Follow up with ad5bc21cb50535d66d628a309d60128db96c8851

Contribution from Zlatko Drmac

1) LWORK query added;

2) few modifications in pure one sided Jacobi (XGESVJ) to remove possible error
in the really extreme cases (sigma_max close to overflow and sigma_min close to
underflow)  - note that XGESVJ is deigned to compute the singular values in the
full range; I used it (double complex) to compute SVD of certain factored
Hankel matrices with the condition number 1.0e616;

3) in the preconditioned Jacobi SVD (XGEJSV), the code I sent before to Julie
had one experimental modification that I had forgotten to remove before sending
- now this is done (the idea was to extend the computational range, but that
brings to much too risky dependence on how other lapack routines behave under
those extreme conditions).

2 files changed, 4472 insertions, 3748 deletions

diff --git a/SRC/cgejsv.f b/SRC/cgejsv.f
index 0641e42c..02794332 100644
--- a/SRC/cgejsv.f
+++ b/SRC/cgejsv.f
@@ -1,1872 +1,2235 @@
-*> \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 deficient. 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 :  successful 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 deficient cases. It will be available in the SIGMA library [4].
-*>  If M is much larger than N, it is obvious that the initial 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.
-*> \endverbatim
-*
-*> \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-deficient.
-         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
-*
+*> \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=B*D. If A has heavily weighted

+*>              rows, then using this condition number gives too pessimistic

+*>              error bound.

+*>       = 'A': Small singular values are not well determined by the data 

+*>              and are considered as noisy; 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, if JOBT .EQ. 'N'.

+*>       = '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.

+*>         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.

+*>         The option 'T' 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.

+*>         In general, this option is considered experimental, and 'N'; should

+*>         be preferred. This is subject to changes in the future.

+*> \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.

+*>          If the call to CGEJSV is a workspace query (indicated by LWORK=-1 or

+*>          LRWORK=-1), then on exit CWORK(1) contains the required length of 

+*>          CWORK for the job parameters used in the call.

+*> \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), LWORK(CGESVJ)).        

+*>            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 + 2*N.

+*>               ->> For optimal performance (blocked code) the optimal value

+*>               is LWORK >= max(N+(N+1)*NB, N*N+2*N)=N**2+2*N.

+*>               In general, the optimal length LWORK is computed as

+*>               LWORK >= max(N+LWORK(CGEQP3),N+LWORK(CGEQRF), LWORK(CGESVJ),

+*>                            N*N+LWORK(CPOCON)).

+*>          2. If SIGMA and the right singular vectors are needed (JOBV.EQ.'V'),

+*>             (JOBU.EQ.'N')

+*>            2.1   .. no scaled condition estimate requested (JOBE.EQ.'N'):    

+*>            -> the minimal requirement is LWORK >= 3*N.

+*>            -> For optimal performance, 

+*>               LWORK >= max(N+(N+1)*NB, 2*N+N*NB)=2*N+N*NB,

+*>               where NB is the optimal block size for CGEQP3, CGEQRF, CGELQ,

+*>               CUNMLQ. In general, the optimal length LWORK is computed as

+*>               LWORK >= max(N+LWORK(CGEQP3), N+LWORK(CGESVJ),

+*>                       N+LWORK(CGELQF), 2*N+LWORK(CGEQRF), N+LWORK(CUNMLQ)).

+*>            2.2 .. an estimate of the scaled condition number of A is

+*>               required (JOBA='E', or 'G').

+*>            -> the minimal requirement is LWORK >= 3*N.      

+*>            -> For optimal performance, 

+*>               LWORK >= max(N+(N+1)*NB, 2*N,2*N+N*NB)=2*N+N*NB,

+*>               where NB is the optimal block size for CGEQP3, CGEQRF, CGELQ,

+*>               CUNMLQ. In general, the optimal length LWORK is computed as

+*>               LWORK >= max(N+LWORK(CGEQP3), 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

+*>            3.1  .. no scaled condition estimate requested (JOBE.EQ.'N'):

+*>            -> 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)=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), 2*N+LWORK(CGEQRF), N+LWORK(CUNMQR)). 

+*>            3.2  .. an estimate of the scaled condition number of A is

+*>               required (JOBA='E', or 'G').

+*>            -> 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)=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.

+*> 

+*>          If the call to CGEJSV is a workspace query (indicated by LWORK=-1 or

+*>          LRWORK=-1), then on exit CWORK(1) contains the optimal and CWORK(2) contains the

+*>          minimal length of CWORK for the job parameters used in the call.        

+*> \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).)

+*>          If the call to CGEJSV is a workspace query (indicated by LWORK=-1 or

+*>          LRWORK=-1), then on exit RWORK(1) contains the required length of

+*>          RWORK for the job parameters used in the call.

+*> \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 the 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, 2 * M ).

+*>          1.2. Otherwise, LRWORK  = max( 7,  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, 2 * M ).

+*>          2.2. Otherwise, LRWORK  = max( 7,  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, 2 * M ).

+*>          3.2. Otherwise, LRWORK  = max( 7,  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, 2 * M ).

+*>          4.2. Otherwise, LRWORK  = max( 7, N ).

+*> 

+*>          If, on entry, LRWORK = -1 ot LWORK=-1, a workspace query is assumed and 

+*>          the length of RWORK is returned in RWORK(1). 

+*> \endverbatim

+*>

+*> \param[out] IWORK

+*> \verbatim

+*>          IWORK is INTEGER array, of dimension at least 4, that further depends

+*>          on the job:

+*> 

+*>          1. If only the singular values are requested then:

+*>             If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') ) 

+*>             then the length of IWORK is N+M; otherwise the length of IWORK is N.

+*>          2. If the singular values and the right singular vectors are requested then:

+*>             If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') ) 

+*>             then the length of IWORK is N+M; otherwise the length of IWORK is N. 

+*>          3. If the singular values and the left singular vectors are requested then:

+*>             If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') ) 

+*>             then the length of IWORK is N+M; otherwise the length of IWORK is N. 

+*>          4. If the singular values with both the left and the right singular vectors

+*>             are requested, then:      

+*>             4.1. If LSAME(JOBV,'J') the length of IWORK is determined as follows:

+*>                  If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') ) 

+*>                  then the length of IWORK is N+M; otherwise the length of IWORK is N. 

+*>             4.2. If LSAME(JOBV,'V') the length of IWORK is determined as follows:

+*>                  If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') ) 

+*>                  then the length of IWORK is 2*N+M; otherwise the length of IWORK is 2*N.

+*>        

+*>          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.

+*>          IWORK(4) = 1 or -1. If IWORK(4) .EQ. 1, then the procedure used A^* to

+*>                     do the job as specified by the JOB parameters.

+*>          If the call to CGEJSV is a workspace query (indicated by LWORK .EQ. -1 and 

+*>          LRWORK .EQ. -1), then on exit IWORK(1) contains the required length of 

+*>          IWORK for the job parameters used in the call.

+*> \endverbatim

+*>

+*> \param[out] INFO

+*> \verbatim

+*>          INFO is INTEGER

+*>           < 0  : if INFO = -i, then the i-th argument had an illegal value.

+*>           = 0 :  successful 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 deficient cases. It will be available in the SIGMA library [4].

+*>  If M is much larger than N, it is obvious that the initial 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.

+*> \endverbatim

+*

+*> \par Contributor:

+*  ==================

+*>

+*>  Zlatko Drmac (Zagreb, Croatia)

+*

+*> \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, 2016.

+*> \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 3.7.0) --

+*  -- LAPACK is a software package provided by Univ. of Tennessee,    --

+*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--

+*     December 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( LRWORK )

+      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,   LQUERY,

+     $        LSVEC,  L2ABER, L2KILL, L2PERT,  L2RANK, L2TRAN, NOSCAL,

+     $        ROWPIV, RSVEC,  TRANSP

+*

+      INTEGER OPTWRK, MINWRK, MINRWRK, MINIWRK

+      INTEGER LWCON,  LWLQF, LWQP3, LWQRF, LWUNMLQ, LWUNMQR, LWUNMQRM,

+     $        LWSVDJ, LWSVDJV, LRWQP3, LRWCON, LRWSVDJ, IWOFF

+      INTEGER LWRK_CGELQF, LWRK_CGEQP3,  LWRK_CGEQP3N, LWRK_CGEQRF,  

+     $        LWRK_CGESVJ, LWRK_CGESVJV, LWRK_CGESVJU, LWRK_CUNMLQ, 

+     $        LWRK_CUNMQR, LWRK_CUNMQRM    

+*     ..

+*     .. Local Arrays

+      COMPLEX CDUMMY(1)

+      REAL    RDUMMY(1)

+*

+*     .. Intrinsic Functions ..

+      INTRINSIC ABS, CMPLX, CONJG, ALOG, MAX, MIN, REAL, NINT, SQRT

+*     ..

+*     .. External Functions ..

+      REAL      SLAMCH, SCNRM2

+      INTEGER   ISAMAX, ICAMAX

+      LOGICAL   LSAME

+      EXTERNAL  ISAMAX, ICAMAX, LSAME, SLAMCH, SCNRM2

+*     ..

+*     .. External Subroutines ..

+      EXTERNAL  SLASSQ, CCOPY,  CGELQF, CGEQP3, CGEQRF, CLACPY, CLAPMR,

+     $          CLASCL, SLASCL, CLASET, CLASSQ, CLASWP, CUNGQR, CUNMLQ,

+     $          CUNMQR, CPOCON, SSCAL,  CSSCAL, CSWAP,  CTRSM,  CLACGV,

+     $          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' ) .AND. ( M .EQ. N )

+      L2KILL = LSAME( JOBR, 'R' )

+      DEFR   = LSAME( JOBR, 'N' )

+      L2PERT = LSAME( JOBP, 'P' )

+*

+      LQUERY = ( LWORK .EQ. -1 ) .OR. ( LRWORK .EQ. -1 )

+*

+      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' ) .AND. RSVEC .AND. L2TRAN ) ) ) THEN

+         INFO = - 2

+      ELSE IF ( .NOT.( RSVEC .OR. LSAME( JOBV, 'N' ) .OR.

+     $   ( LSAME( JOBV, 'W' ) .AND. LSVEC .AND. L2TRAN ) ) ) THEN

+         INFO = - 3

+      ELSE IF ( .NOT. ( L2KILL .OR. DEFR ) )    THEN

+         INFO = - 4

+      ELSE IF ( .NOT. ( LSAME(JOBT,'T') .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

+*        #:)

+         INFO = 0

+      END IF

+*

+      IF ( INFO .EQ. 0 ) THEN 

+*         .. compute the minimal and the optimal workspace lengths 

+*         [[The expressions for computing the minimal and the optimal

+*         values of LCWORK, LRWORK are written with a lot of redundancy and

+*         can be simplified. However, this verbose form is useful for

+*         maintenance and modifications of the code.]]

+*

+*        .. minimal workspace length for CGEQP3 of an M x N matrix,

+*         CGEQRF of an N x N matrix, CGELQF of an N x N matrix,

+*         CUNMLQ for computing N x N matrix, CUNMQR for computing N x N

+*         matrix, CUNMQR for computing M x N matrix, respectively.

+          LWQP3 = N+1   

+          LWQRF = MAX( 1, N )

+          LWLQF = MAX( 1, N )

+          LWUNMLQ  = MAX( 1, N )

+          LWUNMQR  = MAX( 1, N )

+          LWUNMQRM = MAX( 1, M )

+*        .. minimal workspace length for CPOCON of an N x N matrix

+          LWCON = 2 * N 

+*        .. minimal workspace length for CGESVJ of an N x N matrix,

+*         without and with explicit accumulation of Jacobi rotations

+          LWSVDJ  = MAX( 2 * N, 1 )         

+          LWSVDJV = MAX( 2 * N, 1 )

+*         .. minimal REAL workspace length for CGEQP3, CPOCON, CGESVJ

+          LRWQP3  = N 

+          LRWCON  = N 

+          LRWSVDJ = N 

+          IF ( LQUERY ) THEN 

+              CALL CGEQP3( M, N, A, LDA, IWORK, CDUMMY, CDUMMY, -1, 

+     $             RDUMMY, IERR )

+              LWRK_CGEQP3 = CDUMMY(1)

+              CALL CGEQRF( N, N, A, LDA, CDUMMY, CDUMMY,-1, IERR )

+              LWRK_CGEQRF = CDUMMY(1)

+              CALL CGELQF( N, N, A, LDA, CDUMMY, CDUMMY,-1, IERR )

+              LWRK_CGELQF = CDUMMY(1)             

+          END IF

+          MINWRK  = 2

+          OPTWRK  = 2

+          MINIWRK = N 

+          IF ( .NOT. (LSVEC .OR. RSVEC ) ) THEN

+*             .. minimal and optimal sizes of the complex workspace if

+*             only the singular values are requested

+              IF ( ERREST ) THEN 

+                  MINWRK = MAX( N+LWQP3, N**2+LWCON, N+LWQRF, LWSVDJ )

+              ELSE

+                  MINWRK = MAX( N+LWQP3, N+LWQRF, LWSVDJ )

+              END IF

+              IF ( LQUERY ) THEN 

+                  CALL CGESVJ( 'L', 'N', 'N', N, N, A, LDA, SVA, N, V, 

+     $                 LDV, CDUMMY, -1, RDUMMY, -1, IERR )

+                  LWRK_CGESVJ = CDUMMY(1)

+                  IF ( ERREST ) THEN 

+                      OPTWRK = MAX( N+LWRK_CGEQP3, N**2+LWCON, 

+     $                              N+LWRK_CGEQRF, LWRK_CGESVJ )

+                  ELSE

+                      OPTWRK = MAX( N+LWRK_CGEQP3, N+LWRK_CGEQRF, 

+     $                              LWRK_CGESVJ )

+                  END IF

+              END IF

+              IF ( L2TRAN .OR. ROWPIV ) THEN 

+                  IF ( ERREST ) THEN 

+                     MINRWRK = MAX( 7, 2*M,  LRWQP3, LRWCON, LRWSVDJ )

+                  ELSE

+                     MINRWRK = MAX( 7, 2*M,  LRWQP3, LRWSVDJ )

+                  END IF                 

+              ELSE

+                  IF ( ERREST ) THEN 

+                     MINRWRK = MAX( 7, LRWQP3, LRWCON, LRWSVDJ )

+                  ELSE

+                     MINRWRK = MAX( 7, LRWQP3, LRWSVDJ )

+                  END IF

+              END IF   

+              IF ( ROWPIV .OR. L2TRAN ) MINIWRK = MINIWRK + M 

+          ELSE IF ( RSVEC .AND. (.NOT.LSVEC) ) THEN

+*            .. minimal and optimal sizes of the complex workspace if the

+*            singular values and the right singular vectors are requested

+             IF ( ERREST ) THEN 

+                 MINWRK = MAX( N+LWQP3, LWCON, LWSVDJ, N+LWLQF,  

+     $                         2*N+LWQRF, N+LWSVDJ, N+LWUNMLQ )

+             ELSE

+                 MINWRK = MAX( N+LWQP3, LWSVDJ, N+LWLQF, 2*N+LWQRF, 

+     $                         N+LWSVDJ, N+LWUNMLQ )

