cgejsv.f

Go to the documentation of this file.

*> \brief \b CGEJSV
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at 
*            http://www.netlib.org/lapack/explore-html/ 
*
*> \htmlonly
*> Download CGEJSV + dependencies 
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cgejsv.f"> 
*> [TGZ]</a> 
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cgejsv.f"> 
*> [ZIP]</a> 
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cgejsv.f"> 
*> [TXT]</a>
*> \endhtmlonly 
*
*  Definition:
*  ===========
*
*     SUBROUTINE CGEJSV( JOBA, JOBU, JOBV, JOBR, JOBT, JOBP,
*                         M, N, A, LDA, SVA, U, LDU, V, LDV,
*                         CWORK, LWORK, RWORK, LRWORK, IWORK, INFO )
* 
*     .. Scalar Arguments ..
*     IMPLICIT    NONE
*     INTEGER     INFO, LDA, LDU, LDV, LWORK, M, N
*     ..
*     .. Array Arguments ..
*     COMPLEX     A( LDA, * ),  U( LDU, * ), V( LDV, * ), CWORK( LWORK )
*     REAL        SVA( N ), RWORK( LRWORK )      
*     INTEGER     IWORK( * )
*     CHARACTER*1 JOBA, JOBP, JOBR, JOBT, JOBU, JOBV
*       ..
*  
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> CGEJSV computes the singular value decomposition (SVD) of a complex M-by-N
*> matrix [A], where M >= N. The SVD of [A] is written as
*>
*>              [A] = [U] * [SIGMA] * [V]^*,
*>
*> where [SIGMA] is an N-by-N (M-by-N) matrix which is zero except for its N
*> diagonal elements, [U] is an M-by-N (or M-by-M) unitary 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. The matrices [U] and [V]
*> are computed and stored in the arrays U and V, respectively. The diagonal
*> of [SIGMA] is computed and stored in the array SVA.
*> \endverbatim
*>
*>  Arguments:
*>  ==========
*>
*> \param[in] JOBA
*> \verbatim
*>          JOBA is CHARACTER*1
*>         Specifies the level of accuracy:
*>       = 'C': This option works well (high relative accuracy) if A = B * D,
*>              with well-conditioned B and arbitrary diagonal matrix D.
*>              The accuracy cannot be spoiled by COLUMN scaling. The
*>              accuracy of the computed output depends on the condition of
*>              B, and the procedure aims at the best theoretical accuracy.
*>              The relative error max_{i=1:N}|d sigma_i| / sigma_i is
*>              bounded by f(M,N)*epsilon* cond(B), independent of D.
*>              The input matrix is preprocessed with the QRF with column
*>              pivoting. This initial preprocessing and preconditioning by
*>              a rank revealing QR factorization is common for all values of
*>              JOBA. Additional actions are specified as follows:
*>       = 'E': Computation as with 'C' with an additional estimate of the
*>              condition number of B. It provides a realistic error bound.
*>       = 'F': If A = D1 * C * D2 with ill-conditioned diagonal scalings
*>              D1, D2, and well-conditioned matrix C, this option gives
*>              higher accuracy than the 'C' option. If the structure of the
*>              input matrix is not known, and relative accuracy is
*>              desirable, then this option is advisable. The input matrix A
*>              is preprocessed with QR factorization with FULL (row and
*>              column) pivoting.
*>       = 'G'  Computation as with 'F' with an additional estimate of the
*>              condition number of B, where A=D*B. If A has heavily weighted
*>              rows, then using this condition number gives too pessimistic
*>              error bound.
*>       = 'A': Small singular values are the noise and the matrix is treated
*>              as numerically rank defficient. The error in the computed
*>              singular values is bounded by f(m,n)*epsilon*||A||.
*>              The computed SVD A = U * S * V^* restores A up to
*>              f(m,n)*epsilon*||A||.
*>              This gives the procedure the licence to discard (set to zero)
*>              all singular values below N*epsilon*||A||.
*>       = 'R': Similar as in 'A'. Rank revealing property of the initial
*>              QR factorization is used do reveal (using triangular factor)
*>              a gap sigma_{r+1} < epsilon * sigma_r in which case the
*>              numerical RANK is declared to be r. The SVD is computed with
*>              absolute error bounds, but more accurately than with 'A'.
*> \endverbatim
*> 
*> \param[in] JOBU
*> \verbatim
*>          JOBU is CHARACTER*1
*>         Specifies whether to compute the columns of U:
*>       = 'U': N columns of U are returned in the array U.
*>       = 'F': full set of M left sing. vectors is returned in the array U.
*>       = 'W': U may be used as workspace of length M*N. See the description
*>              of U.
*>       = 'N': U is not computed.
*> \endverbatim
*> 
*> \param[in] JOBV
*> \verbatim
*>          JOBV is CHARACTER*1
*>         Specifies whether to compute the matrix V:
*>       = 'V': N columns of V are returned in the array V; Jacobi rotations
*>              are not explicitly accumulated.
*>       = 'J': N columns of V are returned in the array V, but they are
*>              computed as the product of Jacobi rotations. This option is
*>              allowed only if JOBU .NE. 'N', i.e. in computing the full SVD.
*>       = 'W': V may be used as workspace of length N*N. See the description
*>              of V.
*>       = 'N': V is not computed.
*> \endverbatim
*> 
*> \param[in] JOBR
*> \verbatim
*>          JOBR is CHARACTER*1
*>         Specifies the RANGE for the singular values. Issues the licence to
*>         set to zero small positive singular values if they are outside
*>         specified range. If A .NE. 0 is scaled so that the largest singular
*>         value of c*A is around SQRT(BIG), BIG=SLAMCH('O'), then JOBR issues
*>         the licence to kill columns of A whose norm in c*A is less than
*>         SQRT(SFMIN) (for JOBR.EQ.'R'), or less than SMALL=SFMIN/EPSLN,
*>         where SFMIN=SLAMCH('S'), EPSLN=SLAMCH('E').
*>       = 'N': Do not kill small columns of c*A. This option assumes that
*>              BLAS and QR factorizations and triangular solvers are
*>              implemented to work in that range. If the condition of A
*>              is greater than BIG, use CGESVJ.
*>       = 'R': RESTRICTED range for sigma(c*A) is [SQRT(SFMIN), SQRT(BIG)]
*>              (roughly, as described above). This option is recommended.
*>                                             ===========================
*>         For computing the singular values in the FULL range [SFMIN,BIG]
*>         use CGESVJ.
*> \endverbatim
*> 
*> \param[in] JOBT
*> \verbatim
*>          JOBT is CHARACTER*1
*>         If the matrix is square then the procedure may determine to use
*>         transposed A if A^* seems to be better with respect to convergence.
*>         If the matrix is not square, JOBT is ignored. This is subject to
*>         changes in the future.
*>         The decision is based on two values of entropy over the adjoint
*>         orbit of A^* * A. See the descriptions of WORK(6) and WORK(7).
*>       = 'T': transpose if entropy test indicates possibly faster
*>         convergence of Jacobi process if A^* is taken as input. If A is
*>         replaced with A^*, then the row pivoting is included automatically.
*>       = 'N': do not speculate.
*>         This option can be used to compute only the singular values, or the
*>         full SVD (U, SIGMA and V). For only one set of singular vectors
*>         (U or V), the caller should provide both U and V, as one of the
*>         matrices is used as workspace if the matrix A is transposed.
*>         The implementer can easily remove this constraint and make the
*>         code more complicated. See the descriptions of U and V.
*> \endverbatim
*> 
*> \param[in] JOBP
*> \verbatim
*>          JOBP is CHARACTER*1
*>         Issues the licence to introduce structured perturbations to drown
*>         denormalized numbers. This licence should be active if the
*>         denormals are poorly implemented, causing slow computation,
*>         especially in cases of fast convergence (!). For details see [1,2].
*>         For the sake of simplicity, this perturbations are included only
*>         when the full SVD or only the singular values are requested. The
*>         implementer/user can easily add the perturbation for the cases of
*>         computing one set of singular vectors.
*>       = 'P': introduce perturbation
*>       = 'N': do not perturb
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>         The number of rows of the input matrix A.  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.
*> \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,
*>          - For WORK(1)/WORK(2) = ONE: The singular values of A. During the
*>            computation SVA contains Euclidean column norms of the
*>            iterated matrices in the array A.
*>          - For WORK(1) .NE. WORK(2): The singular values of A are
*>            (WORK(1)/WORK(2)) * SVA(1:N). This factored form is used if
*>            sigma_max(A) overflows or if small singular values have been
*>            saved from underflow by scaling the input matrix A.
*>          - If JOBR='R' then some of the singular values may be returned
*>            as exact zeros obtained by "set to zero" because they are
*>            below the numerical rank threshold or are denormalized numbers.
*> \endverbatim
*>
*> \param[out] U
*> \verbatim
*>          U is COMPLEX array, dimension ( LDU, N ) or ( LDU, M )
*>          If JOBU = 'U', then U contains on exit the M-by-N matrix of
*>                         the left singular vectors.
*>          If JOBU = 'F', then U contains on exit the M-by-M matrix of
*>                         the left singular vectors, including an ONB
*>                         of the orthogonal complement of the Range(A).
*>          If JOBU = 'W'  .AND. (JOBV.EQ.'V' .AND. JOBT.EQ.'T' .AND. M.EQ.N),
*>                         then U is used as workspace if the procedure
*>                         replaces A with A^*. In that case, [V] is computed
*>                         in U as left singular vectors of A^* and then
*>                         copied back to the V array. This 'W' option is just
*>                         a reminder to the caller that in this case U is
*>                         reserved as workspace of length N*N.
*>          If JOBU = 'N'  U is not referenced, unless JOBT='T'.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*>          LDU is INTEGER
*>          The leading dimension of the array U,  LDU >= 1.
*>          IF  JOBU = 'U' or 'F' or 'W',  then LDU >= M.
*> \endverbatim
*>
*> \param[out] V
*> \verbatim
*>          V is COMPLEX array, dimension ( LDV, N )
*>          If JOBV = 'V', 'J' then V contains on exit the N-by-N matrix of
*>                         the right singular vectors;
*>          If JOBV = 'W', AND (JOBU.EQ.'U' AND JOBT.EQ.'T' AND M.EQ.N),
*>                         then V is used as workspace if the pprocedure
*>                         replaces A with A^*. In that case, [U] is computed
*>                         in V as right singular vectors of A^* and then
*>                         copied back to the U array. This 'W' option is just
*>                         a reminder to the caller that in this case V is
*>                         reserved as workspace of length N*N.
