--- 
:name: dgejsv
:md5sum: 48e427f846d8774d81a35aecade0f887
:category: :subroutine
:arguments: 
- joba: 
    :type: char
    :intent: input
- jobu: 
    :type: char
    :intent: input
- jobv: 
    :type: char
    :intent: input
- jobr: 
    :type: char
    :intent: input
- jobt: 
    :type: char
    :intent: input
- jobp: 
    :type: char
    :intent: input
- m: 
    :type: integer
    :intent: input
- n: 
    :type: integer
    :intent: input
- a: 
    :type: doublereal
    :intent: input
    :dims: 
    - lda
    - n
- lda: 
    :type: integer
    :intent: input
- sva: 
    :type: doublereal
    :intent: output
    :dims: 
    - n
- u: 
    :type: doublereal
    :intent: output
    :dims: 
    - ldu
    - n
- ldu: 
    :type: integer
    :intent: input
- v: 
    :type: doublereal
    :intent: output
    :dims: 
    - ldv
    - n
- ldv: 
    :type: integer
    :intent: input
- work: 
    :type: doublereal
    :intent: input/output
    :dims: 
    - lwork
- lwork: 
    :type: integer
    :intent: input
    :option: true
    :default: "(lsame_(&jobu,\"N\")&&lsame_(&jobv,\"N\")) ? MAX(MAX(2*m+n,4*n+n*n),7) : lsame_(&jobv,\"V\") ? MAX(2*n+m,7) : ((lsame_(&jobu,\"U\")||lsame_(&jobu,\"F\"))&&lsame_(&jobv,\"V\")) ? MAX(MAX(6*n+2*n*n,m+3*n+n*n),7) : MAX(2*n+m,7)"
- iwork: 
    :type: integer
    :intent: output
    :dims: 
    - m+3*n
- info: 
    :type: integer
    :intent: output
:substitutions: 
  ldu: "(lsame_(&jobu,\"U\")||lsame_(&jobu,\"F\")||lsame_(&jobu,\"W\")) ? m : 1"
  ldv: "(lsame_(&jobu,\"U\")||lsame_(&jobu,\"F\")||lsame_(&jobu,\"W\")) ? n : 1"
:fortran_help: "      SUBROUTINE DGEJSV( JOBA, JOBU, JOBV, JOBR, JOBT, JOBP, M, N, A, LDA, SVA, U, LDU, V, LDV, WORK, LWORK, IWORK, INFO )\n\n\
  *  Purpose\n\
  *  =======\n\
  *\n\
  *  DGEJSV computes the singular value decomposition (SVD) of a real M-by-N\n\
  *  matrix [A], where M >= N. The SVD of [A] is written as\n\
  *\n\
  *               [A] = [U] * [SIGMA] * [V]^t,\n\
  *\n\
  *  where [SIGMA] is an N-by-N (M-by-N) matrix which is zero except for its N\n\
  *  diagonal elements, [U] is an M-by-N (or M-by-M) orthonormal matrix, and\n\
  *  [V] is an N-by-N orthogonal matrix. The diagonal elements of [SIGMA] are\n\
  *  the singular values of [A]. The columns of [U] and [V] are the left and\n\
  *  the right singular vectors of [A], respectively. The matrices [U] and [V]\n\
  *  are computed and stored in the arrays U and V, respectively. The diagonal\n\
  *  of [SIGMA] is computed and stored in the array SVA.\n\
  *\n\n\
  *  Arguments\n\
  *  =========\n\
  *\n\
  *  JOBA    (input) CHARACTER*1\n\
  *        Specifies the level of accuracy:\n\
  *       = 'C': This option works well (high relative accuracy) if A = B * D,\n\
  *             with well-conditioned B and arbitrary diagonal matrix D.\n\
  *             The accuracy cannot be spoiled by COLUMN scaling. The\n\
  *             accuracy of the computed output depends on the condition of\n\
  *             B, and the procedure aims at the best theoretical accuracy.\n\
  *             The relative error max_{i=1:N}|d sigma_i| / sigma_i is\n\
  *             bounded by f(M,N)*epsilon* cond(B), independent of D.\n\
  *             The input matrix is preprocessed with the QRF with column\n\
  *             pivoting. This initial preprocessing and preconditioning by\n\
  *             a rank revealing QR factorization is common for all values of\n\
  *             JOBA. Additional actions are specified as follows:\n\
  *       = 'E': Computation as with 'C' with an additional estimate of the\n\
  *             condition number of B. It provides a realistic error bound.\n\
  *       = 'F': If A = D1 * C * D2 with ill-conditioned diagonal scalings\n\
  *             D1, D2, and well-conditioned matrix C, this option gives\n\
  *             higher accuracy than the 'C' option. If the structure of the\n\
  *             input matrix is not known, and relative accuracy is\n\
  *             desirable, then this option is advisable. The input matrix A\n\
  *             is preprocessed with QR factorization with FULL (row and\n\
  *             column) pivoting.\n\
  *       = 'G'  Computation as with 'F' with an additional estimate of the\n\
  *             condition number of B, where A=D*B. If A has heavily weighted\n\
  *             rows, then using this condition number gives too pessimistic\n\
  *             error bound.\n\
  *       = 'A': Small singular values are the noise and the matrix is treated\n\
  *             as numerically rank defficient. The error in the computed\n\
  *             singular values is bounded by f(m,n)*epsilon*||A||.\n\
  *             The computed SVD A = U * S * V^t restores A up to\n\
  *             f(m,n)*epsilon*||A||.\n\
  *             This gives the procedure the licence to discard (set to zero)\n\
  *             all singular values below N*epsilon*||A||.\n\
  *       = 'R': Similar as in 'A'. Rank revealing property of the initial\n\
  *             QR factorization is used do reveal (using triangular factor)\n\
  *             a gap sigma_{r+1} < epsilon * sigma_r in which case the\n\
  *             numerical RANK is declared to be r. The SVD is computed with\n\
  *             absolute error bounds, but more accurately than with 'A'.\n\
  *\n\
  *  JOBU    (input) CHARACTER*1\n\
  *        Specifies whether to compute the columns of U:\n\
  *       = 'U': N columns of U are returned in the array U.\n\
  *       = 'F': full set of M left sing. vectors is returned in the array U.\n\
  *       = 'W': U may be used as workspace of length M*N. See the description\n\
  *             of U.\n\
  *       = 'N': U is not computed.\n\
  *\n\
  *  JOBV    (input) CHARACTER*1\n\
  *        Specifies whether to compute the matrix V:\n\
  *       = 'V': N columns of V are returned in the array V; Jacobi rotations\n\
  *             are not explicitly accumulated.\n\
  *       = 'J': N columns of V are returned in the array V, but they are\n\
  *             computed as the product of Jacobi rotations. This option is\n\
  *             allowed only if JOBU .NE. 'N', i.e. in computing the full SVD.\n\
  *       = 'W': V may be used as workspace of length N*N. See the description\n\
  *             of V.\n\
  *       = 'N': V is not computed.\n\
  *\n\
  *  JOBR    (input) CHARACTER*1\n\
  *        Specifies the RANGE for the singular values. Issues the licence to\n\
  *        set to zero small positive singular values if they are outside\n\
  *        specified range. If A .NE. 0 is scaled so that the largest singular\n\
  *        value of c*A is around DSQRT(BIG), BIG=SLAMCH('O'), then JOBR issues\n\
  *        the licence to kill columns of A whose norm in c*A is less than\n\
  *        DSQRT(SFMIN) (for JOBR.EQ.'R'), or less than SMALL=SFMIN/EPSLN,\n\
  *        where SFMIN=SLAMCH('S'), EPSLN=SLAMCH('E').\n\
  *       = 'N': Do not kill small columns of c*A. This option assumes that\n\
  *             BLAS and QR factorizations and triangular solvers are\n\
  *             implemented to work in that range. If the condition of A\n\
  *             is greater than BIG, use DGESVJ.\n\
  *       = 'R': RESTRICTED range for sigma(c*A) is [DSQRT(SFMIN), DSQRT(BIG)]\n\
  *             (roughly, as described above). This option is recommended.\n\
  *                                            ~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\
  *        For computing the singular values in the FULL range [SFMIN,BIG]\n\
  *        use DGESVJ.\n\
  *\n\
  *  JOBT    (input) CHARACTER*1\n\
  *        If the matrix is square then the procedure may determine to use\n\
  *        transposed A if A^t seems to be better with respect to convergence.\n\
  *        If the matrix is not square, JOBT is ignored. This is subject to\n\
  *        changes in the future.\n\
  *        The decision is based on two values of entropy over the adjoint\n\
  *        orbit of A^t * A. See the descriptions of WORK(6) and WORK(7).\n\
  *       = 'T': transpose if entropy test indicates possibly faster\n\
  *        convergence of Jacobi process if A^t is taken as input. If A is\n\
  *        replaced with A^t, then the row pivoting is included automatically.\n\
  *       = 'N': do not speculate.\n\
  *        This option can be used to compute only the singular values, or the\n\
  *        full SVD (U, SIGMA and V). For only one set of singular vectors\n\
  *        (U or V), the caller should provide both U and V, as one of the\n\
  *        matrices is used as workspace if the matrix A is transposed.\n\
  *        The implementer can easily remove this constraint and make the\n\
  *        code more complicated. See the descriptions of U and V.\n\
  *\n\
  *  JOBP    (input) CHARACTER*1\n\
  *        Issues the licence to introduce structured perturbations to drown\n\
  *        denormalized numbers. This licence should be active if the\n\
  *        denormals are poorly implemented, causing slow computation,\n\
  *        especially in cases of fast convergence (!). For details see [1,2].\n\
  *        For the sake of simplicity, this perturbations are included only\n\
  *        when the full SVD or only the singular values are requested. The\n\
