cgesvdq

 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.

          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 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'.

          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.

          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 = 'N'.
       = 'W': V may be used as workspace of length N*N. See the description
              of V.
       = 'N': V is not computed.

          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 = '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.

          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.

          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

          M is INTEGER
         The number of rows of the input matrix A.  M >= 0.

          N is INTEGER
         The number of columns of the input matrix A. M >= N >= 0.

          A is COMPLEX array, dimension (LDA,N)
          On entry, the M-by-N matrix A.

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

          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.

          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 = 'V' .AND. JOBT = 'T' .AND. M = 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'.

          LDU is INTEGER
          The leading dimension of the array U,  LDU >= 1.
          IF  JOBU = 'U' or 'F' or 'W',  then LDU >= M.

          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 = 'U' AND JOBT = 'T' AND M = 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'.

          LDV is INTEGER
          The leading dimension of the array V,  LDV >= 1.
          If JOBV = 'V' or 'J' or 'W', then LDV >= N.

          CWORK is COMPLEX array, dimension (MAX(2,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.

          LWORK is INTEGER
          Length of CWORK to confirm proper allocation of workspace.
          LWORK depends on the job:

          1. If only SIGMA is needed ( JOBU = 'N', JOBV = '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 = 'V'),
             (JOBU = 'N')
            2.1   .. no scaled condition estimate requested (JOBE = '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 = 'N'):
            -> the minimal requirement is LWORK >= 3*N.
            -> For optimal performance:
               if JOBU = '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 = '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 = 'U' or JOBU = 'F') and
            4.1. if JOBV = 'V'
               the minimal requirement is LWORK >= 5*N+2*N*N.
            4.2. if JOBV = '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.        

          RWORK is REAL array, dimension (MAX(7,LWORK))
          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 = 'E' or 'G')
                    SCONDA is an estimate of SQRT(||(R^* * R)^(-1)||_1).
                    It is computed using CPOCON. 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 provided 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 = '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.

          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 or LWORK=-1, a workspace query is assumed and 
          the length of RWORK is returned in RWORK(1). 

          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) = 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) = 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 = -1 and 
          LRWORK = -1), then on exit IWORK(1) contains the required length of 
          IWORK for the job parameters used in the call.

          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.

  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.

 [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.

 CGESDD computes the singular value decomposition (SVD) of a complex
 M-by-N matrix A, optionally computing the left and/or right singular
 vectors, by using divide-and-conquer method. The SVD is written

      A = U * SIGMA * conjugate-transpose(V)

 where SIGMA is an M-by-N matrix which is zero except for its
 min(m,n) diagonal elements, U is an 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; they are real and non-negative, and
 are returned in descending order.  The first min(m,n) columns of
 U and V are the left and right singular vectors of A.

 Note that the routine returns VT = V**H, not V.

 The divide and conquer algorithm makes very mild assumptions about
 floating point arithmetic. It will work on machines with a guard
 digit in add/subtract, or on those binary machines without guard
 digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
 Cray-2. It could conceivably fail on hexadecimal or decimal machines
 without guard digits, but we know of none.

          JOBZ is CHARACTER*1
          Specifies options for computing all or part of the matrix U:
          = 'A':  all M columns of U and all N rows of V**H are
                  returned in the arrays U and VT;
          = 'S':  the first min(M,N) columns of U and the first
                  min(M,N) rows of V**H are returned in the arrays U
                  and VT;
          = 'O':  If M >= N, the first N columns of U are overwritten
                  in the array A and all rows of V**H are returned in
                  the array VT;
                  otherwise, all columns of U are returned in the
                  array U and the first M rows of V**H are overwritten
                  in the array A;
          = 'N':  no columns of U or rows of V**H are computed.

          M is INTEGER
          The number of rows of the input matrix A.  M >= 0.

          N is INTEGER
          The number of columns of the input matrix A.  N >= 0.

          A is COMPLEX array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit,
          if JOBZ = 'O',  A is overwritten with the first N columns
                          of U (the left singular vectors, stored
                          columnwise) if M >= N;
                          A is overwritten with the first M rows
                          of V**H (the right singular vectors, stored
                          rowwise) otherwise.
          if JOBZ .ne. 'O', the contents of A are destroyed.

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

          S is REAL array, dimension (min(M,N))
          The singular values of A, sorted so that S(i) >= S(i+1).

          U is COMPLEX array, dimension (LDU,UCOL)
          UCOL = M if JOBZ = 'A' or JOBZ = 'O' and M < N;
          UCOL = min(M,N) if JOBZ = 'S'.
          If JOBZ = 'A' or JOBZ = 'O' and M < N, U contains the M-by-M
          unitary matrix U;
          if JOBZ = 'S', U contains the first min(M,N) columns of U
          (the left singular vectors, stored columnwise);
          if JOBZ = 'O' and M >= N, or JOBZ = 'N', U is not referenced.