*>          If JOBV = 'N'  V is not referenced, unless JOBT='T'.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*>          LDV is INTEGER
*>          The leading dimension of the array V,  LDV >= 1.
*>          If JOBV = 'V' or 'J' or 'W', then LDV >= N.
*> \endverbatim
*>
*> \param[out] CWORK
*> \verbatim
*>          CWORK is COMPLEX array, dimension at least LWORK.     
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          Length of CWORK to confirm proper allocation of workspace.
*>          LWORK depends on the job:
*>
*>          1. If only SIGMA is needed ( JOBU.EQ.'N', JOBV.EQ.'N' ) and
*>            1.1 .. no scaled condition estimate required (JOBA.NE.'E'.AND.JOBA.NE.'G'):
*>               LWORK >= 2*N+1. This is the minimal requirement.
*>               ->> For optimal performance (blocked code) the optimal value
*>               is LWORK >= N + (N+1)*NB. Here NB is the optimal
*>               block size for CGEQP3 and CGEQRF.
*>               In general, optimal LWORK is computed as 
*>               LWORK >= max(N+LWORK(CGEQP3),N+LWORK(CGEQRF)).        
*>            1.2. .. an estimate of the scaled condition number of A is
*>               required (JOBA='E', or 'G'). In this case, LWORK the minimal
*>               requirement is LWORK >= N*N + 3*N.
*>               ->> For optimal performance (blocked code) the optimal value 
*>               is LWORK >= max(N+(N+1)*NB, N*N+3*N).
*>               In general, the optimal length LWORK is computed as
*>               LWORK >= max(N+LWORK(CGEQP3),N+LWORK(CGEQRF), 
*>                                                     N+N*N+LWORK(CPOCON)).
*>
*>          2. If SIGMA and the right singular vectors are needed (JOBV.EQ.'V'),
*>             (JOBU.EQ.'N')
*>            -> the minimal requirement is LWORK >= 3*N.
*>            -> For optimal performance, LWORK >= max(N+(N+1)*NB, 3*N,2*N+N*NB),
*>               where NB is the optimal block size for CGEQP3, CGEQRF, CGELQF,
*>               CUNMLQ. In general, the optimal length LWORK is computed as
*>               LWORK >= max(N+LWORK(CGEQP3), N+LWORK(CPOCON), N+LWORK(CGESVJ),
*>                       N+LWORK(CGELQF), 2*N+LWORK(CGEQRF), N+LWORK(CUNMLQ)).
*>
*>          3. If SIGMA and the left singular vectors are needed
*>            -> the minimal requirement is LWORK >= 3*N.
*>            -> For optimal performance:
*>               if JOBU.EQ.'U' :: LWORK >= max(3*N, N+(N+1)*NB, 2*N+N*NB),
*>               where NB is the optimal block size for CGEQP3, CGEQRF, CUNMQR.
*>               In general, the optimal length LWORK is computed as
*>               LWORK >= max(N+LWORK(CGEQP3),N+LWORK(CPOCON),
*>                        2*N+LWORK(CGEQRF), N+LWORK(CUNMQR)). 
*>               
*>          4. If the full SVD is needed: (JOBU.EQ.'U' or JOBU.EQ.'F') and 
*>            4.1. if JOBV.EQ.'V'  
*>               the minimal requirement is LWORK >= 5*N+2*N*N. 
*>            4.2. if JOBV.EQ.'J' the minimal requirement is 
*>               LWORK >= 4*N+N*N.
*>            In both cases, the allocated CWORK can accommodate blocked runs
*>            of CGEQP3, CGEQRF, CGELQF, CUNMQR, CUNMLQ.
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*>          RWORK is REAL array, dimension at least LRWORK.
*>          On exit,
*>          RWORK(1) = Determines the scaling factor SCALE = RWORK(2) / RWORK(1)
*>                    such that SCALE*SVA(1:N) are the computed singular values
*>                    of A. (See the description of SVA().)
*>          RWORK(2) = See the description of RWORK(1).
*>          RWORK(3) = SCONDA is an estimate for the condition number of
*>                    column equilibrated A. (If JOBA .EQ. 'E' or 'G')
*>                    SCONDA is an estimate of SQRT(||(R^* * R)^(-1)||_1).
*>                    It is computed using SPOCON. It holds
*>                    N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA
*>                    where R is the triangular factor from the QRF of A.
*>                    However, if R is truncated and the numerical rank is
*>                    determined to be strictly smaller than N, SCONDA is
*>                    returned as -1, thus indicating that the smallest
*>                    singular values might be lost.
*>
*>          If full SVD is needed, the following two condition numbers are
*>          useful for the analysis of the algorithm. They are provied for
*>          a developer/implementer who is familiar with the details of
*>          the method.
*>
*>          RWORK(4) = an estimate of the scaled condition number of the
*>                    triangular factor in the first QR factorization.
*>          RWORK(5) = an estimate of the scaled condition number of the
*>                    triangular factor in the second QR factorization.
*>          The following two parameters are computed if JOBT .EQ. 'T'.
*>          They are provided for a developer/implementer who is familiar
*>          with the details of the method.
*>          RWORK(6) = the entropy of A^* * A :: this is the Shannon entropy
*>                    of diag(A^* * A) / Trace(A^* * A) taken as point in the
*>                    probability simplex.
*>          RWORK(7) = the entropy of A * A^*. (See the description of RWORK(6).)
*> \endverbatim
*>
*> \param[in] LRWORK
*> \verbatim
*>          LRWORK is INTEGER
*>          Length of RWORK to confirm proper allocation of workspace.
*>          LRWORK depends on the job:
*>
*>       1. If only singular values are requested i.e. if 
*>          LSAME(JOBU,'N') .AND. LSAME(JOBV,'N') 
*>          then:
*>          1.1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'),
*>          then LRWORK = max( 7, N + 2 * M ). 
*>          1.2. Otherwise, LRWORK  = max( 7, 2 * N ).
*>       2. If singular values with the right singular vectors are requested
*>          i.e. if 
*>          (LSAME(JOBV,'V').OR.LSAME(JOBV,'J')) .AND. 
*>          .NOT.(LSAME(JOBU,'U').OR.LSAME(JOBU,'F'))
*>          then:
*>          2.1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'),
*>          then LRWORK = max( 7, N + 2 * M ). 
*>          2.2. Otherwise, LRWORK  = max( 7, 2 * N ).      
*>       3. If singular values with the left singular vectors are requested, i.e. if    
*>          (LSAME(JOBU,'U').OR.LSAME(JOBU,'F')) .AND.
*>          .NOT.(LSAME(JOBV,'V').OR.LSAME(JOBV,'J'))
*>          then:
*>          3.1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'),
*>          then LRWORK = max( 7, N + 2 * M ). 
*>          3.2. Otherwise, LRWORK  = max( 7, 2 * N ).    
*>       4. If singular values with both the left and the right singular vectors 
*>          are requested, i.e. if     
*>          (LSAME(JOBU,'U').OR.LSAME(JOBU,'F')) .AND.
*>          (LSAME(JOBV,'V').OR.LSAME(JOBV,'J'))
*>          then:
*>          4.1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'),
*>          then LRWORK = max( 7, N + 2 * M ). 
*>          4.2. Otherwise, LRWORK  = max( 7, 2 * N ).    
*> \endverbatim
*>          
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, of dimension:
*>                If LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'), then 
*>                the dimension of IWORK is max( 3, 2 * N + M ).
*>                Otherwise, the dimension of IWORK is 
*>                -> max( 3, 2*N ) for full SVD
*>                -> max( 3, N ) for singular values only or singular
*>                   values with one set of singular vectors (left or right)
*>          On exit,
*>          IWORK(1) = the numerical rank determined after the initial
*>                     QR factorization with pivoting. See the descriptions
*>                     of JOBA and JOBR.
*>          IWORK(2) = the number of the computed nonzero singular values
*>          IWORK(3) = if nonzero, a warning message:
*>                     If IWORK(3).EQ.1 then some of the column norms of A
*>                     were denormalized floats. The requested high accuracy
*>                     is not warranted by the data.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>           < 0  : if INFO = -i, then the i-th argument had an illegal value.
*>           = 0 :  successfull exit;
*>           > 0 :  CGEJSV  did not converge in the maximal allowed number
*>                  of sweeps. The computed values may be inaccurate.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee 
*> \author Univ. of California Berkeley 
*> \author Univ. of Colorado Denver 
*> \author NAG Ltd. 
*
*> \date June 2016
*
*> \ingroup complexGEsing
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  CGEJSV implements a preconditioned Jacobi SVD algorithm. It uses CGEQP3,
*>  CGEQRF, and CGELQF as preprocessors and preconditioners. Optionally, an
*>  additional row pivoting can be used as a preprocessor, which in some
*>  cases results in much higher accuracy. An example is matrix A with the
*>  structure A = D1 * C * D2, where D1, D2 are arbitrarily ill-conditioned
*>  diagonal matrices and C is well-conditioned matrix. In that case, complete
*>  pivoting in the first QR factorizations provides accuracy dependent on the
*>  condition number of C, and independent of D1, D2. Such higher accuracy is
*>  not completely understood theoretically, but it works well in practice.
*>  Further, if A can be written as A = B*D, with well-conditioned B and some
*>  diagonal D, then the high accuracy is guaranteed, both theoretically and
*>  in software, independent of D. For more details see [1], [2].
*>     The computational range for the singular values can be the full range
*>  ( UNDERFLOW,OVERFLOW ), provided that the machine arithmetic and the BLAS
*>  & LAPACK routines called by CGEJSV are implemented to work in that range.
*>  If that is not the case, then the restriction for safe computation with
*>  the singular values in the range of normalized IEEE numbers is that the
*>  spectral condition number kappa(A)=sigma_max(A)/sigma_min(A) does not
*>  overflow. This code (CGEJSV) is best used in this restricted range,
*>  meaning that singular values of magnitude below ||A||_2 / SLAMCH('O') are
*>  returned as zeros. See JOBR for details on this.