+             END IF

+             IF ( LQUERY ) THEN

+                 CALL CGESVJ( 'L', 'U', 'N', N,N, U, LDU, SVA, N, A,

+     $                LDA, CDUMMY, -1, RDUMMY, -1, IERR )

+                 LWRK_CGESVJ = CDUMMY(1)

+                 CALL CUNMLQ( 'L', 'C', N, N, N, A, LDA, CDUMMY,

+     $                V, LDV, CDUMMY, -1, IERR )

+                 LWRK_CUNMLQ = CDUMMY(1)                

+                 IF ( ERREST ) THEN 

+                 OPTWRK = MAX( N+LWRK_CGEQP3, LWCON, LWRK_CGESVJ, 

+     $                         N+LWRK_CGELQF, 2*N+LWRK_CGEQRF,

+     $                         N+LWRK_CGESVJ,  N+LWRK_CUNMLQ )

+                 ELSE

+                 OPTWRK = MAX( N+LWRK_CGEQP3, LWRK_CGESVJ,N+LWRK_CGELQF,

+     $                         2*N+LWRK_CGEQRF, N+LWRK_CGESVJ, 

+     $                         N+LWRK_CUNMLQ )

+                 END IF

+             END IF

+             IF ( L2TRAN .OR. ROWPIV ) THEN 

+                  IF ( ERREST ) THEN 

+                     MINRWRK = MAX( 7, 2*M,  LRWQP3, LRWSVDJ, LRWCON )

+                  ELSE

+                     MINRWRK = MAX( 7, 2*M,  LRWQP3, LRWSVDJ ) 

+                  END IF                  

+             ELSE

+                  IF ( ERREST ) THEN 

+                     MINRWRK = MAX( 7, LRWQP3, LRWSVDJ, LRWCON )

+                  ELSE

+                     MINRWRK = MAX( 7, LRWQP3, LRWSVDJ ) 

+                  END IF                 

+             END IF

+             IF ( ROWPIV .OR. L2TRAN ) MINIWRK = MINIWRK + M

+          ELSE IF ( LSVEC .AND. (.NOT.RSVEC) ) THEN  

+*            .. minimal and optimal sizes of the complex workspace if the

+*            singular values and the left singular vectors are requested

+             IF ( ERREST ) THEN

+                 MINWRK = N + MAX( LWQP3,LWCON,N+LWQRF,LWSVDJ,LWUNMQRM )

+             ELSE

+                 MINWRK = N + MAX( LWQP3, N+LWQRF, LWSVDJ, LWUNMQRM )

+             END IF

+             IF ( LQUERY ) THEN

+                 CALL CGESVJ( 'L', 'U', 'N', N,N, U, LDU, SVA, N, A,

+     $                LDA, CDUMMY, -1, RDUMMY, -1, IERR )

+                 LWRK_CGESVJ = CDUMMY(1)

+                 CALL CUNMQR( 'L', 'N', M, N, N, A, LDA, CDUMMY, U,

+     $               LDU, CDUMMY, -1, IERR )

+                 LWRK_CUNMQRM = CDUMMY(1)

+                 IF ( ERREST ) THEN

+                 OPTWRK = N + MAX( LWRK_CGEQP3, LWCON, N+LWRK_CGEQRF,

+     $                             LWRK_CGESVJ, LWRK_CUNMQRM )

+                 ELSE

+                 OPTWRK = N + MAX( LWRK_CGEQP3, N+LWRK_CGEQRF,

+     $                             LWRK_CGESVJ, LWRK_CUNMQRM )

+                 END IF

+             END IF

+             IF ( L2TRAN .OR. ROWPIV ) THEN 

+                 IF ( ERREST ) THEN 

+                    MINRWRK = MAX( 7, 2*M,  LRWQP3, LRWSVDJ, LRWCON )

+                 ELSE

+                    MINRWRK = MAX( 7, 2*M,  LRWQP3, LRWSVDJ )

+                 END IF                 

+             ELSE

+                 IF ( ERREST ) THEN 

+                    MINRWRK = MAX( 7, LRWQP3, LRWSVDJ, LRWCON )

+                 ELSE

+                    MINRWRK = MAX( 7, LRWQP3, LRWSVDJ )

+                 END IF                

+             END IF 

+             IF ( ROWPIV .OR. L2TRAN ) MINIWRK = MINIWRK + M

+          ELSE

+*            .. minimal and optimal sizes of the complex workspace if the

+*            full SVD is requested

+             IF ( .NOT. JRACC ) THEN                

+                 IF ( ERREST ) THEN 

+                    MINWRK = MAX( N+LWQP3, N+LWCON,  2*N+N**2+LWCON, 

+     $                         2*N+LWQRF,         2*N+LWQP3, 

+     $                         2*N+N**2+N+LWLQF,  2*N+N**2+N+N**2+LWCON,

+     $                         2*N+N**2+N+LWSVDJ, 2*N+N**2+N+LWSVDJV, 

+     $                         2*N+N**2+N+LWUNMQR,2*N+N**2+N+LWUNMLQ, 

+     $                         N+N**2+LWSVDJ,   N+LWUNMQRM )

+                 ELSE

+                    MINWRK = MAX( N+LWQP3,        2*N+N**2+LWCON, 

+     $                         2*N+LWQRF,         2*N+LWQP3, 

+     $                         2*N+N**2+N+LWLQF,  2*N+N**2+N+N**2+LWCON,

+     $                         2*N+N**2+N+LWSVDJ, 2*N+N**2+N+LWSVDJV,

+     $                         2*N+N**2+N+LWUNMQR,2*N+N**2+N+LWUNMLQ,

+     $                         N+N**2+LWSVDJ,      N+LWUNMQRM ) 

+                 END IF 

+                 MINIWRK = MINIWRK + N 

+                 IF ( ROWPIV .OR. L2TRAN ) MINIWRK = MINIWRK + M

+             ELSE

+                 IF ( ERREST ) THEN 

+                    MINWRK = MAX( N+LWQP3, N+LWCON, 2*N+LWQRF, 

+     $                         2*N+N**2+LWSVDJV, 2*N+N**2+N+LWUNMQR, 

+     $                         N+LWUNMQRM )

+                 ELSE

+                    MINWRK = MAX( N+LWQP3, 2*N+LWQRF, 

+     $                         2*N+N**2+LWSVDJV, 2*N+N**2+N+LWUNMQR, 

+     $                         N+LWUNMQRM ) 

+                 END IF   

+                 IF ( ROWPIV .OR. L2TRAN ) MINIWRK = MINIWRK + M

+             END IF

+             IF ( LQUERY ) THEN

+                 CALL CUNMQR( 'L', 'N', M, N, N, A, LDA, CDUMMY, U,

+     $                LDU, CDUMMY, -1, IERR )

+                 LWRK_CUNMQRM = CDUMMY(1)

+                 CALL CUNMQR( 'L', 'N', N, N, N, A, LDA, CDUMMY, U,

+     $                LDU, CDUMMY, -1, IERR )

+                 LWRK_CUNMQR = CDUMMY(1)

+                 IF ( .NOT. JRACC ) THEN

+                     CALL CGEQP3( N,N, A, LDA, IWORK, CDUMMY,CDUMMY, -1,

+     $                    RDUMMY, IERR )

+                     LWRK_CGEQP3N = CDUMMY(1)

+                     CALL CGESVJ( 'L', 'U', 'N', N, N, U, LDU, SVA,

+     $                    N, V, LDV, CDUMMY, -1, RDUMMY, -1, IERR )

+                     LWRK_CGESVJ = CDUMMY(1)

+                     CALL CGESVJ( 'U', 'U', 'N', N, N, U, LDU, SVA,

+     $                    N, V, LDV, CDUMMY, -1, RDUMMY, -1, IERR )

+                     LWRK_CGESVJU = CDUMMY(1)

+                     CALL CGESVJ( 'L', 'U', 'V', N, N, U, LDU, SVA,

+     $                    N, V, LDV, CDUMMY, -1, RDUMMY, -1, IERR )

+                     LWRK_CGESVJV = CDUMMY(1)

+                     CALL CUNMLQ( 'L', 'C', N, N, N, A, LDA, CDUMMY,

+     $                    V, LDV, CDUMMY, -1, IERR )

+                     LWRK_CUNMLQ = CDUMMY(1)

+                     IF ( ERREST ) THEN 

+                       OPTWRK = MAX( N+LWRK_CGEQP3, N+LWCON, 

+     $                          2*N+N**2+LWCON, 2*N+LWRK_CGEQRF, 

+     $                          2*N+LWRK_CGEQP3N, 

+     $                          2*N+N**2+N+LWRK_CGELQF,  

+     $                          2*N+N**2+N+N**2+LWCON,

+     $                          2*N+N**2+N+LWRK_CGESVJ, 

+     $                          2*N+N**2+N+LWRK_CGESVJV,               

+     $                          2*N+N**2+N+LWRK_CUNMQR,

+     $                          2*N+N**2+N+LWRK_CUNMLQ, 

+     $                          N+N**2+LWRK_CGESVJU,                  

+     $                          N+LWRK_CUNMQRM )

+                     ELSE

+                       OPTWRK = MAX( N+LWRK_CGEQP3,  

+     $                          2*N+N**2+LWCON, 2*N+LWRK_CGEQRF, 

+     $                          2*N+LWRK_CGEQP3N, 

+     $                          2*N+N**2+N+LWRK_CGELQF,  

+     $                          2*N+N**2+N+N**2+LWCON,

+     $                          2*N+N**2+N+LWRK_CGESVJ,               

+     $                          2*N+N**2+N+LWRK_CGESVJV, 

+     $                          2*N+N**2+N+LWRK_CUNMQR,

+     $                          2*N+N**2+N+LWRK_CUNMLQ, 

+     $                          N+N**2+LWRK_CGESVJU,

+     $                          N+LWRK_CUNMQRM )

+                     END IF                    

+                 ELSE

+                     CALL CGESVJ( 'L', 'U', 'V', N, N, U, LDU, SVA,

+     $                    N, V, LDV, CDUMMY, -1, RDUMMY, -1, IERR )

+                     LWRK_CGESVJV = CDUMMY(1)

+                     CALL CUNMQR( 'L', 'N', N, N, N, CDUMMY, N, CDUMMY,

+     $                    V, LDV, CDUMMY, -1, IERR )

+                     LWRK_CUNMQR = CDUMMY(1)

+                     CALL CUNMQR( 'L', 'N', M, N, N, A, LDA, CDUMMY, U,

+     $                    LDU, CDUMMY, -1, IERR )

+                     LWRK_CUNMQRM = CDUMMY(1)   

+                     IF ( ERREST ) THEN 

+                        OPTWRK = MAX( N+LWRK_CGEQP3, N+LWCON,   

+     $                           2*N+LWRK_CGEQRF, 2*N+N**2,  

+     $                           2*N+N**2+LWRK_CGESVJV,  

+     $                           2*N+N**2+N+LWRK_CUNMQR,N+LWRK_CUNMQRM )

+                     ELSE

+                        OPTWRK = MAX( N+LWRK_CGEQP3, 2*N+LWRK_CGEQRF,  

+     $                           2*N+N**2, 2*N+N**2+LWRK_CGESVJV, 

+     $                           2*N+N**2+N+LWRK_CUNMQR, 

+     $                           N+LWRK_CUNMQRM )   

+                     END IF                  

+                 END IF               

+             END IF 

+             IF ( L2TRAN .OR. ROWPIV ) THEN 

+                 MINRWRK = MAX( 7, 2*M,  LRWQP3, LRWSVDJ, LRWCON )

+             ELSE

+                 MINRWRK = MAX( 7, LRWQP3, LRWSVDJ, LRWCON )

+             END IF 

+          END IF

+          MINWRK = MAX( 2, MINWRK )

+          OPTWRK = MAX( 2, OPTWRK )

+          IF ( LWORK  .LT. MINWRK  .AND. (.NOT.LQUERY) ) INFO = - 17

+          IF ( LRWORK .LT. MINRWRK .AND. (.NOT.LQUERY) ) INFO = - 19   

+      END IF

+*      

+      IF ( INFO .NE. 0 ) THEN

+*       #:(

+         CALL XERBLA( 'CGEJSV', - INFO )

+         RETURN

+      ELSE IF ( LQUERY ) THEN

+          CWORK(1) = OPTWRK

+          CWORK(2) = MINWRK

+          RWORK(1) = MINRWRK

+          IWORK(1) = MAX( 4, MINIWRK )

+          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(REAL(M)*REAL(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 = MAX( AAPP, SVA(p) )

+         IF ( SVA(p) .NE. ZERO ) AAQQ = MIN( 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

+         IWORK(4) = -1

+         RETURN

+      END IF

+*

+*     Issue warning if denormalized column norms detected. Override the

+*     high relative accuracy request. Issue licence to kill nonzero 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

+         IWORK(4) = -1

+         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.

+*

+      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+p)  = XSC * SCALEM

+               RWORK(p)    = XSC * (SCALEM*SQRT(TEMP1))

+               AATMAX = MAX( AATMAX, RWORK(p) )

+               IF (RWORK(p) .NE. ZERO) 

+     $            AATMIN = MIN(AATMIN,RWORK(p))

+ 1950       CONTINUE

+         ELSE

+            DO 1904 p = 1, M

+               RWORK(M+p) = SCALEM*ABS( A(p,ICAMAX(N,A(p,1),LDA)) )

+               AATMAX = MAX( AATMAX, RWORK(M+p) )

+               AATMIN = MIN( AATMIN, RWORK(M+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(REAL(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 = 1, M

+            BIG1 = ( ( RWORK(p) / XSC )**2 ) * TEMP1

+            IF ( BIG1 .NE. ZERO ) ENTRAT = ENTRAT + BIG1 * ALOG(BIG1)

+ 1114    CONTINUE

+         ENTRAT = - ENTRAT / ALOG(REAL(M))

+*

+*        Analyze the entropies and decide A or A^*. Smaller entropy

+*        usually means better input for the algorithm.

+*

+         TRANSP = ( ENTRAT .LT. ENTRA )

+* 

+*        If A^* is better than A, take the adjoint of A. This is allowed

+*        only for square matrices, M=N.  

+         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+p) = SVA(p)

+               SVA(p) = RWORK(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 / REAL(N) )

+*     >> for future updates: allow bigger range, i.e. the largest column

+*     will be allowed up to BIG/N and CGESVJ will do the rest. However, for

+*     this all other (LAPACK) components must allow such a range.      

+*     TEMP1  = BIG/REAL(N)

+*     TEMP1  = BIG * EPSLN  this should 'almost' work with current LAPACK components

+      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.

