OpenBLAS/lapack-netlib/SRC/cgesvj.f

*> \brief <b> CGESVJ </b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
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*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE CGESVJ( 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            A( LDA, * ),  V( LDV, * ), CWORK( LWORK )
*       REAL               RWORK( LRWORK ),  SVA( N )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> CGESVJ 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' or 'F': 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=SQRT(M), EPS=SLAMCH('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' or 'J': 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/SLAMCH('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 array, dimension (LDA,N)
*>          On entry, the M-by-N matrix A.
*>          On exit,
*>          If JOBU = 'U' .OR. JOBU = 'C':
*>                 If INFO = 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 SLAMCH('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 = 'C'),
*>                 see the description of JOBU.
*>                 If INFO > 0,
*>                 the procedure CGESVJ 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 = 'N':
*>                 If INFO = 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 > 0 :
*>                 the procedure CGESVJ 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 REAL array, dimension (N)
*>          On exit,
*>          If INFO = 0 :
*>          depending on the value SCALE = RWORK(1), we have:
*>                 If SCALE = 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 > 0 :
*>          the procedure CGESVJ 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 = 'A', then the product of Jacobi rotations in CGESVJ
*>          is applied to the first MV rows of V. See the description of JOBV.
*> \endverbatim
*>
*> \param[in,out] V
*> \verbatim
*>          V is COMPLEX 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 >= 1.
*>          If JOBV = 'V', then LDV >= max(1,N).
*>          If JOBV = 'A', then LDV >= max(1,MV) .
*> \endverbatim
*>
*> \param[in,out] CWORK
*> \verbatim
*>          CWORK is COMPLEX array, dimension (max(1,LWORK))
*>          Used as workspace.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER.
*>          Length of CWORK.
*>          LWORK >= 1, if MIN(M,N) = 0, and LWORK >= M+N, otherwise.
*>
*>          If on entry LWORK = -1, then a workspace query is assumed and
*>          no computation is done; CWORK(1) is set to the minial (and optimal)
*>          length of CWORK.
*> \endverbatim
*>
*> \param[in,out] RWORK
*> \verbatim
*>          RWORK is REAL array, dimension (max(6,LRWORK))
*>          On entry,
*>          If JOBU = '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=SLAMCH('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 CGESVJ did
*>                    not converge, as it can be used to estimate whether
*>                    the output is still 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 >= 1, if MIN(M,N) = 0, and LRWORK >= MAX(6,N), otherwise
*>
*>         If on entry LRWORK = -1, then a workspace query is assumed and
*>         no computation is done; RWORK(1) is set to the minial (and optimal)
*>         length of RWORK.
*> \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:  CGESVJ 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.
*
*> \ingroup gesvj
*
*> \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 tuning 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 Contributor:
*  ==================
*>
*> \verbatim
*>
*>  ============
*>
*>  Zlatko Drmac (Zagreb, Croatia)
*>
*> \endverbatim
*
*> \par References:
*  ================
*>
*> \verbatim
*>
*> [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 CGESVJ( JOBA, JOBU, JOBV, M, N, A, LDA, SVA, MV, V,
     $                   LDV, CWORK, LWORK, RWORK, LRWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
      IMPLICIT NONE
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, LDV, LWORK, LRWORK, M, MV, N
      CHARACTER*1        JOBA, JOBU, JOBV
*     ..
*     .. Array Arguments ..
      COMPLEX            A( LDA, * ),  V( LDV, * ), CWORK( LWORK )
      REAL               RWORK( LRWORK ), SVA( N )
*     ..
*
*  =====================================================================
*
*     .. Local Parameters ..
      REAL         ZERO,         HALF,         ONE
      PARAMETER  ( ZERO = 0.0E0, HALF = 0.5E0, ONE = 1.0E0)
      COMPLEX      CZERO,                  CONE
      PARAMETER  ( CZERO = (0.0E0, 0.0E0), CONE = (1.0E0, 0.0E0) )
      INTEGER      NSWEEP
      PARAMETER  ( NSWEEP = 30 )
*     ..