  *        implementer/user can easily add the perturbation for the cases of\n\
  *        computing one set of singular vectors.\n\
  *       = 'P': introduce perturbation\n\
  *       = 'N': do not perturb\n\
  *\n\
  *  M       (input) INTEGER\n\
  *         The number of rows of the input matrix A.  M >= 0.\n\
  *\n\
  *  N       (input) INTEGER\n\
  *         The number of columns of the input matrix A. M >= N >= 0.\n\
  *\n\
  *  A       (input/workspace) DOUBLE PRECISION array, dimension (LDA,N)\n\
  *          On entry, the M-by-N matrix A.\n\
  *\n\
  *  LDA     (input) INTEGER\n\
  *          The leading dimension of the array A.  LDA >= max(1,M).\n\
  *\n\
  *  SVA     (workspace/output) DOUBLE PRECISION array, dimension (N)\n\
  *          On exit,\n\
  *          - For WORK(1)/WORK(2) = ONE: The singular values of A. During the\n\
  *            computation SVA contains Euclidean column norms of the\n\
  *            iterated matrices in the array A.\n\
  *          - For WORK(1) .NE. WORK(2): The singular values of A are\n\
  *            (WORK(1)/WORK(2)) * SVA(1:N). This factored form is used if\n\
  *            sigma_max(A) overflows or if small singular values have been\n\
  *            saved from underflow by scaling the input matrix A.\n\
  *          - If JOBR='R' then some of the singular values may be returned\n\
  *            as exact zeros obtained by \"set to zero\" because they are\n\
  *            below the numerical rank threshold or are denormalized numbers.\n\
  *\n\
  *  U       (workspace/output) DOUBLE PRECISION array, dimension ( LDU, N )\n\
  *          If JOBU = 'U', then U contains on exit the M-by-N matrix of\n\
  *                         the left singular vectors.\n\
  *          If JOBU = 'F', then U contains on exit the M-by-M matrix of\n\
  *                         the left singular vectors, including an ONB\n\
  *                         of the orthogonal complement of the Range(A).\n\
  *          If JOBU = 'W'  .AND. (JOBV.EQ.'V' .AND. JOBT.EQ.'T' .AND. M.EQ.N),\n\
  *                         then U is used as workspace if the procedure\n\
  *                         replaces A with A^t. In that case, [V] is computed\n\
  *                         in U as left singular vectors of A^t and then\n\
  *                         copied back to the V array. This 'W' option is just\n\
  *                         a reminder to the caller that in this case U is\n\
  *                         reserved as workspace of length N*N.\n\
  *          If JOBU = 'N'  U is not referenced.\n\
  *\n\
  * LDU      (input) INTEGER\n\
  *          The leading dimension of the array U,  LDU >= 1.\n\
  *          IF  JOBU = 'U' or 'F' or 'W',  then LDU >= M.\n\
  *\n\
  *  V       (workspace/output) DOUBLE PRECISION array, dimension ( LDV, N )\n\
  *          If JOBV = 'V', 'J' then V contains on exit the N-by-N matrix of\n\
  *                         the right singular vectors;\n\
  *          If JOBV = 'W', AND (JOBU.EQ.'U' AND JOBT.EQ.'T' AND M.EQ.N),\n\
  *                         then V is used as workspace if the pprocedure\n\
  *                         replaces A with A^t. In that case, [U] is computed\n\
  *                         in V as right singular vectors of A^t and then\n\
  *                         copied back to the U array. This 'W' option is just\n\
  *                         a reminder to the caller that in this case V is\n\
  *                         reserved as workspace of length N*N.\n\
  *          If JOBV = 'N'  V is not referenced.\n\
  *\n\
  *  LDV     (input) INTEGER\n\
  *          The leading dimension of the array V,  LDV >= 1.\n\
  *          If JOBV = 'V' or 'J' or 'W', then LDV >= N.\n\
  *\n\
  *  WORK    (workspace/output) DOUBLE PRECISION array, dimension at least LWORK.\n\
  *          On exit,\n\
  *          WORK(1) = SCALE = WORK(2) / WORK(1) is the scaling factor such\n\
  *                    that SCALE*SVA(1:N) are the computed singular values\n\
  *                    of A. (See the description of SVA().)\n\
  *          WORK(2) = See the description of WORK(1).\n\
  *          WORK(3) = SCONDA is an estimate for the condition number of\n\
  *                    column equilibrated A. (If JOBA .EQ. 'E' or 'G')\n\
  *                    SCONDA is an estimate of DSQRT(||(R^t * R)^(-1)||_1).\n\
  *                    It is computed using DPOCON. It holds\n\
  *                    N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA\n\
  *                    where R is the triangular factor from the QRF of A.\n\
  *                    However, if R is truncated and the numerical rank is\n\
  *                    determined to be strictly smaller than N, SCONDA is\n\
  *                    returned as -1, thus indicating that the smallest\n\
  *                    singular values might be lost.\n\
  *\n\
  *          If full SVD is needed, the following two condition numbers are\n\
  *          useful for the analysis of the algorithm. They are provied for\n\
  *          a developer/implementer who is familiar with the details of\n\
  *          the method.\n\
  *\n\
  *          WORK(4) = an estimate of the scaled condition number of the\n\
  *                    triangular factor in the first QR factorization.\n\
  *          WORK(5) = an estimate of the scaled condition number of the\n\
  *                    triangular factor in the second QR factorization.\n\
  *          The following two parameters are computed if JOBT .EQ. 'T'.\n\
  *          They are provided for a developer/implementer who is familiar\n\
  *          with the details of the method.\n\
  *\n\
  *          WORK(6) = the entropy of A^t*A :: this is the Shannon entropy\n\
  *                    of diag(A^t*A) / Trace(A^t*A) taken as point in the\n\
  *                    probability simplex.\n\
  *          WORK(7) = the entropy of A*A^t.\n\
  *\n\
  *  LWORK   (input) INTEGER\n\
  *          Length of WORK to confirm proper allocation of work space.\n\
  *          LWORK depends on the job:\n\
  *\n\
  *          If only SIGMA is needed ( JOBU.EQ.'N', JOBV.EQ.'N' ) and\n\
  *            -> .. no scaled condition estimate required ( JOBE.EQ.'N'):\n\
  *               LWORK >= max(2*M+N,4*N+1,7). This is the minimal requirement.\n\
  *               For optimal performance (blocked code) the optimal value\n\
  *               is LWORK >= max(2*M+N,3*N+(N+1)*NB,7). Here NB is the optimal\n\
  *               block size for xGEQP3/xGEQRF.\n\
  *            -> .. an estimate of the scaled condition number of A is\n\
  *               required (JOBA='E', 'G'). In this case, LWORK is the maximum\n\
  *               of the above and N*N+4*N, i.e. LWORK >= max(2*M+N,N*N+4N,7).\n\
  *\n\
  *          If SIGMA and the right singular vectors are needed (JOBV.EQ.'V'),\n\
  *            -> the minimal requirement is LWORK >= max(2*N+M,7).\n\
  *            -> For optimal performance, LWORK >= max(2*N+M,2*N+N*NB,7),\n\
  *               where NB is the optimal block size.\n\
  *\n\
  *          If SIGMA and the left singular vectors are needed\n\
  *            -> the minimal requirement is LWORK >= max(2*N+M,7).\n\
  *            -> For optimal performance, LWORK >= max(2*N+M,2*N+N*NB,7),\n\
  *               where NB is the optimal block size.\n\
  *\n\
  *          If full SVD is needed ( JOBU.EQ.'U' or 'F', JOBV.EQ.'V' ) and\n\
  *            -> .. the singular vectors are computed without explicit\n\
  *               accumulation of the Jacobi rotations, LWORK >= 6*N+2*N*N\n\
  *            -> .. in the iterative part, the Jacobi rotations are\n\
  *               explicitly accumulated (option, see the description of JOBV),\n\
  *               then the minimal requirement is LWORK >= max(M+3*N+N*N,7).\n\
  *               For better performance, if NB is the optimal block size,\n\
  *               LWORK >= max(3*N+N*N+M,3*N+N*N+N*NB,7).\n\
  *\n\
  *  IWORK   (workspace/output) INTEGER array, dimension M+3*N.\n\
  *          On exit,\n\
  *          IWORK(1) = the numerical rank determined after the initial\n\
  *                     QR factorization with pivoting. See the descriptions\n\
  *                     of JOBA and JOBR.\n\
  *          IWORK(2) = the number of the computed nonzero singular values\n\
  *          IWORK(3) = if nonzero, a warning message:\n\
  *                     If IWORK(3).EQ.1 then some of the column norms of A\n\
  *                     were denormalized floats. The requested high accuracy\n\
  *                     is not warranted by the data.\n\
  *\n\
  *  INFO    (output) INTEGER\n\
  *           < 0  : if INFO = -i, then the i-th argument had an illegal value.\n\
  *           = 0 :  successfull exit;\n\
  *           > 0 :  DGEJSV  did not converge in the maximal allowed number\n\
  *                  of sweeps. The computed values may be inaccurate.\n\
  *\n\n\
  *  Further Details\n\
  *  ===============\n\
  *\n\
  *  DGEJSV implements a preconditioned Jacobi SVD algorithm. It uses SGEQP3,\n\
  *  SGEQRF, and SGELQF as preprocessors and preconditioners. Optionally, an\n\
  *  additional row pivoting can be used as a preprocessor, which in some\n\
  *  cases results in much higher accuracy. An example is matrix A with the\n\
