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version 1.4, 2010/12/21 13:48:05 version 1.13, 2015/11/26 11:44:15
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       SUBROUTINE DGEJSV( JOBA, JOBU, JOBV, JOBR, JOBT, JOBP,  *> \brief \b DGEJSV
      &                   M, N, A, LDA, SVA, U, LDU, V, LDV,  *
      &                   WORK, LWORK, IWORK, INFO )  *  =========== DOCUMENTATION ===========
 *  *
 *  -- LAPACK routine (version 3.3.0)                                    --  * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
 *  *
 *  -- Contributed by Zlatko Drmac of the University of Zagreb and     --  *> \htmlonly
 *  -- Kresimir Veselic of the Fernuniversitaet Hagen                  --  *> Download DGEJSV + dependencies 
 *     November 2010  *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgejsv.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgejsv.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgejsv.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DGEJSV( JOBA, JOBU, JOBV, JOBR, JOBT, JOBP,
   *                          M, N, A, LDA, SVA, U, LDU, V, LDV,
   *                          WORK, LWORK, IWORK, INFO )
   * 
   *       .. Scalar Arguments ..
   *       IMPLICIT    NONE
   *       INTEGER     INFO, LDA, LDU, LDV, LWORK, M, N
   *       ..
   *       .. Array Arguments ..
   *       DOUBLE PRECISION A( LDA, * ), SVA( N ), U( LDU, * ), V( LDV, * ),
   *      $            WORK( LWORK )
   *       INTEGER     IWORK( * )
   *       CHARACTER*1 JOBA, JOBP, JOBR, JOBT, JOBU, JOBV
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> DGEJSV computes the singular value decomposition (SVD) of a real M-by-N
   *> matrix [A], where M >= N. The SVD of [A] is written as
   *>
   *>              [A] = [U] * [SIGMA] * [V]^t,
   *>
   *> 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) orthonormal matrix, and
   *> [V] is an N-by-N orthogonal 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.
   *> DGEJSV can sometimes compute tiny singular values and their singular vectors much
   *> more accurately than other SVD routines, see below under Further Details.*> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] JOBA
   *> \verbatim
   *>          JOBA is CHARACTER*1
   *>        Specifies the level of accuracy:
   *>       = 'C': This option works well (high relative accuracy) if A = B * D,
   *>             with well-conditioned B and arbitrary diagonal matrix D.
   *>             The accuracy cannot be spoiled by COLUMN scaling. The
   *>             accuracy of the computed output depends on the condition of
   *>             B, and the procedure aims at the best theoretical accuracy.
   *>             The relative error max_{i=1:N}|d sigma_i| / sigma_i is
   *>             bounded by f(M,N)*epsilon* cond(B), independent of D.
   *>             The input matrix is preprocessed with the QRF with column
   *>             pivoting. This initial preprocessing and preconditioning by
   *>             a rank revealing QR factorization is common for all values of
   *>             JOBA. Additional actions are specified as follows:
   *>       = 'E': Computation as with 'C' with an additional estimate of the
   *>             condition number of B. It provides a realistic error bound.
   *>       = 'F': If A = D1 * C * D2 with ill-conditioned diagonal scalings
   *>             D1, D2, and well-conditioned matrix C, this option gives
   *>             higher accuracy than the 'C' option. If the structure of the
   *>             input matrix is not known, and relative accuracy is
   *>             desirable, then this option is advisable. The input matrix A
   *>             is preprocessed with QR factorization with FULL (row and
   *>             column) pivoting.
   *>       = 'G'  Computation as with 'F' with an additional estimate of the
   *>             condition number of B, where A=D*B. If A has heavily weighted
   *>             rows, then using this condition number gives too pessimistic
   *>             error bound.
   *>       = 'A': Small singular values are the noise and the matrix is treated
   *>             as numerically rank defficient. The error in the computed
   *>             singular values is bounded by f(m,n)*epsilon*||A||.
   *>             The computed SVD A = U * S * V^t 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 DSQRT(BIG), BIG=SLAMCH('O'), then JOBR issues
   *>        the licence to kill columns of A whose norm in c*A is less than
   *>        DSQRT(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 DGESVJ.
   *>       = 'R': RESTRICTED range for sigma(c*A) is [DSQRT(SFMIN), DSQRT(BIG)]
   *>             (roughly, as described above). This option is recommended.
   *>                                            ~~~~~~~~~~~~~~~~~~~~~~~~~~~
   *>        For computing the singular values in the FULL range [SFMIN,BIG]
   *>        use DGESVJ.
   *> \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^t 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^t * A. See the descriptions of WORK(6) and WORK(7).
   *>       = 'T': transpose if entropy test indicates possibly faster
   *>        convergence of Jacobi process if A^t is taken as input. If A is
   *>        replaced with A^t, 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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^t. In that case, [V] is computed
   *>                         in U as left singular vectors of A^t 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.
   *> \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 DOUBLE PRECISION 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^t. In that case, [U] is computed
   *>                         in V as right singular vectors of A^t 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.
   *> \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] WORK
   *> \verbatim
   *>          WORK is DOUBLE PRECISION array, dimension at least LWORK.
   *>          On exit, if N.GT.0 .AND. M.GT.0 (else not referenced), 
   *>          WORK(1) = SCALE = WORK(2) / WORK(1) is the scaling factor such
   *>                    that SCALE*SVA(1:N) are the computed singular values
   *>                    of A. (See the description of SVA().)
   *>          WORK(2) = See the description of WORK(1).
   *>          WORK(3) = SCONDA is an estimate for the condition number of
   *>                    column equilibrated A. (If JOBA .EQ. 'E' or 'G')
   *>                    SCONDA is an estimate of DSQRT(||(R^t * R)^(-1)||_1).
   *>                    It is computed using DPOCON. 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.
   *>
   *>          WORK(4) = an estimate of the scaled condition number of the
   *>                    triangular factor in the first QR factorization.
   *>          WORK(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.
   *>
   *>          WORK(6) = the entropy of A^t*A :: this is the Shannon entropy
   *>                    of diag(A^t*A) / Trace(A^t*A) taken as point in the
   *>                    probability simplex.
   *>          WORK(7) = the entropy of A*A^t.
   *> \endverbatim
   *>
   *> \param[in] LWORK
   *> \verbatim
   *>          LWORK is INTEGER
   *>          Length of WORK to confirm proper allocation of work space.
   *>          LWORK depends on the job:
   *>
   *>          If only SIGMA is needed ( JOBU.EQ.'N', JOBV.EQ.'N' ) and
   *>            -> .. no scaled condition estimate required (JOBE.EQ.'N'):
   *>               LWORK >= max(2*M+N,4*N+1,7). This is the minimal requirement.
   *>               ->> For optimal performance (blocked code) the optimal value
   *>               is LWORK >= max(2*M+N,3*N+(N+1)*NB,7). Here NB is the optimal
   *>               block size for DGEQP3 and DGEQRF.
   *>               In general, optimal LWORK is computed as 
   *>               LWORK >= max(2*M+N,N+LWORK(DGEQP3),N+LWORK(DGEQRF), 7).        
   *>            -> .. an estimate of the scaled condition number of A is
   *>               required (JOBA='E', 'G'). In this case, LWORK is the maximum
   *>               of the above and N*N+4*N, i.e. LWORK >= max(2*M+N,N*N+4*N,7).
   *>               ->> For optimal performance (blocked code) the optimal value 
   *>               is LWORK >= max(2*M+N,3*N+(N+1)*NB, N*N+4*N, 7).
   *>               In general, the optimal length LWORK is computed as
   *>               LWORK >= max(2*M+N,N+LWORK(DGEQP3),N+LWORK(DGEQRF), 
   *>                                                     N+N*N+LWORK(DPOCON),7).
   *>
   *>          If SIGMA and the right singular vectors are needed (JOBV.EQ.'V'),
   *>            -> the minimal requirement is LWORK >= max(2*M+N,4*N+1,7).
   *>            -> For optimal performance, LWORK >= max(2*M+N,3*N+(N+1)*NB,7),
   *>               where NB is the optimal block size for DGEQP3, DGEQRF, DGELQ,
   *>               DORMLQ. In general, the optimal length LWORK is computed as
   *>               LWORK >= max(2*M+N,N+LWORK(DGEQP3), N+LWORK(DPOCON), 
   *>                       N+LWORK(DGELQ), 2*N+LWORK(DGEQRF), N+LWORK(DORMLQ)).
   *>
   *>          If SIGMA and the left singular vectors are needed
   *>            -> the minimal requirement is LWORK >= max(2*M+N,4*N+1,7).
   *>            -> For optimal performance:
   *>               if JOBU.EQ.'U' :: LWORK >= max(2*M+N,3*N+(N+1)*NB,7),
   *>               if JOBU.EQ.'F' :: LWORK >= max(2*M+N,3*N+(N+1)*NB,N+M*NB,7),
   *>               where NB is the optimal block size for DGEQP3, DGEQRF, DORMQR.
   *>               In general, the optimal length LWORK is computed as
   *>               LWORK >= max(2*M+N,N+LWORK(DGEQP3),N+LWORK(DPOCON),
   *>                        2*N+LWORK(DGEQRF), N+LWORK(DORMQR)). 