*>     Further, this implementation is somewhat slower than the one described
*>  in [1,2] due to replacement of some non-LAPACK components, and because
*>  the choice of some tuning parameters in the iterative part (CGESVJ) is
*>  left to the implementer on a particular machine.
*>     The rank revealing QR factorization (in this code: CGEQP3) should be
*>  implemented as in [3]. We have a new version of CGEQP3 under development
*>  that is more robust than the current one in LAPACK, with a cleaner cut in
*>  rank defficient cases. It will be available in the SIGMA library [4].
*>  If M is much larger than N, it is obvious that the inital QRF with
*>  column pivoting can be preprocessed by the QRF without pivoting. That
*>  well known trick is not used in CGEJSV because in some cases heavy row
*>  weighting can be treated with complete pivoting. The overhead in cases
*>  M much larger than N is then only due to pivoting, but the benefits in
*>  terms of accuracy have prevailed. The implementer/user can incorporate
*>  this extra QRF step easily. The implementer can also improve data movement
*>  (matrix transpose, matrix copy, matrix transposed copy) - this
*>  implementation of CGEJSV uses only the simplest, naive data movement.
*
*> \par Contributors:
*  ==================
*>
*>  Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)
*
*> \par References:
*  ================
*>
*> \verbatim
*>
*> [1] 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.
*> [2] 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.
*> [3] Z. Drmac and Z. Bujanovic: On the failure of rank-revealing QR
*>     factorization software - a case study.
*>     ACM Trans. Math. Softw. Vol. 35, No 2 (2008), pp. 1-28.
*>     LAPACK Working note 176.
*> [4] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV,
*>     QSVD, (H,K)-SVD computations.
*>     Department of Mathematics, University of Zagreb, 2008.
*> \endverbatim
*
*>  \par Bugs, examples and comments:
*   =================================
*>
*>  Please report all bugs and send interesting examples and/or comments to
*>  drmac@math.hr. Thank you.
*>
*  =====================================================================
      SUBROUTINE cgejsv( JOBA, JOBU, JOBV, JOBR, JOBT, JOBP,
     $                   m, n, a, lda, sva, u, ldu, v, ldv,
     $                   cwork, lwork, rwork, lrwork, iwork, info )
*
*  -- LAPACK computational routine (version 3.6.1) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     June 2016
*
*     .. Scalar Arguments ..
      IMPLICIT    NONE
      INTEGER     INFO, LDA, LDU, LDV, LWORK, LRWORK, M, N
*     ..
*     .. Array Arguments ..
      COMPLEX     A( lda, * ), U( ldu, * ), V( ldv, * ), CWORK( lwork )
      REAL        SVA( n ), RWORK( * )
      INTEGER     IWORK( * )
      CHARACTER*1 JOBA, JOBP, JOBR, JOBT, JOBU, JOBV
*     ..
*
*  ===========================================================================
*
*     .. Local Parameters ..
      REAL        ZERO,         ONE
      parameter( zero = 0.0e0, one = 1.0e0 )
      COMPLEX     CZERO,                    CONE
      parameter( czero = ( 0.0e0, 0.0e0 ), cone = ( 1.0e0, 0.0e0 ) )
*     ..
*     .. Local Scalars ..
      COMPLEX CTEMP
      REAL    AAPP,   AAQQ,   AATMAX, AATMIN, BIG,    BIG1,   COND_OK,
     $        condr1, condr2, entra,  entrat, epsln,  maxprj, scalem,
     $        sconda, sfmin,  small,  temp1,  uscal1, uscal2, xsc
      INTEGER IERR,   N1,     NR,     NUMRANK,        p, q,   WARNING
      LOGICAL ALMORT, DEFR,   ERREST, GOSCAL, JRACC,  KILL,   LSVEC,
     $        l2aber, l2kill, l2pert, l2rank, l2tran,
     $        noscal, rowpiv, rsvec,  transp
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC abs,  conjg, alog, amax1, amin1, cmplx, float,
     $          max0, min0, nint,  sign,  sqrt
*     ..
*     .. External Functions ..
      REAL      SLAMCH, SCNRM2
      INTEGER   ISAMAX, ICAMAX
      LOGICAL   LSAME
      EXTERNAL  isamax, icamax, lsame, slamch, scnrm2
*     ..
*     .. External Subroutines ..
      EXTERNAL  ccopy,  cgelqf, cgeqp3, cgeqrf, clacpy, clascl,
     $          slascl, claset, classq, slassq, claswp, cungqr, cunmlq,
     $          cunmqr, cpocon, sscal, csscal, cswap,  ctrsm,  xerbla
*
      EXTERNAL  cgesvj
*     ..
*
*     Test the input arguments
*
      lsvec  = lsame( jobu, 'U' ) .OR. lsame( jobu, 'F' )
      jracc  = lsame( jobv, 'J' )
      rsvec  = lsame( jobv, 'V' ) .OR. jracc
      rowpiv = lsame( joba, 'F' ) .OR. lsame( joba, 'G' )
      l2rank = lsame( joba, 'R' )
      l2aber = lsame( joba, 'A' )
      errest = lsame( joba, 'E' ) .OR. lsame( joba, 'G' )
      l2tran = lsame( jobt, 'T' )
      l2kill = lsame( jobr, 'R' )
      defr   = lsame( jobr, 'N' )
      l2pert = lsame( jobp, 'P' )
*
      IF ( .NOT.(rowpiv .OR. l2rank .OR. l2aber .OR.
     $     errest .OR. lsame( joba, 'C' ) )) THEN
         info = - 1
      ELSE IF ( .NOT.( lsvec  .OR. lsame( jobu, 'N' ) .OR.
     $                             lsame( jobu, 'W' )) ) THEN
         info = - 2
      ELSE IF ( .NOT.( rsvec .OR. lsame( jobv, 'N' ) .OR.
     $   lsame( jobv, 'W' )) .OR. ( jracc .AND. (.NOT.lsvec) ) ) THEN
         info = - 3
      ELSE IF ( .NOT. ( l2kill .OR. defr ) )    THEN
         info = - 4
      ELSE IF ( .NOT. ( l2tran .OR. lsame( jobt, 'N' ) ) ) THEN
         info = - 5
      ELSE IF ( .NOT. ( l2pert .OR. lsame( jobp, 'N' ) ) ) THEN
         info = - 6
      ELSE IF ( m .LT. 0 ) THEN
         info = - 7
      ELSE IF ( ( n .LT. 0 ) .OR. ( n .GT. m ) ) THEN
         info = - 8
      ELSE IF ( lda .LT. m ) THEN
         info = - 10
      ELSE IF ( lsvec .AND. ( ldu .LT. m ) ) THEN
         info = - 13
      ELSE IF ( rsvec .AND. ( ldv .LT. n ) ) THEN
         info = - 15
      ELSE IF ( (.NOT.(lsvec .OR. rsvec .OR. errest).AND.
     $                           (lwork .LT. 2*n+1)) .OR.
     $ (.NOT.(lsvec .OR. rsvec) .AND. errest .AND.
     $                         (lwork .LT. n*n+3*n)) .OR.
     $ (lsvec .AND. (.NOT.rsvec) .AND. (lwork .LT. 3*n))
     $ .OR.
     $ (rsvec .AND. (.NOT.lsvec) .AND. (lwork .LT. 3*n))
     $ .OR.
     $ (lsvec .AND. rsvec .AND. (.NOT.jracc) .AND. 
     $                          (lwork.LT.5*n+2*n*n))
     $ .OR. (lsvec .AND. rsvec .AND. jracc .AND.
     $                          lwork.LT.4*n+n*n))
     $   THEN
         info = - 17
      ELSE IF ( lrwork.LT. max0(n+2*m,7)) THEN
         info = -19 
      ELSE
*        #:)
         info = 0
      END IF
*
      IF ( info .NE. 0 ) THEN
*       #:(
         CALL xerbla( 'CGEJSV', - info )
         RETURN
      END IF
*
*     Quick return for void matrix (Y3K safe)
* #:)
      IF ( ( m .EQ. 0 ) .OR. ( n .EQ. 0 ) ) THEN
         iwork(1:3) = 0
         rwork(1:7) = 0
         RETURN
      ENDIF
*
*     Determine whether the matrix U should be M x N or M x M
*
      IF ( lsvec ) THEN
         n1 = n
         IF ( lsame( jobu, 'F' ) ) n1 = m
      END IF
*
*     Set numerical parameters
*
*!    NOTE: Make sure SLAMCH() does not fail on the target architecture.
*
      epsln = slamch('Epsilon')
      sfmin = slamch('SafeMinimum')
      small = sfmin / epsln
      big   = slamch('O')
*     BIG   = ONE / SFMIN
*
*     Initialize SVA(1:N) = diag( ||A e_i||_2 )_1^N
*
*(!)  If necessary, scale SVA() to protect the largest norm from
*     overflow. It is possible that this scaling pushes the smallest
*     column norm left from the underflow threshold (extreme case).
*
      scalem  = one / sqrt(float(m)*float(n))
      noscal  = .true.
      goscal  = .true.
      DO 1874 p = 1, n
         aapp = zero
         aaqq = one
         CALL classq( m, a(1,p), 1, aapp, aaqq )
         IF ( aapp .GT. big ) THEN
            info = - 9
            CALL xerbla( 'CGEJSV', -info )
            RETURN
         END IF
         aaqq = sqrt(aaqq)
         IF ( ( aapp .LT. (big / aaqq) ) .AND. noscal  ) THEN
            sva(p)  = aapp * aaqq
         ELSE
            noscal  = .false.
            sva(p)  = aapp * ( aaqq * scalem )
            IF ( goscal ) THEN
               goscal = .false.
               CALL sscal( p-1, scalem, sva, 1 )
            END IF
         END IF
 1874 CONTINUE
*
      IF ( noscal ) scalem = one
*
      aapp = zero
      aaqq = big
      DO 4781 p = 1, n
         aapp = amax1( aapp, sva(p) )
         IF ( sva(p) .NE. zero ) aaqq = amin1( aaqq, sva(p) )
 4781 CONTINUE
*
*     Quick return for zero M x N matrix
* #:)
      IF ( aapp .EQ. zero ) THEN
         IF ( lsvec ) CALL claset( 'G', m, n1, czero, cone, u, ldu )
         IF ( rsvec ) CALL claset( 'G', n, n,  czero, cone, v, ldv )
         rwork(1) = one
         rwork(2) = one
         IF ( errest ) rwork(3) = one
         IF ( lsvec .AND. rsvec ) THEN
            rwork(4) = one
            rwork(5) = one
         END IF
         IF ( l2tran ) THEN
            rwork(6) = zero
            rwork(7) = zero
         END IF
         iwork(1) = 0
         iwork(2) = 0
         iwork(3) = 0
         RETURN
      END IF
*
*     Issue warning if denormalized column norms detected. Override the
*     high relative accuracy request. Issue licence to kill columns
*     (set them to zero) whose norm is less than sigma_max / BIG (roughly).