          LDU is INTEGER
          The leading dimension of the array U.  LDU >= 1;
          if JOBZ = 'S' or 'A' or JOBZ = 'O' and M < N, LDU >= M.

          VT is COMPLEX array, dimension (LDVT,N)
          If JOBZ = 'A' or JOBZ = 'O' and M >= N, VT contains the
          N-by-N unitary matrix V**H;
          if JOBZ = 'S', VT contains the first min(M,N) rows of
          V**H (the right singular vectors, stored rowwise);
          if JOBZ = 'O' and M < N, or JOBZ = 'N', VT is not referenced.

          LDVT is INTEGER
          The leading dimension of the array VT.  LDVT >= 1;
          if JOBZ = 'A' or JOBZ = 'O' and M >= N, LDVT >= N;
          if JOBZ = 'S', LDVT >= min(M,N).

          WORK is COMPLEX array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

          LWORK is INTEGER
          The dimension of the array WORK. LWORK >= 1.
          If LWORK = -1, a workspace query is assumed.  The optimal
          size for the WORK array is calculated and stored in WORK(1),
          and no other work except argument checking is performed.

          Let mx = max(M,N) and mn = min(M,N).
          If JOBZ = 'N', LWORK >= 2*mn + mx.
          If JOBZ = 'O', LWORK >= 2*mn*mn + 2*mn + mx.
          If JOBZ = 'S', LWORK >=   mn*mn + 3*mn.
          If JOBZ = 'A', LWORK >=   mn*mn + 2*mn + mx.
          These are not tight minimums in all cases; see comments inside code.
          For good performance, LWORK should generally be larger;
          a query is recommended.

          RWORK is REAL array, dimension (MAX(1,LRWORK))
          Let mx = max(M,N) and mn = min(M,N).
          If JOBZ = 'N',    LRWORK >= 5*mn (LAPACK <= 3.6 needs 7*mn);
          else if mx >> mn, LRWORK >= 5*mn*mn + 5*mn;
          else              LRWORK >= max( 5*mn*mn + 5*mn,
                                           2*mx*mn + 2*mn*mn + mn ).

          IWORK is INTEGER array, dimension (8*min(M,N))

          INFO is INTEGER
          <  0:  if INFO = -i, the i-th argument had an illegal value.
          = -4:  if A had a NAN entry.
          >  0:  The updating process of SBDSDC did not converge.
          =  0:  successful exit.

 CGESVD computes the singular value decomposition (SVD) of a complex
 M-by-N matrix A, optionally computing the left and/or right singular
 vectors. The SVD is written

      A = U * SIGMA * conjugate-transpose(V)

 where SIGMA is an M-by-N matrix which is zero except for its
 min(m,n) diagonal elements, U is an 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; they are real and non-negative, and
 are returned in descending order.  The first min(m,n) columns of
 U and V are the left and right singular vectors of A.

 Note that the routine returns V**H, not V.

          JOBU is CHARACTER*1
          Specifies options for computing all or part of the matrix U:
          = 'A':  all M columns of U are returned in array U:
          = 'S':  the first min(m,n) columns of U (the left singular
                  vectors) are returned in the array U;
          = 'O':  the first min(m,n) columns of U (the left singular
                  vectors) are overwritten on the array A;
          = 'N':  no columns of U (no left singular vectors) are
                  computed.

          JOBVT is CHARACTER*1
          Specifies options for computing all or part of the matrix
          V**H:
          = 'A':  all N rows of V**H are returned in the array VT;
          = 'S':  the first min(m,n) rows of V**H (the right singular
                  vectors) are returned in the array VT;
          = 'O':  the first min(m,n) rows of V**H (the right singular
                  vectors) are overwritten on the array A;
          = 'N':  no rows of V**H (no right singular vectors) are
                  computed.

          JOBVT and JOBU cannot both be 'O'.

          M is INTEGER
          The number of rows of the input matrix A.  M >= 0.

          N is INTEGER
          The number of columns of the input matrix A.  N >= 0.

          A is COMPLEX array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit,
          if JOBU = 'O',  A is overwritten with the first min(m,n)
                          columns of U (the left singular vectors,
                          stored columnwise);
          if JOBVT = 'O', A is overwritten with the first min(m,n)
                          rows of V**H (the right singular vectors,
                          stored rowwise);
          if JOBU .ne. 'O' and JOBVT .ne. 'O', the contents of A
                          are destroyed.