+         IF ( ( LSVEC .AND. RSVEC ) .AND. .NOT.( JRACC ) ) THEN 

+              IWOFF = 2*N

+         ELSE

+              IWOFF = N

+         END IF

+         DO 1952 p = 1, M - 1

+            q = ISAMAX( M-p+1, RWORK(M+p), 1 ) + p - 1

+            IWORK(IWOFF+p) = q

+            IF ( p .NE. q ) THEN

+               TEMP1      = RWORK(M+p)

+               RWORK(M+p) = RWORK(M+q)

+               RWORK(M+q) = TEMP1

+            END IF

+ 1952    CONTINUE

+         CALL CLASWP( N, A, LDA, 1, M-1, IWORK(IWOFF+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(REAL(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 = MIN( MAXPRJ, TEMP1 )

+ 3051    CONTINUE

+         IF ( MAXPRJ**2 .GE. ONE - REAL(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

+               IF ( LSVEC )THEN

+                   CALL CPOCON( 'U', N, V, LDV, ONE, TEMP1,

+     $                  CWORK(N+1), RWORK, IERR )

+               ELSE

+                   CALL CPOCON( 'U', N, V, LDV, ONE, TEMP1,

+     $                  CWORK, RWORK, IERR )

+               END IF               

+*          

+            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 )

+*[]            CALL CLACPY( 'U', N, N, A, LDA, CWORK(N+1), N )

+*              Change: here index shifted by N to the left, CWORK(1:N) 

+*              not needed for SIGMA only computation

+               DO 3052 p = 1, N

+                  TEMP1 = SVA(IWORK(p))

+*[]               CALL CSSCAL( p, ONE/TEMP1, CWORK(N+(p-1)*N+1), 1 )

+                  CALL CSSCAL( p, ONE/TEMP1, CWORK((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 )

+               CALL CPOCON( 'U', N, CWORK, N, ONE, TEMP1,

+     $              CWORK(N*N+1), RWORK, IERR )               

+*              

+            END IF

+            IF ( TEMP1 .NE. ZERO ) THEN 

+               SCONDA = ONE / SQRT(TEMP1)

+            ELSE

+               SCONDA = - ONE

+            END IF

+*           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, MIN( 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 / REAL(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 / REAL(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', 'N', 'N', 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 ) .AND. ( .NOT. JRACC ) ) 

+     $       .OR. 

+     $   ( JRACC .AND. ( .NOT. LSVEC ) .AND. ( NR .NE. N ) ) ) 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( 'U', 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( 'L', 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( 'L', NR, NR, A, LDA, V, LDV )

+            CALL CLASET( 'U', 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('U', NR-1, NR-1, CZERO, CZERO, V(1,2), LDV)

+*

+            CALL CGESVJ( 'L', '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( 'L', 'C', N, N, NR, A, LDA, CWORK,

+     $               V, LDV, CWORK(N+1), LWORK-N, IERR )

+*

+         END IF

+*         .. permute the rows of V

+*         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 )

+         CALL CLAPMR( .FALSE., N, N, V, LDV, IWORK )

+*

+          IF ( TRANSP ) THEN

+            CALL CLACPY( 'A', N, N, V, LDV, U, LDU )

+          END IF

+*

+      ELSE IF ( JRACC .AND. (.NOT. LSVEC) .AND. ( NR.EQ. N ) ) THEN 

+*          

+         CALL CLASET( 'L', N-1,N-1, CZERO, CZERO, A(2,1), LDA )

+*

+         CALL CGESVJ( 'U','N','V', N, N, A, LDA, SVA, N, V, LDV,

+     $               CWORK, LWORK, RWORK, LRWORK, INFO )

+          SCALEM  = RWORK(1)

+          NUMRANK = NINT(RWORK(2))

+          CALL CLAPMR( .FALSE., N, N, V, LDV, IWORK )

+*

+      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( 'U', 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( 'U', NR-1, NR-1, CZERO, CZERO, U(1,2), LDU )

+*

+         CALL CGESVJ( 'L', '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( 'L', 'N', 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(IWOFF+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( 'A', 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('L',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. REAL(N)

+*           more conservative    <=> CONDR1 .LT. SQRT(REAL(N))

+*

+            COND_OK = SQRT(SQRT(REAL(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*MIN(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*MIN(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*MIN(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) = - 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(REAL(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( 'L', 'N', 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(REAL(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(IWOFF+1), -1 )

+*

+         ELSE

+*

+*        .. the initial matrix A has almost orthogonal columns and

+*        the second QRF is not needed

+*

+            CALL CLACPY( 'U', 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( 'L',N-1,N-1,CZERO,CZERO,CWORK(N+2),N )

+            END IF

+*

+            CALL CGESVJ( 'U', '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( 'L', 'U', 'N', 'N', 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(REAL(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( 'L', 'N', M, N1, N, A, LDA, CWORK, U,

+     $           LDU, CWORK(N+1), LWORK-N, IERR )

+            TEMP1 = SQRT(REAL(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(IWOFF+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, e.g. when the singular values spread from

+*        the underflow to the overflow threshold. 

+*

+         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 * MIN(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(REAL(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( 'L', 'N', 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(IWOFF+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

+      IF ( TRANSP ) THEN

+          IWORK(4) =  1 

+      ELSE

+          IWORK(4) = -1

+      END IF 

+      

+*

+      RETURN

+*     ..

+*     .. END OF CGEJSV

+*     ..