*     .. Local Scalars ..
      COMPLEX    AAPQ, OMPQ
      REAL       AAPP, AAPP0, AAPQ1, AAQQ, APOAQ, AQOAP, BIG,
     $           BIGTHETA, CS, CTOL, EPSLN, 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,
     $           MINMN, LWMIN, LRWMIN
      LOGICAL    APPLV, GOSCALE, LOWER, LQUERY, LSVEC, NOSCALE, ROTOK,
     $           RSVEC, UCTOL, UPPER
*     ..
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC ABS, MAX, MIN, CONJG, REAL, SIGN, SQRT
*     ..
*     .. External Functions ..
*     ..
*     from BLAS
      REAL               SCNRM2
      COMPLEX            CDOTC
      EXTERNAL           CDOTC, SCNRM2
      INTEGER            ISAMAX
      EXTERNAL           ISAMAX
*     from LAPACK
      REAL               SLAMCH, SROUNDUP_LWORK
      EXTERNAL           SLAMCH, SROUNDUP_LWORK
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
*     ..
*     from BLAS
      EXTERNAL           CCOPY, CROT, CSSCAL, CSWAP, CAXPY
*     from LAPACK
      EXTERNAL           CLASCL, CLASET, CLASSQ, SLASCL, XERBLA
      EXTERNAL           CGSVJ0, CGSVJ1
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
      LSVEC = LSAME( JOBU, 'U' ) .OR. LSAME( JOBU, 'F' )
      UCTOL = LSAME( JOBU, 'C' )
      RSVEC = LSAME( JOBV, 'V' ) .OR. LSAME( JOBV, 'J' )
      APPLV = LSAME( JOBV, 'A' )
      UPPER = LSAME( JOBA, 'U' )
      LOWER = LSAME( JOBA, 'L' )
*
      MINMN = MIN( M, N )
      IF( MINMN.EQ.0 ) THEN
         LWMIN  = 1
         LRWMIN = 1
      ELSE
         LWMIN  = M + N
         LRWMIN = MAX( 6, N )
      END IF
*
      LQUERY = ( LWORK.EQ.-1 ) .OR. ( LRWORK.EQ.-1 )
      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.LWMIN .AND. ( .NOT.LQUERY ) ) THEN
         INFO = -13
      ELSE IF( LRWORK.LT.LRWMIN .AND. ( .NOT.LQUERY ) ) THEN
         INFO = -15
      ELSE
         INFO = 0
      END IF
*
*     #:(
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'CGESVJ', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         CWORK( 1 ) = SROUNDUP_LWORK( LWMIN )
         RWORK( 1 ) = SROUNDUP_LWORK( LRWMIN )
         RETURN
      END IF
*
* #:) Quick return for void matrix
*
      IF( MINMN.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 = SQRT( REAL( M ) )
         ELSE
            CTOL = REAL( M )
         END IF
      END IF
*     ... and the machine dependent parameters are
*[!]  (Make sure that SLAMCH() works properly on the target machine.)
*
      EPSLN = SLAMCH( 'Epsilon' )
      ROOTEPS = SQRT( EPSLN )
      SFMIN = SLAMCH( 'SafeMinimum' )
      ROOTSFMIN = SQRT( SFMIN )
      SMALL = SFMIN / EPSLN
*      BIG = SLAMCH( 'Overflow' )
      BIG     = ONE  / SFMIN
      ROOTBIG = ONE / ROOTSFMIN
*     LARGE = BIG / SQRT( REAL( M*N ) )
      BIGTHETA = ONE / ROOTEPS
*
      TOL = CTOL*EPSLN
      ROOTTOL = SQRT( TOL )
*
      IF( REAL( M )*EPSLN.GE.ONE ) THEN
         INFO = -4
         CALL XERBLA( 'CGESVJ', -INFO )
         RETURN
      END IF
*
*     Initialize the right singular vector matrix.