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

          S is REAL array, dimension (min(M,N))
          The singular values of A, sorted so that S(i) >= S(i+1).

          U is COMPLEX array, dimension (LDU,UCOL)
          (LDU,M) if JOBU = 'A' or (LDU,min(M,N)) if JOBU = 'S'.
          If JOBU = 'A', U contains the M-by-M unitary matrix U;
          if JOBU = 'S', U contains the first min(m,n) columns of U
          (the left singular vectors, stored columnwise);
          if JOBU = 'N' or 'O', U is not referenced.

          LDU is INTEGER
          The leading dimension of the array U.  LDU >= 1; if
          JOBU = 'S' or 'A', LDU >= M.

          VT is COMPLEX array, dimension (LDVT,N)
          If JOBVT = 'A', VT contains the N-by-N unitary matrix
          V**H;
          if JOBVT = 'S', VT contains the first min(m,n) rows of
          V**H (the right singular vectors, stored rowwise);
          if JOBVT = 'N' or 'O', VT is not referenced.

          LDVT is INTEGER
          The leading dimension of the array VT.  LDVT >= 1; if
          JOBVT = 'A', LDVT >= N; if JOBVT = 'S', LDVT >= min(M,N).

          WORK is COMPLEX array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

          LWORK is INTEGER
          The dimension of the array WORK.
          LWORK >=  MAX(1,2*MIN(M,N)+MAX(M,N)).
          For good performance, LWORK should generally be larger.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

          RWORK is REAL array, dimension (5*min(M,N))
          On exit, if INFO > 0, RWORK(1:MIN(M,N)-1) contains the
          unconverged superdiagonal elements of an upper bidiagonal
          matrix B whose diagonal is in S (not necessarily sorted).
          B satisfies A = U * B * VT, so it has the same singular
          values as A, and singular vectors related by U and VT.

          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if CBDSQR did not converge, INFO specifies how many
                superdiagonals of an intermediate bidiagonal form B
                did not converge to zero. See the description of RWORK
                above for details.

 CGESVDQ computes the singular value decomposition (SVD) of a complex
 M-by-N matrix A, where M >= N. The SVD of A is written as
                                    [++]   [xx]   [x0]   [xx]
              A = U * SIGMA * V^*,  [++] = [xx] * [ox] * [xx]
                                    [++]   [xx]
 where SIGMA is an N-by-N diagonal matrix, U is an M-by-N orthonormal
 matrix, and V is an N-by-N unitary matrix. The diagonal elements
 of SIGMA are the singular values of A. The columns of U and V are the
 left and the right singular vectors of A, respectively.

  JOBA is CHARACTER*1
  Specifies the level of accuracy in the computed SVD
  = 'A' The requested accuracy corresponds to having the backward
        error bounded by || delta A ||_F <= f(m,n) * EPS * || A ||_F,
        where EPS = SLAMCH('Epsilon'). This authorises CGESVDQ to
        truncate the computed triangular factor in a rank revealing
        QR factorization whenever the truncated part is below the
        threshold of the order of EPS * ||A||_F. This is aggressive
        truncation level.
  = 'M' Similarly as with 'A', but the truncation is more gentle: it
        is allowed only when there is a drop on the diagonal of the
        triangular factor in the QR factorization. This is medium
        truncation level.
  = 'H' High accuracy requested. No numerical rank determination based
        on the rank revealing QR factorization is attempted.
  = 'E' Same as 'H', and in addition the condition number of column
        scaled A is estimated and returned in  RWORK(1).
        N^(-1/4)*RWORK(1) <= ||pinv(A_scaled)||_2 <= N^(1/4)*RWORK(1)

  JOBP is CHARACTER*1
  = 'P' The rows of A are ordered in decreasing order with respect to
        ||A(i,:)||_\infty. This enhances numerical accuracy at the cost
        of extra data movement. Recommended for numerical robustness.
  = 'N' No row pivoting.

          JOBR is CHARACTER*1
          = 'T' After the initial pivoted QR factorization, CGESVD is applied to
          the adjoint R**H of the computed triangular factor R. This involves
          some extra data movement (matrix transpositions). Useful for
          experiments, research and development.
          = 'N' The triangular factor R is given as input to CGESVD. This may be
          preferred as it involves less data movement.

          JOBU is CHARACTER*1
          = 'A' All M left singular vectors are computed and returned in the
          matrix U. See the description of U.
          = 'S' or 'U' N = min(M,N) left singular vectors are computed and returned
          in the matrix U. See the description of U.
          = 'R' Numerical rank NUMRANK is determined and only NUMRANK left singular
          vectors are computed and returned in the matrix U.
          = 'F' The N left singular vectors are returned in factored form as the
          product of the Q factor from the initial QR factorization and the
          N left singular vectors of (R**H , 0)**H. If row pivoting is used,
          then the necessary information on the row pivoting is stored in
          IWORK(N+1:N+M-1).
          = 'N' The left singular vectors are not computed.