* #:(
      warning = 0
      IF ( aaqq .LE. sfmin ) THEN
         l2rank = .true.
         l2kill = .true.
         warning = 1
      END IF
*
*     Quick return for one-column matrix
* #:)
      IF ( n .EQ. 1 ) THEN
*
         IF ( lsvec ) THEN
            CALL clascl( 'G',0,0,sva(1),scalem, m,1,a(1,1),lda,ierr )
            CALL clacpy( 'A', m, 1, a, lda, u, ldu )
*           computing all M left singular vectors of the M x 1 matrix
            IF ( n1 .NE. n  ) THEN
              CALL cgeqrf( m, n, u,ldu, cwork, cwork(n+1),lwork-n,ierr )
              CALL cungqr( m,n1,1, u,ldu,cwork,cwork(n+1),lwork-n,ierr )
              CALL ccopy( m, a(1,1), 1, u(1,1), 1 )
            END IF
         END IF
         IF ( rsvec ) THEN
             v(1,1) = cone
         END IF
         IF ( sva(1) .LT. (big*scalem) ) THEN
            sva(1)  = sva(1) / scalem
            scalem  = one
         END IF
         rwork(1) = one / scalem
         rwork(2) = one
         IF ( sva(1) .NE. zero ) THEN
            iwork(1) = 1
            IF ( ( sva(1) / scalem) .GE. sfmin ) THEN
               iwork(2) = 1
            ELSE
               iwork(2) = 0
            END IF
         ELSE
            iwork(1) = 0
            iwork(2) = 0
         END IF
         iwork(3) = 0 
         IF ( errest ) rwork(3) = one
         IF ( lsvec .AND. rsvec ) THEN
            rwork(4) = one
            rwork(5) = one
         END IF
         IF ( l2tran ) THEN
            rwork(6) = zero
            rwork(7) = zero
         END IF
         RETURN
*
      END IF
*
      transp = .false.
      l2tran = l2tran .AND. ( m .EQ. n )
*
      aatmax = -one
      aatmin =  big
      IF ( rowpiv .OR. l2tran ) THEN
*
*     Compute the row norms, needed to determine row pivoting sequence
*     (in the case of heavily row weighted A, row pivoting is strongly
*     advised) and to collect information needed to compare the
*     structures of A * A^* and A^* * A (in the case L2TRAN.EQ..TRUE.).
*
         IF ( l2tran ) THEN
            DO 1950 p = 1, m
               xsc   = zero
               temp1 = one
               CALL classq( n, a(p,1), lda, xsc, temp1 )
*              CLASSQ gets both the ell_2 and the ell_infinity norm
*              in one pass through the vector
               rwork(m+n+p)  = xsc * scalem
               rwork(n+p)    = xsc * (scalem*sqrt(temp1))
               aatmax = amax1( aatmax, rwork(n+p) )
               IF (rwork(n+p) .NE. zero) 
     $            aatmin = amin1(aatmin,rwork(n+p))
 1950       CONTINUE
         ELSE
            DO 1904 p = 1, m
               rwork(m+n+p) = scalem*abs( a(p,icamax(n,a(p,1),lda)) )
               aatmax = amax1( aatmax, rwork(m+n+p) )
               aatmin = amin1( aatmin, rwork(m+n+p) )
 1904       CONTINUE
         END IF
*
      END IF
*
*     For square matrix A try to determine whether A^*  would be  better
*     input for the preconditioned Jacobi SVD, with faster convergence.
*     The decision is based on an O(N) function of the vector of column
*     and row norms of A, based on the Shannon entropy. This should give
*     the right choice in most cases when the difference actually matters.
*     It may fail and pick the slower converging side.
*
      entra  = zero
      entrat = zero
      IF ( l2tran ) THEN
*
         xsc   = zero
         temp1 = one
         CALL slassq( n, sva, 1, xsc, temp1 )
         temp1 = one / temp1
*
         entra = zero
         DO 1113 p = 1, n
            big1  = ( ( sva(p) / xsc )**2 ) * temp1
            IF ( big1 .NE. zero ) entra = entra + big1 * alog(big1)
 1113    CONTINUE
         entra = - entra / alog(float(n))
*
*        Now, SVA().^2/Trace(A^* * A) is a point in the probability simplex.
*        It is derived from the diagonal of  A^* * A.  Do the same with the
*        diagonal of A * A^*, compute the entropy of the corresponding
*        probability distribution. Note that A * A^* and A^* * A have the
*        same trace.
*
         entrat = zero
         DO 1114 p = n+1, n+m
            big1 = ( ( rwork(p) / xsc )**2 ) * temp1
            IF ( big1 .NE. zero ) entrat = entrat + big1 * alog(big1)
 1114    CONTINUE
         entrat = - entrat / alog(float(m))
*
*        Analyze the entropies and decide A or A^*. Smaller entropy
*        usually means better input for the algorithm.
*
         transp = ( entrat .LT. entra )
         transp = .true.
*
*        If A^* is better than A, take the adjoint of A.
*
         IF ( transp ) THEN
*           In an optimal implementation, this trivial transpose
*           should be replaced with faster transpose.
            DO 1115 p = 1, n - 1
               a(p,p) = conjg(a(p,p)) 
               DO 1116 q = p + 1, n
                   ctemp = conjg(a(q,p))
                  a(q,p) = conjg(a(p,q))
                  a(p,q) = ctemp
 1116          CONTINUE
 1115       CONTINUE
            a(n,n) = conjg(a(n,n))
            DO 1117 p = 1, n
               rwork(m+n+p) = sva(p)
               sva(p)      = rwork(n+p)
*              previously computed row 2-norms are now column 2-norms 
*              of the transposed matrix               
 1117       CONTINUE
            temp1  = aapp
            aapp   = aatmax
            aatmax = temp1
            temp1  = aaqq
            aaqq   = aatmin
            aatmin = temp1
            kill   = lsvec
            lsvec  = rsvec
            rsvec  = kill
            IF ( lsvec ) n1 = n 
*
            rowpiv = .true.
         END IF
*
      END IF
*     END IF L2TRAN
*
*     Scale the matrix so that its maximal singular value remains less
*     than SQRT(BIG) -- the matrix is scaled so that its maximal column
*     has Euclidean norm equal to SQRT(BIG/N). The only reason to keep
*     SQRT(BIG) instead of BIG is the fact that CGEJSV uses LAPACK and
*     BLAS routines that, in some implementations, are not capable of
*     working in the full interval [SFMIN,BIG] and that they may provoke
*     overflows in the intermediate results. If the singular values spread
*     from SFMIN to BIG, then CGESVJ will compute them. So, in that case,
*     one should use CGESVJ instead of CGEJSV.
*
      big1   = sqrt( big )
      temp1  = sqrt( big / float(n) )
*
      CALL slascl( 'G', 0, 0, aapp, temp1, n, 1, sva, n, ierr )
      IF ( aaqq .GT. (aapp * sfmin) ) THEN
          aaqq = ( aaqq / aapp ) * temp1
      ELSE
          aaqq = ( aaqq * temp1 ) / aapp
      END IF
      temp1 = temp1 * scalem
      CALL clascl( 'G', 0, 0, aapp, temp1, m, n, a, lda, ierr )
*
*     To undo scaling at the end of this procedure, multiply the
*     computed singular values with USCAL2 / USCAL1.
*
      uscal1 = temp1
      uscal2 = aapp
*
      IF ( l2kill ) THEN
*        L2KILL enforces computation of nonzero singular values in
*        the restricted range of condition number of the initial A,
*        sigma_max(A) / sigma_min(A) approx. SQRT(BIG)/SQRT(SFMIN).
         xsc = sqrt( sfmin )
      ELSE
         xsc = small
*
*        Now, if the condition number of A is too big,
*        sigma_max(A) / sigma_min(A) .GT. SQRT(BIG/N) * EPSLN / SFMIN,
*        as a precaution measure, the full SVD is computed using CGESVJ
*        with accumulated Jacobi rotations. This provides numerically
*        more robust computation, at the cost of slightly increased run
*        time. Depending on the concrete implementation of BLAS and LAPACK
*        (i.e. how they behave in presence of extreme ill-conditioning) the
*        implementor may decide to remove this switch.
         IF ( ( aaqq.LT.sqrt(sfmin) ) .AND. lsvec .AND. rsvec ) THEN
            jracc = .true.
         END IF
*
      END IF
      IF ( aaqq .LT. xsc ) THEN
         DO 700 p = 1, n
            IF ( sva(p) .LT. xsc ) THEN
               CALL claset( 'A', m, 1, czero, czero, a(1,p), lda )
               sva(p) = zero
            END IF
 700     CONTINUE
      END IF
*
*     Preconditioning using QR factorization with pivoting
*
      IF ( rowpiv ) THEN
*        Optional row permutation (Bjoerck row pivoting):
*        A result by Cox and Higham shows that the Bjoerck's
*        row pivoting combined with standard column pivoting
*        has similar effect as Powell-Reid complete pivoting.
*        The ell-infinity norms of A are made nonincreasing.
         DO 1952 p = 1, m - 1
            q = isamax( m-p+1, rwork(m+n+p), 1 ) + p - 1
            iwork(2*n+p) = q
            IF ( p .NE. q ) THEN
               temp1        = rwork(m+n+p)
               rwork(m+n+p) = rwork(m+n+q)
               rwork(m+n+q) = temp1
            END IF
 1952    CONTINUE
         CALL claswp( n, a, lda, 1, m-1, iwork(2*n+1), 1 )
      END IF
*
*     End of the preparation phase (scaling, optional sorting and
*     transposing, optional flushing of small columns).
*
*     Preconditioning
*
*     If the full SVD is needed, the right singular vectors are computed
*     from a matrix equation, and for that we need theoretical analysis
*     of the Businger-Golub pivoting. So we use CGEQP3 as the first RR QRF.