+      END

+*

diff --git a/SRC/zgejsv.f b/SRC/zgejsv.f
index fa85af00..f8b4ba9a 100644
--- a/SRC/zgejsv.f
+++ b/SRC/zgejsv.f
@@ -1,1876 +1,2237 @@
-*> \brief \b ZGEJSV
-*
-*  =========== DOCUMENTATION ===========
-*
-* Online html documentation available at
-*            http://www.netlib.org/lapack/explore-html/
-*
-*> \htmlonly
-*> Download ZGEJSV + dependencies
-*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgejsv.f">
-*> [TGZ]</a>
-*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgejsv.f">
-*> [ZIP]</a>
-*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgejsv.f">
-*> [TXT]</a>
-*> \endhtmlonly
-*
-*  Definition:
-*  ===========
-*
-*     SUBROUTINE ZGEJSV( 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*16     A( LDA, * ),  U( LDU, * ), V( LDV, * ), CWORK( LWORK )
-*     DOUBLE PRECISION   SVA( N ), RWORK( LRWORK )
-*     INTEGER     IWORK( * )
-*     CHARACTER*1 JOBA, JOBP, JOBR, JOBT, JOBU, JOBV
-*       ..
-*
-*
-*> \par Purpose:
-*  =============
-*>
-*> \verbatim
-*>
-*> ZGEJSV 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 deficient. 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=DLAMCH('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=DLAMCH('S'), EPSLN=DLAMCH('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 ZGESVJ.
-*>       = '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 ZGESVJ.
-*> \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*16 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 DOUBLE PRECISION 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*16 array, dimension ( LDU, N )
-*>          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*16 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*16 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 ZGEQP3 and ZGEQRF.
-*>               In general, optimal LWORK is computed as
-*>               LWORK >= max(N+LWORK(ZGEQP3),N+LWORK(ZGEQRF)).
-*>            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(ZGEQP3),N+LWORK(ZGEQRF),
-*>                                                     N+N*N+LWORK(ZPOCON)).
-*>
-*>          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 ZGEQP3, ZGEQRF, ZGELQF,
-*>               ZUNMLQ. In general, the optimal length LWORK is computed as
-*>               LWORK >= max(N+LWORK(ZGEQP3), N+LWORK(ZPOCON), N+LWORK(ZGESVJ),
-*>                       N+LWORK(ZGELQF), 2*N+LWORK(ZGEQRF), N+LWORK(ZUNMLQ)).
-*>
-*>          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 ZGEQP3, ZGEQRF, ZUNMQR.
-*>               In general, the optimal length LWORK is computed as
-*>               LWORK >= max(N+LWORK(ZGEQP3),N+LWORK(ZPOCON),
-*>                        2*N+LWORK(ZGEQRF), N+LWORK(ZUNMQR)).
-*>
-*>          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 ZGEQP3, ZGEQRF, ZGELQF, ZUNMQR, ZUNMLQ.
-*> \endverbatim
-*>
-*> \param[out] RWORK
-*> \verbatim
-*>          RWORK is DOUBLE PRECISION 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 :  successful exit;
-*>           > 0 :  ZGEJSV  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 complex16GEsing
-*
-*> \par Further Details:
-*  =====================
-*>
-*> \verbatim
-*>
-*>  ZGEJSV implements a preconditioned Jacobi SVD algorithm. It uses ZGEQP3,
-*>  ZGEQRF, and ZGELQF 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 ZGEJSV 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 (ZGEJSV) is best used in this restricted range,
-*>  meaning that singular values of magnitude below ||A||_2 / DLAMCH('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 (ZGESVJ) is
-*>  left to the implementer on a particular machine.
-*>     The rank revealing QR factorization (in this code: ZGEQP3) should be
-*>  implemented as in [3]. We have a new version of ZGEQP3 under development
-*>  that is more robust than the current one in LAPACK, with a cleaner cut in
-*>  rank deficient cases. It will be available in the SIGMA library [4].
-*>  If M is much larger than N, it is obvious that the initial QRF with
-*>  column pivoting can be preprocessed by the QRF without pivoting. That
-*>  well known trick is not used in ZGEJSV 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 ZGEJSV uses only the simplest, naive data movement.
-*> \endverbatim
-*
-*> \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 ZGEJSV( 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*16       A( LDA, * ), U( LDU, * ), V( LDV, * ),
-     $                 CWORK( LWORK )
-      DOUBLE PRECISION SVA( N ), RWORK( * )
-      INTEGER          IWORK( * )
-      CHARACTER*1      JOBA, JOBP, JOBR, JOBT, JOBU, JOBV
-*     ..
-*
-*  ===========================================================================
-*
-*     .. Local Parameters ..
-      DOUBLE PRECISION ZERO,         ONE
-      PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
-      COMPLEX*16                CZERO,       CONE
-      PARAMETER  ( CZERO = ( 0.0D0, 0.0D0 ), CONE = ( 1.0D0, 0.0D0 ) )
-*     ..
-*     .. Local Scalars ..
-      COMPLEX*16       CTEMP
-      DOUBLE PRECISION 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,  DCMPLX, DCONJG, DLOG, DMAX1, DMIN1, DBLE,
-     $          MAX0, MIN0, NINT,  DSQRT
-*     ..
-*     .. External Functions ..
-      DOUBLE PRECISION      DLAMCH, DZNRM2
-      INTEGER   IDAMAX, IZAMAX
-      LOGICAL   LSAME
-      EXTERNAL  IDAMAX, IZAMAX, LSAME, DLAMCH, DZNRM2
-*     ..
-*     .. External Subroutines ..
-      EXTERNAL  DLASSQ, ZCOPY,  ZGELQF, ZGEQP3, ZGEQRF, ZLACPY, ZLASCL,
-     $          DLASCL, ZLASET, ZLASSQ, ZLASWP, ZUNGQR, ZUNMLQ,
-     $          ZUNMQR, ZPOCON, DSCAL, ZDSCAL, ZSWAP,  ZTRSM,  XERBLA
-*
-      EXTERNAL  ZGESVJ
-*     ..
-*
-*     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( 'ZGEJSV', - 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 DLAMCH() does not fail on the target architecture.
-*
-      EPSLN = DLAMCH('Epsilon')
-      SFMIN = DLAMCH('SafeMinimum')
-      SMALL = SFMIN / EPSLN
-      BIG   = DLAMCH('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 / DSQRT(DBLE(M)*DBLE(N))
-      NOSCAL  = .TRUE.
-      GOSCAL  = .TRUE.
-      DO 1874 p = 1, N
-         AAPP = ZERO
-         AAQQ = ONE
-         CALL ZLASSQ( M, A(1,p), 1, AAPP, AAQQ )
-         IF ( AAPP .GT. BIG ) THEN
-            INFO = - 9
-            CALL XERBLA( 'ZGEJSV', -INFO )
-            RETURN
-         END IF
-         AAQQ = DSQRT(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 DSCAL( 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 = DMAX1( AAPP, SVA(p) )
-         IF ( SVA(p) .NE. ZERO ) AAQQ = DMIN1( AAQQ, SVA(p) )
- 4781 CONTINUE
-*
-*     Quick return for zero M x N matrix
-* #:)
-      IF ( AAPP .EQ. ZERO ) THEN
-         IF ( LSVEC ) CALL ZLASET( 'G', M, N1, CZERO, CONE, U, LDU )
-         IF ( RSVEC ) CALL ZLASET( '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 ZLASCL( 'G',0,0,SVA(1),SCALEM, M,1,A(1,1),LDA,IERR )
-            CALL ZLACPY( '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 ZGEQRF( M, N, U,LDU, CWORK, CWORK(N+1),LWORK-N,IERR )
-              CALL ZUNGQR( M,N1,1, U,LDU,CWORK,CWORK(N+1),LWORK-N,IERR )
-              CALL ZCOPY( 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 ZLASSQ( N, A(p,1), LDA, XSC, TEMP1 )
-*              ZLASSQ 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*DSQRT(TEMP1))
-               AATMAX = DMAX1( AATMAX, RWORK(N+p) )
-               IF (RWORK(N+p) .NE. ZERO)
-     $            AATMIN = DMIN1(AATMIN,RWORK(N+p))
- 1950       CONTINUE
-         ELSE
-            DO 1904 p = 1, M
-               RWORK(M+N+p) = SCALEM*ABS( A(p,IZAMAX(N,A(p,1),LDA)) )
-               AATMAX = DMAX1( AATMAX, RWORK(M+N+p) )
-               AATMIN = DMIN1( 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 DLASSQ( 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 * DLOG(BIG1)
- 1113    CONTINUE
-         ENTRA = - ENTRA / DLOG(DBLE(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 * DLOG(BIG1)
- 1114    CONTINUE
-         ENTRAT = - ENTRAT / DLOG(DBLE(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) = DCONJG(A(p,p))
-               DO 1116 q = p + 1, N
-                   CTEMP = DCONJG(A(q,p))
-                  A(q,p) = DCONJG(A(p,q))
-                  A(p,q) = CTEMP
- 1116          CONTINUE
- 1115       CONTINUE
-            A(N,N) = DCONJG(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 ZGEJSV 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 ZGESVJ will compute them. So, in that case,
-*     one should use ZGESVJ instead of ZGEJSV.
-*
-      BIG1   = DSQRT( BIG )
-      TEMP1  = DSQRT( BIG / DBLE(N) )
-*
-      CALL DLASCL( '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 ZLASCL( '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 = DSQRT( 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 ZGESVJ
-*        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.DSQRT(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 ZLASET( '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 = IDAMAX( 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 ZLASWP( 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 ZGEQP3 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 ZGEQP3 improves overal performance of ZGEJSV.
-*
-*     A * P1 = Q1 * [ R1^* 0]^*:
-      DO 1963 p = 1, N
-*        .. all columns are free columns
-         IWORK(p) = 0
- 1963 CONTINUE
-      CALL ZGEQP3( 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 ZGEJSV 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 = DSQRT(DBLE(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-deficient.
-         TEMP1 = DSQRT(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  = DSQRT(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 = DMIN1( MAXPRJ, TEMP1 )
- 3051    CONTINUE
-         IF ( MAXPRJ**2 .GE. ONE - DBLE(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 ZLACPY( 'U', N, N, A, LDA, V, LDV )
-               DO 3053 p = 1, N
-                  TEMP1 = SVA(IWORK(p))
-                  CALL ZDSCAL( p, ONE/TEMP1, V(1,p), 1 )
- 3053          CONTINUE
-               CALL ZPOCON( 'U', N, V, LDV, ONE, TEMP1,
-     $              CWORK(N+1), RWORK, IERR )
-*
-            ELSE IF ( LSVEC ) THEN
-*              .. U is available as workspace
-               CALL ZLACPY( 'U', N, N, A, LDA, U, LDU )
-               DO 3054 p = 1, N
-                  TEMP1 = SVA(IWORK(p))
-                  CALL ZDSCAL( p, ONE/TEMP1, U(1,p), 1 )
- 3054          CONTINUE
-               CALL ZPOCON( 'U', N, U, LDU, ONE, TEMP1,
-     $              CWORK(N+1), RWORK, IERR )
-            ELSE
-               CALL ZLACPY( 'U', N, N, A, LDA, CWORK(N+1), N )
-               DO 3052 p = 1, N
-                  TEMP1 = SVA(IWORK(p))
-                  CALL ZDSCAL( p, ONE/TEMP1, CWORK(N+(p-1)*N+1), 1 )
- 3052          CONTINUE
-*           .. the columns of R are scaled to have unit Euclidean lengths.
-               CALL ZPOCON( 'U', N, CWORK(N+1), N, ONE, TEMP1,
-     $              CWORK(N+N*N+1), RWORK, IERR )
-*
-            END IF
-            SCONDA = ONE / DSQRT(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. DSQRT(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 ZCOPY( N-p, A(p,p+1), LDA, A(p+1,p), 1 )
-            CALL ZLACGV( N-p+1, A(p,p), 1 )
- 1946    CONTINUE
-         IF ( NR .EQ. N ) A(N,N) = DCONJG(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 / DBLE(N)
-               DO 4947 q = 1, NR
-                  CTEMP = DCMPLX(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 ZLASET( 'U', NR-1,NR-1, CZERO,CZERO, A(1,2),LDA )
-            END IF
-*
-*            .. second preconditioning using the QR factorization
-*
-            CALL ZGEQRF( N,NR, A,LDA, CWORK, CWORK(N+1),LWORK-N, IERR )
-*
-*           .. and transpose upper to lower triangular
-            DO 1948 p = 1, NR - 1
-               CALL ZCOPY( NR-p, A(p,p+1), LDA, A(p+1,p), 1 )
-               CALL ZLACGV( 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 / DBLE(N)
-               DO 1947 q = 1, NR
-                  CTEMP = DCMPLX(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 ZLASET( '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 ZGESVJ( '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 ZCOPY( N-p+1, A(p,p), LDA, V(p,p), 1 )
-               CALL ZLACGV( N-p+1, V(p,p), 1 )
- 1998       CONTINUE
-            CALL ZLASET( 'Upper', NR-1,NR-1, CZERO, CZERO, V(1,2), LDV )
-*
-            CALL ZGESVJ( '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 ZLASET( 'Lower', NR-1,NR-1, CZERO, CZERO, A(2,1), LDA )
-            CALL ZGELQF( NR,N, A, LDA, CWORK, CWORK(N+1), LWORK-N, IERR)
-            CALL ZLACPY( 'Lower', NR, NR, A, LDA, V, LDV )
-            CALL ZLASET( 'Upper', NR-1,NR-1, CZERO, CZERO, V(1,2), LDV )
-            CALL ZGEQRF( NR, NR, V, LDV, CWORK(N+1), CWORK(2*N+1),
-     $                   LWORK-2*N, IERR )
-            DO 8998 p = 1, NR
-               CALL ZCOPY( NR-p+1, V(p,p), LDV, V(p,p), 1 )
-               CALL ZLACGV( NR-p+1, V(p,p), 1 )
- 8998       CONTINUE
-            CALL ZLASET('Upper', NR-1, NR-1, CZERO, CZERO, V(1,2), LDV)
-*
-            CALL ZGESVJ( '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 ZLASET( 'A',N-NR, NR, CZERO,CZERO, V(NR+1,1),  LDV )
-               CALL ZLASET( 'A',NR, N-NR, CZERO,CZERO, V(1,NR+1),  LDV )
-               CALL ZLASET( 'A',N-NR,N-NR,CZERO,CONE, V(NR+1,NR+1),LDV )
-            END IF
-*
-         CALL ZUNMLQ( 'Left', 'C', N, N, NR, A, LDA, CWORK,
-     $               V, LDV, CWORK(N+1), LWORK-N, IERR )
-*
-         END IF
-*
-         DO 8991 p = 1, N
-            CALL ZCOPY( N, V(p,1), LDV, A(IWORK(p),1), LDA )
- 8991    CONTINUE
-         CALL ZLACPY( 'All', N, N, A, LDA, V, LDV )
-*
-         IF ( TRANSP ) THEN
-            CALL ZLACPY( '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 ZCOPY( N-p+1, A(p,p), LDA, U(p,p), 1 )
-            CALL ZLACGV( N-p+1, U(p,p), 1 )
- 1965    CONTINUE
-         CALL ZLASET( 'Upper', NR-1, NR-1, CZERO, CZERO, U(1,2), LDU )
-*
-         CALL ZGEQRF( N, NR, U, LDU, CWORK(N+1), CWORK(2*N+1),
-     $              LWORK-2*N, IERR )
-*
-         DO 1967 p = 1, NR - 1
-            CALL ZCOPY( NR-p, U(p,p+1), LDU, U(p+1,p), 1 )
-            CALL ZLACGV( N-p+1, U(p,p), 1 )
- 1967    CONTINUE
-         CALL ZLASET( 'Upper', NR-1, NR-1, CZERO, CZERO, U(1,2), LDU )
-*
-         CALL ZGESVJ( '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 ZLASET( 'A',  M-NR, NR,CZERO, CZERO, U(NR+1,1), LDU )
-            IF ( NR .LT. N1 ) THEN
-               CALL ZLASET( 'A',NR, N1-NR, CZERO, CZERO, U(1,NR+1),LDU )
-               CALL ZLASET( 'A',M-NR,N1-NR,CZERO,CONE,U(NR+1,NR+1),LDU )
-            END IF
-         END IF
-*
-         CALL ZUNMQR( 'Left', 'No Tr', M, N1, N, A, LDA, CWORK, U,
-     $               LDU, CWORK(N+1), LWORK-N, IERR )
-*
-         IF ( ROWPIV )
-     $       CALL ZLASWP( N1, U, LDU, 1, M-1, IWORK(2*N+1), -1 )
-*
-         DO 1974 p = 1, N1
-            XSC = ONE / DZNRM2( M, U(1,p), 1 )
-            CALL ZDSCAL( M, XSC, U(1,p), 1 )
- 1974    CONTINUE
-*
-         IF ( TRANSP ) THEN
-            CALL ZLACPY( '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 ZGEJSV.
-*
-            DO 1968 p = 1, NR
-               CALL ZCOPY( N-p+1, A(p,p), LDA, V(p,p), 1 )
-               CALL ZLACGV( 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 = DSQRT(SMALL)
-               DO 2969 q = 1, NR
-                  CTEMP = DCMPLX(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 ZLASET( '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 ZLACPY( 'L', NR, NR, V, LDV, CWORK(2*N+1), NR )
-            DO 3950 p = 1, NR
-               TEMP1 = DZNRM2(NR-p+1,CWORK(2*N+(p-1)*NR+p),1)
-               CALL ZDSCAL(NR-p+1,ONE/TEMP1,CWORK(2*N+(p-1)*NR+p),1)
- 3950       CONTINUE
-            CALL ZPOCON('Lower',NR,CWORK(2*N+1),NR,ONE,TEMP1,
-     $                   CWORK(2*N+NR*NR+1),RWORK,IERR)
-            CONDR1 = ONE / DSQRT(TEMP1)
-*           .. here need a second oppinion on the condition number
-*           .. then assume worst case scenario
-*           R1 is OK for inverse <=> CONDR1 .LT. DBLE(N)
-*           more conservative    <=> CONDR1 .LT. SQRT(DBLE(N))
-*
-            COND_OK = DSQRT(DSQRT(DBLE(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 ZGEQRF( N, NR, V, LDV, CWORK(N+1), CWORK(2*N+1),
-     $              LWORK-2*N, IERR )
-*
-               IF ( L2PERT ) THEN
-                  XSC = DSQRT(SMALL)/EPSLN
-                  DO 3959 p = 2, NR
-                     DO 3958 q = 1, p - 1
-                        CTEMP=DCMPLX(XSC*DMIN1(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 ZLACPY( '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 ZCOPY( NR-p, V(p,p+1), LDV, V(p+1,p), 1 )
-                  CALL ZLACGV(NR-p+1, V(p,p), 1 )
- 1969          CONTINUE
-               V(NR,NR)=DCONJG(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 ZGEQP3
-*              should be replaced with eg. CALL ZGEQPX (ACM TOMS #782)
-*              with properly (carefully) chosen parameters.
-*
-*              R1^* * P2 = Q2 * R2
-               DO 3003 p = 1, NR
-                  IWORK(N+p) = 0
- 3003          CONTINUE
-               CALL ZGEQP3( N, NR, V, LDV, IWORK(N+1), CWORK(N+1),
-     $                  CWORK(2*N+1), LWORK-2*N, RWORK, IERR )
-**               CALL ZGEQRF( N, NR, V, LDV, CWORK(N+1), CWORK(2*N+1),
-**     $              LWORK-2*N, IERR )
-               IF ( L2PERT ) THEN
-                  XSC = DSQRT(SMALL)
-                  DO 3969 p = 2, NR
-                     DO 3968 q = 1, p - 1
-                        CTEMP=DCMPLX(XSC*DMIN1(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 ZLACPY( 'A', N, NR, V, LDV, CWORK(2*N+1), N )
-*
-               IF ( L2PERT ) THEN
-                  XSC = DSQRT(SMALL)
-                  DO 8970 p = 2, NR
-                     DO 8971 q = 1, p - 1
-                        CTEMP=DCMPLX(XSC*DMIN1(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 ZLASET( 'L',NR-1,NR-1,CZERO,CZERO,V(2,1),LDV )
-               END IF
-*              Now, compute R2 = L3 * Q3, the LQ factorization.
-               CALL ZGELQF( 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 ZLACPY( 'L',NR,NR,V,LDV,CWORK(2*N+N*NR+NR+1),NR )
-               DO 4950 p = 1, NR
-                  TEMP1 = DZNRM2( p, CWORK(2*N+N*NR+NR+p), NR )
-                  CALL ZDSCAL( p, ONE/TEMP1, CWORK(2*N+N*NR+NR+p), NR )
- 4950          CONTINUE
-               CALL ZPOCON( '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 / DSQRT(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 ZLACPY( '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 = DSQRT(SMALL)
-               DO 4968 q = 2, NR
-                  CTEMP = XSC * V(q,q)
-                  DO 4969 p = 1, q - 1
-*                     V(p,q) = - TEMP1*( V(p,q) / ABS(V(p,q)) )
-                     V(p,q) = - CTEMP
- 4969             CONTINUE
- 4968          CONTINUE
-            ELSE
-               CALL ZLASET( '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 ZGESVJ( '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 ZCOPY(  NR, V(1,p), 1, U(1,p), 1 )
-                  CALL ZDSCAL( 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 ZGESVJ, premultiplied with the orthogonal matrix
-*                 from the second QR factorization.
-                  CALL ZTRSM('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 ZGESVJ. The Q-factor from the second QR
-*                 factorization is then built in explicitly.
-                  CALL ZTRSM('L','U','C','N',NR,NR,CONE,CWORK(2*N+1),
-     $                 N,V,LDV)
-                  IF ( NR .LT. N ) THEN
-                  CALL ZLASET('A',N-NR,NR,CZERO,CZERO,V(NR+1,1),LDV)
-                  CALL ZLASET('A',NR,N-NR,CZERO,CZERO,V(1,NR+1),LDV)
-                  CALL ZLASET('A',N-NR,N-NR,CZERO,CONE,V(NR+1,NR+1),LDV)
-                  END IF
-                  CALL ZUNMQR('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 ZGESVJ( '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 ZCOPY( NR, V(1,p), 1, U(1,p), 1 )
-                  CALL ZDSCAL( NR, SVA(p),    U(1,p), 1 )
- 3870          CONTINUE
-               CALL ZTRSM('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 ZLASET( 'A',N-NR,NR,CZERO,CZERO,V(NR+1,1),LDV )
-                  CALL ZLASET( 'A',NR,N-NR,CZERO,CZERO,V(1,NR+1),LDV )
-                  CALL ZLASET('A',N-NR,N-NR,CZERO,CONE,V(NR+1,NR+1),LDV)
-               END IF
-               CALL ZUNMQR( '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 ZGEJSV completes the task.
-*              Compute the full SVD of L3 using ZGESVJ with explicit
-*              accumulation of Jacobi rotations.
-               CALL ZGESVJ( '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 ZLASET( 'A',N-NR,NR,CZERO,CZERO,V(NR+1,1),LDV )
-                  CALL ZLASET( 'A',NR,N-NR,CZERO,CZERO,V(1,NR+1),LDV )
-                  CALL ZLASET('A',N-NR,N-NR,CZERO,CONE,V(NR+1,NR+1),LDV)
-               END IF
-               CALL ZUNMQR( '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 ZUNMLQ( '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 = DSQRT(DBLE(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 / DZNRM2( N, V(1,q), 1 )
-               IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
-     $           CALL ZDSCAL( 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 ZLASET('A', M-NR, NR, CZERO, CZERO, U(NR+1,1), LDU)
-               IF ( NR .LT. N1 ) THEN
-                  CALL ZLASET('A',NR,N1-NR,CZERO,CZERO,U(1,NR+1),LDU)
-                  CALL ZLASET('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 ZUNMQR( '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 = DSQRT(DBLE(M)) * EPSLN
-            DO 1973 p = 1, NR
-               XSC = ONE / DZNRM2( M, U(1,p), 1 )
-               IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
-     $          CALL ZDSCAL( 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 ZLASWP( 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 ZLACPY( 'Upper', N, N, A, LDA, CWORK(N+1), N )
-            IF ( L2PERT ) THEN
-               XSC = DSQRT(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 ZLASET( 'Lower',N-1,N-1,CZERO,CZERO,CWORK(N+2),N )
-            END IF
-*
-            CALL ZGESVJ( '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 ZCOPY( N, CWORK(N+(p-1)*N+1), 1, U(1,p), 1 )
-               CALL ZDSCAL( N, SVA(p), CWORK(N+(p-1)*N+1), 1 )
- 6970       CONTINUE
-*
-            CALL ZTRSM( 'Left', 'Upper', 'NoTrans', 'No UD', N, N,
-     $           CONE, A, LDA, CWORK(N+1), N )
-            DO 6972 p = 1, N
-               CALL ZCOPY( N, CWORK(N+p), N, V(IWORK(p),1), LDV )
- 6972       CONTINUE
-            TEMP1 = DSQRT(DBLE(N))*EPSLN
-            DO 6971 p = 1, N
-               XSC = ONE / DZNRM2( N, V(1,p), 1 )
-               IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
-     $            CALL ZDSCAL( N, XSC, V(1,p), 1 )
- 6971       CONTINUE
-*
-*           Assemble the left singular vector matrix U (M x N).
-*
-            IF ( N .LT. M ) THEN
-               CALL ZLASET( 'A',  M-N, N, CZERO, CZERO, U(N+1,1), LDU )
-               IF ( N .LT. N1 ) THEN
-                  CALL ZLASET('A',N,  N1-N, CZERO, CZERO,  U(1,N+1),LDU)
-                  CALL ZLASET( 'A',M-N,N1-N, CZERO, CONE,U(N+1,N+1),LDU)
-               END IF
-            END IF
-            CALL ZUNMQR( 'Left', 'No Tr', M, N1, N, A, LDA, CWORK, U,
-     $           LDU, CWORK(N+1), LWORK-N, IERR )
-            TEMP1 = DSQRT(DBLE(M))*EPSLN
-            DO 6973 p = 1, N1
-               XSC = ONE / DZNRM2( M, U(1,p), 1 )
-               IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
-     $            CALL ZDSCAL( M, XSC, U(1,p), 1 )
- 6973       CONTINUE
-*
-            IF ( ROWPIV )
-     $         CALL ZLASWP( 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 ZCOPY( N-p+1, A(p,p), LDA, V(p,p), 1 )
-            CALL ZLACGV( N-p+1, V(p,p), 1 )
- 7968    CONTINUE
-*
-         IF ( L2PERT ) THEN
-            XSC = DSQRT(SMALL/EPSLN)
-            DO 5969 q = 1, NR
-               CTEMP = DCMPLX(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 ZLASET( 'U', NR-1, NR-1, CZERO, CZERO, V(1,2), LDV )
-         END IF
-
-         CALL ZGEQRF( N, NR, V, LDV, CWORK(N+1), CWORK(2*N+1),
-     $        LWORK-2*N, IERR )
-         CALL ZLACPY( 'L', N, NR, V, LDV, CWORK(2*N+1), N )
-*
-         DO 7969 p = 1, NR
-            CALL ZCOPY( NR-p+1, V(p,p), LDV, U(p,p), 1 )
-            CALL ZLACGV( NR-p+1, U(p,p), 1 )
- 7969    CONTINUE
-
-         IF ( L2PERT ) THEN
-            XSC = DSQRT(SMALL/EPSLN)
-            DO 9970 q = 2, NR
-               DO 9971 p = 1, q - 1
-                  CTEMP = DCMPLX(XSC * DMIN1(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 ZLASET('U', NR-1, NR-1, CZERO, CZERO, U(1,2), LDU )
-         END IF
-
-         CALL ZGESVJ( '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 ZLASET( 'A',N-NR,NR,CZERO,CZERO,V(NR+1,1),LDV )
-            CALL ZLASET( 'A',NR,N-NR,CZERO,CZERO,V(1,NR+1),LDV )
-            CALL ZLASET( 'A',N-NR,N-NR,CZERO,CONE,V(NR+1,NR+1),LDV )
-         END IF
-
-         CALL ZUNMQR( '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 = DSQRT(DBLE(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 / DZNRM2( N, V(1,q), 1 )
-               IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
-     $           CALL ZDSCAL( 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 ZLASET( 'A',  M-NR, NR, CZERO, CZERO, U(NR+1,1), LDU )
-            IF ( NR .LT. N1 ) THEN
-               CALL ZLASET('A',NR,  N1-NR, CZERO, CZERO,  U(1,NR+1),LDU)
-               CALL ZLASET('A',M-NR,N1-NR, CZERO, CONE,U(NR+1,NR+1),LDU)
-            END IF
-         END IF
-*
-         CALL ZUNMQR( 'Left', 'No Tr', M, N1, N, A, LDA, CWORK, U,
-     $        LDU, CWORK(N+1), LWORK-N, IERR )
-*
-            IF ( ROWPIV )
-     $         CALL ZLASWP( 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 ZSWAP( 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 DLASCL( '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 ZGEJSV
-*     ..
-      END
-*
+*> \brief \b ZGEJSV