   *>               Here LWORK(DORMQR) equals N*NB (for JOBU.EQ.'U') or 
   *>               M*NB (for JOBU.EQ.'F').
   *>               
   *>          If the full SVD is needed: (JOBU.EQ.'U' or JOBU.EQ.'F') and 
   *>            -> if JOBV.EQ.'V'  
   *>               the minimal requirement is LWORK >= max(2*M+N,6*N+2*N*N). 
   *>            -> if JOBV.EQ.'J' the minimal requirement is 
   *>               LWORK >= max(2*M+N, 4*N+N*N,2*N+N*N+6).
   *>            -> For optimal performance, LWORK should be additionally
   *>               larger than N+M*NB, where NB is the optimal block size
   *>               for DORMQR.
   *> \endverbatim
   *>
   *> \param[out] IWORK
   *> \verbatim
   *>          IWORK is INTEGER array, dimension M+3*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.
   *> \endverbatim
   *>
   *> \param[out] INFO
   *> \verbatim
   *>          INFO is INTEGER
   *>           < 0  : if INFO = -i, then the i-th argument had an illegal value.
   *>           = 0 :  successfull exit;
   *>           > 0 :  DGEJSV  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 November 2015
   *
   *> \ingroup doubleGEsing
   *
   *> \par Further Details:
   *  =====================
   *>
   *> \verbatim
   *>
   *>  DGEJSV implements a preconditioned Jacobi SVD algorithm. It uses DGEQP3,
   *>  DGEQRF, and DGELQF 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 DGEJSV 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 (DGEJSV) 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 (DGESVJ) is
   *>  left to the implementer on a particular machine.
   *>     The rank revealing QR factorization (in this code: DGEQP3) should be
   *>  implemented as in [3]. We have a new version of DGEQP3 under development
   *>  that is more robust than the current one in LAPACK, with a cleaner cut in
   *>  rank defficient cases. It will be available in the SIGMA library [4].
   *>  If M is much larger than N, it is obvious that the inital QRF with
   *>  column pivoting can be preprocessed by the QRF without pivoting. That
   *>  well known trick is not used in DGEJSV 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 DGEJSV 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 DGEJSV( JOBA, JOBU, JOBV, JOBR, JOBT, JOBP,
        $                   M, N, A, LDA, SVA, U, LDU, V, LDV,
        $                   WORK, LWORK, IWORK, INFO )
 *  *
   *  -- LAPACK computational routine (version 3.6.0) --
 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --  *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--  *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
 *  *     November 2015
 * This routine is also part of SIGMA (version 1.23, October 23. 2008.)  
 * SIGMA is a library of algorithms for highly accurate algorithms for  
 * computation of SVD, PSVD, QSVD, (H,K)-SVD, and for solution of the  
 * eigenvalue problems Hx = lambda M x, H M x = lambda x with H, M > 0.  
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       IMPLICIT    NONE        IMPLICIT    NONE
Line 22 Line 486
 *     ..  *     ..
 *     .. Array Arguments ..  *     .. Array Arguments ..
       DOUBLE PRECISION A( LDA, * ), SVA( N ), U( LDU, * ), V( LDV, * ),        DOUBLE PRECISION A( LDA, * ), SVA( N ), U( LDU, * ), V( LDV, * ),
      &            WORK( LWORK )       $            WORK( LWORK )
       INTEGER     IWORK( * )        INTEGER     IWORK( * )
       CHARACTER*1 JOBA, JOBP, JOBR, JOBT, JOBU, JOBV        CHARACTER*1 JOBA, JOBP, JOBR, JOBT, JOBU, JOBV
 *     ..  *     ..
 *  *
 *  Purpose  *  ===========================================================================
 *  =======  
 *  
 *  DGEJSV computes the singular value decomposition (SVD) of a real M-by-N  
 *  matrix [A], where M >= N. The SVD of [A] is written as  
 *  
 *               [A] = [U] * [SIGMA] * [V]^t,  
 *  
 *  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) orthonormal matrix, and  
 *  [V] is an N-by-N orthogonal 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.  
 *  
 *  Arguments  
 *  =========  
 *  
 *  JOBA    (input) CHARACTER*1  
 *        Specifies the level of accuracy:  
 *       = 'C': This option works well (high relative accuracy) if A = B * D,  
 *             with well-conditioned B and arbitrary diagonal matrix D.  
 *             The accuracy cannot be spoiled by COLUMN scaling. The  
 *             accuracy of the computed output depends on the condition of  
 *             B, and the procedure aims at the best theoretical accuracy.  
 *             The relative error max_{i=1:N}|d sigma_i| / sigma_i is  
 *             bounded by f(M,N)*epsilon* cond(B), independent of D.  
 *             The input matrix is preprocessed with the QRF with column  
 *             pivoting. This initial preprocessing and preconditioning by  
 *             a rank revealing QR factorization is common for all values of  
 *             JOBA. Additional actions are specified as follows:  
 *       = 'E': Computation as with 'C' with an additional estimate of the  
 *             condition number of B. It provides a realistic error bound.  
 *       = 'F': If A = D1 * C * D2 with ill-conditioned diagonal scalings  
 *             D1, D2, and well-conditioned matrix C, this option gives  
 *             higher accuracy than the 'C' option. If the structure of the  
 *             input matrix is not known, and relative accuracy is  
 *             desirable, then this option is advisable. The input matrix A  
 *             is preprocessed with QR factorization with FULL (row and  
 *             column) pivoting.  
 *       = 'G'  Computation as with 'F' with an additional estimate of the  
 *             condition number of B, where A=D*B. If A has heavily weighted  
 *             rows, then using this condition number gives too pessimistic  
 *             error bound.  
 *       = 'A': Small singular values are the noise and the matrix is treated  
 *             as numerically rank defficient. The error in the computed  
 *             singular values is bounded by f(m,n)*epsilon*||A||.  
 *             The computed SVD A = U * S * V^t 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    (input) 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    (input) 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.  
 *  
 *  JOBR    (input) 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 DSQRT(BIG), BIG=SLAMCH('O'), then JOBR issues  
 *        the licence to kill columns of A whose norm in c*A is less than  
 *        DSQRT(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 DGESVJ.  
 *       = 'R': RESTRICTED range for sigma(c*A) is [DSQRT(SFMIN), DSQRT(BIG)]  
 *             (roughly, as described above). This option is recommended.  
 *                                            ~~~~~~~~~~~~~~~~~~~~~~~~~~~  
 *        For computing the singular values in the FULL range [SFMIN,BIG]  
 *        use DGESVJ.  
 *  
 *  JOBT    (input) CHARACTER*1  
 *        If the matrix is square then the procedure may determine to use  
 *        transposed A if A^t 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^t * A. See the descriptions of WORK(6) and WORK(7).  
 *       = 'T': transpose if entropy test indicates possibly faster  
 *        convergence of Jacobi process if A^t is taken as input. If A is  
 *        replaced with A^t, 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.  
 *  
 *  JOBP    (input) 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       (input) INTEGER  
 *         The number of rows of the input matrix A.  M >= 0.  
 *  
 *  N       (input) INTEGER  
 *         The number of columns of the input matrix A. M >= N >= 0.  
 *  
 *  A       (input/workspace) DOUBLE PRECISION array, dimension (LDA,N)  
 *          On entry, the M-by-N matrix A.  
 *  
 *  LDA     (input) INTEGER  
 *          The leading dimension of the array A.  LDA >= max(1,M).  
 *  
 *  SVA     (workspace/output) 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.  
 *  
 *  U       (workspace/output) DOUBLE PRECISION 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^t. In that case, [V] is computed  
 *                         in U as left singular vectors of A^t 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.  
 *  
 * LDU      (input) INTEGER  
 *          The leading dimension of the array U,  LDU >= 1.  
 *          IF  JOBU = 'U' or 'F' or 'W',  then LDU >= M.  
 *  
 *  V       (workspace/output) DOUBLE PRECISION 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^t. In that case, [U] is computed  
 *                         in V as right singular vectors of A^t 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.  
 *  
 *  LDV     (input) INTEGER  
 *          The leading dimension of the array V,  LDV >= 1.  
 *          If JOBV = 'V' or 'J' or 'W', then LDV >= N.  
 *  
 *  WORK    (workspace/output) DOUBLE PRECISION array, dimension at least LWORK.  
 *          On exit,  
 *          WORK(1) = SCALE = WORK(2) / WORK(1) is the scaling factor such  
 *                    that SCALE*SVA(1:N) are the computed singular values  
 *                    of A. (See the description of SVA().)  
 *          WORK(2) = See the description of WORK(1).  
 *          WORK(3) = SCONDA is an estimate for the condition number of  
 *                    column equilibrated A. (If JOBA .EQ. 'E' or 'G')  
 *                    SCONDA is an estimate of DSQRT(||(R^t * R)^(-1)||_1).  
 *                    It is computed using DPOCON. 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.  
 *  
 *          WORK(4) = an estimate of the scaled condition number of the  
 *                    triangular factor in the first QR factorization.  