          JOBV is CHARACTER*1
          = 'A', 'V' All N right singular vectors are computed and returned in
          the matrix V.
          = 'R' Numerical rank NUMRANK is determined and only NUMRANK right singular
          vectors are computed and returned in the matrix V. This option is
          allowed only if JOBU = 'R' or JOBU = 'N'; otherwise it is illegal.
          = 'N' The right singular vectors are not computed.

          M is INTEGER
          The number of rows of the input matrix A.  M >= 0.

          N is INTEGER
          The number of columns of the input matrix A.  M >= N >= 0.

          A is COMPLEX array of dimensions LDA x N
          On entry, the input matrix A.
          On exit, if JOBU .NE. 'N' or JOBV .NE. 'N', the lower triangle of A contains
          the Householder vectors as stored by CGEQP3. If JOBU = 'F', these Householder
          vectors together with CWORK(1:N) can be used to restore the Q factors from
          the initial pivoted QR factorization of A. See the description of U.

          LDA is INTEGER.
          The leading dimension of the array A.  LDA >= max(1,M).

          S is REAL array of dimension N.
          The singular values of A, ordered so that S(i) >= S(i+1).

          U is COMPLEX array, dimension
          LDU x M if JOBU = 'A'; see the description of LDU. In this case,
          on exit, U contains the M left singular vectors.
          LDU x N if JOBU = 'S', 'U', 'R' ; see the description of LDU. In this
          case, U contains the leading N or the leading NUMRANK left singular vectors.
          LDU x N if JOBU = 'F' ; see the description of LDU. In this case U
          contains N x N unitary matrix that can be used to form the left
          singular vectors.
          If JOBU = 'N', U is not referenced.

          LDU is INTEGER.
          The leading dimension of the array U.
          If JOBU = 'A', 'S', 'U', 'R',  LDU >= max(1,M).
          If JOBU = 'F',                 LDU >= max(1,N).
          Otherwise,                     LDU >= 1.

          V is COMPLEX array, dimension
          LDV x N if JOBV = 'A', 'V', 'R' or if JOBA = 'E' .
          If JOBV = 'A', or 'V',  V contains the N-by-N unitary matrix  V**H;
          If JOBV = 'R', V contains the first NUMRANK rows of V**H (the right
          singular vectors, stored rowwise, of the NUMRANK largest singular values).
          If JOBV = 'N' and JOBA = 'E', V is used as a workspace.
          If JOBV = 'N', and JOBA.NE.'E', V is not referenced.

          LDV is INTEGER
          The leading dimension of the array V.
          If JOBV = 'A', 'V', 'R',  or JOBA = 'E', LDV >= max(1,N).
          Otherwise,                               LDV >= 1.

          NUMRANK is INTEGER
          NUMRANK is the numerical rank first determined after the rank
          revealing QR factorization, following the strategy specified by the
          value of JOBA. If JOBV = 'R' and JOBU = 'R', only NUMRANK
          leading singular values and vectors are then requested in the call
          of CGESVD. The final value of NUMRANK might be further reduced if
          some singular values are computed as zeros.

          IWORK is INTEGER array, dimension (max(1, LIWORK)).
          On exit, IWORK(1:N) contains column pivoting permutation of the
          rank revealing QR factorization.
          If JOBP = 'P', IWORK(N+1:N+M-1) contains the indices of the sequence
          of row swaps used in row pivoting. These can be used to restore the
          left singular vectors in the case JOBU = 'F'.

          If LIWORK, LCWORK, or LRWORK = -1, then on exit, if INFO = 0,
          IWORK(1) returns the minimal LIWORK.

          LIWORK is INTEGER
          The dimension of the array IWORK.
          LIWORK >= N + M - 1,  if JOBP = 'P';
          LIWORK >= N           if JOBP = 'N'.

          If LIWORK = -1, then a workspace query is assumed; the routine
          only calculates and returns the optimal and minimal sizes
          for the CWORK, IWORK, and RWORK arrays, and no error
          message related to LCWORK is issued by XERBLA.

          CWORK is COMPLEX array, dimension (max(2, LCWORK)), used as a workspace.
          On exit, if, on entry, LCWORK.NE.-1, CWORK(1:N) contains parameters
          needed to recover the Q factor from the QR factorization computed by
          CGEQP3.