  *  structure A = D1 * C * D2, where D1, D2 are arbitrarily ill-conditioned\n\
  *  diagonal matrices and C is well-conditioned matrix. In that case, complete\n\
  *  pivoting in the first QR factorizations provides accuracy dependent on the\n\
  *  condition number of C, and independent of D1, D2. Such higher accuracy is\n\
  *  not completely understood theoretically, but it works well in practice.\n\
  *  Further, if A can be written as A = B*D, with well-conditioned B and some\n\
  *  diagonal D, then the high accuracy is guaranteed, both theoretically and\n\
  *  in software, independent of D. For more details see [1], [2].\n\
  *     The computational range for the singular values can be the full range\n\
  *  ( UNDERFLOW,OVERFLOW ), provided that the machine arithmetic and the BLAS\n\
  *  & LAPACK routines called by DGEJSV are implemented to work in that range.\n\
  *  If that is not the case, then the restriction for safe computation with\n\
  *  the singular values in the range of normalized IEEE numbers is that the\n\
  *  spectral condition number kappa(A)=sigma_max(A)/sigma_min(A) does not\n\
  *  overflow. This code (DGEJSV) is best used in this restricted range,\n\
  *  meaning that singular values of magnitude below ||A||_2 / SLAMCH('O') are\n\
  *  returned as zeros. See JOBR for details on this.\n\
  *     Further, this implementation is somewhat slower than the one described\n\
  *  in [1,2] due to replacement of some non-LAPACK components, and because\n\
  *  the choice of some tuning parameters in the iterative part (DGESVJ) is\n\
  *  left to the implementer on a particular machine.\n\
  *     The rank revealing QR factorization (in this code: SGEQP3) should be\n\
  *  implemented as in [3]. We have a new version of SGEQP3 under development\n\
  *  that is more robust than the current one in LAPACK, with a cleaner cut in\n\
  *  rank defficient cases. It will be available in the SIGMA library [4].\n\
  *  If M is much larger than N, it is obvious that the inital QRF with\n\
  *  column pivoting can be preprocessed by the QRF without pivoting. That\n\
  *  well known trick is not used in DGEJSV because in some cases heavy row\n\
  *  weighting can be treated with complete pivoting. The overhead in cases\n\
  *  M much larger than N is then only due to pivoting, but the benefits in\n\
  *  terms of accuracy have prevailed. The implementer/user can incorporate\n\
  *  this extra QRF step easily. The implementer can also improve data movement\n\
  *  (matrix transpose, matrix copy, matrix transposed copy) - this\n\
  *  implementation of DGEJSV uses only the simplest, naive data movement.\n\
  *\n\
  *  Contributors\n\
  *\n\
  *  Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)\n\
  *\n\
  *  References\n\
  *\n\
  * [1] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm I.\n\
  *     SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1322-1342.\n\
  *     LAPACK Working note 169.\n\
  * [2] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm II.\n\
  *     SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1343-1362.\n\
  *     LAPACK Working note 170.\n\
  * [3] Z. Drmac and Z. Bujanovic: On the failure of rank-revealing QR\n\
  *     factorization software - a case study.\n\
  *     ACM Trans. Math. Softw. Vol. 35, No 2 (2008), pp. 1-28.\n\
  *     LAPACK Working note 176.\n\
  * [4] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV,\n\
  *     QSVD, (H,K)-SVD computations.\n\
  *     Department of Mathematics, University of Zagreb, 2008.\n\
  *\n\
  *  Bugs, examples and comments\n\
  * \n\
  *  Please report all bugs and send interesting examples and/or comments to\n\
  *  drmac@math.hr. Thank you.\n\
  *\n\
  * ==========================================================================\n\
  *\n\
  *     .. Local Parameters ..\n      DOUBLE PRECISION   ZERO,  ONE\n      PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )\n\
  *     ..\n\
  *     .. Local Scalars ..\n      DOUBLE PRECISION AAPP, AAQQ, AATMAX, AATMIN, BIG, BIG1, COND_OK,\n     &        CONDR1, CONDR2, ENTRA,  ENTRAT, EPSLN,  MAXPRJ, SCALEM,\n     &        SCONDA, SFMIN,  SMALL,  TEMP1,  USCAL1, USCAL2, XSC\n      INTEGER IERR,   N1,     NR,     NUMRANK,        p, q,   WARNING\n      LOGICAL ALMORT, DEFR,   ERREST, GOSCAL, JRACC,  KILL,   LSVEC,\n     &        L2ABER, L2KILL, L2PERT, L2RANK, L2TRAN,\n     &        NOSCAL, ROWPIV, RSVEC,  TRANSP\n\
  *     ..\n\
  *     .. Intrinsic Functions ..\n      INTRINSIC DABS,  DLOG, DMAX1, DMIN1, DBLE,\n     &          MAX0, MIN0, IDNINT,  DSIGN,  DSQRT\n\
  *     ..\n\
  *     .. External Functions ..\n      DOUBLE PRECISION  DLAMCH, DNRM2\n      INTEGER   IDAMAX\n      LOGICAL   LSAME\n      EXTERNAL  IDAMAX, LSAME, DLAMCH, DNRM2\n\
  *     ..\n\
  *     .. External Subroutines ..\n      EXTERNAL  DCOPY,  DGELQF, DGEQP3, DGEQRF, DLACPY, DLASCL,\n     &          DLASET, DLASSQ, DLASWP, DORGQR, DORMLQ,\n     &          DORMQR, DPOCON, DSCAL,  DSWAP,  DTRSM,  XERBLA\n\
  *\n      EXTERNAL  DGESVJ\n\
  *     ..\n\
  *\n\
  *     Test the input arguments\n\
  *\n      LSVEC  = LSAME( JOBU, 'U' ) .OR. LSAME( JOBU, 'F' )\n      JRACC  = LSAME( JOBV, 'J' )\n      RSVEC  = LSAME( JOBV, 'V' ) .OR. JRACC\n      ROWPIV = LSAME( JOBA, 'F' ) .OR. LSAME( JOBA, 'G' )\n      L2RANK = LSAME( JOBA, 'R' )\n      L2ABER = LSAME( JOBA, 'A' )\n      ERREST = LSAME( JOBA, 'E' ) .OR. LSAME( JOBA, 'G' )\n      L2TRAN = LSAME( JOBT, 'T' )\n      L2KILL = LSAME( JOBR, 'R' )\n      DEFR   = LSAME( JOBR, 'N' )\n      L2PERT = LSAME( JOBP, 'P' )\n\
  *\n      IF ( .NOT.(ROWPIV .OR. L2RANK .OR. L2ABER .OR.\n     &     ERREST .OR. LSAME( JOBA, 'C' ) )) THEN\n         INFO = - 1\n      ELSE IF ( .NOT.( LSVEC  .OR. LSAME( JOBU, 'N' ) .OR.\n     &                             LSAME( JOBU, 'W' )) ) THEN\n         INFO = - 2\n      ELSE IF ( .NOT.( RSVEC .OR. LSAME( JOBV, 'N' ) .OR.\n     &   LSAME( JOBV, 'W' )) .OR. ( JRACC .AND. (.NOT.LSVEC) ) ) THEN\n         INFO = - 3\n      ELSE IF ( .NOT. ( L2KILL .OR. DEFR ) )    THEN\n         INFO = - 4\n      ELSE IF ( .NOT. ( L2TRAN .OR. LSAME( JOBT, 'N' ) ) ) THEN\n         INFO = - 5\n      ELSE IF ( .NOT. ( L2PERT .OR. LSAME( JOBP, 'N' ) ) ) THEN\n         INFO = - 6\n      ELSE IF ( M .LT. 0 ) THEN\n         INFO = - 7\n      ELSE IF ( ( N .LT. 0 ) .OR. ( N .GT. M ) ) THEN\n         INFO = - 8\n      ELSE IF ( LDA .LT. M ) THEN\n         INFO = - 10\n      ELSE IF ( LSVEC .AND. ( LDU .LT. M ) ) THEN\n         INFO = - 13\n      ELSE IF ( RSVEC .AND. ( LDV .LT. N ) ) THEN\n         INFO = - 14\n      ELSE IF ( (.NOT.(LSVEC .OR. RSVEC .OR. ERREST).AND.\n     &                           (LWORK .LT. MAX0(7,4*N+1,2*M+N))) .OR.\n     & (.NOT.(LSVEC .OR. LSVEC) .AND. ERREST .AND.\n     &                         (LWORK .LT. MAX0(7,4*N+N*N,2*M+N))) .OR.\n     & (LSVEC .AND. (.NOT.RSVEC) .AND. (LWORK .LT. MAX0(7,2*N+M))) .OR.\n     & (RSVEC .AND. (.NOT.LSVEC) .AND. (LWORK .LT. MAX0(7,2*N+M))) .OR.\n     & (LSVEC .AND. RSVEC .AND. .NOT.JRACC .AND. (LWORK.LT.6*N+2*N*N))\n     & .OR. (LSVEC.AND.RSVEC.AND.JRACC.AND.LWORK.LT.MAX0(7,M+3*N+N*N)))\n     &   THEN\n         INFO = - 17\n      ELSE\n\
  *        #:)\n         INFO = 0\n      END IF\n\
  *\n      IF ( INFO .NE. 0 ) THEN\n\
  *       #:(\n         CALL XERBLA( 'DGEJSV', - INFO )\n      END IF\n\
  *\n\
  *     Quick return for void matrix (Y3K safe)\n\
  * #:)\n      IF ( ( M .EQ. 0 ) .OR. ( N .EQ. 0 ) ) RETURN\n\
  *\n\
  *     Determine whether the matrix U should be M x N or M x M\n\
  *\n      IF ( LSVEC ) THEN\n         N1 = N\n         IF ( LSAME( JOBU, 'F' ) ) N1 = M\n      END IF\n\
  *\n\
  *     Set numerical parameters\n\
  *\n\
  *!    NOTE: Make sure DLAMCH() does not fail on the target architecture.\n\
  *\n\n      EPSLN = DLAMCH('Epsilon')\n      SFMIN = DLAMCH('SafeMinimum')\n      SMALL = SFMIN / EPSLN\n      BIG   = DLAMCH('O')\n\
  *     BIG   = ONE / SFMIN\n\
  *\n\
  *     Initialize SVA(1:N) = diag( ||A e_i||_2 )_1^N\n\
  *\n\
  *(!)  If necessary, scale SVA() to protect the largest norm from\n\
  *     overflow. It is possible that this scaling pushes the smallest\n\
  *     column norm left from the underflow threshold (extreme case).\n\
  *\n      SCALEM  = ONE / DSQRT(DBLE(M)*DBLE(N))\n      NOSCAL  = .TRUE.\n      GOSCAL  = .TRUE.\n      DO 1874 p = 1, N\n         AAPP = ZERO\n         AAQQ = ONE\n         CALL DLASSQ( M, A(1,p), 1, AAPP, AAQQ )\n         IF ( AAPP .GT. BIG ) THEN\n            INFO = - 9\n            CALL XERBLA( 'DGEJSV', -INFO )\n            RETURN\n         END IF\n         AAQQ = DSQRT(AAQQ)\n         IF ( ( AAPP .LT. (BIG / AAQQ) ) .AND. NOSCAL  ) THEN\n            SVA(p)  = AAPP * AAQQ\n         ELSE\n            NOSCAL  = .FALSE.\n            SVA(p)  = AAPP * ( AAQQ * SCALEM )\n            IF ( GOSCAL ) THEN\n               GOSCAL = .FALSE.\n               CALL DSCAL( p-1, SCALEM, SVA, 1 )\n            END IF\n         END IF\n 1874 CONTINUE\n\
  *\n      IF ( NOSCAL ) SCALEM = ONE\n\
  *\n      AAPP = ZERO\n      AAQQ = BIG\n      DO 4781 p = 1, N\n         AAPP = DMAX1( AAPP, SVA(p) )\n         IF ( SVA(p) .NE. ZERO ) AAQQ = DMIN1( AAQQ, SVA(p) )\n 4781 CONTINUE\n\
  *\n\
  *     Quick return for zero M x N matrix\n\