*     In all other cases the first RR QRF can be chosen by other criteria
*     (eg speed by replacing global with restricted window pivoting, such
*     as in xGEQPX from TOMS # 782). Good results will be obtained using
*     xGEQPX with properly (!) chosen numerical parameters.
*     Any improvement of CGEQP3 improves overal performance of CGEJSV.
*
*     A * P1 = Q1 * [ R1^* 0]^*:
      DO 1963 p = 1, n
*        .. all columns are free columns
         iwork(p) = 0
 1963 CONTINUE
      CALL cgeqp3( m, n, a, lda, iwork, cwork, cwork(n+1), lwork-n, 
     $             rwork, ierr )
*
*     The upper triangular matrix R1 from the first QRF is inspected for
*     rank deficiency and possibilities for deflation, or possible
*     ill-conditioning. Depending on the user specified flag L2RANK,
*     the procedure explores possibilities to reduce the numerical
*     rank by inspecting the computed upper triangular factor. If
*     L2RANK or L2ABER are up, then CGEJSV will compute the SVD of
*     A + dA, where ||dA|| <= f(M,N)*EPSLN.
*
      nr = 1
      IF ( l2aber ) THEN
*        Standard absolute error bound suffices. All sigma_i with
*        sigma_i < N*EPSLN*||A|| are flushed to zero. This is an
*        agressive enforcement of lower numerical rank by introducing a
*        backward error of the order of N*EPSLN*||A||.
         temp1 = sqrt(float(n))*epsln
         DO 3001 p = 2, n
            IF ( abs(a(p,p)) .GE. (temp1*abs(a(1,1))) ) THEN
               nr = nr + 1
            ELSE
               GO TO 3002
            END IF
 3001    CONTINUE
 3002    CONTINUE
      ELSE IF ( l2rank ) THEN
*        .. similarly as above, only slightly more gentle (less agressive).
*        Sudden drop on the diagonal of R1 is used as the criterion for
*        close-to-rank-defficient.
         temp1 = sqrt(sfmin)
         DO 3401 p = 2, n
            IF ( ( abs(a(p,p)) .LT. (epsln*abs(a(p-1,p-1))) ) .OR.
     $           ( abs(a(p,p)) .LT. small ) .OR.
     $           ( l2kill .AND. (abs(a(p,p)) .LT. temp1) ) ) GO TO 3402
            nr = nr + 1
 3401    CONTINUE
 3402    CONTINUE
*
      ELSE
*        The goal is high relative accuracy. However, if the matrix
*        has high scaled condition number the relative accuracy is in
*        general not feasible. Later on, a condition number estimator
*        will be deployed to estimate the scaled condition number.
*        Here we just remove the underflowed part of the triangular
*        factor. This prevents the situation in which the code is
*        working hard to get the accuracy not warranted by the data.
         temp1  = sqrt(sfmin)
         DO 3301 p = 2, n
            IF ( ( abs(a(p,p)) .LT. small ) .OR.
     $           ( l2kill .AND. (abs(a(p,p)) .LT. temp1) ) ) GO TO 3302
            nr = nr + 1
 3301    CONTINUE
 3302    CONTINUE
*
      END IF
*
      almort = .false.
      IF ( nr .EQ. n ) THEN
         maxprj = one
         DO 3051 p = 2, n
            temp1  = abs(a(p,p)) / sva(iwork(p))
            maxprj = amin1( maxprj, temp1 )
 3051    CONTINUE
         IF ( maxprj**2 .GE. one - float(n)*epsln ) almort = .true.
      END IF
*
*
      sconda = - one
      condr1 = - one
      condr2 = - one
*
      IF ( errest ) THEN
         IF ( n .EQ. nr ) THEN
            IF ( rsvec ) THEN
*              .. V is available as workspace
               CALL clacpy( 'U', n, n, a, lda, v, ldv )
               DO 3053 p = 1, n
                  temp1 = sva(iwork(p))
                  CALL csscal( p, one/temp1, v(1,p), 1 )
 3053          CONTINUE
               CALL cpocon( 'U', n, v, ldv, one, temp1,
     $              cwork(n+1), rwork, ierr )
*          
            ELSE IF ( lsvec ) THEN
*              .. U is available as workspace
               CALL clacpy( 'U', n, n, a, lda, u, ldu )
               DO 3054 p = 1, n
                  temp1 = sva(iwork(p))
                  CALL csscal( p, one/temp1, u(1,p), 1 )
 3054          CONTINUE
               CALL cpocon( 'U', n, u, ldu, one, temp1,
     $              cwork(n+1), rwork, ierr )
            ELSE
               CALL clacpy( 'U', n, n, a, lda, cwork(n+1), n )
               DO 3052 p = 1, n
                  temp1 = sva(iwork(p))
                  CALL csscal( p, one/temp1, cwork(n+(p-1)*n+1), 1 )
 3052          CONTINUE
*           .. the columns of R are scaled to have unit Euclidean lengths.
               CALL cpocon( 'U', n, cwork(n+1), n, one, temp1,
     $              cwork(n+n*n+1), rwork, ierr )
*              
            END IF
            sconda = one / sqrt(temp1)
*           SCONDA is an estimate of SQRT(||(R^* * R)^(-1)||_1).
*           N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA
         ELSE
            sconda = - one
         END IF
      END IF
*
      l2pert = l2pert .AND. ( abs( a(1,1)/a(nr,nr) ) .GT. sqrt(big1) )
*     If there is no violent scaling, artificial perturbation is not needed.
*
*     Phase 3:
*
      IF ( .NOT. ( rsvec .OR. lsvec ) ) THEN
*
*         Singular Values only
*
*         .. transpose A(1:NR,1:N)
         DO 1946 p = 1, min0( n-1, nr )
            CALL ccopy( n-p, a(p,p+1), lda, a(p+1,p), 1 )
            CALL clacgv( n-p+1, a(p,p), 1 )
 1946    CONTINUE
         IF ( nr .EQ. n ) a(n,n) = conjg(a(n,n))        
*
*        The following two DO-loops introduce small relative perturbation
*        into the strict upper triangle of the lower triangular matrix.
*        Small entries below the main diagonal are also changed.
*        This modification is useful if the computing environment does not
*        provide/allow FLUSH TO ZERO underflow, for it prevents many
*        annoying denormalized numbers in case of strongly scaled matrices.
*        The perturbation is structured so that it does not introduce any
*        new perturbation of the singular values, and it does not destroy
*        the job done by the preconditioner.
*        The licence for this perturbation is in the variable L2PERT, which
*        should be .FALSE. if FLUSH TO ZERO underflow is active.
*
         IF ( .NOT. almort ) THEN
*
            IF ( l2pert ) THEN
*              XSC = SQRT(SMALL)
               xsc = epsln / float(n)
               DO 4947 q = 1, nr
                  ctemp = cmplx(xsc*abs(a(q,q)),zero)
                  DO 4949 p = 1, n
                     IF ( ( (p.GT.q) .AND. (abs(a(p,q)).LE.temp1) )
     $                    .OR. ( p .LT. q ) )
*     $                     A(p,q) = TEMP1 * ( A(p,q) / ABS(A(p,q)) )
     $                     a(p,q) = ctemp
 4949             CONTINUE
 4947          CONTINUE
            ELSE
               CALL claset( 'U', nr-1,nr-1, czero,czero, a(1,2),lda )
            END IF
*
*            .. second preconditioning using the QR factorization
*
            CALL cgeqrf( n,nr, a,lda, cwork, cwork(n+1),lwork-n, ierr )
*
*           .. and transpose upper to lower triangular
            DO 1948 p = 1, nr - 1
               CALL ccopy( nr-p, a(p,p+1), lda, a(p+1,p), 1 )
               CALL clacgv( nr-p+1, a(p,p), 1 )
 1948       CONTINUE
*
         END IF
*
*           Row-cyclic Jacobi SVD algorithm with column pivoting
*
*           .. again some perturbation (a "background noise") is added
*           to drown denormals
            IF ( l2pert ) THEN
*              XSC = SQRT(SMALL)
               xsc = epsln / float(n)
               DO 1947 q = 1, nr
                  ctemp = cmplx(xsc*abs(a(q,q)),zero)
                  DO 1949 p = 1, nr
                     IF ( ( (p.GT.q) .AND. (abs(a(p,q)).LE.temp1) )
     $                       .OR. ( p .LT. q ) )
*     $                   A(p,q) = TEMP1 * ( A(p,q) / ABS(A(p,q)) )
     $                   a(p,q) = ctemp 
 1949             CONTINUE
 1947          CONTINUE
            ELSE
               CALL claset( 'U', nr-1, nr-1, czero, czero, a(1,2), lda )
            END IF
*
*           .. and one-sided Jacobi rotations are started on a lower
*           triangular matrix (plus perturbation which is ignored in
*           the part which destroys triangular form (confusing?!))