*
      IF( RSVEC ) THEN
         MVL = N
         CALL CLASET( '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 / SQRT( REAL( M )*REAL( 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 CLASSQ( M-p+1, A( p, p ), 1, AAPP, AAQQ )
            IF( AAPP.GT.BIG ) THEN
               INFO = -6
               CALL XERBLA( 'CGESVJ', -INFO )
               RETURN
            END IF
            AAQQ = SQRT( 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 CLASSQ( p, A( 1, p ), 1, AAPP, AAQQ )
            IF( AAPP.GT.BIG ) THEN
               INFO = -6
               CALL XERBLA( 'CGESVJ', -INFO )
               RETURN
            END IF
            AAQQ = SQRT( 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 CLASSQ( M, A( 1, p ), 1, AAPP, AAQQ )
            IF( AAPP.GT.BIG ) THEN
               INFO = -6
               CALL XERBLA( 'CGESVJ', -INFO )
               RETURN
            END IF
            AAQQ = SQRT( 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 = MIN( AAQQ, SVA( p ) )
         AAPP = MAX( AAPP, SVA( p ) )
 4781 CONTINUE
*
* #:) Quick return for zero matrix
*
      IF( AAPP.EQ.ZERO ) THEN
         IF( LSVEC )CALL CLASET( '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 CLASCL( '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 = SQRT( SFMIN / EPSLN )
      TEMP1 = SQRT( BIG / REAL( N ) )
      IF( ( AAPP.LE.SN ) .OR. ( AAQQ.GE.TEMP1 ) .OR.
     $    ( ( SN.LE.AAQQ ) .AND. ( AAPP.LE.TEMP1 ) ) ) THEN
         TEMP1 = MIN( BIG, TEMP1 / AAPP )
*         AAQQ  = AAQQ*TEMP1
*         AAPP  = AAPP*TEMP1
      ELSE IF( ( AAQQ.LE.SN ) .AND. ( AAPP.LE.TEMP1 ) ) THEN
         TEMP1 = MIN( SN / AAQQ, BIG / ( AAPP*SQRT( REAL( N ) ) ) )
*         AAQQ  = AAQQ*TEMP1
*         AAPP  = AAPP*TEMP1
      ELSE IF( ( AAQQ.GE.SN ) .AND. ( AAPP.GE.TEMP1 ) ) THEN
         TEMP1 = MAX( SN / AAQQ, TEMP1 / AAPP )
*         AAQQ  = AAQQ*TEMP1
*         AAPP  = AAPP*TEMP1
      ELSE IF( ( AAQQ.LE.SN ) .AND. ( AAPP.GE.TEMP1 ) ) THEN
         TEMP1 = MIN( SN / AAQQ, BIG / ( SQRT( REAL( N ) )*AAPP ) )
*         AAQQ  = AAQQ*TEMP1
*         AAPP  = AAPP*TEMP1
      ELSE
         TEMP1 = ONE
      END IF
*
*     Scale, if necessary
*
      IF( TEMP1.NE.ONE ) THEN
         CALL SLASCL( 'G', 0, 0, ONE, TEMP1, N, 1, SVA, N, IERR )
      END IF
      SKL = TEMP1*SKL
      IF( SKL.NE.ONE ) THEN
         CALL CLASCL( 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 CGESVJ is used as a computational routine in the preconditioned
*     Jacobi SVD algorithm CGEJSV. 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 = MIN( 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 = MIN( 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.MAX( 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 CGSVJ0( 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 CGSVJ0( 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 CGSVJ1( 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 CGSVJ0( 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 CGSVJ0( JOBV, M, N4, A, LDA, CWORK, SVA, MVL, V, LDV,
     $                   EPSLN, SFMIN, TOL, 1, CWORK( N+1 ), LWORK-N,
     $                   IERR )
*
            CALL CGSVJ1( 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 CGSVJ0( JOBV, N4, N4, A, LDA, CWORK, SVA, MVL, V, LDV,
     $                   EPSLN, SFMIN, TOL, 2, CWORK( N+1 ), LWORK-N,
     $                   IERR )
*
            CALL CGSVJ0( 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 CGSVJ1( JOBV, N2, N2, N4, A, LDA, CWORK, SVA, MVL, V,
     $                   LDV, EPSLN, SFMIN, TOL, 1, CWORK( N+1 ),
     $                   LWORK-N, IERR )
*
            CALL CGSVJ0( 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, MIN( LKAHEAD, NBL-ibr )
*
               igl = igl + ir1*KBL
*
               DO 2001 p = igl, MIN( igl+KBL-1, N-1 )
*
*     .. de Rijk's pivoting
*
                  q = ISAMAX( N-p+1, SVA( p ), 1 ) + p - 1
                  IF( p.NE.q ) THEN
                     CALL CSWAP( M, A( 1, p ), 1, A( 1, q ), 1 )
                     IF( RSVEC )CALL CSWAP( 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 SCNRM2(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, SCNRM2 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 = SCNRM2( M, A(1,p), 1 )".