  * #:)\n      IF ( AAPP .EQ. ZERO ) THEN\n         IF ( LSVEC ) CALL DLASET( 'G', M, N1, ZERO, ONE, U, LDU )\n         IF ( RSVEC ) CALL DLASET( 'G', N, N,  ZERO, ONE, V, LDV )\n         WORK(1) = ONE\n         WORK(2) = ONE\n         IF ( ERREST ) WORK(3) = ONE\n         IF ( LSVEC .AND. RSVEC ) THEN\n            WORK(4) = ONE\n            WORK(5) = ONE\n         END IF\n         IF ( L2TRAN ) THEN\n            WORK(6) = ZERO\n            WORK(7) = ZERO\n         END IF\n         IWORK(1) = 0\n         IWORK(2) = 0\n         RETURN\n      END IF\n\
  *\n\
  *     Issue warning if denormalized column norms detected. Override the\n\
  *     high relative accuracy request. Issue licence to kill columns\n\
  *     (set them to zero) whose norm is less than sigma_max / BIG (roughly).\n\
  * #:(\n      WARNING = 0\n      IF ( AAQQ .LE. SFMIN ) THEN\n         L2RANK = .TRUE.\n         L2KILL = .TRUE.\n         WARNING = 1\n      END IF\n\
  *\n\
  *     Quick return for one-column matrix\n\
  * #:)\n      IF ( N .EQ. 1 ) THEN\n\
  *\n         IF ( LSVEC ) THEN\n            CALL DLASCL( 'G',0,0,SVA(1),SCALEM, M,1,A(1,1),LDA,IERR )\n            CALL DLACPY( 'A', M, 1, A, LDA, U, LDU )\n\
  *           computing all M left singular vectors of the M x 1 matrix\n            IF ( N1 .NE. N  ) THEN\n               CALL DGEQRF( M, N, U,LDU, WORK, WORK(N+1),LWORK-N,IERR )\n               CALL DORGQR( M,N1,1, U,LDU,WORK,WORK(N+1),LWORK-N,IERR )\n               CALL DCOPY( M, A(1,1), 1, U(1,1), 1 )\n            END IF\n         END IF\n         IF ( RSVEC ) THEN\n             V(1,1) = ONE\n         END IF\n         IF ( SVA(1) .LT. (BIG*SCALEM) ) THEN\n            SVA(1)  = SVA(1) / SCALEM\n            SCALEM  = ONE\n         END IF\n         WORK(1) = ONE / SCALEM\n         WORK(2) = ONE\n         IF ( SVA(1) .NE. ZERO ) THEN\n            IWORK(1) = 1\n            IF ( ( SVA(1) / SCALEM) .GE. SFMIN ) THEN\n               IWORK(2) = 1\n            ELSE\n               IWORK(2) = 0\n            END IF\n         ELSE\n            IWORK(1) = 0\n            IWORK(2) = 0\n         END IF\n         IF ( ERREST ) WORK(3) = ONE\n         IF ( LSVEC .AND. RSVEC ) THEN\n            WORK(4) = ONE\n            WORK(5) = ONE\n         END IF\n         IF ( L2TRAN ) THEN\n            WORK(6) = ZERO\n            WORK(7) = ZERO\n         END IF\n         RETURN\n\
  *\n      END IF\n\
  *\n      TRANSP = .FALSE.\n      L2TRAN = L2TRAN .AND. ( M .EQ. N )\n\
  *\n      AATMAX = -ONE\n      AATMIN =  BIG\n      IF ( ROWPIV .OR. L2TRAN ) THEN\n\
  *\n\
  *     Compute the row norms, needed to determine row pivoting sequence\n\
  *     (in the case of heavily row weighted A, row pivoting is strongly\n\
  *     advised) and to collect information needed to compare the\n\
  *     structures of A * A^t and A^t * A (in the case L2TRAN.EQ..TRUE.).\n\
  *\n         IF ( L2TRAN ) THEN\n            DO 1950 p = 1, M\n               XSC   = ZERO\n               TEMP1 = ONE\n               CALL DLASSQ( N, A(p,1), LDA, XSC, TEMP1 )\n\
  *              DLASSQ gets both the ell_2 and the ell_infinity norm\n\
  *              in one pass through the vector\n               WORK(M+N+p)  = XSC * SCALEM\n               WORK(N+p)    = XSC * (SCALEM*DSQRT(TEMP1))\n               AATMAX = DMAX1( AATMAX, WORK(N+p) )\n               IF (WORK(N+p) .NE. ZERO) AATMIN = DMIN1(AATMIN,WORK(N+p))\n 1950       CONTINUE\n         ELSE\n            DO 1904 p = 1, M\n               WORK(M+N+p) = SCALEM*DABS( A(p,IDAMAX(N,A(p,1),LDA)) )\n               AATMAX = DMAX1( AATMAX, WORK(M+N+p) )\n               AATMIN = DMIN1( AATMIN, WORK(M+N+p) )\n 1904       CONTINUE\n         END IF\n\
  *\n      END IF\n\
  *\n\
  *     For square matrix A try to determine whether A^t  would be  better\n\
  *     input for the preconditioned Jacobi SVD, with faster convergence.\n\
  *     The decision is based on an O(N) function of the vector of column\n\
  *     and row norms of A, based on the Shannon entropy. This should give\n\
  *     the right choice in most cases when the difference actually matters.\n\
  *     It may fail and pick the slower converging side.\n\
  *\n      ENTRA  = ZERO\n      ENTRAT = ZERO\n      IF ( L2TRAN ) THEN\n\
  *\n         XSC   = ZERO\n         TEMP1 = ONE\n         CALL DLASSQ( N, SVA, 1, XSC, TEMP1 )\n         TEMP1 = ONE / TEMP1\n\
  *\n         ENTRA = ZERO\n         DO 1113 p = 1, N\n            BIG1  = ( ( SVA(p) / XSC )**2 ) * TEMP1\n            IF ( BIG1 .NE. ZERO ) ENTRA = ENTRA + BIG1 * DLOG(BIG1)\n 1113    CONTINUE\n         ENTRA = - ENTRA / DLOG(DBLE(N))\n\
  *\n\
  *        Now, SVA().^2/Trace(A^t * A) is a point in the probability simplex.\n\
  *        It is derived from the diagonal of  A^t * A.  Do the same with the\n\
  *        diagonal of A * A^t, compute the entropy of the corresponding\n\
  *        probability distribution. Note that A * A^t and A^t * A have the\n\
  *        same trace.\n\
  *\n         ENTRAT = ZERO\n         DO 1114 p = N+1, N+M\n            BIG1 = ( ( WORK(p) / XSC )**2 ) * TEMP1\n            IF ( BIG1 .NE. ZERO ) ENTRAT = ENTRAT + BIG1 * DLOG(BIG1)\n 1114    CONTINUE\n         ENTRAT = - ENTRAT / DLOG(DBLE(M))\n\
  *\n\
  *        Analyze the entropies and decide A or A^t. Smaller entropy\n\
  *        usually means better input for the algorithm.\n\
  *\n         TRANSP = ( ENTRAT .LT. ENTRA )\n\
  *\n\
  *        If A^t is better than A, transpose A.\n\
  *\n         IF ( TRANSP ) THEN\n\
  *           In an optimal implementation, this trivial transpose\n\
  *           should be replaced with faster transpose.\n            DO 1115 p = 1, N - 1\n               DO 1116 q = p + 1, N\n                   TEMP1 = A(q,p)\n                  A(q,p) = A(p,q)\n                  A(p,q) = TEMP1\n 1116          CONTINUE\n 1115       CONTINUE\n            DO 1117 p = 1, N\n               WORK(M+N+p) = SVA(p)\n               SVA(p)      = WORK(N+p)\n 1117       CONTINUE\n            TEMP1  = AAPP\n            AAPP   = AATMAX\n            AATMAX = TEMP1\n            TEMP1  = AAQQ\n            AAQQ   = AATMIN\n            AATMIN = TEMP1\n            KILL   = LSVEC\n            LSVEC  = RSVEC\n            RSVEC  = KILL\n            IF ( LSVEC ) N1 = N\n\
  *\n            ROWPIV = .TRUE.\n         END IF\n\
  *\n      END IF\n\
  *     END IF L2TRAN\n\
  *\n\
  *     Scale the matrix so that its maximal singular value remains less\n\
  *     than DSQRT(BIG) -- the matrix is scaled so that its maximal column\n\
  *     has Euclidean norm equal to DSQRT(BIG/N). The only reason to keep\n\
  *     DSQRT(BIG) instead of BIG is the fact that DGEJSV uses LAPACK and\n\
  *     BLAS routines that, in some implementations, are not capable of\n\
  *     working in the full interval [SFMIN,BIG] and that they may provoke\n\
  *     overflows in the intermediate results. If the singular values spread\n\
  *     from SFMIN to BIG, then DGESVJ will compute them. So, in that case,\n\
  *     one should use DGESVJ instead of DGEJSV.\n\
  *\n      BIG1   = DSQRT( BIG )\n      TEMP1  = DSQRT( BIG / DBLE(N) )\n\
  *\n      CALL DLASCL( 'G', 0, 0, AAPP, TEMP1, N, 1, SVA, N, IERR )\n      IF ( AAQQ .GT. (AAPP * SFMIN) ) THEN\n          AAQQ = ( AAQQ / AAPP ) * TEMP1\n      ELSE\n          AAQQ = ( AAQQ * TEMP1 ) / AAPP\n      END IF\n      TEMP1 = TEMP1 * SCALEM\n      CALL DLASCL( 'G', 0, 0, AAPP, TEMP1, M, N, A, LDA, IERR )\n\
  *\n\
  *     To undo scaling at the end of this procedure, multiply the\n\
  *     computed singular values with USCAL2 / USCAL1.\n\
  *\n      USCAL1 = TEMP1\n      USCAL2 = AAPP\n\
  *\n      IF ( L2KILL ) THEN\n\
  *        L2KILL enforces computation of nonzero singular values in\n\
  *        the restricted range of condition number of the initial A,\n\
  *        sigma_max(A) / sigma_min(A) approx. DSQRT(BIG)/DSQRT(SFMIN).\n         XSC = DSQRT( SFMIN )\n      ELSE\n         XSC = SMALL\n\
  *\n\
  *        Now, if the condition number of A is too big,\n\
  *        sigma_max(A) / sigma_min(A) .GT. DSQRT(BIG/N) * EPSLN / SFMIN,\n\
  *        as a precaution measure, the full SVD is computed using DGESVJ\n\
  *        with accumulated Jacobi rotations. This provides numerically\n\
  *        more robust computation, at the cost of slightly increased run\n\
  *        time. Depending on the concrete implementation of BLAS and LAPACK\n\
  *        (i.e. how they behave in presence of extreme ill-conditioning) the\n\
  *        implementor may decide to remove this switch.\n         IF ( ( AAQQ.LT.DSQRT(SFMIN) ) .AND. LSVEC .AND. RSVEC ) THEN\n            JRACC = .TRUE.\n         END IF\n\
  *\n      END IF\n      IF ( AAQQ .LT. XSC ) THEN\n         DO 700 p = 1, N\n            IF ( SVA(p) .LT. XSC ) THEN\n               CALL DLASET( 'A', M, 1, ZERO, ZERO, A(1,p), LDA )\n               SVA(p) = ZERO\n            END IF\n 700     CONTINUE\n      END IF\n\
  *\n\
  *     Preconditioning using QR factorization with pivoting\n\
  *\n      IF ( ROWPIV ) THEN\n\
  *        Optional row permutation (Bjoerck row pivoting):\n\
  *        A result by Cox and Higham shows that the Bjoerck's\n\
  *        row pivoting combined with standard column pivoting\n\
  *        has similar effect as Powell-Reid complete pivoting.\n\
