]      SKIPPED  = SKIPPED  + 1
                              pskipped = pskipped + 1
                              ijblsk = ijblsk + 1
                           END IF
                        ELSE
                           notrot = notrot + 1
                           pskipped = pskipped + 1
                           ijblsk = ijblsk + 1
                        END IF
*
                        IF( ( i.LE.swband ) .AND. ( ijblsk.GE.blskip ) )
     $                      THEN
                           sva( p ) = aapp
                           notrot = 0
                           GO TO 2011
                        END IF
                        IF( ( i.LE.swband ) .AND.
     $                      ( pskipped.GT.rowskip ) ) THEN
                           aapp = -aapp
                           notrot = 0
                           GO TO 2203
                        END IF
*
 2200                CONTINUE
*        end of the q-loop
 2203                CONTINUE
*
                     sva( p ) = aapp
*
                  ELSE
*
                     IF( aapp.EQ.zero )notrot = notrot +
     $                   min0( jgl+kbl-1, n ) - jgl + 1
                     IF( aapp.LT.zero )notrot = 0
*
                  END IF
*
 2100          CONTINUE
*     end of the p-loop
 2010       CONTINUE
*     end of the jbc-loop
 2011       CONTINUE
*2011 bailed out of the jbc-loop
            DO 2012 p = igl, min0( igl+kbl-1, n )
               sva( p ) = abs( sva( p ) )
 2012       CONTINUE
***
 2000    CONTINUE
*2000 :: end of the ibr-loop
*
*     .. update SVA(N)
         IF( ( sva( n ).LT.rootbig ) .AND. ( sva( n ).GT.rootsfmin ) )
     $       THEN
            sva( n ) = dznrm2( m, a( 1, n ), 1 )
         ELSE
            t = zero
            aapp = one
            CALL zlassq( m, a( 1, n ), 1, t, aapp )
            sva( n ) = t*dsqrt( aapp )
         END IF
*
*     Additional steering devices
*
         IF( ( i.LT.swband ) .AND. ( ( mxaapq.LE.roottol ) .OR.
     $       ( iswrot.LE.n ) ) )swband = i
*
         IF( ( i.GT.swband+1 ) .AND. ( mxaapq.LT.dsqrt( dble( n ) )*
     $       tol ) .AND. ( dble( n )*mxaapq*mxsinj.LT.tol ) ) THEN
            GO TO 1994
         END IF
*
         IF( notrot.GE.emptsw )GO TO 1994
*
 1993 CONTINUE
*     end i=1:NSWEEP loop
*
* #:( Reaching this point means that the procedure has not converged.
      info = nsweep - 1
      GO TO 1995
*
 1994 CONTINUE
* #:) Reaching this point means numerical convergence after the i-th
*     sweep.
*
      info = 0
* #:) INFO = 0 confirms successful iterations.
 1995 CONTINUE
*
*     Sort the singular values and find how many are above
*     the underflow threshold.
*
      n2 = 0
      n4 = 0
      DO 5991 p = 1, n - 1
         q = idamax( n-p+1, sva( p ), 1 ) + p - 1
         IF( p.NE.q ) THEN
            temp1 = sva( p )
            sva( p ) = sva( q )
            sva( q ) = temp1
            CALL zswap( m, a( 1, p ), 1, a( 1, q ), 1 )
            IF( rsvec )CALL zswap( mvl, v( 1, p ), 1, v( 1, q ), 1 )
         END IF
         IF( sva( p ).NE.zero ) THEN
            n4 = n4 + 1
            IF( sva( p )*skl.GT.sfmin )n2 = n2 + 1
         END IF
 5991 CONTINUE
      IF( sva( n ).NE.zero ) THEN
         n4 = n4 + 1
         IF( sva( n )*skl.GT.sfmin )n2 = n2 + 1
      END IF
*
*     Normalize the left singular vectors.
*
      IF( lsvec .OR. uctol ) THEN
         DO 1998 p = 1, n2
            CALL zdscal( m, one / sva( p ), a( 1, p ), 1 )
 1998    CONTINUE
      END IF
*
*     Scale the product of Jacobi rotations.
*
      IF( rsvec ) THEN
            DO 2399 p = 1, n
               temp1 = one / dznrm2( mvl, v( 1, p ), 1 )
               CALL zdscal( mvl, temp1, v( 1, p ), 1 )
 2399       CONTINUE
      END IF
*
*     Undo scaling, if necessary (and possible).
      IF( ( ( skl.GT.one ) .AND. ( sva( 1 ).LT.( big / skl ) ) ) 
     $    .OR. ( ( skl.LT.one ) .AND. ( sva( max( n2, 1 ) ) .GT.
     $    ( sfmin / skl ) ) ) ) THEN
         DO 2400 p = 1, n
            sva( p ) = skl*sva( p )
 2400    CONTINUE
         skl = one
      END IF
*
      rwork( 1 ) = skl
*     The singular values of A are SKL*SVA(1:N). If SKL.NE.ONE
*     then some of the singular values may overflow or underflow and
*     the spectrum is given in this factored representation.
*
      rwork( 2 ) = dble( n4 )
*     N4 is the number of computed nonzero singular values of A.
*
      rwork( 3 ) = dble( n2 )
*     N2 is the number of singular values of A greater than SFMIN.
*     If N2<N, SVA(N2:N) contains ZEROS and/or denormalized numbers
*     that may carry some information.
*
      rwork( 4 ) = dble( i )
*     i is the index of the last sweep before declaring convergence.
*
      rwork( 5 ) = mxaapq
*     MXAAPQ is the largest absolute value of scaled pivots in the
*     last sweep
*
      rwork( 6 ) = mxsinj
*     MXSINJ is the largest absolute value of the sines of Jacobi angles
*     in the last sweep
*
      RETURN
*     ..
*     .. END OF ZGESVJ
*     ..
      END