*
                     IF( ( SVA( p ).LT.ROOTBIG ) .AND.
     $                    ( SVA( p ).GT.ROOTSFMIN ) ) THEN
                        SVA( p ) = SCNRM2( M, A( 1, p ), 1 )
                     ELSE
                        TEMP1 = ZERO
                        AAPP = ONE
                        CALL CLASSQ( M, A( 1, p ), 1, TEMP1, AAPP )
                        SVA( p ) = TEMP1*SQRT( AAPP )
                     END IF
                     AAPP = SVA( p )
                  ELSE
                     AAPP = SVA( p )
                  END IF
*
                  IF( AAPP.GT.ZERO ) THEN
*
                     PSKIPPED = 0
*
                     DO 2002 q = p + 1, MIN( 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 = ( CDOTC( M, A( 1, p ), 1,
     $                                   A( 1, q ), 1 ) / AAQQ ) / AAPP
                              ELSE
                                 CALL CCOPY( M, A( 1, p ), 1,
     $                                        CWORK(N+1), 1 )
                                 CALL CLASCL( 'G', 0, 0, AAPP, ONE,
     $                                M, 1, CWORK(N+1), LDA, IERR )
                                 AAPQ = CDOTC( 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 = ( CDOTC( M, A( 1, p ), 1,
     $                                    A( 1, q ), 1 ) / AAPP ) / AAQQ
                              ELSE
                                 CALL CCOPY( M, A( 1, q ), 1,
     $                                        CWORK(N+1), 1 )
                                 CALL CLASCL( 'G', 0, 0, AAQQ,
     $                                         ONE, M, 1,
     $                                         CWORK(N+1), LDA, IERR )
                                 AAPQ = CDOTC( M, A(1, p ), 1,
     $                                   CWORK(N+1), 1 ) / AAPP
                              END IF
                           END IF
*
*                           AAPQ = AAPQ * CONJG( CWORK(p) ) * CWORK(q)
                           AAPQ1  = -ABS(AAPQ)
                           MXAAPQ = MAX( MXAAPQ, -AAPQ1 )
*
*        TO rotate or NOT to rotate, THAT is the question ...