*
            CALL cgesvj( 'L', 'NoU', 'NoV', nr, nr, a, lda, sva,
     $                n, v, ldv, cwork, lwork, rwork, lrwork, info )
*
            scalem  = rwork(1)
            numrank = nint(rwork(2))
*
*
      ELSE IF ( rsvec .AND. ( .NOT. lsvec ) ) THEN
*
*        -> Singular Values and Right Singular Vectors <-
*
         IF ( almort ) THEN
*
*           .. in this case NR equals N
            DO 1998 p = 1, nr
               CALL ccopy( n-p+1, a(p,p), lda, v(p,p), 1 )
               CALL clacgv( n-p+1, v(p,p), 1 )
 1998       CONTINUE
            CALL claset( 'Upper', nr-1,nr-1, czero, czero, v(1,2), ldv )
*
            CALL cgesvj( 'L','U','N', n, nr, v,ldv, sva, nr, a,lda,
     $                  cwork, lwork, rwork, lrwork, info )
            scalem  = rwork(1)
            numrank = nint(rwork(2))

         ELSE
*
*        .. two more QR factorizations ( one QRF is not enough, two require
*        accumulated product of Jacobi rotations, three are perfect )
*
            CALL claset( 'Lower', nr-1,nr-1, czero, czero, a(2,1), lda )
            CALL cgelqf( nr,n, a, lda, cwork, cwork(n+1), lwork-n, ierr)
            CALL clacpy( 'Lower', nr, nr, a, lda, v, ldv )
            CALL claset( 'Upper', nr-1,nr-1, czero, czero, v(1,2), ldv )
            CALL cgeqrf( nr, nr, v, ldv, cwork(n+1), cwork(2*n+1),
     $                   lwork-2*n, ierr )
            DO 8998 p = 1, nr
               CALL ccopy( nr-p+1, v(p,p), ldv, v(p,p), 1 )
               CALL clacgv( nr-p+1, v(p,p), 1 ) 
 8998       CONTINUE
            CALL claset('Upper', nr-1, nr-1, czero, czero, v(1,2), ldv)
*
            CALL cgesvj( 'Lower', 'U','N', nr, nr, v,ldv, sva, nr, u,
     $                  ldu, cwork(n+1), lwork-n, rwork, lrwork, info )
            scalem  = rwork(1)
            numrank = nint(rwork(2))
            IF ( nr .LT. n ) THEN
               CALL claset( 'A',n-nr, nr, czero,czero, v(nr+1,1),  ldv )
               CALL claset( 'A',nr, n-nr, czero,czero, v(1,nr+1),  ldv )
               CALL claset( 'A',n-nr,n-nr,czero,cone, v(nr+1,nr+1),ldv )
            END IF
*
         CALL cunmlq( 'Left', 'C', n, n, nr, a, lda, cwork,
     $               v, ldv, cwork(n+1), lwork-n, ierr )
*
         END IF
*
         DO 8991 p = 1, n
            CALL ccopy( n, v(p,1), ldv, a(iwork(p),1), lda )
 8991    CONTINUE
         CALL clacpy( 'All', n, n, a, lda, v, ldv )
*
         IF ( transp ) THEN
            CALL clacpy( 'All', n, n, v, ldv, u, ldu )
         END IF
*
      ELSE IF ( lsvec .AND. ( .NOT. rsvec ) ) THEN
*
*        .. Singular Values and Left Singular Vectors                 ..
*
*        .. second preconditioning step to avoid need to accumulate
*        Jacobi rotations in the Jacobi iterations.
         DO 1965 p = 1, nr
            CALL ccopy( n-p+1, a(p,p), lda, u(p,p), 1 )
            CALL clacgv( n-p+1, u(p,p), 1 )
 1965    CONTINUE
         CALL claset( 'Upper', nr-1, nr-1, czero, czero, u(1,2), ldu )
*
         CALL cgeqrf( n, nr, u, ldu, cwork(n+1), cwork(2*n+1),
     $              lwork-2*n, ierr )
*
         DO 1967 p = 1, nr - 1
            CALL ccopy( nr-p, u(p,p+1), ldu, u(p+1,p), 1 )
            CALL clacgv( n-p+1, u(p,p), 1 )            
 1967    CONTINUE
         CALL claset( 'Upper', nr-1, nr-1, czero, czero, u(1,2), ldu )
*
         CALL cgesvj( 'Lower', 'U', 'N', nr,nr, u, ldu, sva, nr, a,
     $        lda, cwork(n+1), lwork-n, rwork, lrwork, info )
         scalem  = rwork(1)
         numrank = nint(rwork(2))
*
         IF ( nr .LT. m ) THEN
            CALL claset( 'A',  m-nr, nr,czero, czero, u(nr+1,1), ldu )
            IF ( nr .LT. n1 ) THEN
               CALL claset( 'A',nr, n1-nr, czero, czero, u(1,nr+1),ldu )
               CALL claset( 'A',m-nr,n1-nr,czero,cone,u(nr+1,nr+1),ldu )
            END IF
         END IF
*
         CALL cunmqr( 'Left', 'No Tr', m, n1, n, a, lda, cwork, u,
     $               ldu, cwork(n+1), lwork-n, ierr )
*
         IF ( rowpiv )
     $       CALL claswp( n1, u, ldu, 1, m-1, iwork(2*n+1), -1 )
*
         DO 1974 p = 1, n1
            xsc = one / scnrm2( m, u(1,p), 1 )
            CALL csscal( m, xsc, u(1,p), 1 )
 1974    CONTINUE
*
         IF ( transp ) THEN
            CALL clacpy( 'All', n, n, u, ldu, v, ldv )
         END IF
*
      ELSE
*
*        .. Full SVD ..
*
         IF ( .NOT. jracc ) THEN
*
         IF ( .NOT. almort ) THEN
*
*           Second Preconditioning Step (QRF [with pivoting])
*           Note that the composition of TRANSPOSE, QRF and TRANSPOSE is
*           equivalent to an LQF CALL. Since in many libraries the QRF
*           seems to be better optimized than the LQF, we do explicit
*           transpose and use the QRF. This is subject to changes in an
*           optimized implementation of CGEJSV.
*
            DO 1968 p = 1, nr
               CALL ccopy( n-p+1, a(p,p), lda, v(p,p), 1 )
               CALL clacgv( n-p+1, v(p,p), 1 )
 1968       CONTINUE
*
*           .. the following two loops perturb small entries to avoid
*           denormals in the second QR factorization, where they are
*           as good as zeros. This is done to avoid painfully slow
*           computation with denormals. The relative size of the perturbation
*           is a parameter that can be changed by the implementer.
*           This perturbation device will be obsolete on machines with
*           properly implemented arithmetic.
*           To switch it off, set L2PERT=.FALSE. To remove it from  the
*           code, remove the action under L2PERT=.TRUE., leave the ELSE part.
*           The following two loops should be blocked and fused with the
*           transposed copy above.
*
            IF ( l2pert ) THEN
               xsc = sqrt(small)
               DO 2969 q = 1, nr
                  ctemp = cmplx(xsc*abs( v(q,q) ),zero)
                  DO 2968 p = 1, n
                     IF ( ( p .GT. q ) .AND. ( abs(v(p,q)) .LE. temp1 )
     $                   .OR. ( p .LT. q ) )
*     $                   V(p,q) = TEMP1 * ( V(p,q) / ABS(V(p,q)) )
     $                   v(p,q) = ctemp      
                     IF ( p .LT. q ) v(p,q) = - v(p,q)
 2968             CONTINUE
 2969          CONTINUE
            ELSE
               CALL claset( 'U', nr-1, nr-1, czero, czero, v(1,2), ldv )
            END IF
*
*           Estimate the row scaled condition number of R1
*           (If R1 is rectangular, N > NR, then the condition number
*           of the leading NR x NR submatrix is estimated.)
*
            CALL clacpy( 'L', nr, nr, v, ldv, cwork(2*n+1), nr )
            DO 3950 p = 1, nr
               temp1 = scnrm2(nr-p+1,cwork(2*n+(p-1)*nr+p),1)
               CALL csscal(nr-p+1,one/temp1,cwork(2*n+(p-1)*nr+p),1)
 3950       CONTINUE
            CALL cpocon('Lower',nr,cwork(2*n+1),nr,one,temp1,
     $                   cwork(2*n+nr*nr+1),rwork,ierr)
            condr1 = one / sqrt(temp1)
*           .. here need a second oppinion on the condition number
*           .. then assume worst case scenario
*           R1 is OK for inverse <=> CONDR1 .LT. FLOAT(N)
*           more conservative    <=> CONDR1 .LT. SQRT(FLOAT(N))
*
            cond_ok = sqrt(sqrt(float(nr)))
*[TP]       COND_OK is a tuning parameter.
*
            IF ( condr1 .LT. cond_ok ) THEN
*              .. the second QRF without pivoting. Note: in an optimized
*              implementation, this QRF should be implemented as the QRF
*              of a lower triangular matrix.
*              R1^* = Q2 * R2
               CALL cgeqrf( n, nr, v, ldv, cwork(n+1), cwork(2*n+1),
     $              lwork-2*n, ierr )
*
               IF ( l2pert ) THEN
                  xsc = sqrt(small)/epsln
                  DO 3959 p = 2, nr
                     DO 3958 q = 1, p - 1
                        ctemp=cmplx(xsc*amin1(abs(v(p,p)),abs(v(q,q))),
     $                              zero)
                        IF ( abs(v(q,p)) .LE. temp1 )
*     $                     V(q,p) = TEMP1 * ( V(q,p) / ABS(V(q,p)) )
     $                     v(q,p) = ctemp    
 3958                CONTINUE
 3959             CONTINUE
               END IF
*
               IF ( nr .NE. n )
     $         CALL clacpy( 'A', n, nr, v, ldv, cwork(2*n+1), n )
*              .. save ...
*
*           .. this transposed copy should be better than naive
               DO 1969 p = 1, nr - 1
                  CALL ccopy( nr-p, v(p,p+1), ldv, v(p+1,p), 1 )
                  CALL clacgv(nr-p+1, v(p,p), 1 )
 1969          CONTINUE
               v(nr,nr)=conjg(v(nr,nr))   
*
               condr2 = condr1
*
            ELSE
*
*              .. ill-conditioned case: second QRF with pivoting
*              Note that windowed pivoting would be equaly good
*              numerically, and more run-time efficient. So, in
*              an optimal implementation, the next call to CGEQP3
*              should be replaced with eg. CALL CGEQPX (ACM TOMS #782)
*              with properly (carefully) chosen parameters.
*
*              R1^* * P2 = Q2 * R2
               DO 3003 p = 1, nr
                  iwork(n+p) = 0
 3003          CONTINUE
               CALL cgeqp3( n, nr, v, ldv, iwork(n+1), cwork(n+1),
     $                  cwork(2*n+1), lwork-2*n, rwork, ierr )
**               CALL CGEQRF( N, NR, V, LDV, CWORK(N+1), CWORK(2*N+1),
**     $              LWORK-2*N, IERR )
               IF ( l2pert ) THEN
                  xsc = sqrt(small)
                  DO 3969 p = 2, nr
                     DO 3968 q = 1, p - 1
                        ctemp=cmplx(xsc*amin1(abs(v(p,p)),abs(v(q,q))),
     $                               zero)
                        IF ( abs(v(q,p)) .LE. temp1 )
*     $                     V(q,p) = TEMP1 * ( V(q,p) / ABS(V(q,p)) )
     $                     v(q,p) = ctemp                     
 3968                CONTINUE
 3969             CONTINUE
               END IF
*
               CALL clacpy( 'A', n, nr, v, ldv, cwork(2*n+1), n )
*
               IF ( l2pert ) THEN
                  xsc = sqrt(small)
                  DO 8970 p = 2, nr
                     DO 8971 q = 1, p - 1
                        ctemp=cmplx(xsc*amin1(abs(v(p,p)),abs(v(q,q))),
     $                               zero)
*                        V(p,q) = - TEMP1*( V(q,p) / ABS(V(q,p)) )
                        v(p,q) = - ctemp      
 8971                CONTINUE
 8970             CONTINUE
               ELSE
                  CALL claset( 'L',nr-1,nr-1,czero,czero,v(2,1),ldv )
               END IF
*              Now, compute R2 = L3 * Q3, the LQ factorization.
