cgesvj.f

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*> \brief <b> CGESVJ </b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
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*
*  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.
*>          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] LWORK
*> \verbatim
*>          LWORK is INTEGER.
*>          Length of CWORK, LWORK >= M+N.
*> \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.
*>         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[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:  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 complexGEcomputational
*
*> \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
      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
      EXTERNAL           slamch
      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' )
*
      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.( m+n ) .AND. ( .NOT.lquery ) ) THEN
         info = -13
      ELSE IF( lrwork.LT.max( n, 6 ) .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) = m + n
         rwork(1) = max( n, 6 )
         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 = 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)
.LE.                              IF( ( SVA( q ) / AAQQ )**2ROOTEPS )
     $                            THEN
.LT..AND.                                 IF( ( AAQQROOTBIG ) 
.GT.     $                               ( AAQQROOTSFMIN ) ) 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
.LE.                              IF( ( AAPP / AAPP0 )**2ROOTEPS ) THEN
.LT..AND.                                 IF( ( AAPPROOTBIG ) 
.GT.     $                               ( AAPPROOTSFMIN ) ) 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
*
.LE..AND..GE.                        IF( ( iSWBAND )  ( IJBLSKBLSKIP ) )
     $                      THEN
                           SVA( p ) = AAPP
                           NOTROT = 0
                           GO TO 2011
                        END IF
.LE..AND.                        IF( ( iSWBAND ) 
.GT.     $                      ( PSKIPPEDROWSKIP ) ) THEN
                           AAPP = -AAPP
                           NOTROT = 0
                           GO TO 2203
                        END IF
*
 2200                CONTINUE
*        end of the q-loop
 2203                CONTINUE
*
                     SVA( p ) = AAPP
*
                  ELSE
*
.EQ.                     IF( AAPPZERO )NOTROT = NOTROT +
     $                   MIN( jgl+KBL-1, N ) - jgl + 1
.LT.                     IF( AAPPZERO )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)
.LT..AND..GT.         IF( ( SVA( N )ROOTBIG )  ( SVA( N )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
*
.LT..AND..LE..OR.         IF( ( iSWBAND )  ( ( MXAAPQROOTTOL ) 
.LE.     $       ( ISWROTN ) ) )SWBAND = i
*
.GT..AND..LT.         IF( ( iSWBAND+1 )  ( MXAAPQSQRT( REAL( N ) )*
.AND..LT.     $       TOL )  ( REAL( N )*MXAAPQ*MXSINJTOL ) ) THEN
            GO TO 1994
         END IF
*
.GE.         IF( NOTROTEMPTSW )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
.NE.         IF( pq ) 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
.NE.         IF( SVA( p )ZERO ) THEN
            N4 = N4 + 1
.GT.            IF( SVA( p )*SKLSFMIN )N2 = N2 + 1
         END IF
 5991 CONTINUE
.NE.      IF( SVA( N )ZERO ) THEN
         N4 = N4 + 1
.GT.         IF( SVA( N )*SKLSFMIN )N2 = N2 + 1
      END IF
*
*     Normalize the left singular vectors.
*
.OR.      IF( LSVEC  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).
.GT..AND..LT.      IF( ( ( SKLONE )  ( SVA( 1 )( BIG / SKL ) ) )
.OR..LT..AND..GT.     $     ( ( SKLONE )  ( SVA( MAX( N2, 1 ) ) 
     $    ( 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
*