 *          WORK(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.  
 *  
 *          WORK(6) = the entropy of A^t*A :: this is the Shannon entropy  
 *                    of diag(A^t*A) / Trace(A^t*A) taken as point in the  
 *                    probability simplex.  
 *          WORK(7) = the entropy of A*A^t.  
 *  
 *  LWORK   (input) INTEGER  
 *          Length of WORK to confirm proper allocation of work space.  
 *          LWORK depends on the job:  
 *  
 *          If only SIGMA is needed ( JOBU.EQ.'N', JOBV.EQ.'N' ) and  
 *            -> .. no scaled condition estimate required ( JOBE.EQ.'N'):  
 *               LWORK >= max(2*M+N,4*N+1,7). This is the minimal requirement.  
 *               For optimal performance (blocked code) the optimal value  
 *               is LWORK >= max(2*M+N,3*N+(N+1)*NB,7). Here NB is the optimal  
 *               block size for xGEQP3/xGEQRF.  
 *            -> .. an estimate of the scaled condition number of A is  
 *               required (JOBA='E', 'G'). In this case, LWORK is the maximum  
 *               of the above and N*N+4*N, i.e. LWORK >= max(2*M+N,N*N+4N,7).  
 *  
 *          If SIGMA and the right singular vectors are needed (JOBV.EQ.'V'),  
 *            -> the minimal requirement is LWORK >= max(2*N+M,7).  
 *            -> For optimal performance, LWORK >= max(2*N+M,2*N+N*NB,7),  
 *               where NB is the optimal block size.  
 *  
 *          If SIGMA and the left singular vectors are needed  
 *            -> the minimal requirement is LWORK >= max(2*N+M,7).  
 *            -> For optimal performance, LWORK >= max(2*N+M,2*N+N*NB,7),  
 *               where NB is the optimal block size.  
 *  
 *          If full SVD is needed ( JOBU.EQ.'U' or 'F', JOBV.EQ.'V' ) and  
 *            -> .. the singular vectors are computed without explicit  
 *               accumulation of the Jacobi rotations, LWORK >= 6*N+2*N*N  
 *            -> .. in the iterative part, the Jacobi rotations are  
 *               explicitly accumulated (option, see the description of JOBV),  
 *               then the minimal requirement is LWORK >= max(M+3*N+N*N,7).  
 *               For better performance, if NB is the optimal block size,  
 *               LWORK >= max(3*N+N*N+M,3*N+N*N+N*NB,7).  
 *  
 *  IWORK   (workspace/output) INTEGER array, dimension M+3*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.  
 *  
 *  INFO    (output) INTEGER  
 *           < 0  : if INFO = -i, then the i-th argument had an illegal value.  
 *           = 0 :  successfull exit;  
 *           > 0 :  DGEJSV  did not converge in the maximal allowed number  
 *                  of sweeps. The computed values may be inaccurate.  
 *  
 *  Further Details  
 *  ===============  
 *  
 *  DGEJSV implements a preconditioned Jacobi SVD algorithm. It uses SGEQP3,  
 *  SGEQRF, and SGELQF 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 DGEJSV 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 (DGEJSV) 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 (DGESVJ) is  
 *  left to the implementer on a particular machine.  
 *     The rank revealing QR factorization (in this code: SGEQP3) should be  
 *  implemented as in [3]. We have a new version of SGEQP3 under development  
 *  that is more robust than the current one in LAPACK, with a cleaner cut in  
 *  rank defficient cases. It will be available in the SIGMA library [4].  
 *  If M is much larger than N, it is obvious that the inital QRF with  
 *  column pivoting can be preprocessed by the QRF without pivoting. That  
 *  well known trick is not used in DGEJSV 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 DGEJSV uses only the simplest, naive data movement.  
 *  
 *  Contributors  
 *  
 *  Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)  
 *  
 *  References  
 *  
 * [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.  
 *  
 *  Bugs, examples and comments  
 *   
 *  Please report all bugs and send interesting examples and/or comments to  
 *  drmac@math.hr. Thank you.  
 *  
 * ==========================================================================  
 *  *
 *     .. Local Parameters ..  *     .. Local Parameters ..
       DOUBLE PRECISION   ZERO,  ONE        DOUBLE PRECISION   ZERO,  ONE
Line 372 Line 499
 *     ..  *     ..
 *     .. Local Scalars ..  *     .. Local Scalars ..
       DOUBLE PRECISION AAPP, AAQQ, AATMAX, AATMIN, BIG, BIG1, COND_OK,        DOUBLE PRECISION AAPP, AAQQ, AATMAX, AATMIN, BIG, BIG1, COND_OK,
      &        CONDR1, CONDR2, ENTRA,  ENTRAT, EPSLN,  MAXPRJ, SCALEM,       $        CONDR1, CONDR2, ENTRA,  ENTRAT, EPSLN,  MAXPRJ, SCALEM,
      &        SCONDA, SFMIN,  SMALL,  TEMP1,  USCAL1, USCAL2, XSC       $        SCONDA, SFMIN,  SMALL,  TEMP1,  USCAL1, USCAL2, XSC
       INTEGER IERR,   N1,     NR,     NUMRANK,        p, q,   WARNING        INTEGER IERR,   N1,     NR,     NUMRANK,        p, q,   WARNING
       LOGICAL ALMORT, DEFR,   ERREST, GOSCAL, JRACC,  KILL,   LSVEC,        LOGICAL ALMORT, DEFR,   ERREST, GOSCAL, JRACC,  KILL,   LSVEC,
      &        L2ABER, L2KILL, L2PERT, L2RANK, L2TRAN,       $        L2ABER, L2KILL, L2PERT, L2RANK, L2TRAN,
      &        NOSCAL, ROWPIV, RSVEC,  TRANSP       $        NOSCAL, ROWPIV, RSVEC,  TRANSP
 *     ..  *     ..
 *     .. Intrinsic Functions ..  *     .. Intrinsic Functions ..
       INTRINSIC DABS,  DLOG, DMAX1, DMIN1, DBLE,        INTRINSIC DABS, DLOG, MAX, MIN, DBLE, IDNINT, DSIGN, DSQRT
      &          MAX0, MIN0, IDNINT,  DSIGN,  DSQRT  
 *     ..  *     ..
 *     .. External Functions ..  *     .. External Functions ..
       DOUBLE PRECISION  DLAMCH, DNRM2        DOUBLE PRECISION  DLAMCH, DNRM2
Line 391 Line 517
 *     ..  *     ..
 *     .. External Subroutines ..  *     .. External Subroutines ..
       EXTERNAL  DCOPY,  DGELQF, DGEQP3, DGEQRF, DLACPY, DLASCL,        EXTERNAL  DCOPY,  DGELQF, DGEQP3, DGEQRF, DLACPY, DLASCL,
      &          DLASET, DLASSQ, DLASWP, DORGQR, DORMLQ,       $          DLASET, DLASSQ, DLASWP, DORGQR, DORMLQ,
      &          DORMQR, DPOCON, DSCAL,  DSWAP,  DTRSM,  XERBLA       $          DORMQR, DPOCON, DSCAL,  DSWAP,  DTRSM,  XERBLA
 *  *
       EXTERNAL  DGESVJ        EXTERNAL  DGESVJ
 *     ..  *     ..
Line 412 Line 538
       L2PERT = LSAME( JOBP, 'P' )        L2PERT = LSAME( JOBP, 'P' )
 *  *
       IF ( .NOT.(ROWPIV .OR. L2RANK .OR. L2ABER .OR.        IF ( .NOT.(ROWPIV .OR. L2RANK .OR. L2ABER .OR.
      &     ERREST .OR. LSAME( JOBA, 'C' ) )) THEN       $     ERREST .OR. LSAME( JOBA, 'C' ) )) THEN
          INFO = - 1           INFO = - 1
       ELSE IF ( .NOT.( LSVEC  .OR. LSAME( JOBU, 'N' ) .OR.        ELSE IF ( .NOT.( LSVEC  .OR. LSAME( JOBU, 'N' ) .OR.
      &                             LSAME( JOBU, 'W' )) ) THEN       $                             LSAME( JOBU, 'W' )) ) THEN
          INFO = - 2           INFO = - 2
       ELSE IF ( .NOT.( RSVEC .OR. LSAME( JOBV, 'N' ) .OR.        ELSE IF ( .NOT.( RSVEC .OR. LSAME( JOBV, 'N' ) .OR.
      &   LSAME( JOBV, 'W' )) .OR. ( JRACC .AND. (.NOT.LSVEC) ) ) THEN       $   LSAME( JOBV, 'W' )) .OR. ( JRACC .AND. (.NOT.LSVEC) ) ) THEN
          INFO = - 3           INFO = - 3
       ELSE IF ( .NOT. ( L2KILL .OR. DEFR ) )    THEN        ELSE IF ( .NOT. ( L2KILL .OR. DEFR ) )    THEN
          INFO = - 4           INFO = - 4
Line 437 Line 563
       ELSE IF ( RSVEC .AND. ( LDV .LT. N ) ) THEN        ELSE IF ( RSVEC .AND. ( LDV .LT. N ) ) THEN
          INFO = - 14           INFO = - 14
       ELSE IF ( (.NOT.(LSVEC .OR. RSVEC .OR. ERREST).AND.        ELSE IF ( (.NOT.(LSVEC .OR. RSVEC .OR. ERREST).AND.