+*

+*  =========== DOCUMENTATION ===========

+*

+* Online html documentation available at

+*            http://www.netlib.org/lapack/explore-html/

+*

+*> \htmlonly

+*> Download ZGEJSV + dependencies

+*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgejsv.f">

+*> [TGZ]</a>

+*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgejsv.f">

+*> [ZIP]</a>

+*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgejsv.f">

+*> [TXT]</a>

+*> \endhtmlonly

+*

+*  Definition:

+*  ===========

+*

+*     SUBROUTINE ZGEJSV( 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*16     A( LDA, * ),  U( LDU, * ), V( LDV, * ), CWORK( LWORK )

+*     DOUBLE PRECISION   SVA( N ), RWORK( LRWORK )

+*     INTEGER     IWORK( * )

+*     CHARACTER*1 JOBA, JOBP, JOBR, JOBT, JOBU, JOBV

+*       ..

+*

+*

+*> \par Purpose:

+*  =============

+*>

+*> \verbatim

+*>

+*> ZGEJSV 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=B*D. If A has heavily weighted

+*>              rows, then using this condition number gives too pessimistic

+*>              error bound.

+*>       = 'A': Small singular values are not well determined by the data 

+*>              and are considered as noisy; 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, if JOBT .EQ. 'N'.

+*>       = '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=DLAMCH('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=DLAMCH('S'), EPSLN=DLAMCH('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 ZGESVJ.

+*>       = '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 ZGESVJ.

+*> \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. 

+*>         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.

+*>         The option 'T' 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.

+*>         In general, this option is considered experimental, and 'N'; should

+*>         be preferred. This is subject to changes in the future.

+*> \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*16 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 DOUBLE PRECISION 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*16 array, dimension ( LDU, N )

+*>          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*16 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*16 array, dimension at least LWORK.

+*>          If the call to ZGEJSV is a workspace query (indicated by LWORK=-1 or

+*>          LRWORK=-1), then on exit CWORK(1) contains the required length of

+*>          CWORK for the job parameters used in the call.

+*> \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 ZGEQP3 and ZGEQRF.

+*>               In general, optimal LWORK is computed as

+*>               LWORK >= max(N+LWORK(ZGEQP3),N+LWORK(ZGEQRF), LWORK(ZGESVJ)).

+*>            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 + 2*N.

+*>               ->> For optimal performance (blocked code) the optimal value

+*>               is LWORK >= max(N+(N+1)*NB, N*N+2*N)=N**2+2*N.

+*>               In general, the optimal length LWORK is computed as

+*>               LWORK >= max(N+LWORK(ZGEQP3),N+LWORK(ZGEQRF), LWORK(ZGESVJ),

+*>                            N*N+LWORK(ZPOCON)).

+*>          2. If SIGMA and the right singular vectors are needed (JOBV.EQ.'V'),

+*>             (JOBU.EQ.'N')

+*>            2.1   .. no scaled condition estimate requested (JOBE.EQ.'N'):    

+*>            -> the minimal requirement is LWORK >= 3*N.

+*>            -> For optimal performance, 

+*>               LWORK >= max(N+(N+1)*NB, 2*N+N*NB)=2*N+N*NB,

+*>               where NB is the optimal block size for ZGEQP3, ZGEQRF, ZGELQ,

+*>               ZUNMLQ. In general, the optimal length LWORK is computed as

+*>               LWORK >= max(N+LWORK(ZGEQP3), N+LWORK(ZGESVJ),

+*>                       N+LWORK(ZGELQF), 2*N+LWORK(ZGEQRF), N+LWORK(ZUNMLQ)).

+*>            2.2 .. an estimate of the scaled condition number of A is

+*>               required (JOBA='E', or 'G').

+*>            -> the minimal requirement is LWORK >= 3*N.      

+*>            -> For optimal performance, 

+*>               LWORK >= max(N+(N+1)*NB, 2*N,2*N+N*NB)=2*N+N*NB,

+*>               where NB is the optimal block size for ZGEQP3, ZGEQRF, ZGELQ,

+*>               ZUNMLQ. In general, the optimal length LWORK is computed as

+*>               LWORK >= max(N+LWORK(ZGEQP3), LWORK(ZPOCON), N+LWORK(ZGESVJ),

+*>                       N+LWORK(ZGELQF), 2*N+LWORK(ZGEQRF), N+LWORK(ZUNMLQ)).   

+*>          3. If SIGMA and the left singular vectors are needed

+*>            3.1  .. no scaled condition estimate requested (JOBE.EQ.'N'):

+*>            -> 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)=2*N+N*NB,

+*>               where NB is the optimal block size for ZGEQP3, ZGEQRF, ZUNMQR.

+*>               In general, the optimal length LWORK is computed as

+*>               LWORK >= max(N+LWORK(ZGEQP3), 2*N+LWORK(ZGEQRF), N+LWORK(ZUNMQR)). 

+*>            3.2  .. an estimate of the scaled condition number of A is

+*>               required (JOBA='E', or 'G').

+*>            -> 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)=2*N+N*NB,

+*>               where NB is the optimal block size for ZGEQP3, ZGEQRF, ZUNMQR.

+*>               In general, the optimal length LWORK is computed as

+*>               LWORK >= max(N+LWORK(ZGEQP3),N+LWORK(ZPOCON),

+*>                        2*N+LWORK(ZGEQRF), N+LWORK(ZUNMQR)).

+*>          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 accomodate blocked runs

+*>            of ZGEQP3, ZGEQRF, ZGELQF, SUNMQR, ZUNMLQ.

+*>

+*>          If the call to ZGEJSV is a workspace query (indicated by LWORK=-1 or

+*>          LRWORK=-1), then on exit CWORK(1) contains the optimal and CWORK(2) contains the

+*>          minimal length of CWORK for the job parameters used in the call.

+*> \endverbatim

+*>

+*> \param[out] RWORK

+*> \verbatim

+*>          RWORK is DOUBLE PRECISION 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).)

+*>          If the call to ZGEJSV is a workspace query (indicated by LWORK=-1 or

+*>          LRWORK=-1), then on exit RWORK(1) contains the required length of

+*>          RWORK for the job parameters used in the call.

+*> \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 the 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, 2 * M ).

+*>          1.2. Otherwise, LRWORK  = max( 7,  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, 2 * M ).

+*>          2.2. Otherwise, LRWORK  = max( 7,  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, 2 * M ).

+*>          3.2. Otherwise, LRWORK  = max( 7,  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, 2 * M ).

+*>          4.2. Otherwise, LRWORK  = max( 7, N ).

+*>

+*>          If, on entry, LRWORK = -1 ot LWORK=-1, a workspace query is assumed and 

+*>          the length of RWORK is returned in RWORK(1). 

+*> \endverbatim

+*>

+*> \param[out] IWORK

+*> \verbatim

+*>          IWORK is INTEGER array, of dimension at least 4, that further depends 

+*>          on the job:

+*>

+*>          1. If only the singular values are requested then:

+*>             If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') ) 

+*>             then the length of IWORK is N+M; otherwise the length of IWORK is N.

+*>          2. If the singular values and the right singular vectors are requested then:

+*>             If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') ) 

+*>             then the length of IWORK is N+M; otherwise the length of IWORK is N. 

+*>          3. If the singular values and the left singular vectors are requested then:

+*>             If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') ) 

+*>             then the length of IWORK is N+M; otherwise the length of IWORK is N. 

+*>          4. If the singular values with both the left and the right singular vectors

+*>             are requested, then:      

+*>             4.1. If LSAME(JOBV,'J') the length of IWORK is determined as follows:

+*>                  If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') ) 

+*>                  then the length of IWORK is N+M; otherwise the length of IWORK is N. 

+*>             4.2. If LSAME(JOBV,'V') the length of IWORK is determined as follows:

+*>                  If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') ) 

+*>                  then the length of IWORK is 2*N+M; otherwise the length of IWORK is 2*N.

+*>        

+*>          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.

+*>          IWORK(4) = 1 or -1. If IWORK(4) .EQ. 1, then the procedure used A^* to

+*>                     do the job as specified by the JOB parameters.

+*>          If the call to ZGEJSV is a workspace query (indicated by LWORK .EQ. -1 or

+*>          LRWORK .EQ. -1), then on exit IWORK(1) contains the required length of 

+*>          IWORK for the job parameters used in the call.

+*> \endverbatim

+*>

+*> \param[out] INFO

+*> \verbatim

+*>          INFO is INTEGER

+*>           < 0  : if INFO = -i, then the i-th argument had an illegal value.

+*>           = 0 :  successful exit;

+*>           > 0 :  ZGEJSV  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 complex16GEsing

+*

+*> \par Further Details:

+*  =====================

+*>

+*> \verbatim

+*>

+*>  ZGEJSV implements a preconditioned Jacobi SVD algorithm. It uses ZGEQP3,

+*>  ZGEQRF, and ZGELQF 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 ZGEJSV 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 (ZGEJSV) is best used in this restricted range,

+*>  meaning that singular values of magnitude below ||A||_2 / DLAMCH('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 (ZGESVJ) is

+*>  left to the implementer on a particular machine.

+*>     The rank revealing QR factorization (in this code: ZGEQP3) should be

+*>  implemented as in [3]. We have a new version of ZGEQP3 under development

+*>  that is more robust than the current one in LAPACK, with a cleaner cut in

+*>  rank deficient cases. It will be available in the SIGMA library [4].

+*>  If M is much larger than N, it is obvious that the initial QRF with

+*>  column pivoting can be preprocessed by the QRF without pivoting. That

+*>  well known trick is not used in ZGEJSV 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 ZGEJSV uses only the simplest, naive data movement.

+*> \endverbatim

+*

+*> \par Contributor:

+*  ==================

+*>

+*>  Zlatko Drmac, Department of Mathematics, Faculty of Science,

+*>  University of Zagreb (Zagreb, Croatia); drmac@math.hr

+*

+*> \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, 2016.

+*> \endverbatim

+*

+*>  \par Bugs, examples and comments:

+*   =================================

+*>

+*>  Please report all bugs and send interesting examples and/or comments to

+*>  drmac@math.hr. Thank you.

+*>

+*  =====================================================================

+      SUBROUTINE ZGEJSV( 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 3.7.0) --

+*  -- LAPACK is a software package provided by Univ. of Tennessee,    --

+*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--

+*     December 2016

+*

+*     .. Scalar Arguments ..

+      IMPLICIT    NONE

+      INTEGER     INFO, LDA, LDU, LDV, LWORK, LRWORK, M, N

+*     ..

+*     .. Array Arguments ..

+      COMPLEX*16       A( LDA, * ), U( LDU, * ), V( LDV, * ),

+     $                 CWORK( LWORK )

+      DOUBLE PRECISION SVA( N ), RWORK( LRWORK )

+      INTEGER          IWORK( * )

+      CHARACTER*1      JOBA, JOBP, JOBR, JOBT, JOBU, JOBV

+*     ..

+*

+*  ===========================================================================

+*

+*     .. Local Parameters ..

+      DOUBLE PRECISION ZERO, ONE

+      PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )

+      COMPLEX*16 CZERO, CONE

+      PARAMETER ( CZERO = ( 0.0D0, 0.0D0 ), CONE = ( 1.0D0, 0.0D0 ) )

+*     ..

+*     .. Local Scalars ..