  *        The ell-infinity norms of A are made nonincreasing.\n         DO 1952 p = 1, M - 1\n            q = IDAMAX( M-p+1, WORK(M+N+p), 1 ) + p - 1\n            IWORK(2*N+p) = q\n            IF ( p .NE. q ) THEN\n               TEMP1       = WORK(M+N+p)\n               WORK(M+N+p) = WORK(M+N+q)\n               WORK(M+N+q) = TEMP1\n            END IF\n 1952    CONTINUE\n         CALL DLASWP( N, A, LDA, 1, M-1, IWORK(2*N+1), 1 )\n      END IF\n\
  *\n\
  *     End of the preparation phase (scaling, optional sorting and\n\
  *     transposing, optional flushing of small columns).\n\
  *\n\
  *     Preconditioning\n\
  *\n\
  *     If the full SVD is needed, the right singular vectors are computed\n\
  *     from a matrix equation, and for that we need theoretical analysis\n\
  *     of the Businger-Golub pivoting. So we use DGEQP3 as the first RR QRF.\n\
  *     In all other cases the first RR QRF can be chosen by other criteria\n\
  *     (eg speed by replacing global with restricted window pivoting, such\n\
  *     as in SGEQPX from TOMS # 782). Good results will be obtained using\n\
  *     SGEQPX with properly (!) chosen numerical parameters.\n\
  *     Any improvement of DGEQP3 improves overal performance of DGEJSV.\n\
  *\n\
  *     A * P1 = Q1 * [ R1^t 0]^t:\n      DO 1963 p = 1, N\n\
  *        .. all columns are free columns\n         IWORK(p) = 0\n 1963 CONTINUE\n      CALL DGEQP3( M,N,A,LDA, IWORK,WORK, WORK(N+1),LWORK-N, IERR )\n\
  *\n\
  *     The upper triangular matrix R1 from the first QRF is inspected for\n\
  *     rank deficiency and possibilities for deflation, or possible\n\
  *     ill-conditioning. Depending on the user specified flag L2RANK,\n\
  *     the procedure explores possibilities to reduce the numerical\n\
  *     rank by inspecting the computed upper triangular factor. If\n\
  *     L2RANK or L2ABER are up, then DGEJSV will compute the SVD of\n\
  *     A + dA, where ||dA|| <= f(M,N)*EPSLN.\n\
  *\n      NR = 1\n      IF ( L2ABER ) THEN\n\
  *        Standard absolute error bound suffices. All sigma_i with\n\
  *        sigma_i < N*EPSLN*||A|| are flushed to zero. This is an\n\
  *        agressive enforcement of lower numerical rank by introducing a\n\
  *        backward error of the order of N*EPSLN*||A||.\n         TEMP1 = DSQRT(DBLE(N))*EPSLN\n         DO 3001 p = 2, N\n            IF ( DABS(A(p,p)) .GE. (TEMP1*DABS(A(1,1))) ) THEN\n               NR = NR + 1\n            ELSE\n               GO TO 3002\n            END IF\n 3001    CONTINUE\n 3002    CONTINUE\n      ELSE IF ( L2RANK ) THEN\n\
  *        .. similarly as above, only slightly more gentle (less agressive).\n\
  *        Sudden drop on the diagonal of R1 is used as the criterion for\n\
  *        close-to-rank-defficient.\n         TEMP1 = DSQRT(SFMIN)\n         DO 3401 p = 2, N\n            IF ( ( DABS(A(p,p)) .LT. (EPSLN*DABS(A(p-1,p-1))) ) .OR.\n     &           ( DABS(A(p,p)) .LT. SMALL ) .OR.\n     &           ( L2KILL .AND. (DABS(A(p,p)) .LT. TEMP1) ) ) GO TO 3402\n            NR = NR + 1\n 3401    CONTINUE\n 3402    CONTINUE\n\
  *\n      ELSE\n\
  *        The goal is high relative accuracy. However, if the matrix\n\
  *        has high scaled condition number the relative accuracy is in\n\
  *        general not feasible. Later on, a condition number estimator\n\
  *        will be deployed to estimate the scaled condition number.\n\
  *        Here we just remove the underflowed part of the triangular\n\
  *        factor. This prevents the situation in which the code is\n\
  *        working hard to get the accuracy not warranted by the data.\n         TEMP1  = DSQRT(SFMIN)\n         DO 3301 p = 2, N\n            IF ( ( DABS(A(p,p)) .LT. SMALL ) .OR.\n     &          ( L2KILL .AND. (DABS(A(p,p)) .LT. TEMP1) ) ) GO TO 3302\n            NR = NR + 1\n 3301    CONTINUE\n 3302    CONTINUE\n\
  *\n      END IF\n\
  *\n      ALMORT = .FALSE.\n      IF ( NR .EQ. N ) THEN\n         MAXPRJ = ONE\n         DO 3051 p = 2, N\n            TEMP1  = DABS(A(p,p)) / SVA(IWORK(p))\n            MAXPRJ = DMIN1( MAXPRJ, TEMP1 )\n 3051    CONTINUE\n         IF ( MAXPRJ**2 .GE. ONE - DBLE(N)*EPSLN ) ALMORT = .TRUE.\n      END IF\n\
  *\n\
  *\n      SCONDA = - ONE\n      CONDR1 = - ONE\n      CONDR2 = - ONE\n\
  *\n      IF ( ERREST ) THEN\n         IF ( N .EQ. NR ) THEN\n            IF ( RSVEC ) THEN\n\
  *              .. V is available as workspace\n               CALL DLACPY( 'U', N, N, A, LDA, V, LDV )\n               DO 3053 p = 1, N\n                  TEMP1 = SVA(IWORK(p))\n                  CALL DSCAL( p, ONE/TEMP1, V(1,p), 1 )\n 3053          CONTINUE\n               CALL DPOCON( 'U', N, V, LDV, ONE, TEMP1,\n     &              WORK(N+1), IWORK(2*N+M+1), IERR )\n            ELSE IF ( LSVEC ) THEN\n\
  *              .. U is available as workspace\n               CALL DLACPY( 'U', N, N, A, LDA, U, LDU )\n               DO 3054 p = 1, N\n                  TEMP1 = SVA(IWORK(p))\n                  CALL DSCAL( p, ONE/TEMP1, U(1,p), 1 )\n 3054          CONTINUE\n               CALL DPOCON( 'U', N, U, LDU, ONE, TEMP1,\n     &              WORK(N+1), IWORK(2*N+M+1), IERR )\n            ELSE\n               CALL DLACPY( 'U', N, N, A, LDA, WORK(N+1), N )\n               DO 3052 p = 1, N\n                  TEMP1 = SVA(IWORK(p))\n                  CALL DSCAL( p, ONE/TEMP1, WORK(N+(p-1)*N+1), 1 )\n 3052          CONTINUE\n\
  *           .. the columns of R are scaled to have unit Euclidean lengths.\n               CALL DPOCON( 'U', N, WORK(N+1), N, ONE, TEMP1,\n     &              WORK(N+N*N+1), IWORK(2*N+M+1), IERR )\n            END IF\n            SCONDA = ONE / DSQRT(TEMP1)\n\
  *           SCONDA is an estimate of DSQRT(||(R^t * R)^(-1)||_1).\n\
  *           N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA\n         ELSE\n            SCONDA = - ONE\n         END IF\n      END IF\n\
  *\n      L2PERT = L2PERT .AND. ( DABS( A(1,1)/A(NR,NR) ) .GT. DSQRT(BIG1) )\n\
  *     If there is no violent scaling, artificial perturbation is not needed.\n\
  *\n\
  *     Phase 3:\n\
  *\n\n      IF ( .NOT. ( RSVEC .OR. LSVEC ) ) THEN\n\
  *\n\
  *         Singular Values only\n\
  *\n\
  *         .. transpose A(1:NR,1:N)\n         DO 1946 p = 1, MIN0( N-1, NR )\n            CALL DCOPY( N-p, A(p,p+1), LDA, A(p+1,p), 1 )\n 1946    CONTINUE\n\
  *\n\
  *        The following two DO-loops introduce small relative perturbation\n\
  *        into the strict upper triangle of the lower triangular matrix.\n\
  *        Small entries below the main diagonal are also changed.\n\
  *        This modification is useful if the computing environment does not\n\
  *        provide/allow FLUSH TO ZERO underflow, for it prevents many\n\
  *        annoying denormalized numbers in case of strongly scaled matrices.\n\
  *        The perturbation is structured so that it does not introduce any\n\
  *        new perturbation of the singular values, and it does not destroy\n\
  *        the job done by the preconditioner.\n\
  *        The licence for this perturbation is in the variable L2PERT, which\n\
  *        should be .FALSE. if FLUSH TO ZERO underflow is active.\n\
  *\n         IF ( .NOT. ALMORT ) THEN\n\
  *\n            IF ( L2PERT ) THEN\n\
  *              XSC = DSQRT(SMALL)\n               XSC = EPSLN / DBLE(N)\n               DO 4947 q = 1, NR\n                  TEMP1 = XSC*DABS(A(q,q))\n                  DO 4949 p = 1, N\n                     IF ( ( (p.GT.q) .AND. (DABS(A(p,q)).LE.TEMP1) )\n     &                    .OR. ( p .LT. q ) )\n     &                     A(p,q) = DSIGN( TEMP1, A(p,q) )\n 4949             CONTINUE\n 4947          CONTINUE\n            ELSE\n               CALL DLASET( 'U', NR-1,NR-1, ZERO,ZERO, A(1,2),LDA )\n            END IF\n\
  *\n\
  *            .. second preconditioning using the QR factorization\n\
  *\n            CALL DGEQRF( N,NR, A,LDA, WORK, WORK(N+1),LWORK-N, IERR )\n\
  *\n\
  *           .. and transpose upper to lower triangular\n            DO 1948 p = 1, NR - 1\n               CALL DCOPY( NR-p, A(p,p+1), LDA, A(p+1,p), 1 )\n 1948       CONTINUE\n\
  *\n         END IF\n\
  *\n\
  *           Row-cyclic Jacobi SVD algorithm with column pivoting\n\
  *\n\
  *           .. again some perturbation (a \"background noise\") is added\n\
  *           to drown denormals\n            IF ( L2PERT ) THEN\n\
  *              XSC = DSQRT(SMALL)\n               XSC = EPSLN / DBLE(N)\n               DO 1947 q = 1, NR\n                  TEMP1 = XSC*DABS(A(q,q))\n                  DO 1949 p = 1, NR\n                     IF ( ( (p.GT.q) .AND. (DABS(A(p,q)).LE.TEMP1) )\n     &                       .OR. ( p .LT. q ) )\n     &                   A(p,q) = DSIGN( TEMP1, A(p,q) )\n 1949             CONTINUE\n 1947          CONTINUE\n            ELSE\n               CALL DLASET( 'U', NR-1, NR-1, ZERO, ZERO, A(1,2), LDA )\n            END IF\n\
  *\n\
  *           .. and one-sided Jacobi rotations are started on a lower\n\
  *           triangular matrix (plus perturbation which is ignored in\n\
  *           the part which destroys triangular form (confusing?!))\n\
  *\n            CALL DGESVJ( 'L', 'NoU', 'NoV', NR, NR, A, LDA, SVA,\n     &                      N, V, LDV, WORK, LWORK, INFO )\n\