      &                           (LWORK .LT. MAX0(7,4*N+1,2*M+N))) .OR.       &                           (LWORK .LT. MAX(7,4*N+1,2*M+N))) .OR.
      & (.NOT.(LSVEC .OR. LSVEC) .AND. ERREST .AND.       & (.NOT.(LSVEC .OR. RSVEC) .AND. ERREST .AND.
      &                         (LWORK .LT. MAX0(7,4*N+N*N,2*M+N))) .OR.       &                         (LWORK .LT. MAX(7,4*N+N*N,2*M+N))) .OR.
      & (LSVEC .AND. (.NOT.RSVEC) .AND. (LWORK .LT. MAX0(7,2*N+M))) .OR.       & (LSVEC .AND. (.NOT.RSVEC) .AND. (LWORK .LT. MAX(7,2*M+N,4*N+1)))
      & (RSVEC .AND. (.NOT.LSVEC) .AND. (LWORK .LT. MAX0(7,2*N+M))) .OR.       & .OR.
      & (LSVEC .AND. RSVEC .AND. .NOT.JRACC .AND. (LWORK.LT.6*N+2*N*N))       & (RSVEC .AND. (.NOT.LSVEC) .AND. (LWORK .LT. MAX(7,2*M+N,4*N+1)))
      & .OR. (LSVEC.AND.RSVEC.AND.JRACC.AND.LWORK.LT.MAX0(7,M+3*N+N*N)))       & .OR.
        & (LSVEC .AND. RSVEC .AND. (.NOT.JRACC) .AND. 
        &                          (LWORK.LT.MAX(2*M+N,6*N+2*N*N)))
        & .OR. (LSVEC .AND. RSVEC .AND. JRACC .AND.
        &                          LWORK.LT.MAX(2*M+N,4*N+N*N,2*N+N*N+6)))
      &   THEN       &   THEN
          INFO = - 17           INFO = - 17
       ELSE        ELSE
Line 454 Line 584
       IF ( INFO .NE. 0 ) THEN        IF ( INFO .NE. 0 ) THEN
 *       #:(  *       #:(
          CALL XERBLA( 'DGEJSV', - INFO )           CALL XERBLA( 'DGEJSV', - INFO )
            RETURN
       END IF        END IF
 *  *
 *     Quick return for void matrix (Y3K safe)  *     Quick return for void matrix (Y3K safe)
Line 471 Line 602
 *  *
 *!    NOTE: Make sure DLAMCH() does not fail on the target architecture.  *!    NOTE: Make sure DLAMCH() does not fail on the target architecture.
 *  *
   
       EPSLN = DLAMCH('Epsilon')        EPSLN = DLAMCH('Epsilon')
       SFMIN = DLAMCH('SafeMinimum')        SFMIN = DLAMCH('SafeMinimum')
       SMALL = SFMIN / EPSLN        SMALL = SFMIN / EPSLN
Line 514 Line 644
       AAPP = ZERO        AAPP = ZERO
       AAQQ = BIG        AAQQ = BIG
       DO 4781 p = 1, N        DO 4781 p = 1, N
          AAPP = DMAX1( AAPP, SVA(p) )           AAPP = MAX( AAPP, SVA(p) )
          IF ( SVA(p) .NE. ZERO ) AAQQ = DMIN1( AAQQ, SVA(p) )           IF ( SVA(p) .NE. ZERO ) AAQQ = MIN( AAQQ, SVA(p) )
  4781 CONTINUE   4781 CONTINUE
 *  *
 *     Quick return for zero M x N matrix  *     Quick return for zero M x N matrix
Line 536 Line 666
          END IF           END IF
          IWORK(1) = 0           IWORK(1) = 0
          IWORK(2) = 0           IWORK(2) = 0
            IWORK(3) = 0
          RETURN           RETURN
       END IF        END IF
 *  *
Line 618 Line 749
 *              in one pass through the vector  *              in one pass through the vector
                WORK(M+N+p)  = XSC * SCALEM                 WORK(M+N+p)  = XSC * SCALEM
                WORK(N+p)    = XSC * (SCALEM*DSQRT(TEMP1))                 WORK(N+p)    = XSC * (SCALEM*DSQRT(TEMP1))
                AATMAX = DMAX1( AATMAX, WORK(N+p) )                 AATMAX = MAX( AATMAX, WORK(N+p) )
                IF (WORK(N+p) .NE. ZERO) AATMIN = DMIN1(AATMIN,WORK(N+p))                 IF (WORK(N+p) .NE. ZERO) AATMIN = MIN(AATMIN,WORK(N+p))
  1950       CONTINUE   1950       CONTINUE
          ELSE           ELSE
             DO 1904 p = 1, M              DO 1904 p = 1, M
                WORK(M+N+p) = SCALEM*DABS( A(p,IDAMAX(N,A(p,1),LDA)) )                 WORK(M+N+p) = SCALEM*DABS( A(p,IDAMAX(N,A(p,1),LDA)) )
                AATMAX = DMAX1( AATMAX, WORK(M+N+p) )                 AATMAX = MAX( AATMAX, WORK(M+N+p) )
                AATMIN = DMIN1( AATMIN, WORK(M+N+p) )                 AATMIN = MIN( AATMIN, WORK(M+N+p) )
  1904       CONTINUE   1904       CONTINUE
          END IF           END IF
 *  *
Line 834 Line 965
          TEMP1 = DSQRT(SFMIN)           TEMP1 = DSQRT(SFMIN)
          DO 3401 p = 2, N           DO 3401 p = 2, N
             IF ( ( DABS(A(p,p)) .LT. (EPSLN*DABS(A(p-1,p-1))) ) .OR.              IF ( ( DABS(A(p,p)) .LT. (EPSLN*DABS(A(p-1,p-1))) ) .OR.
      &           ( DABS(A(p,p)) .LT. SMALL ) .OR.       $           ( DABS(A(p,p)) .LT. SMALL ) .OR.
      &           ( L2KILL .AND. (DABS(A(p,p)) .LT. TEMP1) ) ) GO TO 3402       $           ( L2KILL .AND. (DABS(A(p,p)) .LT. TEMP1) ) ) GO TO 3402
             NR = NR + 1              NR = NR + 1
  3401    CONTINUE   3401    CONTINUE
  3402    CONTINUE   3402    CONTINUE
Line 851 Line 982
          TEMP1  = DSQRT(SFMIN)           TEMP1  = DSQRT(SFMIN)
          DO 3301 p = 2, N           DO 3301 p = 2, N
             IF ( ( DABS(A(p,p)) .LT. SMALL ) .OR.              IF ( ( DABS(A(p,p)) .LT. SMALL ) .OR.
      &          ( L2KILL .AND. (DABS(A(p,p)) .LT. TEMP1) ) ) GO TO 3302       $          ( L2KILL .AND. (DABS(A(p,p)) .LT. TEMP1) ) ) GO TO 3302
             NR = NR + 1              NR = NR + 1
  3301    CONTINUE   3301    CONTINUE
  3302    CONTINUE   3302    CONTINUE
Line 863 Line 994
          MAXPRJ = ONE           MAXPRJ = ONE
          DO 3051 p = 2, N           DO 3051 p = 2, N
             TEMP1  = DABS(A(p,p)) / SVA(IWORK(p))              TEMP1  = DABS(A(p,p)) / SVA(IWORK(p))
             MAXPRJ = DMIN1( MAXPRJ, TEMP1 )              MAXPRJ = MIN( MAXPRJ, TEMP1 )
  3051    CONTINUE   3051    CONTINUE
          IF ( MAXPRJ**2 .GE. ONE - DBLE(N)*EPSLN ) ALMORT = .TRUE.           IF ( MAXPRJ**2 .GE. ONE - DBLE(N)*EPSLN ) ALMORT = .TRUE.
       END IF        END IF
Line 883 Line 1014
                   CALL DSCAL( p, ONE/TEMP1, V(1,p), 1 )                    CALL DSCAL( p, ONE/TEMP1, V(1,p), 1 )
  3053          CONTINUE   3053          CONTINUE
                CALL DPOCON( 'U', N, V, LDV, ONE, TEMP1,                 CALL DPOCON( 'U', N, V, LDV, ONE, TEMP1,
      &              WORK(N+1), IWORK(2*N+M+1), IERR )       $              WORK(N+1), IWORK(2*N+M+1), IERR )
             ELSE IF ( LSVEC ) THEN              ELSE IF ( LSVEC ) THEN
 *              .. U is available as workspace  *              .. U is available as workspace
                CALL DLACPY( 'U', N, N, A, LDA, U, LDU )                 CALL DLACPY( 'U', N, N, A, LDA, U, LDU )
Line 892 Line 1023
                   CALL DSCAL( p, ONE/TEMP1, U(1,p), 1 )                    CALL DSCAL( p, ONE/TEMP1, U(1,p), 1 )
  3054          CONTINUE   3054          CONTINUE
                CALL DPOCON( 'U', N, U, LDU, ONE, TEMP1,                 CALL DPOCON( 'U', N, U, LDU, ONE, TEMP1,
      &              WORK(N+1), IWORK(2*N+M+1), IERR )       $              WORK(N+1), IWORK(2*N+M+1), IERR )
             ELSE              ELSE
                CALL DLACPY( 'U', N, N, A, LDA, WORK(N+1), N )                 CALL DLACPY( 'U', N, N, A, LDA, WORK(N+1), N )
                DO 3052 p = 1, N                 DO 3052 p = 1, N
Line 901 Line 1032
  3052          CONTINUE   3052          CONTINUE
 *           .. the columns of R are scaled to have unit Euclidean lengths.  *           .. the columns of R are scaled to have unit Euclidean lengths.