          If LIWORK, LCWORK, or LRWORK = -1, then on exit, if INFO = 0,
          CWORK(1) returns the optimal LCWORK, and
          CWORK(2) returns the minimal LCWORK.

          LCWORK is INTEGER
          The dimension of the array CWORK. It is determined as follows:
          Let  LWQP3 = N+1,  LWCON = 2*N, and let
          LWUNQ = { MAX( N, 1 ),  if JOBU = 'R', 'S', or 'U'
                  { MAX( M, 1 ),  if JOBU = 'A'
          LWSVD = MAX( 3*N, 1 )
          LWLQF = MAX( N/2, 1 ), LWSVD2 = MAX( 3*(N/2), 1 ), LWUNLQ = MAX( N, 1 ),
          LWQRF = MAX( N/2, 1 ), LWUNQ2 = MAX( N, 1 )
          Then the minimal value of LCWORK is:
          = MAX( N + LWQP3, LWSVD )        if only the singular values are needed;
          = MAX( N + LWQP3, LWCON, LWSVD ) if only the singular values are needed,
                                   and a scaled condition estimate requested;

          = N + MAX( LWQP3, LWSVD, LWUNQ ) if the singular values and the left
                                   singular vectors are requested;
          = N + MAX( LWQP3, LWCON, LWSVD, LWUNQ ) if the singular values and the left
                                   singular vectors are requested, and also
                                   a scaled condition estimate requested;

          = N + MAX( LWQP3, LWSVD )        if the singular values and the right
                                   singular vectors are requested;
          = N + MAX( LWQP3, LWCON, LWSVD ) if the singular values and the right
                                   singular vectors are requested, and also
                                   a scaled condition etimate requested;

          = N + MAX( LWQP3, LWSVD, LWUNQ ) if the full SVD is requested with JOBV = 'R';
                                   independent of JOBR;
          = N + MAX( LWQP3, LWCON, LWSVD, LWUNQ ) if the full SVD is requested,
                                   JOBV = 'R' and, also a scaled condition
                                   estimate requested; independent of JOBR;
          = MAX( N + MAX( LWQP3, LWSVD, LWUNQ ),
         N + MAX( LWQP3, N/2+LWLQF, N/2+LWSVD2, N/2+LWUNLQ, LWUNQ) ) if the
                         full SVD is requested with JOBV = 'A' or 'V', and
                         JOBR ='N'
          = MAX( N + MAX( LWQP3, LWCON, LWSVD, LWUNQ ),
         N + MAX( LWQP3, LWCON, N/2+LWLQF, N/2+LWSVD2, N/2+LWUNLQ, LWUNQ ) )
                         if the full SVD is requested with JOBV = 'A' or 'V', and
                         JOBR ='N', and also a scaled condition number estimate
                         requested.
          = MAX( N + MAX( LWQP3, LWSVD, LWUNQ ),
         N + MAX( LWQP3, N/2+LWQRF, N/2+LWSVD2, N/2+LWUNQ2, LWUNQ ) ) if the
                         full SVD is requested with JOBV = 'A', 'V', and JOBR ='T'
          = MAX( N + MAX( LWQP3, LWCON, LWSVD, LWUNQ ),
         N + MAX( LWQP3, LWCON, N/2+LWQRF, N/2+LWSVD2, N/2+LWUNQ2, LWUNQ ) )
                         if the full SVD is requested with JOBV = 'A', 'V' and
                         JOBR ='T', and also a scaled condition number estimate
                         requested.
          Finally, LCWORK must be at least two: LCWORK = MAX( 2, LCWORK ).

          If LCWORK = -1, then a workspace query is assumed; the routine
          only calculates and returns the optimal and minimal sizes
          for the CWORK, IWORK, and RWORK arrays, and no error
          message related to LCWORK is issued by XERBLA.

          RWORK is REAL array, dimension (max(1, LRWORK)).
          On exit,
          1. If JOBA = 'E', RWORK(1) contains an estimate of the condition
          number of column scaled A. If A = C * D where D is diagonal and C
          has unit columns in the Euclidean norm, then, assuming full column rank,
          N^(-1/4) * RWORK(1) <= ||pinv(C)||_2 <= N^(1/4) * RWORK(1).
          Otherwise, RWORK(1) = -1.
          2. RWORK(2) contains the number of singular values computed as
          exact zeros in CGESVD applied to the upper triangular or trapezoidal
          R (from the initial QR factorization). In case of early exit (no call to
          CGESVD, such as in the case of zero matrix) RWORK(2) = -1.

          If LIWORK, LCWORK, or LRWORK = -1, then on exit, if INFO = 0,
          RWORK(1) returns the minimal LRWORK.