  *\n            SCALEM  = WORK(1)\n            NUMRANK = IDNINT(WORK(2))\n\
  *\n\
  *\n      ELSE IF ( RSVEC .AND. ( .NOT. LSVEC ) ) THEN\n\
  *\n\
  *        -> Singular Values and Right Singular Vectors <-\n\
  *\n         IF ( ALMORT ) THEN\n\
  *\n\
  *           .. in this case NR equals N\n            DO 1998 p = 1, NR\n               CALL DCOPY( N-p+1, A(p,p), LDA, V(p,p), 1 )\n 1998       CONTINUE\n            CALL DLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, V(1,2), LDV )\n\
  *\n            CALL DGESVJ( 'L','U','N', N, NR, V,LDV, SVA, NR, A,LDA,\n     &                  WORK, LWORK, INFO )\n            SCALEM  = WORK(1)\n            NUMRANK = IDNINT(WORK(2))\n\n         ELSE\n\
  *\n\
  *        .. two more QR factorizations ( one QRF is not enough, two require\n\
  *        accumulated product of Jacobi rotations, three are perfect )\n\
  *\n            CALL DLASET( 'Lower', NR-1, NR-1, ZERO, ZERO, A(2,1), LDA )\n            CALL DGELQF( NR, N, A, LDA, WORK, WORK(N+1), LWORK-N, IERR)\n            CALL DLACPY( 'Lower', NR, NR, A, LDA, V, LDV )\n            CALL DLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, V(1,2), LDV )\n            CALL DGEQRF( NR, NR, V, LDV, WORK(N+1), WORK(2*N+1),\n     &                   LWORK-2*N, IERR )\n            DO 8998 p = 1, NR\n               CALL DCOPY( NR-p+1, V(p,p), LDV, V(p,p), 1 )\n 8998       CONTINUE\n            CALL DLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, V(1,2), LDV )\n\
  *\n            CALL DGESVJ( 'Lower', 'U','N', NR, NR, V,LDV, SVA, NR, U,\n     &                  LDU, WORK(N+1), LWORK, INFO )\n            SCALEM  = WORK(N+1)\n            NUMRANK = IDNINT(WORK(N+2))\n            IF ( NR .LT. N ) THEN\n               CALL DLASET( 'A',N-NR, NR, ZERO,ZERO, V(NR+1,1),   LDV )\n               CALL DLASET( 'A',NR, N-NR, ZERO,ZERO, V(1,NR+1),   LDV )\n               CALL DLASET( 'A',N-NR,N-NR,ZERO,ONE, V(NR+1,NR+1), LDV )\n            END IF\n\
  *\n         CALL DORMLQ( 'Left', 'Transpose', N, N, NR, A, LDA, WORK,\n     &               V, LDV, WORK(N+1), LWORK-N, IERR )\n\
  *\n         END IF\n\
  *\n         DO 8991 p = 1, N\n            CALL DCOPY( N, V(p,1), LDV, A(IWORK(p),1), LDA )\n 8991    CONTINUE\n         CALL DLACPY( 'All', N, N, A, LDA, V, LDV )\n\
  *\n         IF ( TRANSP ) THEN\n            CALL DLACPY( 'All', N, N, V, LDV, U, LDU )\n         END IF\n\
  *\n      ELSE IF ( LSVEC .AND. ( .NOT. RSVEC ) ) THEN\n\
  *\n\
  *        .. Singular Values and Left Singular Vectors                 ..\n\
  *\n\
  *        .. second preconditioning step to avoid need to accumulate\n\
  *        Jacobi rotations in the Jacobi iterations.\n         DO 1965 p = 1, NR\n            CALL DCOPY( N-p+1, A(p,p), LDA, U(p,p), 1 )\n 1965    CONTINUE\n         CALL DLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, U(1,2), LDU )\n\
  *\n         CALL DGEQRF( N, NR, U, LDU, WORK(N+1), WORK(2*N+1),\n     &              LWORK-2*N, IERR )\n\
  *\n         DO 1967 p = 1, NR - 1\n            CALL DCOPY( NR-p, U(p,p+1), LDU, U(p+1,p), 1 )\n 1967    CONTINUE\n         CALL DLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, U(1,2), LDU )\n\
  *\n         CALL DGESVJ( 'Lower', 'U', 'N', NR,NR, U, LDU, SVA, NR, A,\n     &        LDA, WORK(N+1), LWORK-N, INFO )\n         SCALEM  = WORK(N+1)\n         NUMRANK = IDNINT(WORK(N+2))\n\
  *\n         IF ( NR .LT. M ) THEN\n            CALL DLASET( 'A',  M-NR, NR,ZERO, ZERO, U(NR+1,1), LDU )\n            IF ( NR .LT. N1 ) THEN\n               CALL DLASET( 'A',NR, N1-NR, ZERO, ZERO, U(1,NR+1), LDU )\n               CALL DLASET( 'A',M-NR,N1-NR,ZERO,ONE,U(NR+1,NR+1), LDU )\n            END IF\n         END IF\n\
  *\n         CALL DORMQR( 'Left', 'No Tr', M, N1, N, A, LDA, WORK, U,\n     &               LDU, WORK(N+1), LWORK-N, IERR )\n\
  *\n         IF ( ROWPIV )\n     &       CALL DLASWP( N1, U, LDU, 1, M-1, IWORK(2*N+1), -1 )\n\
  *\n         DO 1974 p = 1, N1\n            XSC = ONE / DNRM2( M, U(1,p), 1 )\n            CALL DSCAL( M, XSC, U(1,p), 1 )\n 1974    CONTINUE\n\
  *\n         IF ( TRANSP ) THEN\n            CALL DLACPY( 'All', N, N, U, LDU, V, LDV )\n         END IF\n\
  *\n      ELSE\n\
  *\n\
  *        .. Full SVD ..\n\
  *\n         IF ( .NOT. JRACC ) THEN\n\
  *\n         IF ( .NOT. ALMORT ) THEN\n\
  *\n\
  *           Second Preconditioning Step (QRF [with pivoting])\n\
  *           Note that the composition of TRANSPOSE, QRF and TRANSPOSE is\n\
  *           equivalent to an LQF CALL. Since in many libraries the QRF\n\
  *           seems to be better optimized than the LQF, we do explicit\n\
  *           transpose and use the QRF. This is subject to changes in an\n\
  *           optimized implementation of DGEJSV.\n\
  *\n            DO 1968 p = 1, NR\n               CALL DCOPY( N-p+1, A(p,p), LDA, V(p,p), 1 )\n 1968       CONTINUE\n\
  *\n\
  *           .. the following two loops perturb small entries to avoid\n\
  *           denormals in the second QR factorization, where they are\n\
  *           as good as zeros. This is done to avoid painfully slow\n\
  *           computation with denormals. The relative size of the perturbation\n\
  *           is a parameter that can be changed by the implementer.\n\
  *           This perturbation device will be obsolete on machines with\n\
  *           properly implemented arithmetic.\n\
  *           To switch it off, set L2PERT=.FALSE. To remove it from  the\n\
  *           code, remove the action under L2PERT=.TRUE., leave the ELSE part.\n\
  *           The following two loops should be blocked and fused with the\n\
  *           transposed copy above.\n\
  *\n            IF ( L2PERT ) THEN\n               XSC = DSQRT(SMALL)\n               DO 2969 q = 1, NR\n                  TEMP1 = XSC*DABS( V(q,q) )\n                  DO 2968 p = 1, N\n                     IF ( ( p .GT. q ) .AND. ( DABS(V(p,q)) .LE. TEMP1 )\n     &                   .OR. ( p .LT. q ) )\n     &                   V(p,q) = DSIGN( TEMP1, V(p,q) )\n                     IF ( p. LT. q ) V(p,q) = - V(p,q)\n 2968             CONTINUE\n 2969          CONTINUE\n            ELSE\n               CALL DLASET( 'U', NR-1, NR-1, ZERO, ZERO, V(1,2), LDV )\n            END IF\n\
  *\n\
  *           Estimate the row scaled condition number of R1\n\
  *           (If R1 is rectangular, N > NR, then the condition number\n\
  *           of the leading NR x NR submatrix is estimated.)\n\
  *\n            CALL DLACPY( 'L', NR, NR, V, LDV, WORK(2*N+1), NR )\n            DO 3950 p = 1, NR\n               TEMP1 = DNRM2(NR-p+1,WORK(2*N+(p-1)*NR+p),1)\n               CALL DSCAL(NR-p+1,ONE/TEMP1,WORK(2*N+(p-1)*NR+p),1)\n 3950       CONTINUE\n            CALL DPOCON('Lower',NR,WORK(2*N+1),NR,ONE,TEMP1,\n     &                   WORK(2*N+NR*NR+1),IWORK(M+2*N+1),IERR)\n            CONDR1 = ONE / DSQRT(TEMP1)\n\
  *           .. here need a second oppinion on the condition number\n\
  *           .. then assume worst case scenario\n\
  *           R1 is OK for inverse <=> CONDR1 .LT. DBLE(N)\n\
  *           more conservative    <=> CONDR1 .LT. DSQRT(DBLE(N))\n\
  *\n            COND_OK = DSQRT(DBLE(NR))\n\
  *[TP]       COND_OK is a tuning parameter.\n\n            IF ( CONDR1 .LT. COND_OK ) THEN\n\
  *              .. the second QRF without pivoting. Note: in an optimized\n\
  *              implementation, this QRF should be implemented as the QRF\n\
  *              of a lower triangular matrix.\n\
  *              R1^t = Q2 * R2\n               CALL DGEQRF( N, NR, V, LDV, WORK(N+1), WORK(2*N+1),\n     &              LWORK-2*N, IERR )\n\
  *\n               IF ( L2PERT ) THEN\n                  XSC = DSQRT(SMALL)/EPSLN\n                  DO 3959 p = 2, NR\n                     DO 3958 q = 1, p - 1\n                        TEMP1 = XSC * DMIN1(DABS(V(p,p)),DABS(V(q,q)))\n                        IF ( DABS(V(q,p)) .LE. TEMP1 )\n     &                     V(q,p) = DSIGN( TEMP1, V(q,p) )\n 3958                CONTINUE\n 3959             CONTINUE\n               END IF\n\
  *\n               IF ( NR .NE. N )\n\
  *              .. save ...\n     &         CALL DLACPY( 'A', N, NR, V, LDV, WORK(2*N+1), N )\n\
  *\n\
  *           .. this transposed copy should be better than naive\n               DO 1969 p = 1, NR - 1\n                  CALL DCOPY( NR-p, V(p,p+1), LDV, V(p+1,p), 1 )\n 1969          CONTINUE\n\
  *\n               CONDR2 = CONDR1\n\
  *\n            ELSE\n\
  *\n\
  *              .. ill-conditioned case: second QRF with pivoting\n\
  *              Note that windowed pivoting would be equaly good\n\
  *              numerically, and more run-time efficient. So, in\n\
  *              an optimal implementation, the next call to DGEQP3\n\
  *              should be replaced with eg. CALL SGEQPX (ACM TOMS #782)\n\
  *              with properly (carefully) chosen parameters.\n\
  *\n\
  *              R1^t * P2 = Q2 * R2\n               DO 3003 p = 1, NR\n                  IWORK(N+p) = 0\n 3003          CONTINUE\n               CALL DGEQP3( N, NR, V, LDV, IWORK(N+1), WORK(N+1),\n     &                  WORK(2*N+1), LWORK-2*N, IERR )\n\