                CALL DPOCON( 'U', N, WORK(N+1), N, ONE, TEMP1,                 CALL DPOCON( 'U', N, WORK(N+1), N, ONE, TEMP1,
      &              WORK(N+N*N+1), IWORK(2*N+M+1), IERR )       $              WORK(N+N*N+1), IWORK(2*N+M+1), IERR )
             END IF              END IF
             SCONDA = ONE / DSQRT(TEMP1)              SCONDA = ONE / DSQRT(TEMP1)
 *           SCONDA is an estimate of DSQRT(||(R^t * R)^(-1)||_1).  *           SCONDA is an estimate of DSQRT(||(R^t * R)^(-1)||_1).
Line 916 Line 1047
 *  *
 *     Phase 3:  *     Phase 3:
 *  *
   
       IF ( .NOT. ( RSVEC .OR. LSVEC ) ) THEN        IF ( .NOT. ( RSVEC .OR. LSVEC ) ) THEN
 *  *
 *         Singular Values only  *         Singular Values only
 *  *
 *         .. transpose A(1:NR,1:N)  *         .. transpose A(1:NR,1:N)
          DO 1946 p = 1, MIN0( N-1, NR )           DO 1946 p = 1, MIN( N-1, NR )
             CALL DCOPY( N-p, A(p,p+1), LDA, A(p+1,p), 1 )              CALL DCOPY( N-p, A(p,p+1), LDA, A(p+1,p), 1 )
  1946    CONTINUE   1946    CONTINUE
 *  *
Line 947 Line 1077
                   TEMP1 = XSC*DABS(A(q,q))                    TEMP1 = XSC*DABS(A(q,q))
                   DO 4949 p = 1, N                    DO 4949 p = 1, N
                      IF ( ( (p.GT.q) .AND. (DABS(A(p,q)).LE.TEMP1) )                       IF ( ( (p.GT.q) .AND. (DABS(A(p,q)).LE.TEMP1) )
      &                    .OR. ( p .LT. q ) )       $                    .OR. ( p .LT. q ) )
      &                     A(p,q) = DSIGN( TEMP1, A(p,q) )       $                     A(p,q) = DSIGN( TEMP1, A(p,q) )
  4949             CONTINUE   4949             CONTINUE
  4947          CONTINUE   4947          CONTINUE
             ELSE              ELSE
Line 977 Line 1107
                   TEMP1 = XSC*DABS(A(q,q))                    TEMP1 = XSC*DABS(A(q,q))
                   DO 1949 p = 1, NR                    DO 1949 p = 1, NR
                      IF ( ( (p.GT.q) .AND. (DABS(A(p,q)).LE.TEMP1) )                       IF ( ( (p.GT.q) .AND. (DABS(A(p,q)).LE.TEMP1) )
      &                       .OR. ( p .LT. q ) )       $                       .OR. ( p .LT. q ) )
      &                   A(p,q) = DSIGN( TEMP1, A(p,q) )       $                   A(p,q) = DSIGN( TEMP1, A(p,q) )
  1949             CONTINUE   1949             CONTINUE
  1947          CONTINUE   1947          CONTINUE
             ELSE              ELSE
Line 990 Line 1120
 *           the part which destroys triangular form (confusing?!))  *           the part which destroys triangular form (confusing?!))
 *  *
             CALL DGESVJ( 'L', 'NoU', 'NoV', NR, NR, A, LDA, SVA,              CALL DGESVJ( 'L', 'NoU', 'NoV', NR, NR, A, LDA, SVA,
      &                      N, V, LDV, WORK, LWORK, INFO )       $                      N, V, LDV, WORK, LWORK, INFO )
 *  *
             SCALEM  = WORK(1)              SCALEM  = WORK(1)
             NUMRANK = IDNINT(WORK(2))              NUMRANK = IDNINT(WORK(2))
Line 1009 Line 1139
             CALL DLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, V(1,2), LDV )              CALL DLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, V(1,2), LDV )
 *  *
             CALL DGESVJ( 'L','U','N', N, NR, V,LDV, SVA, NR, A,LDA,              CALL DGESVJ( 'L','U','N', N, NR, V,LDV, SVA, NR, A,LDA,
      &                  WORK, LWORK, INFO )       $                  WORK, LWORK, INFO )
             SCALEM  = WORK(1)              SCALEM  = WORK(1)
             NUMRANK = IDNINT(WORK(2))              NUMRANK = IDNINT(WORK(2))
   
Line 1023 Line 1153
             CALL DLACPY( 'Lower', NR, NR, A, LDA, V, LDV )              CALL DLACPY( 'Lower', NR, NR, A, LDA, V, LDV )
             CALL DLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, V(1,2), LDV )              CALL DLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, V(1,2), LDV )
             CALL DGEQRF( NR, NR, V, LDV, WORK(N+1), WORK(2*N+1),              CALL DGEQRF( NR, NR, V, LDV, WORK(N+1), WORK(2*N+1),
      &                   LWORK-2*N, IERR )       $                   LWORK-2*N, IERR )
             DO 8998 p = 1, NR              DO 8998 p = 1, NR
                CALL DCOPY( NR-p+1, V(p,p), LDV, V(p,p), 1 )                 CALL DCOPY( NR-p+1, V(p,p), LDV, V(p,p), 1 )
  8998       CONTINUE   8998       CONTINUE
             CALL DLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, V(1,2), LDV )              CALL DLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, V(1,2), LDV )
 *  *
             CALL DGESVJ( 'Lower', 'U','N', NR, NR, V,LDV, SVA, NR, U,              CALL DGESVJ( 'Lower', 'U','N', NR, NR, V,LDV, SVA, NR, U,
      &                  LDU, WORK(N+1), LWORK, INFO )       $                  LDU, WORK(N+1), LWORK, INFO )
             SCALEM  = WORK(N+1)              SCALEM  = WORK(N+1)
             NUMRANK = IDNINT(WORK(N+2))              NUMRANK = IDNINT(WORK(N+2))
             IF ( NR .LT. N ) THEN              IF ( NR .LT. N ) THEN
Line 1040 Line 1170
             END IF              END IF
 *  *
          CALL DORMLQ( 'Left', 'Transpose', N, N, NR, A, LDA, WORK,           CALL DORMLQ( 'Left', 'Transpose', N, N, NR, A, LDA, WORK,
      &               V, LDV, WORK(N+1), LWORK-N, IERR )       $               V, LDV, WORK(N+1), LWORK-N, IERR )
 *  *
          END IF           END IF
 *  *
Line 1065 Line 1195
          CALL DLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, U(1,2), LDU )           CALL DLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, U(1,2), LDU )
 *  *
          CALL DGEQRF( N, NR, U, LDU, WORK(N+1), WORK(2*N+1),           CALL DGEQRF( N, NR, U, LDU, WORK(N+1), WORK(2*N+1),
      &              LWORK-2*N, IERR )       $              LWORK-2*N, IERR )
 *  *
          DO 1967 p = 1, NR - 1           DO 1967 p = 1, NR - 1
             CALL DCOPY( NR-p, U(p,p+1), LDU, U(p+1,p), 1 )              CALL DCOPY( NR-p, U(p,p+1), LDU, U(p+1,p), 1 )
Line 1073 Line 1203
          CALL DLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, U(1,2), LDU )           CALL DLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, U(1,2), LDU )
 *  *
          CALL DGESVJ( 'Lower', 'U', 'N', NR,NR, U, LDU, SVA, NR, A,           CALL DGESVJ( 'Lower', 'U', 'N', NR,NR, U, LDU, SVA, NR, A,
      &        LDA, WORK(N+1), LWORK-N, INFO )       $        LDA, WORK(N+1), LWORK-N, INFO )
          SCALEM  = WORK(N+1)           SCALEM  = WORK(N+1)
          NUMRANK = IDNINT(WORK(N+2))           NUMRANK = IDNINT(WORK(N+2))
 *  *
Line 1086 Line 1216
          END IF           END IF
 *  *
          CALL DORMQR( 'Left', 'No Tr', M, N1, N, A, LDA, WORK, U,           CALL DORMQR( 'Left', 'No Tr', M, N1, N, A, LDA, WORK, U,
      &               LDU, WORK(N+1), LWORK-N, IERR )       $               LDU, WORK(N+1), LWORK-N, IERR )
 *  *
          IF ( ROWPIV )           IF ( ROWPIV )
      &       CALL DLASWP( N1, U, LDU, 1, M-1, IWORK(2*N+1), -1 )       $       CALL DLASWP( N1, U, LDU, 1, M-1, IWORK(2*N+1), -1 )
 *  *
          DO 1974 p = 1, N1           DO 1974 p = 1, N1
             XSC = ONE / DNRM2( M, U(1,p), 1 )              XSC = ONE / DNRM2( M, U(1,p), 1 )
Line 1137 Line 1267
                   TEMP1 = XSC*DABS( V(q,q) )                    TEMP1 = XSC*DABS( V(q,q) )
                   DO 2968 p = 1, N                    DO 2968 p = 1, N
                      IF ( ( p .GT. q ) .AND. ( DABS(V(p,q)) .LE. TEMP1 )                       IF ( ( p .GT. q ) .AND. ( DABS(V(p,q)) .LE. TEMP1 )
      &                   .OR. ( p .LT. q ) )       $                   .OR. ( p .LT. q ) )
      &                   V(p,q) = DSIGN( TEMP1, V(p,q) )       $                   V(p,q) = DSIGN( TEMP1, V(p,q) )
                      IF ( p. LT. q ) V(p,q) = - V(p,q)                       IF ( p .LT. q ) V(p,q) = - V(p,q)
  2968             CONTINUE   2968             CONTINUE
  2969          CONTINUE   2969          CONTINUE
             ELSE              ELSE
Line 1156 Line 1286
                CALL DSCAL(NR-p+1,ONE/TEMP1,WORK(2*N+(p-1)*NR+p),1)                 CALL DSCAL(NR-p+1,ONE/TEMP1,WORK(2*N+(p-1)*NR+p),1)
  3950       CONTINUE   3950       CONTINUE
             CALL DPOCON('Lower',NR,WORK(2*N+1),NR,ONE,TEMP1,              CALL DPOCON('Lower',NR,WORK(2*N+1),NR,ONE,TEMP1,
      &                   WORK(2*N+NR*NR+1),IWORK(M+2*N+1),IERR)       $                   WORK(2*N+NR*NR+1),IWORK(M+2*N+1),IERR)
             CONDR1 = ONE / DSQRT(TEMP1)              CONDR1 = ONE / DSQRT(TEMP1)
 *           .. here need a second oppinion on the condition number  *           .. here need a second oppinion on the condition number
 *           .. then assume worst case scenario  *           .. then assume worst case scenario
Line 1172 Line 1302
 *              of a lower triangular matrix.  *              of a lower triangular matrix.