+      COMPLEX*16       CTEMP

+      DOUBLE PRECISION 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,   LQUERY,

+     $        LSVEC,  L2ABER, L2KILL, L2PERT,  L2RANK, L2TRAN, NOSCAL,

+     $        ROWPIV, RSVEC,  TRANSP

+*

+      INTEGER OPTWRK, MINWRK, MINRWRK, MINIWRK

+      INTEGER LWCON,  LWLQF, LWQP3, LWQRF, LWUNMLQ, LWUNMQR, LWUNMQRM,

+     $        LWSVDJ, LWSVDJV, LRWQP3, LRWCON, LRWSVDJ, IWOFF

+      INTEGER LWRK_ZGELQF, LWRK_ZGEQP3,  LWRK_ZGEQP3N, LWRK_ZGEQRF,  

+     $        LWRK_ZGESVJ, LWRK_ZGESVJV, LWRK_ZGESVJU, LWRK_ZUNMLQ, 

+     $        LWRK_ZUNMQR, LWRK_ZUNMQRM    

+*     ..

+*     .. Local Arrays

+      COMPLEX*16         CDUMMY(1)

+      DOUBLE PRECISION   RDUMMY(1)

+*

+*     .. Intrinsic Functions ..

+      INTRINSIC ABS, DCMPLX, CONJG, DLOG, MAX, MIN, DBLE, NINT, SQRT

+*     ..

+*     .. External Functions ..

+      DOUBLE PRECISION      DLAMCH, DZNRM2

+      INTEGER   IDAMAX, IZAMAX

+      LOGICAL   LSAME

+      EXTERNAL  IDAMAX, IZAMAX, LSAME, DLAMCH, DZNRM2

+*     ..

+*     .. External Subroutines ..

+      EXTERNAL  DLASSQ, ZCOPY,  ZGELQF, ZGEQP3, ZGEQRF, ZLACPY, ZLAPMR,

+     $          ZLASCL, DLASCL, ZLASET, ZLASSQ, ZLASWP, ZUNGQR, ZUNMLQ,

+     $          ZUNMQR, ZPOCON, DSCAL,  ZDSCAL, ZSWAP,  ZTRSM,  ZLACGV,

+     $          XERBLA

+*

+      EXTERNAL  ZGESVJ

+*     ..

+*

+*     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' ) .AND. ( M .EQ. N )

+      L2KILL = LSAME( JOBR, 'R' )

+      DEFR   = LSAME( JOBR, 'N' )

+      L2PERT = LSAME( JOBP, 'P' )

+*

+      LQUERY = ( LWORK .EQ. -1 ) .OR. ( LRWORK .EQ. -1 )

+*

+      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' ) .AND. RSVEC .AND. L2TRAN ) ) ) THEN

+         INFO = - 2

+      ELSE IF ( .NOT.( RSVEC .OR. LSAME( JOBV, 'N' ) .OR.

+     $   ( LSAME( JOBV, 'W' ) .AND. LSVEC .AND. L2TRAN ) ) ) THEN

+         INFO = - 3

+      ELSE IF ( .NOT. ( L2KILL .OR. DEFR ) )    THEN

+         INFO = - 4

+      ELSE IF ( .NOT. ( LSAME(JOBT,'T') .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

+*        #:)

+         INFO = 0

+      END IF

+*

+      IF ( INFO .EQ. 0 ) THEN 

+*         .. compute the minimal and the optimal workspace lengths 

+*         [[The expressions for computing the minimal and the optimal

+*         values of LCWORK, LRWORK are written with a lot of redundancy and

+*         can be simplified. However, this verbose form is useful for

+*         maintenance and modifications of the code.]]

+*

+*        .. minimal workspace length for ZGEQP3 of an M x N matrix,

+*         ZGEQRF of an N x N matrix, ZGELQF of an N x N matrix,

+*         ZUNMLQ for computing N x N matrix, ZUNMQR for computing N x N

+*         matrix, ZUNMQR for computing M x N matrix, respectively.

+          LWQP3 = N+1   

+          LWQRF = MAX( 1, N )

+          LWLQF = MAX( 1, N )

+          LWUNMLQ  = MAX( 1, N )

+          LWUNMQR  = MAX( 1, N )

+          LWUNMQRM = MAX( 1, M )

+*        .. minimal workspace length for ZPOCON of an N x N matrix

+          LWCON = 2 * N 

+*        .. minimal workspace length for ZGESVJ of an N x N matrix,

+*         without and with explicit accumulation of Jacobi rotations

+          LWSVDJ  = MAX( 2 * N, 1 )         

+          LWSVDJV = MAX( 2 * N, 1 )

+*         .. minimal REAL workspace length for ZGEQP3, ZPOCON, ZGESVJ

+          LRWQP3  = N 

+          LRWCON  = N 

+          LRWSVDJ = N 

+          IF ( LQUERY ) THEN 

+              CALL ZGEQP3( M, N, A, LDA, IWORK, CDUMMY, CDUMMY, -1, 

+     $             RDUMMY, IERR )

+              LWRK_ZGEQP3 = CDUMMY(1)

+              CALL ZGEQRF( N, N, A, LDA, CDUMMY, CDUMMY,-1, IERR )

+              LWRK_ZGEQRF = CDUMMY(1)

+              CALL ZGELQF( N, N, A, LDA, CDUMMY, CDUMMY,-1, IERR )

+              LWRK_ZGELQF = CDUMMY(1)             

+          END IF

+          MINWRK  = 2

+          OPTWRK  = 2

+          MINIWRK = N 

+          IF ( .NOT. (LSVEC .OR. RSVEC ) ) THEN

+*             .. minimal and optimal sizes of the complex workspace if

+*             only the singular values are requested

+              IF ( ERREST ) THEN 

+                  MINWRK = MAX( N+LWQP3, N**2+LWCON, N+LWQRF, LWSVDJ )

+              ELSE

+                  MINWRK = MAX( N+LWQP3, N+LWQRF, LWSVDJ )

+              END IF

+              IF ( LQUERY ) THEN 

+                  CALL ZGESVJ( 'L', 'N', 'N', N, N, A, LDA, SVA, N, V, 

+     $                 LDV, CDUMMY, -1, RDUMMY, -1, IERR )

+                  LWRK_ZGESVJ = CDUMMY(1)

+                  IF ( ERREST ) THEN 

+                      OPTWRK = MAX( N+LWRK_ZGEQP3, N**2+LWCON, 

+     $                              N+LWRK_ZGEQRF, LWRK_ZGESVJ )

+                  ELSE

+                      OPTWRK = MAX( N+LWRK_ZGEQP3, N+LWRK_ZGEQRF, 

+     $                              LWRK_ZGESVJ )

+                  END IF

+              END IF

+              IF ( L2TRAN .OR. ROWPIV ) THEN 

+                  IF ( ERREST ) THEN 

+                     MINRWRK = MAX( 7, 2*M,  LRWQP3, LRWCON, LRWSVDJ )

+                  ELSE

+                     MINRWRK = MAX( 7, 2*M,  LRWQP3, LRWSVDJ )

+                  END IF                 

+              ELSE

+                  IF ( ERREST ) THEN 

+                     MINRWRK = MAX( 7, LRWQP3, LRWCON, LRWSVDJ )

+                  ELSE

+                     MINRWRK = MAX( 7, LRWQP3, LRWSVDJ )

+                  END IF

+              END IF   

+              IF ( ROWPIV .OR. L2TRAN ) MINIWRK = MINIWRK + M 

+          ELSE IF ( RSVEC .AND. (.NOT.LSVEC) ) THEN

+*            .. minimal and optimal sizes of the complex workspace if the

+*            singular values and the right singular vectors are requested

+             IF ( ERREST ) THEN 

+                 MINWRK = MAX( N+LWQP3, LWCON, LWSVDJ, N+LWLQF,  

+     $                         2*N+LWQRF, N+LWSVDJ, N+LWUNMLQ )

+             ELSE

+                 MINWRK = MAX( N+LWQP3, LWSVDJ, N+LWLQF, 2*N+LWQRF, 

+     $                         N+LWSVDJ, N+LWUNMLQ )

+             END IF

+             IF ( LQUERY ) THEN

+                 CALL ZGESVJ( 'L', 'U', 'N', N,N, U, LDU, SVA, N, A,

+     $                LDA, CDUMMY, -1, RDUMMY, -1, IERR )

+                 LWRK_ZGESVJ = CDUMMY(1)

+                 CALL ZUNMLQ( 'L', 'C', N, N, N, A, LDA, CDUMMY,

+     $                V, LDV, CDUMMY, -1, IERR )

+                 LWRK_ZUNMLQ = CDUMMY(1)                

+                 IF ( ERREST ) THEN 

+                 OPTWRK = MAX( N+LWRK_ZGEQP3, LWCON, LWRK_ZGESVJ, 

+     $                         N+LWRK_ZGELQF, 2*N+LWRK_ZGEQRF,

+     $                         N+LWRK_ZGESVJ,  N+LWRK_ZUNMLQ )

+                 ELSE

+                 OPTWRK = MAX( N+LWRK_ZGEQP3, LWRK_ZGESVJ,N+LWRK_ZGELQF,

+     $                         2*N+LWRK_ZGEQRF, N+LWRK_ZGESVJ, 

+     $                         N+LWRK_ZUNMLQ )

+                 END IF

+             END IF

+             IF ( L2TRAN .OR. ROWPIV ) THEN 

+                  IF ( ERREST ) THEN 

+                     MINRWRK = MAX( 7, 2*M,  LRWQP3, LRWSVDJ, LRWCON )

+                  ELSE

+                     MINRWRK = MAX( 7, 2*M,  LRWQP3, LRWSVDJ ) 

+                  END IF                  

+             ELSE

+                  IF ( ERREST ) THEN 

+                     MINRWRK = MAX( 7, LRWQP3, LRWSVDJ, LRWCON )

+                  ELSE

+                     MINRWRK = MAX( 7, LRWQP3, LRWSVDJ ) 

+                  END IF                 

+             END IF

+             IF ( ROWPIV .OR. L2TRAN ) MINIWRK = MINIWRK + M

+          ELSE IF ( LSVEC .AND. (.NOT.RSVEC) ) THEN  

+*            .. minimal and optimal sizes of the complex workspace if the

+*            singular values and the left singular vectors are requested

+             IF ( ERREST ) THEN

+                 MINWRK = N + MAX( LWQP3,LWCON,N+LWQRF,LWSVDJ,LWUNMQRM )

+             ELSE

+                 MINWRK = N + MAX( LWQP3, N+LWQRF, LWSVDJ, LWUNMQRM )

+             END IF

+             IF ( LQUERY ) THEN

+                 CALL ZGESVJ( 'L', 'U', 'N', N,N, U, LDU, SVA, N, A,

+     $                LDA, CDUMMY, -1, RDUMMY, -1, IERR )

+                 LWRK_ZGESVJ = CDUMMY(1)

+                 CALL ZUNMQR( 'L', 'N', M, N, N, A, LDA, CDUMMY, U,

+     $               LDU, CDUMMY, -1, IERR )

+                 LWRK_ZUNMQRM = CDUMMY(1)

+                 IF ( ERREST ) THEN

+                 OPTWRK = N + MAX( LWRK_ZGEQP3, LWCON, N+LWRK_ZGEQRF,

+     $                             LWRK_ZGESVJ, LWRK_ZUNMQRM )

+                 ELSE

+                 OPTWRK = N + MAX( LWRK_ZGEQP3, N+LWRK_ZGEQRF,

+     $                             LWRK_ZGESVJ, LWRK_ZUNMQRM )

+                 END IF

+             END IF

+             IF ( L2TRAN .OR. ROWPIV ) THEN 

+                 IF ( ERREST ) THEN 

+                    MINRWRK = MAX( 7, 2*M,  LRWQP3, LRWSVDJ, LRWCON )

+                 ELSE

+                    MINRWRK = MAX( 7, 2*M,  LRWQP3, LRWSVDJ )

+                 END IF                 

+             ELSE

+                 IF ( ERREST ) THEN 

+                    MINRWRK = MAX( 7, LRWQP3, LRWSVDJ, LRWCON )

+                 ELSE

+                    MINRWRK = MAX( 7, LRWQP3, LRWSVDJ )

+                 END IF                

+             END IF 

+             IF ( ROWPIV .OR. L2TRAN ) MINIWRK = MINIWRK + M

+          ELSE

+*            .. minimal and optimal sizes of the complex workspace if the

+*            full SVD is requested

+             IF ( .NOT. JRACC ) THEN                

+                 IF ( ERREST ) THEN 

+                    MINWRK = MAX( N+LWQP3, N+LWCON,  2*N+N**2+LWCON, 

+     $                         2*N+LWQRF,         2*N+LWQP3, 

+     $                         2*N+N**2+N+LWLQF,  2*N+N**2+N+N**2+LWCON,

+     $                         2*N+N**2+N+LWSVDJ, 2*N+N**2+N+LWSVDJV, 

+     $                         2*N+N**2+N+LWUNMQR,2*N+N**2+N+LWUNMLQ, 

+     $                         N+N**2+LWSVDJ,   N+LWUNMQRM )

+                 ELSE

+                    MINWRK = MAX( N+LWQP3,        2*N+N**2+LWCON, 

+     $                         2*N+LWQRF,         2*N+LWQP3, 

+     $                         2*N+N**2+N+LWLQF,  2*N+N**2+N+N**2+LWCON,

+     $                         2*N+N**2+N+LWSVDJ, 2*N+N**2+N+LWSVDJV,

+     $                         2*N+N**2+N+LWUNMQR,2*N+N**2+N+LWUNMLQ,

+     $                         N+N**2+LWSVDJ,      N+LWUNMQRM ) 

+                 END IF 

+                 MINIWRK = MINIWRK + N 

+                 IF ( ROWPIV .OR. L2TRAN ) MINIWRK = MINIWRK + M

+             ELSE

+                 IF ( ERREST ) THEN 

+                    MINWRK = MAX( N+LWQP3, N+LWCON, 2*N+LWQRF, 

+     $                         2*N+N**2+LWSVDJV, 2*N+N**2+N+LWUNMQR, 

+     $                         N+LWUNMQRM )

+                 ELSE

+                    MINWRK = MAX( N+LWQP3, 2*N+LWQRF, 

+     $                         2*N+N**2+LWSVDJV, 2*N+N**2+N+LWUNMQR, 

+     $                         N+LWUNMQRM ) 

+                 END IF   

+                 IF ( ROWPIV .OR. L2TRAN ) MINIWRK = MINIWRK + M

+             END IF

+             IF ( LQUERY ) THEN

+                 CALL ZUNMQR( 'L', 'N', M, N, N, A, LDA, CDUMMY, U,

+     $                LDU, CDUMMY, -1, IERR )

+                 LWRK_ZUNMQRM = CDUMMY(1)

+                 CALL ZUNMQR( 'L', 'N', N, N, N, A, LDA, CDUMMY, U,

+     $                LDU, CDUMMY, -1, IERR )

+                 LWRK_ZUNMQR = CDUMMY(1)

+                 IF ( .NOT. JRACC ) THEN

+                     CALL ZGEQP3( N,N, A, LDA, IWORK, CDUMMY,CDUMMY, -1,

+     $                    RDUMMY, IERR )

+                     LWRK_ZGEQP3N = CDUMMY(1)

+                     CALL ZGESVJ( 'L', 'U', 'N', N, N, U, LDU, SVA,

+     $                    N, V, LDV, CDUMMY, -1, RDUMMY, -1, IERR )

+                     LWRK_ZGESVJ = CDUMMY(1)