*
                           IF( ABS( AAPQ1 ).GT.TOL ) THEN
                               OMPQ = AAPQ / ABS(AAPQ)
*
*           .. rotate
*[RTD]      ROTATED = ROTATED + ONE
*
                              IF( ir1.EQ.0 ) THEN
                                 NOTROT = 0
                                 PSKIPPED = 0
                                 ISWROT = ISWROT + 1
                              END IF
*
                              IF( ROTOK ) THEN
*
                                 AQOAP = AAQQ / AAPP
                                 APOAQ = AAPP / AAQQ
                                 THETA = -HALF*ABS( AQOAP-APOAQ )/AAPQ1
*
                                 IF( ABS( THETA ).GT.BIGTHETA ) THEN
*
                                    T  = HALF / THETA
                                    CS = ONE

                                    CALL CROT( M, A(1,p), 1, A(1,q), 1,
     $                                          CS, CONJG(OMPQ)*T )
                                    IF ( RSVEC ) THEN
                                        CALL CROT( MVL, V(1,p), 1,
     $                                  V(1,q), 1, CS, CONJG(OMPQ)*T )
                                    END IF

                                    SVA( q ) = AAQQ*SQRT( MAX( ZERO,
     $                                          ONE+T*APOAQ*AAPQ1 ) )
                                    AAPP = AAPP*SQRT( MAX( ZERO,
     $                                          ONE-T*AQOAP*AAPQ1 ) )
                                    MXSINJ = MAX( MXSINJ, ABS( T ) )
*
                                 ELSE
*
*                 .. choose correct signum for THETA and rotate
*
                                    THSIGN = -SIGN( ONE, AAPQ1 )
                                    T = ONE / ( THETA+THSIGN*
     $                                   SQRT( ONE+THETA*THETA ) )
                                    CS = SQRT( ONE / ( ONE+T*T ) )
                                    SN = T*CS
*
                                    MXSINJ = MAX( MXSINJ, ABS( SN ) )
                                    SVA( q ) = AAQQ*SQRT( MAX( ZERO,
     $                                          ONE+T*APOAQ*AAPQ1 ) )
                                    AAPP = AAPP*SQRT( MAX( ZERO,
     $                                      ONE-T*AQOAP*AAPQ1 ) )
*
                                    CALL CROT( M, A(1,p), 1, A(1,q), 1,
     $                                          CS, CONJG(OMPQ)*SN )
                                    IF ( RSVEC ) THEN
                                        CALL CROT( MVL, V(1,p), 1,
     $                                  V(1,q), 1, CS, CONJG(OMPQ)*SN )
                                    END IF
                                 END IF
                                 CWORK(p) = -CWORK(q) * OMPQ
*
                                 ELSE
*              .. have to use modified Gram-Schmidt like transformation
                                 CALL CCOPY( M, A( 1, p ), 1,
     $                                       CWORK(N+1), 1 )
                                 CALL CLASCL( 'G', 0, 0, AAPP, ONE, M,
     $                                        1, CWORK(N+1), LDA,
     $                                        IERR )
                                 CALL CLASCL( 'G', 0, 0, AAQQ, ONE, M,
     $                                        1, A( 1, q ), LDA, IERR )
                                 CALL CAXPY( M, -AAPQ, CWORK(N+1), 1,
     $                                       A( 1, q ), 1 )
                                 CALL CLASCL( 'G', 0, 0, ONE, AAQQ, M,
     $                                        1, A( 1, q ), LDA, IERR )
                                 SVA( q ) = AAQQ*SQRT( MAX( ZERO,
     $                                      ONE-AAPQ1*AAPQ1 ) )
                                 MXSINJ = MAX( 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 ) = SCNRM2( M, A( 1, q ), 1 )
                                 ELSE
                                    T = ZERO
                                    AAQQ = ONE
                                    CALL CLASSQ( M, A( 1, q ), 1, T,
     $                                           AAQQ )
                                    SVA( q ) = T*SQRT( AAQQ )
                                 END IF
                              END IF
                              IF( ( AAPP / AAPP0 ).LE.ROOTEPS ) THEN
                                 IF( ( AAPP.LT.ROOTBIG ) .AND.