 *              R1^t = Q2 * R2  *              R1^t = Q2 * R2
                CALL DGEQRF( N, NR, V, LDV, WORK(N+1), WORK(2*N+1),                 CALL DGEQRF( N, NR, V, LDV, WORK(N+1), WORK(2*N+1),
      &              LWORK-2*N, IERR )       $              LWORK-2*N, IERR )
 *  *
                IF ( L2PERT ) THEN                 IF ( L2PERT ) THEN
                   XSC = DSQRT(SMALL)/EPSLN                    XSC = DSQRT(SMALL)/EPSLN
                   DO 3959 p = 2, NR                    DO 3959 p = 2, NR
                      DO 3958 q = 1, p - 1                       DO 3958 q = 1, p - 1
                         TEMP1 = XSC * DMIN1(DABS(V(p,p)),DABS(V(q,q)))                          TEMP1 = XSC * MIN(DABS(V(p,p)),DABS(V(q,q)))
                         IF ( DABS(V(q,p)) .LE. TEMP1 )                          IF ( DABS(V(q,p)) .LE. TEMP1 )
      &                     V(q,p) = DSIGN( TEMP1, V(q,p) )       $                     V(q,p) = DSIGN( TEMP1, V(q,p) )
  3958                CONTINUE   3958                CONTINUE
  3959             CONTINUE   3959             CONTINUE
                END IF                 END IF
 *  *
                IF ( NR .NE. N )                 IF ( NR .NE. N )
        $         CALL DLACPY( 'A', N, NR, V, LDV, WORK(2*N+1), N )
 *              .. save ...  *              .. save ...
      &         CALL DLACPY( 'A', N, NR, V, LDV, WORK(2*N+1), N )  
 *  *
 *           .. this transposed copy should be better than naive  *           .. this transposed copy should be better than naive
                DO 1969 p = 1, NR - 1                 DO 1969 p = 1, NR - 1
Line 1210 Line 1340
                   IWORK(N+p) = 0                    IWORK(N+p) = 0
  3003          CONTINUE   3003          CONTINUE
                CALL DGEQP3( N, NR, V, LDV, IWORK(N+1), WORK(N+1),                 CALL DGEQP3( N, NR, V, LDV, IWORK(N+1), WORK(N+1),
      &                  WORK(2*N+1), LWORK-2*N, IERR )       $                  WORK(2*N+1), LWORK-2*N, IERR )
 **               CALL DGEQRF( N, NR, V, LDV, WORK(N+1), WORK(2*N+1),  **               CALL DGEQRF( N, NR, V, LDV, WORK(N+1), WORK(2*N+1),
 **     &              LWORK-2*N, IERR )  **     $              LWORK-2*N, IERR )
                IF ( L2PERT ) THEN                 IF ( L2PERT ) THEN
                   XSC = DSQRT(SMALL)                    XSC = DSQRT(SMALL)
                   DO 3969 p = 2, NR                    DO 3969 p = 2, NR
                      DO 3968 q = 1, p - 1                       DO 3968 q = 1, p - 1
                         TEMP1 = XSC * DMIN1(DABS(V(p,p)),DABS(V(q,q)))                          TEMP1 = XSC * MIN(DABS(V(p,p)),DABS(V(q,q)))
                         IF ( DABS(V(q,p)) .LE. TEMP1 )                          IF ( DABS(V(q,p)) .LE. TEMP1 )
      &                     V(q,p) = DSIGN( TEMP1, V(q,p) )       $                     V(q,p) = DSIGN( TEMP1, V(q,p) )
  3968                CONTINUE   3968                CONTINUE
  3969             CONTINUE   3969             CONTINUE
                END IF                 END IF
Line 1230 Line 1360
                   XSC = DSQRT(SMALL)                    XSC = DSQRT(SMALL)
                   DO 8970 p = 2, NR                    DO 8970 p = 2, NR
                      DO 8971 q = 1, p - 1                       DO 8971 q = 1, p - 1
                         TEMP1 = XSC * DMIN1(DABS(V(p,p)),DABS(V(q,q)))                          TEMP1 = XSC * MIN(DABS(V(p,p)),DABS(V(q,q)))
                         V(p,q) = - DSIGN( TEMP1, V(q,p) )                          V(p,q) = - DSIGN( TEMP1, V(q,p) )
  8971                CONTINUE   8971                CONTINUE
  8970             CONTINUE   8970             CONTINUE
Line 1239 Line 1369
                END IF                 END IF
 *              Now, compute R2 = L3 * Q3, the LQ factorization.  *              Now, compute R2 = L3 * Q3, the LQ factorization.
                CALL DGELQF( NR, NR, V, LDV, WORK(2*N+N*NR+1),                 CALL DGELQF( NR, NR, V, LDV, WORK(2*N+N*NR+1),
      &               WORK(2*N+N*NR+NR+1), LWORK-2*N-N*NR-NR, IERR )       $               WORK(2*N+N*NR+NR+1), LWORK-2*N-N*NR-NR, IERR )
 *              .. and estimate the condition number  *              .. and estimate the condition number
                CALL DLACPY( 'L',NR,NR,V,LDV,WORK(2*N+N*NR+NR+1),NR )                 CALL DLACPY( 'L',NR,NR,V,LDV,WORK(2*N+N*NR+NR+1),NR )
                DO 4950 p = 1, NR                 DO 4950 p = 1, NR
Line 1247 Line 1377
                   CALL DSCAL( p, ONE/TEMP1, WORK(2*N+N*NR+NR+p), NR )                    CALL DSCAL( p, ONE/TEMP1, WORK(2*N+N*NR+NR+p), NR )
  4950          CONTINUE   4950          CONTINUE
                CALL DPOCON( 'L',NR,WORK(2*N+N*NR+NR+1),NR,ONE,TEMP1,                 CALL DPOCON( 'L',NR,WORK(2*N+N*NR+NR+1),NR,ONE,TEMP1,
      &              WORK(2*N+N*NR+NR+NR*NR+1),IWORK(M+2*N+1),IERR )       $              WORK(2*N+N*NR+NR+NR*NR+1),IWORK(M+2*N+1),IERR )
                CONDR2 = ONE / DSQRT(TEMP1)                 CONDR2 = ONE / DSQRT(TEMP1)
 *  *
                IF ( CONDR2 .GE. COND_OK ) THEN                 IF ( CONDR2 .GE. COND_OK ) THEN
Line 1284 Line 1414
             IF ( CONDR1 .LT. COND_OK ) THEN              IF ( CONDR1 .LT. COND_OK ) THEN
 *  *
                CALL DGESVJ( 'L','U','N',NR,NR,V,LDV,SVA,NR,U,                 CALL DGESVJ( 'L','U','N',NR,NR,V,LDV,SVA,NR,U,
      &              LDU,WORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,INFO )       $              LDU,WORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,INFO )
                SCALEM  = WORK(2*N+N*NR+NR+1)                 SCALEM  = WORK(2*N+N*NR+NR+1)
                NUMRANK = IDNINT(WORK(2*N+N*NR+NR+2))                 NUMRANK = IDNINT(WORK(2*N+N*NR+NR+2))
                DO 3970 p = 1, NR                 DO 3970 p = 1, NR
Line 1294 Line 1424
   
 *        .. pick the right matrix equation and solve it  *        .. pick the right matrix equation and solve it
 *  *
                IF ( NR. EQ. N ) THEN                 IF ( NR .EQ. N ) THEN
 * :))             .. best case, R1 is inverted. The solution of this matrix  * :))             .. best case, R1 is inverted. The solution of this matrix
 *                 equation is Q2*V2 = the product of the Jacobi rotations  *                 equation is Q2*V2 = the product of the Jacobi rotations
 *                 used in DGESVJ, premultiplied with the orthogonal matrix  *                 used in DGESVJ, premultiplied with the orthogonal matrix
Line 1306 Line 1436
 *                 used in DGESVJ. The Q-factor from the second QR  *                 used in DGESVJ. The Q-factor from the second QR
 *                 factorization is then built in explicitly.  *                 factorization is then built in explicitly.