+                     CALL ZGESVJ( 'U', 'U', 'N', N, N, U, LDU, SVA,

+     $                    N, V, LDV, CDUMMY, -1, RDUMMY, -1, IERR )

+                     LWRK_ZGESVJU = CDUMMY(1)

+                     CALL ZGESVJ( 'L', 'U', 'V', N, N, U, LDU, SVA,

+     $                    N, V, LDV, CDUMMY, -1, RDUMMY, -1, IERR )

+                     LWRK_ZGESVJV = CDUMMY(1)

+                     CALL ZUNMLQ( 'L', 'C', N, N, N, A, LDA, CDUMMY,

+     $                    V, LDV, CDUMMY, -1, IERR )

+                     LWRK_ZUNMLQ = CDUMMY(1)

+                     IF ( ERREST ) THEN 

+                       OPTWRK = MAX( N+LWRK_ZGEQP3, N+LWCON, 

+     $                          2*N+N**2+LWCON, 2*N+LWRK_ZGEQRF, 

+     $                          2*N+LWRK_ZGEQP3N, 

+     $                          2*N+N**2+N+LWRK_ZGELQF,  

+     $                          2*N+N**2+N+N**2+LWCON,

+     $                          2*N+N**2+N+LWRK_ZGESVJ, 

+     $                          2*N+N**2+N+LWRK_ZGESVJV,               

+     $                          2*N+N**2+N+LWRK_ZUNMQR,

+     $                          2*N+N**2+N+LWRK_ZUNMLQ, 

+     $                          N+N**2+LWRK_ZGESVJU,                  

+     $                          N+LWRK_ZUNMQRM )

+                     ELSE

+                       OPTWRK = MAX( N+LWRK_ZGEQP3,  

+     $                          2*N+N**2+LWCON, 2*N+LWRK_ZGEQRF, 

+     $                          2*N+LWRK_ZGEQP3N, 

+     $                          2*N+N**2+N+LWRK_ZGELQF,  

+     $                          2*N+N**2+N+N**2+LWCON,

+     $                          2*N+N**2+N+LWRK_ZGESVJ,               

+     $                          2*N+N**2+N+LWRK_ZGESVJV, 

+     $                          2*N+N**2+N+LWRK_ZUNMQR,

+     $                          2*N+N**2+N+LWRK_ZUNMLQ, 

+     $                          N+N**2+LWRK_ZGESVJU,

+     $                          N+LWRK_ZUNMQRM )

+                     END IF                    

+                 ELSE

+                     CALL ZGESVJ( 'L', 'U', 'V', N, N, U, LDU, SVA,

+     $                    N, V, LDV, CDUMMY, -1, RDUMMY, -1, IERR )

+                     LWRK_ZGESVJV = CDUMMY(1)

+                     CALL ZUNMQR( 'L', 'N', N, N, N, CDUMMY, N, CDUMMY,

+     $                    V, LDV, CDUMMY, -1, IERR )

+                     LWRK_ZUNMQR = CDUMMY(1)

+                     CALL ZUNMQR( 'L', 'N', M, N, N, A, LDA, CDUMMY, U,

+     $                    LDU, CDUMMY, -1, IERR )

+                     LWRK_ZUNMQRM = CDUMMY(1)   

+                     IF ( ERREST ) THEN 

+                        OPTWRK = MAX( N+LWRK_ZGEQP3, N+LWCON,   

+     $                           2*N+LWRK_ZGEQRF, 2*N+N**2,  

+     $                           2*N+N**2+LWRK_ZGESVJV,  

+     $                           2*N+N**2+N+LWRK_ZUNMQR,N+LWRK_ZUNMQRM )

+                     ELSE

+                        OPTWRK = MAX( N+LWRK_ZGEQP3, 2*N+LWRK_ZGEQRF,  

+     $                           2*N+N**2, 2*N+N**2+LWRK_ZGESVJV, 

+     $                           2*N+N**2+N+LWRK_ZUNMQR, 

+     $                           N+LWRK_ZUNMQRM )   

+                     END IF                  

+                 END IF               

+             END IF 

+             IF ( L2TRAN .OR. ROWPIV ) THEN 

+                 MINRWRK = MAX( 7, 2*M,  LRWQP3, LRWSVDJ, LRWCON )

+             ELSE

+                 MINRWRK = MAX( 7, LRWQP3, LRWSVDJ, LRWCON )

+             END IF 

+          END IF

+          MINWRK = MAX( 2, MINWRK )

+          OPTWRK = MAX( 2, OPTWRK )

+          IF ( LWORK  .LT. MINWRK  .AND. (.NOT.LQUERY) ) INFO = - 17

+          IF ( LRWORK .LT. MINRWRK .AND. (.NOT.LQUERY) ) INFO = - 19   

+      END IF

+*      

+      IF ( INFO .NE. 0 ) THEN

+*       #:(

+         CALL XERBLA( 'ZGEJSV', - INFO )

+         RETURN

+      ELSE IF ( LQUERY ) THEN

+          CWORK(1) = OPTWRK

+          CWORK(2) = MINWRK

+          RWORK(1) = MINRWRK

+          IWORK(1) = MAX( 4, MINIWRK )

+          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 DLAMCH() does not fail on the target architecture.

+*

+      EPSLN = DLAMCH('Epsilon')

+      SFMIN = DLAMCH('SafeMinimum')

+      SMALL = SFMIN / EPSLN

+      BIG   = DLAMCH('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(DBLE(M)*DBLE(N))

+      NOSCAL  = .TRUE.

+      GOSCAL  = .TRUE.

+      DO 1874 p = 1, N

+         AAPP = ZERO

+         AAQQ = ONE

+         CALL ZLASSQ( M, A(1,p), 1, AAPP, AAQQ )

+         IF ( AAPP .GT. BIG ) THEN

+            INFO = - 9

+            CALL XERBLA( 'ZGEJSV', -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 DSCAL( 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 = MAX( AAPP, SVA(p) )

+         IF ( SVA(p) .NE. ZERO ) AAQQ = MIN( AAQQ, SVA(p) )

+ 4781 CONTINUE

+*

+*     Quick return for zero M x N matrix

+* #:)

+      IF ( AAPP .EQ. ZERO ) THEN

+         IF ( LSVEC ) CALL ZLASET( 'G', M, N1, CZERO, CONE, U, LDU )

+         IF ( RSVEC ) CALL ZLASET( '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

+         IWORK(4) = -1

+         RETURN

+      END IF

+*

+*     Issue warning if denormalized column norms detected. Override the

+*     high relative accuracy request. Issue licence to kill nonzero 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 ZLASCL( 'G',0,0,SVA(1),SCALEM, M,1,A(1,1),LDA,IERR )

+            CALL ZLACPY( '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 ZGEQRF( M, N, U,LDU, CWORK, CWORK(N+1),LWORK-N,IERR )

+              CALL ZUNGQR( M,N1,1, U,LDU,CWORK,CWORK(N+1),LWORK-N,IERR )

+              CALL ZCOPY( 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

+         IWORK(4) = -1

+         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.

+*

+      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 ZLASSQ( N, A(p,1), LDA, XSC, TEMP1 )

+*              ZLASSQ gets both the ell_2 and the ell_infinity norm

+*              in one pass through the vector

+               RWORK(M+p)  = XSC * SCALEM

+               RWORK(p)    = XSC * (SCALEM*SQRT(TEMP1))

+               AATMAX = MAX( AATMAX, RWORK(p) )

+               IF (RWORK(p) .NE. ZERO) 

+     $            AATMIN = MIN(AATMIN,RWORK(p))

+ 1950       CONTINUE

+         ELSE

+            DO 1904 p = 1, M

+               RWORK(M+p) = SCALEM*ABS( A(p,IZAMAX(N,A(p,1),LDA)) )

+               AATMAX = MAX( AATMAX, RWORK(M+p) )

+               AATMIN = MIN( AATMIN, RWORK(M+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 DLASSQ( 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 * DLOG(BIG1)

+ 1113    CONTINUE

+         ENTRA = - ENTRA / DLOG(DBLE(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 = 1, M

+            BIG1 = ( ( RWORK(p) / XSC )**2 ) * TEMP1

+            IF ( BIG1 .NE. ZERO ) ENTRAT = ENTRAT + BIG1 * DLOG(BIG1)

+ 1114    CONTINUE

+         ENTRAT = - ENTRAT / DLOG(DBLE(M))

+*

+*        Analyze the entropies and decide A or A^*. Smaller entropy

+*        usually means better input for the algorithm.

+*

+         TRANSP = ( ENTRAT .LT. ENTRA )

+* 

+*        If A^* is better than A, take the adjoint of A. This is allowed

+*        only for square matrices, M=N.

+         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+p) = SVA(p)

+               SVA(p)     = RWORK(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 ZGEJSV 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 ZGESVJ will compute them. So, in that case,

+*     one should use ZGESVJ instead of ZGEJSV.

+*     >> change in the April 2016 update: allow bigger range, i.e. the

+*     largest column is allowed up to BIG/N and ZGESVJ will do the rest.

+      BIG1   = SQRT( BIG )

+      TEMP1  = SQRT( BIG / DBLE(N) ) 

+*      TEMP1  = BIG/DBLE(N)

+*

+      CALL DLASCL( '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 ZLASCL( '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 ZGESVJ

+*        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 ZLASET( '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.

+         IF ( ( LSVEC .AND. RSVEC ) .AND. .NOT.( JRACC ) ) THEN 

+              IWOFF = 2*N

+         ELSE

+              IWOFF = N

+         END IF

+         DO 1952 p = 1, M - 1

+            q = IDAMAX( M-p+1, RWORK(M+p), 1 ) + p - 1

+            IWORK(IWOFF+p) = q

+            IF ( p .NE. q ) THEN

+               TEMP1      = RWORK(M+p)

+               RWORK(M+p) = RWORK(M+q)

+               RWORK(M+q) = TEMP1

+            END IF

+ 1952    CONTINUE

+         CALL ZLASWP( N, A, LDA, 1, M-1, IWORK(IWOFF+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 ZGEQP3 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 ZGEQP3 improves overal performance of ZGEJSV.

+*

+*     A * P1 = Q1 * [ R1^* 0]^*:

+      DO 1963 p = 1, N

+*        .. all columns are free columns

+         IWORK(p) = 0

+ 1963 CONTINUE

+      CALL ZGEQP3( 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 ZGEJSV 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(DBLE(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-deficient.

+         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 = MIN( MAXPRJ, TEMP1 )

+ 3051    CONTINUE

+         IF ( MAXPRJ**2 .GE. ONE - DBLE(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 ZLACPY( 'U', N, N, A, LDA, V, LDV )

+               DO 3053 p = 1, N

+                  TEMP1 = SVA(IWORK(p))

+                  CALL ZDSCAL( p, ONE/TEMP1, V(1,p), 1 )

+ 3053          CONTINUE

+               IF ( LSVEC )THEN

+                   CALL ZPOCON( 'U', N, V, LDV, ONE, TEMP1,

+     $                  CWORK(N+1), RWORK, IERR )

+               ELSE

+                   CALL ZPOCON( 'U', N, V, LDV, ONE, TEMP1,

+     $                  CWORK, RWORK, IERR )

+               END IF               

+*          

+            ELSE IF ( LSVEC ) THEN

+*              .. U is available as workspace

+               CALL ZLACPY( 'U', N, N, A, LDA, U, LDU )

+               DO 3054 p = 1, N

+                  TEMP1 = SVA(IWORK(p))

+                  CALL ZDSCAL( p, ONE/TEMP1, U(1,p), 1 )

+ 3054          CONTINUE

+               CALL ZPOCON( 'U', N, U, LDU, ONE, TEMP1,

+     $              CWORK(N+1), RWORK, IERR )

+            ELSE

+               CALL ZLACPY( 'U', N, N, A, LDA, CWORK, N )

+*[]            CALL ZLACPY( 'U', N, N, A, LDA, CWORK(N+1), N )

+*              Change: here index shifted by N to the left, CWORK(1:N) 

+*              not needed for SIGMA only computation

+               DO 3052 p = 1, N

+                  TEMP1 = SVA(IWORK(p))

+*[]               CALL ZDSCAL( p, ONE/TEMP1, CWORK(N+(p-1)*N+1), 1 )

+                  CALL ZDSCAL( p, ONE/TEMP1, CWORK((p-1)*N+1), 1 )

+ 3052          CONTINUE

+*           .. the columns of R are scaled to have unit Euclidean lengths.

+*[]               CALL ZPOCON( 'U', N, CWORK(N+1), N, ONE, TEMP1,

+*[]     $              CWORK(N+N*N+1), RWORK, IERR )

+               CALL ZPOCON( 'U', N, CWORK, N, ONE, TEMP1,

+     $              CWORK(N*N+1), RWORK, IERR )               

+*              

+            END IF

+            IF ( TEMP1 .NE. ZERO ) THEN 

+               SCONDA = ONE / SQRT(TEMP1)

+            ELSE

+               SCONDA = - ONE

+            END IF

+*           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, MIN( N-1, NR )

+            CALL ZCOPY( N-p, A(p,p+1), LDA, A(p+1,p), 1 )

+            CALL ZLACGV( 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 / DBLE(N)

+               DO 4947 q = 1, NR

+                  CTEMP = DCMPLX(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 ZLASET( 'U', NR-1,NR-1, CZERO,CZERO, A(1,2),LDA )

+            END IF

+*

+*            .. second preconditioning using the QR factorization

+*

+            CALL ZGEQRF( N,NR, A,LDA, CWORK, CWORK(N+1),LWORK-N, IERR )

+*

+*           .. and transpose upper to lower triangular

+            DO 1948 p = 1, NR - 1

+               CALL ZCOPY( NR-p, A(p,p+1), LDA, A(p+1,p), 1 )

+               CALL ZLACGV( 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 / DBLE(N)

+               DO 1947 q = 1, NR

+                  CTEMP = DCMPLX(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 ZLASET( '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 ZGESVJ( 'L', 'N', 'N', 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 ) .AND. ( .NOT. JRACC ) )

+     $       .OR. 