               CALL cgelqf( nr, nr, v, ldv, cwork(2*n+n*nr+1),
     $               cwork(2*n+n*nr+nr+1), lwork-2*n-n*nr-nr, ierr )
*              .. and estimate the condition number
               CALL clacpy( 'L',nr,nr,v,ldv,cwork(2*n+n*nr+nr+1),nr )
               DO 4950 p = 1, nr
                  temp1 = scnrm2( p, cwork(2*n+n*nr+nr+p), nr )
                  CALL csscal( p, one/temp1, cwork(2*n+n*nr+nr+p), nr )
 4950          CONTINUE
               CALL cpocon( 'L',nr,cwork(2*n+n*nr+nr+1),nr,one,temp1,
     $              cwork(2*n+n*nr+nr+nr*nr+1),rwork,ierr ) 
               condr2 = one / sqrt(temp1)
*
*
               IF ( condr2 .GE. cond_ok ) THEN
*                 .. save the Householder vectors used for Q3
*                 (this overwrittes the copy of R2, as it will not be
*                 needed in this branch, but it does not overwritte the
*                 Huseholder vectors of Q2.).
                  CALL clacpy( 'U', nr, nr, v, ldv, cwork(2*n+1), n )
*                 .. and the rest of the information on Q3 is in
*                 WORK(2*N+N*NR+1:2*N+N*NR+N)
               END IF
*
            END IF
*
            IF ( l2pert ) THEN
               xsc = sqrt(small)
               DO 4968 q = 2, nr
                  ctemp = xsc * v(q,q)
                  DO 4969 p = 1, q - 1
*                    V(p,q) = - SIGN( TEMP1, V(q,p) )
*                     V(p,q) = - TEMP1*( V(p,q) / ABS(V(p,q)) )
                     v(p,q) = - ctemp
 4969             CONTINUE
 4968          CONTINUE
            ELSE
               CALL claset( 'U', nr-1,nr-1, czero,czero, v(1,2), ldv )
            END IF
*
*        Second preconditioning finished; continue with Jacobi SVD
*        The input matrix is lower trinagular.
*
*        Recover the right singular vectors as solution of a well
*        conditioned triangular matrix equation.
*
            IF ( condr1 .LT. cond_ok ) THEN
*
               CALL cgesvj( 'L','U','N',nr,nr,v,ldv,sva,nr,u, ldu,
     $              cwork(2*n+n*nr+nr+1),lwork-2*n-n*nr-nr,rwork,
     $              lrwork, info )
               scalem  = rwork(1)
               numrank = nint(rwork(2))
               DO 3970 p = 1, nr
                  CALL ccopy(  nr, v(1,p), 1, u(1,p), 1 )
                  CALL csscal( nr, sva(p),    v(1,p), 1 )
 3970          CONTINUE

*        .. pick the right matrix equation and solve it
*
               IF ( nr .EQ. n ) THEN
* :))             .. best case, R1 is inverted. The solution of this matrix
*                 equation is Q2*V2 = the product of the Jacobi rotations
*                 used in CGESVJ, premultiplied with the orthogonal matrix
*                 from the second QR factorization.
                  CALL ctrsm('L','U','N','N', nr,nr,cone, a,lda, v,ldv)
               ELSE
*                 .. R1 is well conditioned, but non-square. Adjoint of R2
*                 is inverted to get the product of the Jacobi rotations
*                 used in CGESVJ. The Q-factor from the second QR
*                 factorization is then built in explicitly.
                  CALL ctrsm('L','U','C','N',nr,nr,cone,cwork(2*n+1),
     $                 n,v,ldv)
                  IF ( nr .LT. n ) THEN
                  CALL claset('A',n-nr,nr,czero,czero,v(nr+1,1),ldv)
                  CALL claset('A',nr,n-nr,czero,czero,v(1,nr+1),ldv)
                  CALL claset('A',n-nr,n-nr,czero,cone,v(nr+1,nr+1),ldv)
                  END IF
                  CALL cunmqr('L','N',n,n,nr,cwork(2*n+1),n,cwork(n+1),
     $                v,ldv,cwork(2*n+n*nr+nr+1),lwork-2*n-n*nr-nr,ierr)
               END IF
*
            ELSE IF ( condr2 .LT. cond_ok ) THEN
*
*              The matrix R2 is inverted. The solution of the matrix equation
*              is Q3^* * V3 = the product of the Jacobi rotations (appplied to
*              the lower triangular L3 from the LQ factorization of
*              R2=L3*Q3), pre-multiplied with the transposed Q3.
               CALL cgesvj( 'L', 'U', 'N', nr, nr, v, ldv, sva, nr, u,
     $              ldu, cwork(2*n+n*nr+nr+1), lwork-2*n-n*nr-nr, 
     $          rwork, lrwork, info )
               scalem  = rwork(1)
               numrank = nint(rwork(2))
               DO 3870 p = 1, nr
                  CALL ccopy( nr, v(1,p), 1, u(1,p), 1 )
                  CALL csscal( nr, sva(p),    u(1,p), 1 )
 3870          CONTINUE
               CALL ctrsm('L','U','N','N',nr,nr,cone,cwork(2*n+1),n,
     $                    u,ldu)
*              .. apply the permutation from the second QR factorization
               DO 873 q = 1, nr
                  DO 872 p = 1, nr
                     cwork(2*n+n*nr+nr+iwork(n+p)) = u(p,q)
 872              CONTINUE
                  DO 874 p = 1, nr
                     u(p,q) = cwork(2*n+n*nr+nr+p)
 874              CONTINUE
 873           CONTINUE
               IF ( nr .LT. n ) THEN
                  CALL claset( 'A',n-nr,nr,czero,czero,v(nr+1,1),ldv )
                  CALL claset( 'A',nr,n-nr,czero,czero,v(1,nr+1),ldv )
                  CALL claset('A',n-nr,n-nr,czero,cone,v(nr+1,nr+1),ldv)
               END IF
               CALL cunmqr( 'L','N',n,n,nr,cwork(2*n+1),n,cwork(n+1),
     $              v,ldv,cwork(2*n+n*nr+nr+1),lwork-2*n-n*nr-nr,ierr )
            ELSE
*              Last line of defense.
* #:(          This is a rather pathological case: no scaled condition
*              improvement after two pivoted QR factorizations. Other
*              possibility is that the rank revealing QR factorization
*              or the condition estimator has failed, or the COND_OK
*              is set very close to ONE (which is unnecessary). Normally,
*              this branch should never be executed, but in rare cases of
*              failure of the RRQR or condition estimator, the last line of
*              defense ensures that CGEJSV completes the task.
*              Compute the full SVD of L3 using CGESVJ with explicit
*              accumulation of Jacobi rotations.
               CALL cgesvj( 'L', 'U', 'V', nr, nr, v, ldv, sva, nr, u,
     $              ldu, cwork(2*n+n*nr+nr+1), lwork-2*n-n*nr-nr, 
     $                         rwork, lrwork, info )
               scalem  = rwork(1)
               numrank = nint(rwork(2))
               IF ( nr .LT. n ) THEN
                  CALL claset( 'A',n-nr,nr,czero,czero,v(nr+1,1),ldv )
                  CALL claset( 'A',nr,n-nr,czero,czero,v(1,nr+1),ldv )
                  CALL claset('A',n-nr,n-nr,czero,cone,v(nr+1,nr+1),ldv)
               END IF
               CALL cunmqr( 'L','N',n,n,nr,cwork(2*n+1),n,cwork(n+1),
     $              v,ldv,cwork(2*n+n*nr+nr+1),lwork-2*n-n*nr-nr,ierr )
*
               CALL cunmlq( 'L', 'C', nr, nr, nr, cwork(2*n+1), n,
     $              cwork(2*n+n*nr+1), u, ldu, cwork(2*n+n*nr+nr+1),
     $              lwork-2*n-n*nr-nr, ierr )
               DO 773 q = 1, nr
                  DO 772 p = 1, nr
                     cwork(2*n+n*nr+nr+iwork(n+p)) = u(p,q)
 772              CONTINUE
                  DO 774 p = 1, nr
                     u(p,q) = cwork(2*n+n*nr+nr+p)
 774              CONTINUE
 773           CONTINUE
*
            END IF
*
*           Permute the rows of V using the (column) permutation from the
*           first QRF. Also, scale the columns to make them unit in
*           Euclidean norm. This applies to all cases.
*
            temp1 = sqrt(float(n)) * epsln
            DO 1972 q = 1, n
               DO 972 p = 1, n
                  cwork(2*n+n*nr+nr+iwork(p)) = v(p,q)
  972          CONTINUE
               DO 973 p = 1, n
                  v(p,q) = cwork(2*n+n*nr+nr+p)
  973          CONTINUE
               xsc = one / scnrm2( n, v(1,q), 1 )
               IF ( (xsc .LT. (one-temp1)) .OR. (xsc .GT. (one+temp1)) )
     $           CALL csscal( n, xsc, v(1,q), 1 )
 1972       CONTINUE
*           At this moment, V contains the right singular vectors of A.
*           Next, assemble the left singular vector matrix U (M x N).
            IF ( nr .LT. m ) THEN
               CALL claset('A', m-nr, nr, czero, czero, u(nr+1,1), ldu)
               IF ( nr .LT. n1 ) THEN
                  CALL claset('A',nr,n1-nr,czero,czero,u(1,nr+1),ldu)
                  CALL claset('A',m-nr,n1-nr,czero,cone,
     $                        u(nr+1,nr+1),ldu)
               END IF
            END IF
*
*           The Q matrix from the first QRF is built into the left singular
*           matrix U. This applies to all cases.