     $                               ( AAPP.GT.ROOTSFMIN ) ) THEN
                                    AAPP = SCNRM2( M, A( 1, p ), 1 )
                                 ELSE
                                    T = ZERO
                                    AAPP = ONE
                                    CALL CLASSQ( M, A( 1, p ), 1, T,
     $                                           AAPP )
                                    AAPP = T*SQRT( 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 + MIN( 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, MIN( igl+KBL-1, N )
*
                  AAPP = SVA( p )
                  IF( AAPP.GT.ZERO ) THEN
*
                     PSKIPPED = 0
*
                     DO 2200 q = jgl, MIN( 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 = ( CDOTC( M, A( 1, p ), 1,
     $                                  A( 1, q ), 1 ) / AAQQ ) / AAPP
                              ELSE
                                 CALL CCOPY( M, A( 1, p ), 1,
     $                                       CWORK(N+1), 1 )
                                 CALL CLASCL( 'G', 0, 0, AAPP,
     $                                        ONE, M, 1,
     $                                        CWORK(N+1), LDA, IERR )
                                 AAPQ = CDOTC( 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 = ( CDOTC( M, A( 1, p ), 1,
     $                                 A( 1, q ), 1 ) / MAX(AAQQ,AAPP) )
     $                                                / MIN(AAQQ,AAPP)
                              ELSE
                                 CALL CCOPY( M, A( 1, q ), 1,
     $                                       CWORK(N+1), 1 )
                                 CALL CLASCL( 'G', 0, 0, AAQQ,
     $                                        ONE, M, 1,
     $                                        CWORK(N+1), LDA, IERR )
                                 AAPQ = CDOTC( M, A( 1, p ), 1,
     $                                  CWORK(N+1),  1 ) / AAPP
                              END IF
                           END IF
*
*                           AAPQ = AAPQ * CONJG(CWORK(p))*CWORK(q)
                           AAPQ1  = -ABS(AAPQ)
                           MXAAPQ = MAX( MXAAPQ, -AAPQ1 )
*
*        TO rotate or NOT to rotate, THAT is the question ...
*
                           IF( ABS( AAPQ1 ).GT.TOL ) THEN
                              OMPQ = AAPQ / ABS(AAPQ)
                              NOTROT = 0
*[RTD]      ROTATED  = ROTATED + 1
                              PSKIPPED = 0
                              ISWROT = ISWROT + 1
*
                              IF( ROTOK ) THEN
*
                                 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 CROT( M, A(1,p), 1, A(1,q), 1,
     $                                          CS, CONJG(OMPQ)*T )
                                    IF( RSVEC ) THEN
                                        CALL CROT( MVL, V(1,p), 1,
     $                                  V(1,q), 1, CS, CONJG(OMPQ)*T )
                                    END IF
                                    SVA( q ) = AAQQ*SQRT( MAX( ZERO,
     $                                         ONE+T*APOAQ*AAPQ1 ) )
                                    AAPP = AAPP*SQRT( MAX( ZERO,
     $                                     ONE-T*AQOAP*AAPQ1 ) )
                                    MXSINJ = MAX( MXSINJ, ABS( T ) )
                                 ELSE
*
*                 .. choose correct signum for THETA and rotate
*
                                    THSIGN = -SIGN( ONE, AAPQ1 )
                                    IF( AAQQ.GT.AAPP0 )THSIGN = -THSIGN
                                    T = ONE / ( THETA+THSIGN*
     $                                  SQRT( ONE+THETA*THETA ) )
                                    CS = SQRT( ONE / ( ONE+T*T ) )
                                    SN = T*CS
                                    MXSINJ = MAX( MXSINJ, ABS( SN ) )
                                    SVA( q ) = AAQQ*SQRT( MAX( ZERO,
     $                                         ONE+T*APOAQ*AAPQ1 ) )
                                    AAPP = AAPP*SQRT( MAX( ZERO,
     $                                         ONE-T*AQOAP*AAPQ1 ) )
*
                                    CALL CROT( M, A(1,p), 1, A(1,q), 1,
     $                                          CS, CONJG(OMPQ)*SN )
                                    IF( RSVEC ) THEN
                                        CALL CROT( MVL, V(1,p), 1,
     $                                  V(1,q), 1, CS, CONJG(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 CCOPY( M, A( 1, p ), 1,