                   CALL DTRSM('L','U','T','N',NR,NR,ONE,WORK(2*N+1),                    CALL DTRSM('L','U','T','N',NR,NR,ONE,WORK(2*N+1),
      &                 N,V,LDV)       $                 N,V,LDV)
                   IF ( NR .LT. N ) THEN                    IF ( NR .LT. N ) THEN
                     CALL DLASET('A',N-NR,NR,ZERO,ZERO,V(NR+1,1),LDV)                      CALL DLASET('A',N-NR,NR,ZERO,ZERO,V(NR+1,1),LDV)
                     CALL DLASET('A',NR,N-NR,ZERO,ZERO,V(1,NR+1),LDV)                      CALL DLASET('A',NR,N-NR,ZERO,ZERO,V(1,NR+1),LDV)
                     CALL DLASET('A',N-NR,N-NR,ZERO,ONE,V(NR+1,NR+1),LDV)                      CALL DLASET('A',N-NR,N-NR,ZERO,ONE,V(NR+1,NR+1),LDV)
                   END IF                    END IF
                   CALL DORMQR('L','N',N,N,NR,WORK(2*N+1),N,WORK(N+1),                    CALL DORMQR('L','N',N,N,NR,WORK(2*N+1),N,WORK(N+1),
      &                 V,LDV,WORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR)       $                 V,LDV,WORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR)
                END IF                 END IF
 *  *
             ELSE IF ( CONDR2 .LT. COND_OK ) THEN              ELSE IF ( CONDR2 .LT. COND_OK ) THEN
Line 1325 Line 1455
 *              the lower triangular L3 from the LQ factorization of  *              the lower triangular L3 from the LQ factorization of
 *              R2=L3*Q3), pre-multiplied with the transposed Q3.  *              R2=L3*Q3), pre-multiplied with the transposed Q3.
                CALL DGESVJ( 'L', 'U', 'N', NR, NR, V, LDV, SVA, NR, U,                 CALL DGESVJ( 'L', 'U', 'N', NR, NR, V, LDV, SVA, NR, U,
      &              LDU, WORK(2*N+N*NR+NR+1), LWORK-2*N-N*NR-NR, INFO )       $              LDU, WORK(2*N+N*NR+NR+1), LWORK-2*N-N*NR-NR, INFO )
                SCALEM  = WORK(2*N+N*NR+NR+1)                 SCALEM  = WORK(2*N+N*NR+NR+1)
                NUMRANK = IDNINT(WORK(2*N+N*NR+NR+2))                 NUMRANK = IDNINT(WORK(2*N+N*NR+NR+2))
                DO 3870 p = 1, NR                 DO 3870 p = 1, NR
Line 1348 Line 1478
                   CALL DLASET( 'A',N-NR,N-NR,ZERO,ONE,V(NR+1,NR+1),LDV )                    CALL DLASET( 'A',N-NR,N-NR,ZERO,ONE,V(NR+1,NR+1),LDV )
                END IF                 END IF
                CALL DORMQR( 'L','N',N,N,NR,WORK(2*N+1),N,WORK(N+1),                 CALL DORMQR( 'L','N',N,N,NR,WORK(2*N+1),N,WORK(N+1),
      &              V,LDV,WORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR )       $              V,LDV,WORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR )
             ELSE              ELSE
 *              Last line of defense.  *              Last line of defense.
 * #:(          This is a rather pathological case: no scaled condition  * #:(          This is a rather pathological case: no scaled condition
Line 1362 Line 1492
 *              Compute the full SVD of L3 using DGESVJ with explicit  *              Compute the full SVD of L3 using DGESVJ with explicit
 *              accumulation of Jacobi rotations.  *              accumulation of Jacobi rotations.
                CALL DGESVJ( 'L', 'U', 'V', NR, NR, V, LDV, SVA, NR, U,                 CALL DGESVJ( 'L', 'U', 'V', NR, NR, V, LDV, SVA, NR, U,
      &              LDU, WORK(2*N+N*NR+NR+1), LWORK-2*N-N*NR-NR, INFO )       $              LDU, WORK(2*N+N*NR+NR+1), LWORK-2*N-N*NR-NR, INFO )
                SCALEM  = WORK(2*N+N*NR+NR+1)                 SCALEM  = WORK(2*N+N*NR+NR+1)
                NUMRANK = IDNINT(WORK(2*N+N*NR+NR+2))                 NUMRANK = IDNINT(WORK(2*N+N*NR+NR+2))
                IF ( NR .LT. N ) THEN                 IF ( NR .LT. N ) THEN
Line 1371 Line 1501
                   CALL DLASET( 'A',N-NR,N-NR,ZERO,ONE,V(NR+1,NR+1),LDV )                    CALL DLASET( 'A',N-NR,N-NR,ZERO,ONE,V(NR+1,NR+1),LDV )
                END IF                 END IF
                CALL DORMQR( 'L','N',N,N,NR,WORK(2*N+1),N,WORK(N+1),                 CALL DORMQR( 'L','N',N,N,NR,WORK(2*N+1),N,WORK(N+1),
      &              V,LDV,WORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR )       $              V,LDV,WORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR )
 *  *
                CALL DORMLQ( 'L', 'T', NR, NR, NR, WORK(2*N+1), N,                 CALL DORMLQ( 'L', 'T', NR, NR, NR, WORK(2*N+1), N,
      &              WORK(2*N+N*NR+1), U, LDU, WORK(2*N+N*NR+NR+1),       $              WORK(2*N+N*NR+1), U, LDU, WORK(2*N+N*NR+NR+1),
      &              LWORK-2*N-N*NR-NR, IERR )       $              LWORK-2*N-N*NR-NR, IERR )
                DO 773 q = 1, NR                 DO 773 q = 1, NR
                   DO 772 p = 1, NR                    DO 772 p = 1, NR
                      WORK(2*N+N*NR+NR+IWORK(N+p)) = U(p,q)                       WORK(2*N+N*NR+NR+IWORK(N+p)) = U(p,q)
Line 1401 Line 1531
   973          CONTINUE    973          CONTINUE
                XSC = ONE / DNRM2( N, V(1,q), 1 )                 XSC = ONE / DNRM2( N, V(1,q), 1 )
                IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )                 IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
      &           CALL DSCAL( N, XSC, V(1,q), 1 )       $           CALL DSCAL( N, XSC, V(1,q), 1 )
  1972       CONTINUE   1972       CONTINUE
 *           At this moment, V contains the right singular vectors of A.  *           At this moment, V contains the right singular vectors of A.
 *           Next, assemble the left singular vector matrix U (M x N).  *           Next, assemble the left singular vector matrix U (M x N).
Line 1417 Line 1547
 *           matrix U. This applies to all cases.  *           matrix U. This applies to all cases.
 *  *
             CALL DORMQR( 'Left', 'No_Tr', M, N1, N, A, LDA, WORK, U,              CALL DORMQR( 'Left', 'No_Tr', M, N1, N, A, LDA, WORK, U,
      &           LDU, WORK(N+1), LWORK-N, IERR )       $           LDU, WORK(N+1), LWORK-N, IERR )
   
 *           The columns of U are normalized. The cost is O(M*N) flops.  *           The columns of U are normalized. The cost is O(M*N) flops.