          LRWORK is INTEGER.
          The dimension of the array RWORK.
          If JOBP ='P', then LRWORK >= MAX(2, M, 5*N);
          Otherwise, LRWORK >= MAX(2, 5*N).

          If LRWORK = -1, then a workspace query is assumed; the routine
          only calculates and returns the optimal and minimal sizes
          for the CWORK, IWORK, and RWORK arrays, and no error
          message related to LCWORK is issued by XERBLA.

          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if CBDSQR did not converge, INFO specifies how many superdiagonals
          of an intermediate bidiagonal form B (computed in CGESVD) did not
          converge to zero.

   1. The data movement (matrix transpose) is coded using simple nested
   DO-loops because BLAS and LAPACK do not provide corresponding subroutines.
   Those DO-loops are easily identified in this source code - by the CONTINUE
   statements labeled with 11**. In an optimized version of this code, the
   nested DO loops should be replaced with calls to an optimized subroutine.
   2. This code scales A by 1/SQRT(M) if the largest ABS(A(i,j)) could cause
   column norm overflow. This is the minial precaution and it is left to the
   SVD routine (CGESVD) to do its own preemptive scaling if potential over-
   or underflows are detected. To avoid repeated scanning of the array A,
   an optimal implementation would do all necessary scaling before calling
   CGESVD and the scaling in CGESVD can be switched off.
   3. Other comments related to code optimization are given in comments in the
   code, enlosed in [[double brackets]].

  Please report all bugs and send interesting examples and/or comments to
  [email protected]. Thank you.

  [1] Zlatko Drmac, Algorithm 977: A QR-Preconditioned QR SVD Method for
      Computing the SVD with High Accuracy. ACM Trans. Math. Softw.
      44(1): 11:1-11:30 (2017)

  SIGMA library, xGESVDQ section updated February 2016.
  Developed and coded by Zlatko Drmac, Department of Mathematics
  University of Zagreb, Croatia, [email protected]

 Developed and coded by Zlatko Drmac, Department of Mathematics
  University of Zagreb, Croatia, [email protected]

  CGESVDX computes the singular value decomposition (SVD) of a complex
  M-by-N matrix A, optionally computing the left and/or right singular
  vectors. The SVD is written

      A = U * SIGMA * transpose(V)

  where SIGMA is an M-by-N matrix which is zero except for its
  min(m,n) diagonal elements, U is an 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; they are real and non-negative, and
  are returned in descending order.  The first min(m,n) columns of
  U and V are the left and right singular vectors of A.

  CGESVDX uses an eigenvalue problem for obtaining the SVD, which
  allows for the computation of a subset of singular values and
  vectors. See SBDSVDX for details.

  Note that the routine returns V**T, not V.

          JOBU is CHARACTER*1
          Specifies options for computing all or part of the matrix U:
          = 'V':  the first min(m,n) columns of U (the left singular
                  vectors) or as specified by RANGE are returned in
                  the array U;
          = 'N':  no columns of U (no left singular vectors) are
                  computed.

          JOBVT is CHARACTER*1
           Specifies options for computing all or part of the matrix
           V**T:
           = 'V':  the first min(m,n) rows of V**T (the right singular
                   vectors) or as specified by RANGE are returned in
                   the array VT;
           = 'N':  no rows of V**T (no right singular vectors) are
                   computed.

          RANGE is CHARACTER*1
          = 'A': all singular values will be found.
          = 'V': all singular values in the half-open interval (VL,VU]
                 will be found.
          = 'I': the IL-th through IU-th singular values will be found.

          M is INTEGER
          The number of rows of the input matrix A.  M >= 0.

          N is INTEGER
          The number of columns of the input matrix A.  N >= 0.

          A is COMPLEX array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit, the contents of A are destroyed.

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

          VL is REAL
          If RANGE='V', the lower bound of the interval to
          be searched for singular values. VU > VL.
          Not referenced if RANGE = 'A' or 'I'.

          VU is REAL
          If RANGE='V', the upper bound of the interval to
          be searched for singular values. VU > VL.
          Not referenced if RANGE = 'A' or 'I'.

          IL is INTEGER
          If RANGE='I', the index of the
          smallest singular value to be returned.
          1 <= IL <= IU <= min(M,N), if min(M,N) > 0.
          Not referenced if RANGE = 'A' or 'V'.

          IU is INTEGER
          If RANGE='I', the index of the
          largest singular value to be returned.
          1 <= IL <= IU <= min(M,N), if min(M,N) > 0.
          Not referenced if RANGE = 'A' or 'V'.

          NS is INTEGER
          The total number of singular values found,
          0 <= NS <= min(M,N).
          If RANGE = 'A', NS = min(M,N); if RANGE = 'I', NS = IU-IL+1.