     $                                          CWORK(N+1), 1 )
                                    CALL CLASCL( 'G', 0, 0, AAPP, ONE,
     $                                           M, 1, CWORK(N+1),LDA,
     $                                           IERR )
                                    CALL CLASCL( 'G', 0, 0, AAQQ, ONE,
     $                                           M, 1, A( 1, q ), LDA,
     $                                           IERR )
                                    CALL CAXPY( M, -AAPQ, CWORK(N+1),
     $                                          1, A( 1, q ), 1 )
                                    CALL CLASCL( 'G', 0, 0, ONE, AAQQ,
     $                                           M, 1, A( 1, q ), LDA,
     $                                           IERR )
                                    SVA( q ) = AAQQ*SQRT( MAX( ZERO,
     $                                         ONE-AAPQ1*AAPQ1 ) )
                                    MXSINJ = MAX( MXSINJ, SFMIN )
                               ELSE
                                   CALL CCOPY( M, A( 1, q ), 1,
     $                                          CWORK(N+1), 1 )
                                    CALL CLASCL( 'G', 0, 0, AAQQ, ONE,
     $                                           M, 1, CWORK(N+1),LDA,
     $                                           IERR )
                                    CALL CLASCL( 'G', 0, 0, AAPP, ONE,
     $                                           M, 1, A( 1, p ), LDA,
     $                                           IERR )
                                    CALL CAXPY( M, -CONJG(AAPQ),
     $                                   CWORK(N+1), 1, A( 1, p ), 1 )
                                    CALL CLASCL( 'G', 0, 0, ONE, AAPP,
     $                                           M, 1, A( 1, p ), LDA,
     $                                           IERR )
                                    SVA( p ) = AAPP*SQRT( MAX( ZERO,
     $                                         ONE-AAPQ1*AAPQ1 ) )
                                    MXSINJ = MAX( 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 ) = SCNRM2( M, A( 1, q ), 1)
                                  ELSE
                                    T = ZERO
                                    AAQQ = ONE
                                    CALL CLASSQ( M, A( 1, q ), 1, T,
     $                                           AAQQ )
                                    SVA( q ) = T*SQRT( AAQQ )
                                 END IF
                              END IF
                              IF( ( AAPP / AAPP0 )**2.LE.ROOTEPS ) THEN
                                 IF( ( AAPP.LT.ROOTBIG ) .AND.
     $                               ( AAPP.GT.ROOTSFMIN ) ) THEN
                                    AAPP = SCNRM2( M, A( 1, p ), 1 )
                                 ELSE
                                    T = ZERO
                                    AAPP = ONE
                                    CALL CLASSQ( M, A( 1, p ), 1, T,
     $                                           AAPP )
                                    AAPP = T*SQRT( 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 +
     $                   MIN( 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, MIN( 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 ) = SCNRM2( M, A( 1, N ), 1 )
         ELSE
            T = ZERO
            AAPP = ONE
            CALL CLASSQ( M, A( 1, N ), 1, T, AAPP )
            SVA( N ) = T*SQRT( 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.SQRT( REAL( N ) )*
     $       TOL ) .AND. ( REAL( 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 = ISAMAX( N-p+1, SVA( p ), 1 ) + p - 1
         IF( p.NE.q ) THEN
            TEMP1 = SVA( p )
            SVA( p ) = SVA( q )
            SVA( q ) = TEMP1
            CALL CSWAP( M, A( 1, p ), 1, A( 1, q ), 1 )
            IF( RSVEC )CALL CSWAP( 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, N4
*           CALL CSSCAL( M, ONE / SVA( p ), A( 1, p ), 1 )
            CALL CLASCL( 'G',0,0, SVA(p), ONE, M, 1, A(1,p), M, IERR )
 1998    CONTINUE
      END IF
*
*     Scale the product of Jacobi rotations.
*
      IF( RSVEC ) THEN
            DO 2399 p = 1, N
               TEMP1 = ONE / SCNRM2( MVL, V( 1, p ), 1 )
               CALL CSSCAL( 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 ) = REAL( N4 )
*     N4 is the number of computed nonzero singular values of A.
*
      RWORK( 3 ) = REAL( 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 ) = REAL( 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 CGESVJ
*     ..
      END
*