*
            CALL cunmqr( 'Left', 'No_Tr', m, n1, n, a, lda, cwork, u,
     $           ldu, cwork(n+1), lwork-n, ierr )

*           The columns of U are normalized. The cost is O(M*N) flops.
            temp1 = sqrt(float(m)) * epsln
            DO 1973 p = 1, nr
               xsc = one / scnrm2( m, u(1,p), 1 )
               IF ( (xsc .LT. (one-temp1)) .OR. (xsc .GT. (one+temp1)) )
     $          CALL csscal( m, xsc, u(1,p), 1 )
 1973       CONTINUE
*
*           If the initial QRF is computed with row pivoting, the left
*           singular vectors must be adjusted.
*
            IF ( rowpiv )
     $          CALL claswp( n1, u, ldu, 1, m-1, iwork(2*n+1), -1 )
*
         ELSE
*
*        .. the initial matrix A has almost orthogonal columns and
*        the second QRF is not needed
*
            CALL clacpy( 'Upper', n, n, a, lda, cwork(n+1), n )
            IF ( l2pert ) THEN
               xsc = sqrt(small)
               DO 5970 p = 2, n
                  ctemp = xsc * cwork( n + (p-1)*n + p )
                  DO 5971 q = 1, p - 1
*                     CWORK(N+(q-1)*N+p)=-TEMP1 * ( CWORK(N+(p-1)*N+q) /
*     $                                        ABS(CWORK(N+(p-1)*N+q)) )
                     cwork(n+(q-1)*n+p)=-ctemp           
 5971             CONTINUE
 5970          CONTINUE
            ELSE
               CALL claset( 'Lower',n-1,n-1,czero,czero,cwork(n+2),n )
            END IF
*
            CALL cgesvj( 'Upper', 'U', 'N', n, n, cwork(n+1), n, sva,
     $           n, u, ldu, cwork(n+n*n+1), lwork-n-n*n, rwork, lrwork, 
     $       info )
*
            scalem  = rwork(1)
            numrank = nint(rwork(2))
            DO 6970 p = 1, n
               CALL ccopy( n, cwork(n+(p-1)*n+1), 1, u(1,p), 1 )
               CALL csscal( n, sva(p), cwork(n+(p-1)*n+1), 1 )
 6970       CONTINUE
*
            CALL ctrsm( 'Left', 'Upper', 'NoTrans', 'No UD', n, n,
     $           cone, a, lda, cwork(n+1), n )
            DO 6972 p = 1, n
               CALL ccopy( n, cwork(n+p), n, v(iwork(p),1), ldv )
 6972       CONTINUE
            temp1 = sqrt(float(n))*epsln
            DO 6971 p = 1, n
               xsc = one / scnrm2( n, v(1,p), 1 )
               IF ( (xsc .LT. (one-temp1)) .OR. (xsc .GT. (one+temp1)) )
     $            CALL csscal( n, xsc, v(1,p), 1 )
 6971       CONTINUE
*
*           Assemble the left singular vector matrix U (M x N).
*
            IF ( n .LT. m ) THEN
               CALL claset( 'A',  m-n, n, czero, czero, u(n+1,1), ldu )
               IF ( n .LT. n1 ) THEN
                  CALL claset('A',n,  n1-n, czero, czero,  u(1,n+1),ldu)
                  CALL claset( 'A',m-n,n1-n, czero, cone,u(n+1,n+1),ldu)
               END IF
            END IF
            CALL cunmqr( 'Left', 'No Tr', m, n1, n, a, lda, cwork, u,
     $           ldu, cwork(n+1), lwork-n, ierr )
            temp1 = sqrt(float(m))*epsln
            DO 6973 p = 1, n1
               xsc = one / scnrm2( m, u(1,p), 1 )
               IF ( (xsc .LT. (one-temp1)) .OR. (xsc .GT. (one+temp1)) )
     $            CALL csscal( m, xsc, u(1,p), 1 )
 6973       CONTINUE
*
            IF ( rowpiv )
     $         CALL claswp( n1, u, ldu, 1, m-1, iwork(2*n+1), -1 )
*
         END IF
*
*        end of the  >> almost orthogonal case <<  in the full SVD
*
         ELSE
*
*        This branch deploys a preconditioned Jacobi SVD with explicitly
*        accumulated rotations. It is included as optional, mainly for
*        experimental purposes. It does perfom well, and can also be used.
*        In this implementation, this branch will be automatically activated
*        if the  condition number sigma_max(A) / sigma_min(A) is predicted
*        to be greater than the overflow threshold. This is because the
*        a posteriori computation of the singular vectors assumes robust
*        implementation of BLAS and some LAPACK procedures, capable of working
*        in presence of extreme values. Since that is not always the case, ...
*
         DO 7968 p = 1, nr
            CALL ccopy( n-p+1, a(p,p), lda, v(p,p), 1 )
            CALL clacgv( n-p+1, v(p,p), 1 )
 7968    CONTINUE
*
         IF ( l2pert ) THEN
            xsc = sqrt(small/epsln)
            DO 5969 q = 1, nr
               ctemp = cmplx(xsc*abs( v(q,q) ),zero)
               DO 5968 p = 1, n
                  IF ( ( p .GT. q ) .AND. ( abs(v(p,q)) .LE. temp1 )
     $                .OR. ( p .LT. q ) )
*     $                V(p,q) = TEMP1 * ( V(p,q) / ABS(V(p,q)) )
     $                v(p,q) = ctemp        
                  IF ( p .LT. q ) v(p,q) = - v(p,q)
 5968          CONTINUE
 5969       CONTINUE
         ELSE
            CALL claset( 'U', nr-1, nr-1, czero, czero, v(1,2), ldv )
         END IF

         CALL cgeqrf( n, nr, v, ldv, cwork(n+1), cwork(2*n+1),
     $        lwork-2*n, ierr )
         CALL clacpy( 'L', n, nr, v, ldv, cwork(2*n+1), n )
*
         DO 7969 p = 1, nr
            CALL ccopy( nr-p+1, v(p,p), ldv, u(p,p), 1 )
            CALL clacgv( nr-p+1, u(p,p), 1 )
 7969    CONTINUE

         IF ( l2pert ) THEN
            xsc = sqrt(small/epsln)
            DO 9970 q = 2, nr
               DO 9971 p = 1, q - 1
                  ctemp = cmplx(xsc * amin1(abs(u(p,p)),abs(u(q,q))),
     $                           zero)
*                  U(p,q) = - TEMP1 * ( U(q,p) / ABS(U(q,p)) )
                  u(p,q) = - ctemp     
 9971          CONTINUE
 9970       CONTINUE
         ELSE
            CALL claset('U', nr-1, nr-1, czero, czero, u(1,2), ldu )
         END IF

         CALL cgesvj( 'L', 'U', 'V', nr, nr, u, ldu, sva,
     $        n, v, ldv, cwork(2*n+n*nr+1), lwork-2*n-n*nr, 
     $         rwork, lrwork, info )
         scalem  = rwork(1)
         numrank = nint(rwork(2))

         IF ( nr .LT. n ) THEN
            CALL claset( 'A',n-nr,nr,czero,czero,v(nr+1,1),ldv )
            CALL claset( 'A',nr,n-nr,czero,czero,v(1,nr+1),ldv )
            CALL claset( 'A',n-nr,n-nr,czero,cone,v(nr+1,nr+1),ldv )
         END IF

         CALL cunmqr( 'L','N',n,n,nr,cwork(2*n+1),n,cwork(n+1),
     $        v,ldv,cwork(2*n+n*nr+nr+1),lwork-2*n-n*nr-nr,ierr )
*
*           Permute the rows of V using the (column) permutation from the
*           first QRF. Also, scale the columns to make them unit in
*           Euclidean norm. This applies to all cases.
*
            temp1 = sqrt(float(n)) * epsln
            DO 7972 q = 1, n
               DO 8972 p = 1, n
                  cwork(2*n+n*nr+nr+iwork(p)) = v(p,q)
 8972          CONTINUE
               DO 8973 p = 1, n
                  v(p,q) = cwork(2*n+n*nr+nr+p)
 8973          CONTINUE
               xsc = one / scnrm2( n, v(1,q), 1 )
               IF ( (xsc .LT. (one-temp1)) .OR. (xsc .GT. (one+temp1)) )
     $           CALL csscal( n, xsc, v(1,q), 1 )
 7972       CONTINUE
*
*           At this moment, V contains the right singular vectors of A.
*           Next, assemble the left singular vector matrix U (M x N).
*
         IF ( nr .LT. m ) THEN
            CALL claset( 'A',  m-nr, nr, czero, czero, u(nr+1,1), ldu )
            IF ( nr .LT. n1 ) THEN
               CALL claset('A',nr,  n1-nr, czero, czero,  u(1,nr+1),ldu)
               CALL claset('A',m-nr,n1-nr, czero, cone,u(nr+1,nr+1),ldu)
            END IF
         END IF
*
         CALL cunmqr( 'Left', 'No Tr', m, n1, n, a, lda, cwork, u,
     $        ldu, cwork(n+1), lwork-n, ierr )
*
            IF ( rowpiv )
     $         CALL claswp( n1, u, ldu, 1, m-1, iwork(2*n+1), -1 )
*
*
         END IF
         IF ( transp ) THEN
*           .. swap U and V because the procedure worked on A^*
            DO 6974 p = 1, n
               CALL cswap( n, u(1,p), 1, v(1,p), 1 )
 6974       CONTINUE
         END IF
*
      END IF
*     end of the full SVD
*
*     Undo scaling, if necessary (and possible)
*
      IF ( uscal2 .LE. (big/sva(1))*uscal1 ) THEN
         CALL slascl( 'G', 0, 0, uscal1, uscal2, nr, 1, sva, n, ierr )
         uscal1 = one
         uscal2 = one
      END IF
*
      IF ( nr .LT. n ) THEN
         DO 3004 p = nr+1, n
            sva(p) = zero
 3004    CONTINUE
      END IF
*
      rwork(1) = uscal2 * scalem
      rwork(2) = uscal1
      IF ( errest ) rwork(3) = sconda
      IF ( lsvec .AND. rsvec ) THEN
         rwork(4) = condr1
         rwork(5) = condr2
      END IF
      IF ( l2tran ) THEN
         rwork(6) = entra
         rwork(7) = entrat
      END IF
*
      iwork(1) = nr
      iwork(2) = numrank
      iwork(3) = warning
*
      RETURN
*     ..
*     .. END OF CGEJSV
*     ..
      END
*