          S is REAL array, dimension (min(M,N))
          The singular values of A, sorted so that S(i) >= S(i+1).

          U is COMPLEX array, dimension (LDU,UCOL)
          If JOBU = 'V', U contains columns of U (the left singular
          vectors, stored columnwise) as specified by RANGE; if
          JOBU = 'N', U is not referenced.
          Note: The user must ensure that UCOL >= NS; if RANGE = 'V',
          the exact value of NS is not known in advance and an upper
          bound must be used.

          LDU is INTEGER
          The leading dimension of the array U.  LDU >= 1; if
          JOBU = 'V', LDU >= M.

          VT is COMPLEX array, dimension (LDVT,N)
          If JOBVT = 'V', VT contains the rows of V**T (the right singular
          vectors, stored rowwise) as specified by RANGE; if JOBVT = 'N',
          VT is not referenced.
          Note: The user must ensure that LDVT >= NS; if RANGE = 'V',
          the exact value of NS is not known in advance and an upper
          bound must be used.

          LDVT is INTEGER
          The leading dimension of the array VT.  LDVT >= 1; if
          JOBVT = 'V', LDVT >= NS (see above).

          WORK is COMPLEX array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK;

          LWORK is INTEGER
          The dimension of the array WORK.
          LWORK >= MAX(1,MIN(M,N)*(MIN(M,N)+4)) for the paths (see
          comments inside the code):
             - PATH 1  (M much larger than N)
             - PATH 1t (N much larger than M)
          LWORK >= MAX(1,MIN(M,N)*2+MAX(M,N)) for the other paths.
          For good performance, LWORK should generally be larger.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

          RWORK is REAL array, dimension (MAX(1,LRWORK))
          LRWORK >= MIN(M,N)*(MIN(M,N)*2+15*MIN(M,N)).

          IWORK is INTEGER array, dimension (12*MIN(M,N))
          If INFO = 0, the first NS elements of IWORK are zero. If INFO > 0,
          then IWORK contains the indices of the eigenvectors that failed
          to converge in SBDSVDX/SSTEVX.

     INFO is INTEGER
           = 0:  successful exit
           < 0:  if INFO = -i, the i-th argument had an illegal value
           > 0:  if INFO = i, then i eigenvectors failed to converge
                 in SBDSVDX/SSTEVX.
                 if INFO = N*2 + 1, an internal error occurred in
                 SBDSVDX

 CGGSVD3 computes the generalized singular value decomposition (GSVD)
 of an M-by-N complex matrix A and P-by-N complex matrix B:

       U**H*A*Q = D1*( 0 R ),    V**H*B*Q = D2*( 0 R )

 where U, V and Q are unitary matrices.
 Let K+L = the effective numerical rank of the
 matrix (A**H,B**H)**H, then R is a (K+L)-by-(K+L) nonsingular upper
 triangular matrix, D1 and D2 are M-by-(K+L) and P-by-(K+L) 'diagonal'
 matrices and of the following structures, respectively:

 If M-K-L >= 0,

                     K  L
        D1 =     K ( I  0 )
                 L ( 0  C )
             M-K-L ( 0  0 )

                   K  L
        D2 =   L ( 0  S )
             P-L ( 0  0 )

                 N-K-L  K    L
   ( 0 R ) = K (  0   R11  R12 )
             L (  0    0   R22 )

 where

   C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
   S = diag( BETA(K+1),  ... , BETA(K+L) ),
   C**2 + S**2 = I.

   R is stored in A(1:K+L,N-K-L+1:N) on exit.

 If M-K-L < 0,

                   K M-K K+L-M
        D1 =   K ( I  0    0   )
             M-K ( 0  C    0   )

                     K M-K K+L-M
        D2 =   M-K ( 0  S    0  )
             K+L-M ( 0  0    I  )
               P-L ( 0  0    0  )

                    N-K-L  K   M-K  K+L-M
   ( 0 R ) =     K ( 0    R11  R12  R13  )
               M-K ( 0     0   R22  R23  )
             K+L-M ( 0     0    0   R33  )

 where

   C = diag( ALPHA(K+1), ... , ALPHA(M) ),
   S = diag( BETA(K+1),  ... , BETA(M) ),
   C**2 + S**2 = I.

   (R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored
   ( 0  R22 R23 )
   in B(M-K+1:L,N+M-K-L+1:N) on exit.

 The routine computes C, S, R, and optionally the unitary
 transformation matrices U, V and Q.