  **               CALL DGEQRF( N, NR, V, LDV, WORK(N+1), WORK(2*N+1),\n\
  **     &              LWORK-2*N, IERR )\n               IF ( L2PERT ) THEN\n                  XSC = DSQRT(SMALL)\n                  DO 3969 p = 2, NR\n                     DO 3968 q = 1, p - 1\n                        TEMP1 = XSC * DMIN1(DABS(V(p,p)),DABS(V(q,q)))\n                        IF ( DABS(V(q,p)) .LE. TEMP1 )\n     &                     V(q,p) = DSIGN( TEMP1, V(q,p) )\n 3968                CONTINUE\n 3969             CONTINUE\n               END IF\n\
  *\n               CALL DLACPY( 'A', N, NR, V, LDV, WORK(2*N+1), N )\n\
  *\n               IF ( L2PERT ) THEN\n                  XSC = DSQRT(SMALL)\n                  DO 8970 p = 2, NR\n                     DO 8971 q = 1, p - 1\n                        TEMP1 = XSC * DMIN1(DABS(V(p,p)),DABS(V(q,q)))\n                        V(p,q) = - DSIGN( TEMP1, V(q,p) )\n 8971                CONTINUE\n 8970             CONTINUE\n               ELSE\n                  CALL DLASET( 'L',NR-1,NR-1,ZERO,ZERO,V(2,1),LDV )\n               END IF\n\
  *              Now, compute R2 = L3 * Q3, the LQ factorization.\n               CALL DGELQF( NR, NR, V, LDV, WORK(2*N+N*NR+1),\n     &               WORK(2*N+N*NR+NR+1), LWORK-2*N-N*NR-NR, IERR )\n\
  *              .. and estimate the condition number\n               CALL DLACPY( 'L',NR,NR,V,LDV,WORK(2*N+N*NR+NR+1),NR )\n               DO 4950 p = 1, NR\n                  TEMP1 = DNRM2( p, WORK(2*N+N*NR+NR+p), NR )\n                  CALL DSCAL( p, ONE/TEMP1, WORK(2*N+N*NR+NR+p), NR )\n 4950          CONTINUE\n               CALL DPOCON( 'L',NR,WORK(2*N+N*NR+NR+1),NR,ONE,TEMP1,\n     &              WORK(2*N+N*NR+NR+NR*NR+1),IWORK(M+2*N+1),IERR )\n               CONDR2 = ONE / DSQRT(TEMP1)\n\
  *\n               IF ( CONDR2 .GE. COND_OK ) THEN\n\
  *                 .. save the Householder vectors used for Q3\n\
  *                 (this overwrittes the copy of R2, as it will not be\n\
  *                 needed in this branch, but it does not overwritte the\n\
  *                 Huseholder vectors of Q2.).\n                  CALL DLACPY( 'U', NR, NR, V, LDV, WORK(2*N+1), N )\n\
  *                 .. and the rest of the information on Q3 is in\n\
  *                 WORK(2*N+N*NR+1:2*N+N*NR+N)\n               END IF\n\
  *\n            END IF\n\
  *\n            IF ( L2PERT ) THEN\n               XSC = DSQRT(SMALL)\n               DO 4968 q = 2, NR\n                  TEMP1 = XSC * V(q,q)\n                  DO 4969 p = 1, q - 1\n\
  *                    V(p,q) = - DSIGN( TEMP1, V(q,p) )\n                     V(p,q) = - DSIGN( TEMP1, V(p,q) )\n 4969             CONTINUE\n 4968          CONTINUE\n            ELSE\n               CALL DLASET( 'U', NR-1,NR-1, ZERO,ZERO, V(1,2), LDV )\n            END IF\n\
  *\n\
  *        Second preconditioning finished; continue with Jacobi SVD\n\
  *        The input matrix is lower trinagular.\n\
  *\n\
  *        Recover the right singular vectors as solution of a well\n\
  *        conditioned triangular matrix equation.\n\
  *\n            IF ( CONDR1 .LT. COND_OK ) THEN\n\
  *\n               CALL DGESVJ( 'L','U','N',NR,NR,V,LDV,SVA,NR,U,\n     &              LDU,WORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,INFO )\n               SCALEM  = WORK(2*N+N*NR+NR+1)\n               NUMRANK = IDNINT(WORK(2*N+N*NR+NR+2))\n               DO 3970 p = 1, NR\n                  CALL DCOPY( NR, V(1,p), 1, U(1,p), 1 )\n                  CALL DSCAL( NR, SVA(p),    V(1,p), 1 )\n 3970          CONTINUE\n\n\
  *        .. pick the right matrix equation and solve it\n\
  *\n               IF ( NR. EQ. N ) THEN\n\
  * :))             .. best case, R1 is inverted. The solution of this matrix\n\
  *                 equation is Q2*V2 = the product of the Jacobi rotations\n\
  *                 used in DGESVJ, premultiplied with the orthogonal matrix\n\
  *                 from the second QR factorization.\n                  CALL DTRSM( 'L','U','N','N', NR,NR,ONE, A,LDA, V,LDV )\n               ELSE\n\
  *                 .. R1 is well conditioned, but non-square. Transpose(R2)\n\
  *                 is inverted to get the product of the Jacobi rotations\n\
  *                 used in DGESVJ. The Q-factor from the second QR\n\
  *                 factorization is then built in explicitly.\n                  CALL DTRSM('L','U','T','N',NR,NR,ONE,WORK(2*N+1),\n     &                 N,V,LDV)\n                  IF ( NR .LT. N ) THEN\n                    CALL DLASET('A',N-NR,NR,ZERO,ZERO,V(NR+1,1),LDV)\n                    CALL DLASET('A',NR,N-NR,ZERO,ZERO,V(1,NR+1),LDV)\n                    CALL DLASET('A',N-NR,N-NR,ZERO,ONE,V(NR+1,NR+1),LDV)\n                  END IF\n                  CALL DORMQR('L','N',N,N,NR,WORK(2*N+1),N,WORK(N+1),\n     &                 V,LDV,WORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR)\n               END IF\n\
  *\n            ELSE IF ( CONDR2 .LT. COND_OK ) THEN\n\
  *\n\
  * :)           .. the input matrix A is very likely a relative of\n\
  *              the Kahan matrix :)\n\
  *              The matrix R2 is inverted. The solution of the matrix equation\n\
  *              is Q3^T*V3 = the product of the Jacobi rotations (appplied to\n\
  *              the lower triangular L3 from the LQ factorization of\n\
  *              R2=L3*Q3), pre-multiplied with the transposed Q3.\n               CALL DGESVJ( 'L', 'U', 'N', NR, NR, V, LDV, SVA, NR, U,\n     &              LDU, WORK(2*N+N*NR+NR+1), LWORK-2*N-N*NR-NR, INFO )\n               SCALEM  = WORK(2*N+N*NR+NR+1)\n               NUMRANK = IDNINT(WORK(2*N+N*NR+NR+2))\n               DO 3870 p = 1, NR\n                  CALL DCOPY( NR, V(1,p), 1, U(1,p), 1 )\n                  CALL DSCAL( NR, SVA(p),    U(1,p), 1 )\n 3870          CONTINUE\n               CALL DTRSM('L','U','N','N',NR,NR,ONE,WORK(2*N+1),N,U,LDU)\n\
  *              .. apply the permutation from the second QR factorization\n               DO 873 q = 1, NR\n                  DO 872 p = 1, NR\n                     WORK(2*N+N*NR+NR+IWORK(N+p)) = U(p,q)\n 872              CONTINUE\n                  DO 874 p = 1, NR\n                     U(p,q) = WORK(2*N+N*NR+NR+p)\n 874              CONTINUE\n 873           CONTINUE\n               IF ( NR .LT. N ) THEN\n                  CALL DLASET( 'A',N-NR,NR,ZERO,ZERO,V(NR+1,1),LDV )\n                  CALL DLASET( 'A',NR,N-NR,ZERO,ZERO,V(1,NR+1),LDV )\n                  CALL DLASET( 'A',N-NR,N-NR,ZERO,ONE,V(NR+1,NR+1),LDV )\n               END IF\n               CALL DORMQR( 'L','N',N,N,NR,WORK(2*N+1),N,WORK(N+1),\n     &              V,LDV,WORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR )\n            ELSE\n\
  *              Last line of defense.\n\
  * #:(          This is a rather pathological case: no scaled condition\n\
  *              improvement after two pivoted QR factorizations. Other\n\
  *              possibility is that the rank revealing QR factorization\n\
  *              or the condition estimator has failed, or the COND_OK\n\
  *              is set very close to ONE (which is unnecessary). Normally,\n\
  *              this branch should never be executed, but in rare cases of\n\
  *              failure of the RRQR or condition estimator, the last line of\n\
  *              defense ensures that DGEJSV completes the task.\n\
  *              Compute the full SVD of L3 using DGESVJ with explicit\n\
  *              accumulation of Jacobi rotations.\n               CALL DGESVJ( 'L', 'U', 'V', NR, NR, V, LDV, SVA, NR, U,\n     &              LDU, WORK(2*N+N*NR+NR+1), LWORK-2*N-N*NR-NR, INFO )\n               SCALEM  = WORK(2*N+N*NR+NR+1)\n               NUMRANK = IDNINT(WORK(2*N+N*NR+NR+2))\n               IF ( NR .LT. N ) THEN\n                  CALL DLASET( 'A',N-NR,NR,ZERO,ZERO,V(NR+1,1),LDV )\n                  CALL DLASET( 'A',NR,N-NR,ZERO,ZERO,V(1,NR+1),LDV )\n                  CALL DLASET( 'A',N-NR,N-NR,ZERO,ONE,V(NR+1,NR+1),LDV )\n               END IF\n               CALL DORMQR( 'L','N',N,N,NR,WORK(2*N+1),N,WORK(N+1),\n     &              V,LDV,WORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR )\n\
  *\n               CALL DORMLQ( 'L', 'T', NR, NR, NR, WORK(2*N+1), N,\n     &              WORK(2*N+N*NR+1), U, LDU, WORK(2*N+N*NR+NR+1),\n     &              LWORK-2*N-N*NR-NR, IERR )\n               DO 773 q = 1, NR\n                  DO 772 p = 1, NR\n                     WORK(2*N+N*NR+NR+IWORK(N+p)) = U(p,q)\n 772              CONTINUE\n                  DO 774 p = 1, NR\n                     U(p,q) = WORK(2*N+N*NR+NR+p)\n 774              CONTINUE\n 773           CONTINUE\n\
  *\n            END IF\n\
  *\n\
  *           Permute the rows of V using the (column) permutation from the\n\
  *           first QRF. Also, scale the columns to make them unit in\n\
  *           Euclidean norm. This applies to all cases.\n\
  *\n            TEMP1 = DSQRT(DBLE(N)) * EPSLN\n            DO 1972 q = 1, N\n               DO 972 p = 1, N\n                  WORK(2*N+N*NR+NR+IWORK(p)) = V(p,q)\n  972          CONTINUE\n               DO 973 p = 1, N\n                  V(p,q) = WORK(2*N+N*NR+NR+p)\n  973          CONTINUE\n               XSC = ONE / DNRM2( N, V(1,q), 1 )\n               IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )\n     &           CALL DSCAL( N, XSC, V(1,q), 1 )\n 1972       CONTINUE\n\