+     $   ( JRACC .AND. ( .NOT. LSVEC ) .AND. ( NR .NE. N ) ) ) THEN

+*

+*        -> Singular Values and Right Singular Vectors <-

+*

+         IF ( ALMORT ) THEN

+*

+*           .. in this case NR equals N

+            DO 1998 p = 1, NR

+               CALL ZCOPY( N-p+1, A(p,p), LDA, V(p,p), 1 )

+               CALL ZLACGV( N-p+1, V(p,p), 1 )

+ 1998       CONTINUE

+            CALL ZLASET( 'U', NR-1,NR-1, CZERO, CZERO, V(1,2), LDV )

+*

+            CALL ZGESVJ( '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 ZLASET( 'L', NR-1,NR-1, CZERO, CZERO, A(2,1), LDA )

+            CALL ZGELQF( NR,N, A, LDA, CWORK, CWORK(N+1), LWORK-N, IERR)

+            CALL ZLACPY( 'L', NR, NR, A, LDA, V, LDV )

+            CALL ZLASET( 'U', NR-1,NR-1, CZERO, CZERO, V(1,2), LDV )

+            CALL ZGEQRF( NR, NR, V, LDV, CWORK(N+1), CWORK(2*N+1),

+     $                   LWORK-2*N, IERR )

+            DO 8998 p = 1, NR

+               CALL ZCOPY( NR-p+1, V(p,p), LDV, V(p,p), 1 )

+               CALL ZLACGV( NR-p+1, V(p,p), 1 )

+ 8998       CONTINUE

+            CALL ZLASET('U', NR-1, NR-1, CZERO, CZERO, V(1,2), LDV)

+*

+            CALL ZGESVJ( 'L', '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 ZLASET( 'A',N-NR, NR, CZERO,CZERO, V(NR+1,1),  LDV )

+               CALL ZLASET( 'A',NR, N-NR, CZERO,CZERO, V(1,NR+1),  LDV )

+               CALL ZLASET( 'A',N-NR,N-NR,CZERO,CONE, V(NR+1,NR+1),LDV )

+            END IF

+*

+         CALL ZUNMLQ( 'L', 'C', N, N, NR, A, LDA, CWORK,

+     $               V, LDV, CWORK(N+1), LWORK-N, IERR )

+*

+         END IF

+*         .. permute the rows of V

+*         DO 8991 p = 1, N

+*            CALL ZCOPY( N, V(p,1), LDV, A(IWORK(p),1), LDA )

+* 8991    CONTINUE

+*         CALL ZLACPY( 'All', N, N, A, LDA, V, LDV )

+         CALL ZLAPMR( .FALSE., N, N, V, LDV, IWORK )

+*

+          IF ( TRANSP ) THEN

+            CALL ZLACPY( 'A', N, N, V, LDV, U, LDU )

+          END IF

+*

+      ELSE IF ( JRACC .AND. (.NOT. LSVEC) .AND. ( NR.EQ. N ) ) THEN 

+*          

+         CALL ZLASET( 'L', N-1,N-1, CZERO, CZERO, A(2,1), LDA )

+*

+         CALL ZGESVJ( 'U','N','V', N, N, A, LDA, SVA, N, V, LDV,

+     $               CWORK, LWORK, RWORK, LRWORK, INFO )

+          SCALEM  = RWORK(1)

+          NUMRANK = NINT(RWORK(2))

+          CALL ZLAPMR( .FALSE., N, N, V, LDV, IWORK )

+*

+      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 ZCOPY( N-p+1, A(p,p), LDA, U(p,p), 1 )

+            CALL ZLACGV( N-p+1, U(p,p), 1 )

+ 1965    CONTINUE

+         CALL ZLASET( 'U', NR-1, NR-1, CZERO, CZERO, U(1,2), LDU )

+*

+         CALL ZGEQRF( N, NR, U, LDU, CWORK(N+1), CWORK(2*N+1),

+     $              LWORK-2*N, IERR )

+*

+         DO 1967 p = 1, NR - 1

+            CALL ZCOPY( NR-p, U(p,p+1), LDU, U(p+1,p), 1 )

+            CALL ZLACGV( N-p+1, U(p,p), 1 )

+ 1967    CONTINUE

+         CALL ZLASET( 'U', NR-1, NR-1, CZERO, CZERO, U(1,2), LDU )

+*

+         CALL ZGESVJ( 'L', '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 ZLASET( 'A',  M-NR, NR,CZERO, CZERO, U(NR+1,1), LDU )

+            IF ( NR .LT. N1 ) THEN

+               CALL ZLASET( 'A',NR, N1-NR, CZERO, CZERO, U(1,NR+1),LDU )

+               CALL ZLASET( 'A',M-NR,N1-NR,CZERO,CONE,U(NR+1,NR+1),LDU )

+            END IF

+         END IF

+*

+         CALL ZUNMQR( 'L', 'N', M, N1, N, A, LDA, CWORK, U,

+     $               LDU, CWORK(N+1), LWORK-N, IERR )

+*

+         IF ( ROWPIV )

+     $       CALL ZLASWP( N1, U, LDU, 1, M-1, IWORK(IWOFF+1), -1 )

+*

+         DO 1974 p = 1, N1

+            XSC = ONE / DZNRM2( M, U(1,p), 1 )

+            CALL ZDSCAL( M, XSC, U(1,p), 1 )

+ 1974    CONTINUE

+*

+         IF ( TRANSP ) THEN

+            CALL ZLACPY( 'A', 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 ZGEJSV.

+*

+            DO 1968 p = 1, NR

+               CALL ZCOPY( N-p+1, A(p,p), LDA, V(p,p), 1 )

+               CALL ZLACGV( 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 = DCMPLX(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 ZLASET( '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 ZLACPY( 'L', NR, NR, V, LDV, CWORK(2*N+1), NR )

+            DO 3950 p = 1, NR

+               TEMP1 = DZNRM2(NR-p+1,CWORK(2*N+(p-1)*NR+p),1)

+               CALL ZDSCAL(NR-p+1,ONE/TEMP1,CWORK(2*N+(p-1)*NR+p),1)

+ 3950       CONTINUE

+            CALL ZPOCON('L',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. DBLE(N)

+*           more conservative    <=> CONDR1 .LT. SQRT(DBLE(N))

+*

+            COND_OK = SQRT(SQRT(DBLE(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 ZGEQRF( 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=DCMPLX(XSC*MIN(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 ZLACPY( '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 ZCOPY( NR-p, V(p,p+1), LDV, V(p+1,p), 1 )

+                  CALL ZLACGV(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 ZGEQP3

+*              should be replaced with eg. CALL ZGEQPX (ACM TOMS #782)

+*              with properly (carefully) chosen parameters.

+*

+*              R1^* * P2 = Q2 * R2

+               DO 3003 p = 1, NR

+                  IWORK(N+p) = 0

+ 3003          CONTINUE

+               CALL ZGEQP3( N, NR, V, LDV, IWORK(N+1), CWORK(N+1),

+     $                  CWORK(2*N+1), LWORK-2*N, RWORK, IERR )

+**               CALL ZGEQRF( 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=DCMPLX(XSC*MIN(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 ZLACPY( '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=DCMPLX(XSC*MIN(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 ZLASET( 'L',NR-1,NR-1,CZERO,CZERO,V(2,1),LDV )

+               END IF

+*              Now, compute R2 = L3 * Q3, the LQ factorization.

+               CALL ZGELQF( 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 ZLACPY( 'L',NR,NR,V,LDV,CWORK(2*N+N*NR+NR+1),NR )

+               DO 4950 p = 1, NR

+                  TEMP1 = DZNRM2( p, CWORK(2*N+N*NR+NR+p), NR )

+                  CALL ZDSCAL( p, ONE/TEMP1, CWORK(2*N+N*NR+NR+p), NR )

+ 4950          CONTINUE

+               CALL ZPOCON( '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 ZLACPY( '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) = - TEMP1*( V(p,q) / ABS(V(p,q)) )

+                     V(p,q) = - CTEMP

+ 4969             CONTINUE

+ 4968          CONTINUE

+            ELSE

+               CALL ZLASET( '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 ZGESVJ( '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 ZCOPY(  NR, V(1,p), 1, U(1,p), 1 )

+                  CALL ZDSCAL( 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 ZGESVJ, premultiplied with the orthogonal matrix

+*                 from the second QR factorization.

+                  CALL ZTRSM('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 ZGESVJ. The Q-factor from the second QR

+*                 factorization is then built in explicitly.

+                  CALL ZTRSM('L','U','C','N',NR,NR,CONE,CWORK(2*N+1),

+     $                 N,V,LDV)

+                  IF ( NR .LT. N ) THEN

+                  CALL ZLASET('A',N-NR,NR,CZERO,CZERO,V(NR+1,1),LDV)

+                  CALL ZLASET('A',NR,N-NR,CZERO,CZERO,V(1,NR+1),LDV)

+                  CALL ZLASET('A',N-NR,N-NR,CZERO,CONE,V(NR+1,NR+1),LDV)

+                  END IF

+                  CALL ZUNMQR('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 ZGESVJ( '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 ZCOPY( NR, V(1,p), 1, U(1,p), 1 )

+                  CALL ZDSCAL( NR, SVA(p),    U(1,p), 1 )

+ 3870          CONTINUE

+               CALL ZTRSM('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 ZLASET( 'A',N-NR,NR,CZERO,CZERO,V(NR+1,1),LDV )

+                  CALL ZLASET( 'A',NR,N-NR,CZERO,CZERO,V(1,NR+1),LDV )

+                  CALL ZLASET('A',N-NR,N-NR,CZERO,CONE,V(NR+1,NR+1),LDV)

+               END IF

+               CALL ZUNMQR( '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 ZGEJSV completes the task.

+*              Compute the full SVD of L3 using ZGESVJ with explicit

+*              accumulation of Jacobi rotations.

+               CALL ZGESVJ( '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 ZLASET( 'A',N-NR,NR,CZERO,CZERO,V(NR+1,1),LDV )

+                  CALL ZLASET( 'A',NR,N-NR,CZERO,CZERO,V(1,NR+1),LDV )

+                  CALL ZLASET('A',N-NR,N-NR,CZERO,CONE,V(NR+1,NR+1),LDV)

+               END IF

+               CALL ZUNMQR( '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 ZUNMLQ( '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(DBLE(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 / DZNRM2( N, V(1,q), 1 )

+               IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )

+     $           CALL ZDSCAL( 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 ZLASET('A', M-NR, NR, CZERO, CZERO, U(NR+1,1), LDU)

+               IF ( NR .LT. N1 ) THEN

+                  CALL ZLASET('A',NR,N1-NR,CZERO,CZERO,U(1,NR+1),LDU)

+                  CALL ZLASET('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 ZUNMQR( 'L', 'N', 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(DBLE(M)) * EPSLN

+            DO 1973 p = 1, NR

+               XSC = ONE / DZNRM2( M, U(1,p), 1 )

+               IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )

+     $          CALL ZDSCAL( 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 ZLASWP( N1, U, LDU, 1, M-1, IWORK(IWOFF+1), -1 )

+*

+         ELSE

+*

+*        .. the initial matrix A has almost orthogonal columns and

+*        the second QRF is not needed

+*

+            CALL ZLACPY( 'U', 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 ZLASET( 'L',N-1,N-1,CZERO,CZERO,CWORK(N+2),N )

+            END IF

+*

+            CALL ZGESVJ( 'U', '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 ZCOPY( N, CWORK(N+(p-1)*N+1), 1, U(1,p), 1 )

+               CALL ZDSCAL( N, SVA(p), CWORK(N+(p-1)*N+1), 1 )

+ 6970       CONTINUE

+*

+            CALL ZTRSM( 'L', 'U', 'N', 'N', N, N,

+     $           CONE, A, LDA, CWORK(N+1), N )

+            DO 6972 p = 1, N

+               CALL ZCOPY( N, CWORK(N+p), N, V(IWORK(p),1), LDV )

+ 6972       CONTINUE

+            TEMP1 = SQRT(DBLE(N))*EPSLN

+            DO 6971 p = 1, N

+               XSC = ONE / DZNRM2( N, V(1,p), 1 )

+               IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )

+     $            CALL ZDSCAL( N, XSC, V(1,p), 1 )

+ 6971       CONTINUE

+*

+*           Assemble the left singular vector matrix U (M x N).

+*

+            IF ( N .LT. M ) THEN

+               CALL ZLASET( 'A',  M-N, N, CZERO, CZERO, U(N+1,1), LDU )

+               IF ( N .LT. N1 ) THEN

+                  CALL ZLASET('A',N,  N1-N, CZERO, CZERO,  U(1,N+1),LDU)

+                  CALL ZLASET( 'A',M-N,N1-N, CZERO, CONE,U(N+1,N+1),LDU)

+               END IF

+            END IF

+            CALL ZUNMQR( 'L', 'N', M, N1, N, A, LDA, CWORK, U,

+     $           LDU, CWORK(N+1), LWORK-N, IERR )

+            TEMP1 = SQRT(DBLE(M))*EPSLN

+            DO 6973 p = 1, N1

+               XSC = ONE / DZNRM2( M, U(1,p), 1 )

+               IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )

+     $            CALL ZDSCAL( M, XSC, U(1,p), 1 )

+ 6973       CONTINUE

+*

+            IF ( ROWPIV )

+     $         CALL ZLASWP( N1, U, LDU, 1, M-1, IWORK(IWOFF+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, e.g. when the singular values spread from

+*        the underflow to the overflow threshold. 

+*

+         DO 7968 p = 1, NR

+            CALL ZCOPY( N-p+1, A(p,p), LDA, V(p,p), 1 )

+            CALL ZLACGV( N-p+1, V(p,p), 1 )

+ 7968    CONTINUE

+*

+         IF ( L2PERT ) THEN

+            XSC = SQRT(SMALL/EPSLN)

+            DO 5969 q = 1, NR

+               CTEMP = DCMPLX(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 ZLASET( 'U', NR-1, NR-1, CZERO, CZERO, V(1,2), LDV )

+         END IF

+

+         CALL ZGEQRF( N, NR, V, LDV, CWORK(N+1), CWORK(2*N+1),

+     $        LWORK-2*N, IERR )

+         CALL ZLACPY( 'L', N, NR, V, LDV, CWORK(2*N+1), N )

+*

+         DO 7969 p = 1, NR

+            CALL ZCOPY( NR-p+1, V(p,p), LDV, U(p,p), 1 )

+            CALL ZLACGV( 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 = DCMPLX(XSC * MIN(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 ZLASET('U', NR-1, NR-1, CZERO, CZERO, U(1,2), LDU )

+         END IF

+

+         CALL ZGESVJ( '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 ZLASET( 'A',N-NR,NR,CZERO,CZERO,V(NR+1,1),LDV )

+            CALL ZLASET( 'A',NR,N-NR,CZERO,CZERO,V(1,NR+1),LDV )

+            CALL ZLASET( 'A',N-NR,N-NR,CZERO,CONE,V(NR+1,NR+1),LDV )

+         END IF

+

+         CALL ZUNMQR( '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(DBLE(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 / DZNRM2( N, V(1,q), 1 )

+               IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )

+     $           CALL ZDSCAL( 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 ZLASET( 'A',  M-NR, NR, CZERO, CZERO, U(NR+1,1), LDU )

+            IF ( NR .LT. N1 ) THEN

+               CALL ZLASET('A',NR,  N1-NR, CZERO, CZERO,  U(1,NR+1),LDU)

+               CALL ZLASET('A',M-NR,N1-NR, CZERO, CONE,U(NR+1,NR+1),LDU)

+            END IF

+         END IF

+*

+         CALL ZUNMQR( 'L', 'N', M, N1, N, A, LDA, CWORK, U,

+     $        LDU, CWORK(N+1), LWORK-N, IERR )

+*

+            IF ( ROWPIV )

+     $         CALL ZLASWP( N1, U, LDU, 1, M-1, IWORK(IWOFF+1), -1 )

+*

+*

+         END IF

+         IF ( TRANSP ) THEN

+*           .. swap U and V because the procedure worked on A^*

+            DO 6974 p = 1, N

+               CALL ZSWAP( 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 DLASCL( '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

+      IF ( TRANSP ) THEN

+          IWORK(4) =  1 

+      ELSE

+          IWORK(4) = -1

+      END IF 

+      

+*

+      RETURN

+*     ..

+*     .. END OF ZGEJSV

+*     ..

+      END

+*