 In particular, if B is an N-by-N nonsingular matrix, then the GSVD of
 A and B implicitly gives the SVD of A*inv(B):
                      A*inv(B) = U*(D1*inv(D2))*V**H.
 If ( A**H,B**H)**H has orthonormal columns, then the GSVD of A and B is also
 equal to the CS decomposition of A and B. Furthermore, the GSVD can
 be used to derive the solution of the eigenvalue problem:
                      A**H*A x = lambda* B**H*B x.
 In some literature, the GSVD of A and B is presented in the form
                  U**H*A*X = ( 0 D1 ),   V**H*B*X = ( 0 D2 )
 where U and V are orthogonal and X is nonsingular, and D1 and D2 are
 ``diagonal''.  The former GSVD form can be converted to the latter
 form by taking the nonsingular matrix X as

                       X = Q*(  I   0    )
                             (  0 inv(R) )

          JOBU is CHARACTER*1
          = 'U':  Unitary matrix U is computed;
          = 'N':  U is not computed.

          JOBV is CHARACTER*1
          = 'V':  Unitary matrix V is computed;
          = 'N':  V is not computed.

          JOBQ is CHARACTER*1
          = 'Q':  Unitary matrix Q is computed;
          = 'N':  Q is not computed.

          M is INTEGER
          The number of rows of the matrix A.  M >= 0.

          N is INTEGER
          The number of columns of the matrices A and B.  N >= 0.

          P is INTEGER
          The number of rows of the matrix B.  P >= 0.

          K is INTEGER

          L is INTEGER

          On exit, K and L specify the dimension of the subblocks
          described in Purpose.
          K + L = effective numerical rank of (A**H,B**H)**H.

          A is COMPLEX array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit, A contains the triangular matrix R, or part of R.
          See Purpose for details.

          LDA is INTEGER
          The leading dimension of the array A. LDA >= max(1,M).

          B is COMPLEX array, dimension (LDB,N)
          On entry, the P-by-N matrix B.
          On exit, B contains part of the triangular matrix R if
          M-K-L < 0.  See Purpose for details.

          LDB is INTEGER
          The leading dimension of the array B. LDB >= max(1,P).

          ALPHA is REAL array, dimension (N)

          BETA is REAL array, dimension (N)

          On exit, ALPHA and BETA contain the generalized singular
          value pairs of A and B;
            ALPHA(1:K) = 1,
            BETA(1:K)  = 0,
          and if M-K-L >= 0,
            ALPHA(K+1:K+L) = C,
            BETA(K+1:K+L)  = S,
          or if M-K-L < 0,
            ALPHA(K+1:M)=C, ALPHA(M+1:K+L)=0
            BETA(K+1:M) =S, BETA(M+1:K+L) =1
          and
            ALPHA(K+L+1:N) = 0
            BETA(K+L+1:N)  = 0

          U is COMPLEX array, dimension (LDU,M)
          If JOBU = 'U', U contains the M-by-M unitary matrix U.
          If JOBU = 'N', U is not referenced.

          LDU is INTEGER
          The leading dimension of the array U. LDU >= max(1,M) if
          JOBU = 'U'; LDU >= 1 otherwise.

          V is COMPLEX array, dimension (LDV,P)
          If JOBV = 'V', V contains the P-by-P unitary matrix V.
          If JOBV = 'N', V is not referenced.

          LDV is INTEGER
          The leading dimension of the array V. LDV >= max(1,P) if
          JOBV = 'V'; LDV >= 1 otherwise.

          Q is COMPLEX array, dimension (LDQ,N)
          If JOBQ = 'Q', Q contains the N-by-N unitary matrix Q.
          If JOBQ = 'N', Q is not referenced.

          LDQ is INTEGER
          The leading dimension of the array Q. LDQ >= max(1,N) if
          JOBQ = 'Q'; LDQ >= 1 otherwise.

          WORK is COMPLEX array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

          LWORK is INTEGER
          The dimension of the array WORK.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

          RWORK is REAL array, dimension (2*N)

          IWORK is INTEGER array, dimension (N)
          On exit, IWORK stores the sorting information. More
          precisely, the following loop will sort ALPHA
             for I = K+1, min(M,K+L)
                 swap ALPHA(I) and ALPHA(IWORK(I))
             endfor
          such that ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(N).

          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = 1, the Jacobi-type procedure failed to
                converge.  For further details, see subroutine CTGSJA.

  TOLA    REAL
  TOLB    REAL
          TOLA and TOLB are the thresholds to determine the effective
          rank of (A**H,B**H)**H. Generally, they are set to
                   TOLA = MAX(M,N)*norm(A)*MACHEPS,
                   TOLB = MAX(P,N)*norm(B)*MACHEPS.
          The size of TOLA and TOLB may affect the size of backward
          errors of the decomposition.