  *           At this moment, V contains the right singular vectors of A.\n\
  *           Next, assemble the left singular vector matrix U (M x N).\n            IF ( NR .LT. M ) THEN\n               CALL DLASET( 'A', M-NR, NR, ZERO, ZERO, U(NR+1,1), LDU )\n               IF ( NR .LT. N1 ) THEN\n                  CALL DLASET('A',NR,N1-NR,ZERO,ZERO,U(1,NR+1),LDU)\n                  CALL DLASET('A',M-NR,N1-NR,ZERO,ONE,U(NR+1,NR+1),LDU)\n               END IF\n            END IF\n\
  *\n\
  *           The Q matrix from the first QRF is built into the left singular\n\
  *           matrix U. This applies to all cases.\n\
  *\n            CALL DORMQR( 'Left', 'No_Tr', M, N1, N, A, LDA, WORK, U,\n     &           LDU, WORK(N+1), LWORK-N, IERR )\n\n\
  *           The columns of U are normalized. The cost is O(M*N) flops.\n            TEMP1 = DSQRT(DBLE(M)) * EPSLN\n            DO 1973 p = 1, NR\n               XSC = ONE / DNRM2( M, U(1,p), 1 )\n               IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )\n     &          CALL DSCAL( M, XSC, U(1,p), 1 )\n 1973       CONTINUE\n\
  *\n\
  *           If the initial QRF is computed with row pivoting, the left\n\
  *           singular vectors must be adjusted.\n\
  *\n            IF ( ROWPIV )\n     &          CALL DLASWP( N1, U, LDU, 1, M-1, IWORK(2*N+1), -1 )\n\
  *\n         ELSE\n\
  *\n\
  *        .. the initial matrix A has almost orthogonal columns and\n\
  *        the second QRF is not needed\n\
  *\n            CALL DLACPY( 'Upper', N, N, A, LDA, WORK(N+1), N )\n            IF ( L2PERT ) THEN\n               XSC = DSQRT(SMALL)\n               DO 5970 p = 2, N\n                  TEMP1 = XSC * WORK( N + (p-1)*N + p )\n                  DO 5971 q = 1, p - 1\n                     WORK(N+(q-1)*N+p)=-DSIGN(TEMP1,WORK(N+(p-1)*N+q))\n 5971             CONTINUE\n 5970          CONTINUE\n            ELSE\n               CALL DLASET( 'Lower',N-1,N-1,ZERO,ZERO,WORK(N+2),N )\n            END IF\n\
  *\n            CALL DGESVJ( 'Upper', 'U', 'N', N, N, WORK(N+1), N, SVA,\n     &           N, U, LDU, WORK(N+N*N+1), LWORK-N-N*N, INFO )\n\
  *\n            SCALEM  = WORK(N+N*N+1)\n            NUMRANK = IDNINT(WORK(N+N*N+2))\n            DO 6970 p = 1, N\n               CALL DCOPY( N, WORK(N+(p-1)*N+1), 1, U(1,p), 1 )\n               CALL DSCAL( N, SVA(p), WORK(N+(p-1)*N+1), 1 )\n 6970       CONTINUE\n\
  *\n            CALL DTRSM( 'Left', 'Upper', 'NoTrans', 'No UD', N, N,\n     &           ONE, A, LDA, WORK(N+1), N )\n            DO 6972 p = 1, N\n               CALL DCOPY( N, WORK(N+p), N, V(IWORK(p),1), LDV )\n 6972       CONTINUE\n            TEMP1 = DSQRT(DBLE(N))*EPSLN\n            DO 6971 p = 1, N\n               XSC = ONE / DNRM2( N, V(1,p), 1 )\n               IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )\n     &            CALL DSCAL( N, XSC, V(1,p), 1 )\n 6971       CONTINUE\n\
  *\n\
  *           Assemble the left singular vector matrix U (M x N).\n\
  *\n            IF ( N .LT. M ) THEN\n               CALL DLASET( 'A',  M-N, N, ZERO, ZERO, U(N+1,1), LDU )\n               IF ( N .LT. N1 ) THEN\n                  CALL DLASET( 'A',N,  N1-N, ZERO, ZERO,  U(1,N+1),LDU )\n                  CALL DLASET( 'A',M-N,N1-N, ZERO, ONE,U(N+1,N+1),LDU )\n               END IF\n            END IF\n            CALL DORMQR( 'Left', 'No Tr', M, N1, N, A, LDA, WORK, U,\n     &           LDU, WORK(N+1), LWORK-N, IERR )\n            TEMP1 = DSQRT(DBLE(M))*EPSLN\n            DO 6973 p = 1, N1\n               XSC = ONE / DNRM2( M, U(1,p), 1 )\n               IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )\n     &            CALL DSCAL( M, XSC, U(1,p), 1 )\n 6973       CONTINUE\n\
  *\n            IF ( ROWPIV )\n     &         CALL DLASWP( N1, U, LDU, 1, M-1, IWORK(2*N+1), -1 )\n\
  *\n         END IF\n\
  *\n\
  *        end of the  >> almost orthogonal case <<  in the full SVD\n\
  *\n         ELSE\n\
  *\n\
  *        This branch deploys a preconditioned Jacobi SVD with explicitly\n\
  *        accumulated rotations. It is included as optional, mainly for\n\
  *        experimental purposes. It does perfom well, and can also be used.\n\
  *        In this implementation, this branch will be automatically activated\n\
  *        if the  condition number sigma_max(A) / sigma_min(A) is predicted\n\
  *        to be greater than the overflow threshold. This is because the\n\
  *        a posteriori computation of the singular vectors assumes robust\n\
  *        implementation of BLAS and some LAPACK procedures, capable of working\n\
  *        in presence of extreme values. Since that is not always the case, ...\n\
  *\n         DO 7968 p = 1, NR\n            CALL DCOPY( N-p+1, A(p,p), LDA, V(p,p), 1 )\n 7968    CONTINUE\n\
  *\n         IF ( L2PERT ) THEN\n            XSC = DSQRT(SMALL/EPSLN)\n            DO 5969 q = 1, NR\n               TEMP1 = XSC*DABS( V(q,q) )\n               DO 5968 p = 1, N\n                  IF ( ( p .GT. q ) .AND. ( DABS(V(p,q)) .LE. TEMP1 )\n     &                .OR. ( p .LT. q ) )\n     &                V(p,q) = DSIGN( TEMP1, V(p,q) )\n                  IF ( p. LT. q ) V(p,q) = - V(p,q)\n 5968          CONTINUE\n 5969       CONTINUE\n         ELSE\n            CALL DLASET( 'U', NR-1, NR-1, ZERO, ZERO, V(1,2), LDV )\n         END IF\n\n         CALL DGEQRF( N, NR, V, LDV, WORK(N+1), WORK(2*N+1),\n     &        LWORK-2*N, IERR )\n         CALL DLACPY( 'L', N, NR, V, LDV, WORK(2*N+1), N )\n\
  *\n         DO 7969 p = 1, NR\n            CALL DCOPY( NR-p+1, V(p,p), LDV, U(p,p), 1 )\n 7969    CONTINUE\n\n         IF ( L2PERT ) THEN\n            XSC = DSQRT(SMALL/EPSLN)\n            DO 9970 q = 2, NR\n               DO 9971 p = 1, q - 1\n                  TEMP1 = XSC * DMIN1(DABS(U(p,p)),DABS(U(q,q)))\n                  U(p,q) = - DSIGN( TEMP1, U(q,p) )\n 9971          CONTINUE\n 9970       CONTINUE\n         ELSE\n            CALL DLASET('U', NR-1, NR-1, ZERO, ZERO, U(1,2), LDU )\n         END IF\n\n         CALL DGESVJ( 'G', 'U', 'V', NR, NR, U, LDU, SVA,\n     &        N, V, LDV, WORK(2*N+N*NR+1), LWORK-2*N-N*NR, INFO )\n         SCALEM  = WORK(2*N+N*NR+1)\n         NUMRANK = IDNINT(WORK(2*N+N*NR+2))\n\n         IF ( NR .LT. N ) THEN\n            CALL DLASET( 'A',N-NR,NR,ZERO,ZERO,V(NR+1,1),LDV )\n            CALL DLASET( 'A',NR,N-NR,ZERO,ZERO,V(1,NR+1),LDV )\n            CALL DLASET( 'A',N-NR,N-NR,ZERO,ONE,V(NR+1,NR+1),LDV )\n         END IF\n\n         CALL DORMQR( 'L','N',N,N,NR,WORK(2*N+1),N,WORK(N+1),\n     &        V,LDV,WORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR )\n\
  *\n\
  *           Permute the rows of V using the (column) permutation from the\n\
  *           first QRF. Also, scale the columns to make them unit in\n\
  *           Euclidean norm. This applies to all cases.\n\
  *\n            TEMP1 = DSQRT(DBLE(N)) * EPSLN\n            DO 7972 q = 1, N\n               DO 8972 p = 1, N\n                  WORK(2*N+N*NR+NR+IWORK(p)) = V(p,q)\n 8972          CONTINUE\n               DO 8973 p = 1, N\n                  V(p,q) = WORK(2*N+N*NR+NR+p)\n 8973          CONTINUE\n               XSC = ONE / DNRM2( N, V(1,q), 1 )\n               IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )\n     &           CALL DSCAL( N, XSC, V(1,q), 1 )\n 7972       CONTINUE\n\
  *\n\
  *           At this moment, V contains the right singular vectors of A.\n\
  *           Next, assemble the left singular vector matrix U (M x N).\n\
  *\n         IF ( NR .LT. M ) THEN\n            CALL DLASET( 'A',  M-NR, NR, ZERO, ZERO, U(NR+1,1), LDU )\n            IF ( NR .LT. N1 ) THEN\n               CALL DLASET( 'A',NR,  N1-NR, ZERO, ZERO,  U(1,NR+1),LDU )\n               CALL DLASET( 'A',M-NR,N1-NR, ZERO, ONE,U(NR+1,NR+1),LDU )\n            END IF\n         END IF\n\
  *\n         CALL DORMQR( 'Left', 'No Tr', M, N1, N, A, LDA, WORK, U,\n     &        LDU, WORK(N+1), LWORK-N, IERR )\n\
  *\n            IF ( ROWPIV )\n     &         CALL DLASWP( N1, U, LDU, 1, M-1, IWORK(2*N+1), -1 )\n\
  *\n\
  *\n         END IF\n         IF ( TRANSP ) THEN\n\
  *           .. swap U and V because the procedure worked on A^t\n            DO 6974 p = 1, N\n               CALL DSWAP( N, U(1,p), 1, V(1,p), 1 )\n 6974       CONTINUE\n         END IF\n\
  *\n      END IF\n\
  *     end of the full SVD\n\
  *\n\
  *     Undo scaling, if necessary (and possible)\n\
  *\n      IF ( USCAL2 .LE. (BIG/SVA(1))*USCAL1 ) THEN\n         CALL DLASCL( 'G', 0, 0, USCAL1, USCAL2, NR, 1, SVA, N, IERR )\n         USCAL1 = ONE\n         USCAL2 = ONE\n      END IF\n\
  *\n      IF ( NR .LT. N ) THEN\n         DO 3004 p = NR+1, N\n            SVA(p) = ZERO\n 3004    CONTINUE\n      END IF\n\
  *\n      WORK(1) = USCAL2 * SCALEM\n      WORK(2) = USCAL1\n      IF ( ERREST ) WORK(3) = SCONDA\n      IF ( LSVEC .AND. RSVEC ) THEN\n         WORK(4) = CONDR1\n         WORK(5) = CONDR2\n      END IF\n      IF ( L2TRAN ) THEN\n         WORK(6) = ENTRA\n         WORK(7) = ENTRAT\n      END IF\n\
  *\n      IWORK(1) = NR\n      IWORK(2) = NUMRANK\n      IWORK(3) = WARNING\n\
  *\n      RETURN\n\
  *     ..\n\
  *     .. END OF DGEJSV\n\
  *     ..\n      END\n\
  *\n"