             TEMP1 = DSQRT(DBLE(M)) * EPSLN              TEMP1 = DSQRT(DBLE(M)) * EPSLN
             DO 1973 p = 1, NR              DO 1973 p = 1, NR
                XSC = ONE / DNRM2( M, U(1,p), 1 )                 XSC = ONE / DNRM2( M, U(1,p), 1 )
                IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )                 IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
      &          CALL DSCAL( M, XSC, U(1,p), 1 )       $          CALL DSCAL( M, XSC, U(1,p), 1 )
  1973       CONTINUE   1973       CONTINUE
 *  *
 *           If the initial QRF is computed with row pivoting, the left  *           If the initial QRF is computed with row pivoting, the left
 *           singular vectors must be adjusted.  *           singular vectors must be adjusted.
 *  *
             IF ( ROWPIV )              IF ( ROWPIV )
      &          CALL DLASWP( N1, U, LDU, 1, M-1, IWORK(2*N+1), -1 )       $          CALL DLASWP( N1, U, LDU, 1, M-1, IWORK(2*N+1), -1 )
 *  *
          ELSE           ELSE
 *  *
Line 1452 Line 1582
             END IF              END IF
 *  *
             CALL DGESVJ( 'Upper', 'U', 'N', N, N, WORK(N+1), N, SVA,              CALL DGESVJ( 'Upper', 'U', 'N', N, N, WORK(N+1), N, SVA,
      &           N, U, LDU, WORK(N+N*N+1), LWORK-N-N*N, INFO )       $           N, U, LDU, WORK(N+N*N+1), LWORK-N-N*N, INFO )
 *  *
             SCALEM  = WORK(N+N*N+1)              SCALEM  = WORK(N+N*N+1)
             NUMRANK = IDNINT(WORK(N+N*N+2))              NUMRANK = IDNINT(WORK(N+N*N+2))
Line 1462 Line 1592
  6970       CONTINUE   6970       CONTINUE
 *  *
             CALL DTRSM( 'Left', 'Upper', 'NoTrans', 'No UD', N, N,              CALL DTRSM( 'Left', 'Upper', 'NoTrans', 'No UD', N, N,
      &           ONE, A, LDA, WORK(N+1), N )       $           ONE, A, LDA, WORK(N+1), N )
             DO 6972 p = 1, N              DO 6972 p = 1, N
                CALL DCOPY( N, WORK(N+p), N, V(IWORK(p),1), LDV )                 CALL DCOPY( N, WORK(N+p), N, V(IWORK(p),1), LDV )
  6972       CONTINUE   6972       CONTINUE
Line 1470 Line 1600
             DO 6971 p = 1, N              DO 6971 p = 1, N
                XSC = ONE / DNRM2( N, V(1,p), 1 )                 XSC = ONE / DNRM2( N, V(1,p), 1 )
                IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )                 IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
      &            CALL DSCAL( N, XSC, V(1,p), 1 )       $            CALL DSCAL( N, XSC, V(1,p), 1 )
  6971       CONTINUE   6971       CONTINUE
 *  *
 *           Assemble the left singular vector matrix U (M x N).  *           Assemble the left singular vector matrix U (M x N).
Line 1483 Line 1613
                END IF                 END IF
             END IF              END IF
             CALL DORMQR( 'Left', 'No Tr', M, N1, N, A, LDA, WORK, U,              CALL DORMQR( 'Left', 'No Tr', M, N1, N, A, LDA, WORK, U,
      &           LDU, WORK(N+1), LWORK-N, IERR )       $           LDU, WORK(N+1), LWORK-N, IERR )
             TEMP1 = DSQRT(DBLE(M))*EPSLN              TEMP1 = DSQRT(DBLE(M))*EPSLN
             DO 6973 p = 1, N1              DO 6973 p = 1, N1
                XSC = ONE / DNRM2( M, U(1,p), 1 )                 XSC = ONE / DNRM2( M, U(1,p), 1 )
                IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )                 IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
      &            CALL DSCAL( M, XSC, U(1,p), 1 )       $            CALL DSCAL( M, XSC, U(1,p), 1 )
  6973       CONTINUE   6973       CONTINUE
 *  *
             IF ( ROWPIV )              IF ( ROWPIV )
      &         CALL DLASWP( N1, U, LDU, 1, M-1, IWORK(2*N+1), -1 )       $         CALL DLASWP( N1, U, LDU, 1, M-1, IWORK(2*N+1), -1 )
 *  *
          END IF           END IF
 *  *
Line 1520 Line 1650
                TEMP1 = XSC*DABS( V(q,q) )                 TEMP1 = XSC*DABS( V(q,q) )
                DO 5968 p = 1, N                 DO 5968 p = 1, N
                   IF ( ( p .GT. q ) .AND. ( DABS(V(p,q)) .LE. TEMP1 )                    IF ( ( p .GT. q ) .AND. ( DABS(V(p,q)) .LE. TEMP1 )
      &                .OR. ( p .LT. q ) )       $                .OR. ( p .LT. q ) )
      &                V(p,q) = DSIGN( TEMP1, V(p,q) )       $                V(p,q) = DSIGN( TEMP1, V(p,q) )
                   IF ( p. LT. q ) V(p,q) = - V(p,q)                    IF ( p .LT. q ) V(p,q) = - V(p,q)
  5968          CONTINUE   5968          CONTINUE
  5969       CONTINUE   5969       CONTINUE
          ELSE           ELSE
Line 1530 Line 1660
          END IF           END IF
   
          CALL DGEQRF( N, NR, V, LDV, WORK(N+1), WORK(2*N+1),           CALL DGEQRF( N, NR, V, LDV, WORK(N+1), WORK(2*N+1),
      &        LWORK-2*N, IERR )       $        LWORK-2*N, IERR )
          CALL DLACPY( 'L', N, NR, V, LDV, WORK(2*N+1), N )           CALL DLACPY( 'L', N, NR, V, LDV, WORK(2*N+1), N )
 *  *
          DO 7969 p = 1, NR           DO 7969 p = 1, NR
Line 1541 Line 1671
             XSC = DSQRT(SMALL/EPSLN)              XSC = DSQRT(SMALL/EPSLN)
             DO 9970 q = 2, NR              DO 9970 q = 2, NR
                DO 9971 p = 1, q - 1                 DO 9971 p = 1, q - 1
                   TEMP1 = XSC * DMIN1(DABS(U(p,p)),DABS(U(q,q)))                    TEMP1 = XSC * MIN(DABS(U(p,p)),DABS(U(q,q)))
                   U(p,q) = - DSIGN( TEMP1, U(q,p) )                    U(p,q) = - DSIGN( TEMP1, U(q,p) )
  9971          CONTINUE   9971          CONTINUE
  9970       CONTINUE   9970       CONTINUE
Line 1550 Line 1680
          END IF           END IF
   
          CALL DGESVJ( 'G', 'U', 'V', NR, NR, U, LDU, SVA,           CALL DGESVJ( 'G', 'U', 'V', NR, NR, U, LDU, SVA,
      &        N, V, LDV, WORK(2*N+N*NR+1), LWORK-2*N-N*NR, INFO )       $        N, V, LDV, WORK(2*N+N*NR+1), LWORK-2*N-N*NR, INFO )
          SCALEM  = WORK(2*N+N*NR+1)           SCALEM  = WORK(2*N+N*NR+1)
          NUMRANK = IDNINT(WORK(2*N+N*NR+2))           NUMRANK = IDNINT(WORK(2*N+N*NR+2))
   
Line 1561 Line 1691
          END IF           END IF
   
          CALL DORMQR( 'L','N',N,N,NR,WORK(2*N+1),N,WORK(N+1),           CALL DORMQR( 'L','N',N,N,NR,WORK(2*N+1),N,WORK(N+1),
      &        V,LDV,WORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR )       $        V,LDV,WORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR )
 *  *
 *           Permute the rows of V using the (column) permutation from the  *           Permute the rows of V using the (column) permutation from the
 *           first QRF. Also, scale the columns to make them unit in  *           first QRF. Also, scale the columns to make them unit in
Line 1577 Line 1707
  8973          CONTINUE   8973          CONTINUE
                XSC = ONE / DNRM2( N, V(1,q), 1 )                 XSC = ONE / DNRM2( N, V(1,q), 1 )
                IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )                 IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
      &           CALL DSCAL( N, XSC, V(1,q), 1 )       $           CALL DSCAL( N, XSC, V(1,q), 1 )
  7972       CONTINUE   7972       CONTINUE
 *  *
 *           At this moment, V contains the right singular vectors of A.  *           At this moment, V contains the right singular vectors of A.
Line 1592 Line 1722
          END IF           END IF
 *  *
          CALL DORMQR( 'Left', 'No Tr', M, N1, N, A, LDA, WORK, U,           CALL DORMQR( 'Left', 'No Tr', M, N1, N, A, LDA, WORK, U,
      &        LDU, WORK(N+1), LWORK-N, IERR )       $        LDU, WORK(N+1), LWORK-N, IERR )
 *  *
             IF ( ROWPIV )              IF ( ROWPIV )
      &         CALL DLASWP( N1, U, LDU, 1, M-1, IWORK(2*N+1), -1 )       $         CALL DLASWP( N1, U, LDU, 1, M-1, IWORK(2*N+1), -1 )
 *  *
 *  *
          END IF           END IF
Removed from v.1.4  
changed lines
  Added in v.1.13

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