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version 1.4, 2010/08/06 15:32:30 version 1.19, 2023/08/07 08:38:59
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       SUBROUTINE DLASQ1( N, D, E, WORK, INFO )  *> \brief \b DLASQ1 computes the singular values of a real square bidiagonal matrix. Used by sbdsqr.
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at
   *            http://www.netlib.org/lapack/explore-html/
 *  *
 *  -- LAPACK routine (version 3.2)                                    --  *> \htmlonly
   *> Download DLASQ1 + dependencies
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasq1.f">
   *> [TGZ]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasq1.f">
   *> [ZIP]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasq1.f">
   *> [TXT]</a>
   *> \endhtmlonly
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DLASQ1( N, D, E, WORK, INFO )
   *
   *       .. Scalar Arguments ..
   *       INTEGER            INFO, N
   *       ..
   *       .. Array Arguments ..
   *       DOUBLE PRECISION   D( * ), E( * ), WORK( * )
   *       ..
   *
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> DLASQ1 computes the singular values of a real N-by-N bidiagonal
   *> matrix with diagonal D and off-diagonal E. The singular values
   *> are computed to high relative accuracy, in the absence of
   *> denormalization, underflow and overflow. The algorithm was first
   *> presented in
   *>
   *> "Accurate singular values and differential qd algorithms" by K. V.
   *> Fernando and B. N. Parlett, Numer. Math., Vol-67, No. 2, pp. 191-230,
   *> 1994,
   *>
   *> and the present implementation is described in "An implementation of
   *> the dqds Algorithm (Positive Case)", LAPACK Working Note.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] N
   *> \verbatim
   *>          N is INTEGER
   *>        The number of rows and columns in the matrix. N >= 0.
   *> \endverbatim
   *>
   *> \param[in,out] D
   *> \verbatim
   *>          D is DOUBLE PRECISION array, dimension (N)
   *>        On entry, D contains the diagonal elements of the
   *>        bidiagonal matrix whose SVD is desired. On normal exit,
   *>        D contains the singular values in decreasing order.
   *> \endverbatim
   *>
   *> \param[in,out] E
   *> \verbatim
   *>          E is DOUBLE PRECISION array, dimension (N)
   *>        On entry, elements E(1:N-1) contain the off-diagonal elements
   *>        of the bidiagonal matrix whose SVD is desired.
   *>        On exit, E is overwritten.
   *> \endverbatim
   *>
   *> \param[out] WORK
   *> \verbatim
   *>          WORK is DOUBLE PRECISION array, dimension (4*N)
   *> \endverbatim
   *>
   *> \param[out] INFO
   *> \verbatim
   *>          INFO is INTEGER
   *>        = 0: successful exit
   *>        < 0: if INFO = -i, the i-th argument had an illegal value
   *>        > 0: the algorithm failed
   *>             = 1, a split was marked by a positive value in E
   *>             = 2, current block of Z not diagonalized after 100*N
   *>                  iterations (in inner while loop)  On exit D and E
   *>                  represent a matrix with the same singular values
   *>                  which the calling subroutine could use to finish the
   *>                  computation, or even feed back into DLASQ1
   *>             = 3, termination criterion of outer while loop not met
   *>                  (program created more than N unreduced blocks)
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee
   *> \author Univ. of California Berkeley
   *> \author Univ. of Colorado Denver
   *> \author NAG Ltd.
 *  *
 *  -- Contributed by Osni Marques of the Lawrence Berkeley National   --  *> \ingroup auxOTHERcomputational
 *  -- Laboratory and Beresford Parlett of the Univ. of California at  --  *
 *  -- Berkeley                                                        --  *  =====================================================================
 *  -- November 2008                                                   --        SUBROUTINE DLASQ1( N, D, E, WORK, INFO )
 *  *
   *  -- LAPACK computational routine --
 *  -- 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..--
 *  *
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       DOUBLE PRECISION   D( * ), E( * ), WORK( * )        DOUBLE PRECISION   D( * ), E( * ), WORK( * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  DLASQ1 computes the singular values of a real N-by-N bidiagonal  
 *  matrix with diagonal D and off-diagonal E. The singular values  
 *  are computed to high relative accuracy, in the absence of  
 *  denormalization, underflow and overflow. The algorithm was first  
 *  presented in  
 *  
 *  "Accurate singular values and differential qd algorithms" by K. V.  
 *  Fernando and B. N. Parlett, Numer. Math., Vol-67, No. 2, pp. 191-230,  
 *  1994,  
 *  
 *  and the present implementation is described in "An implementation of  
 *  the dqds Algorithm (Positive Case)", LAPACK Working Note.  
 *  
 *  Arguments  
 *  =========  
 *  
 *  N     (input) INTEGER  
 *        The number of rows and columns in the matrix. N >= 0.  
 *  
 *  D     (input/output) DOUBLE PRECISION array, dimension (N)  
 *        On entry, D contains the diagonal elements of the  
 *        bidiagonal matrix whose SVD is desired. On normal exit,  
 *        D contains the singular values in decreasing order.  
 *  
 *  E     (input/output) DOUBLE PRECISION array, dimension (N)  
 *        On entry, elements E(1:N-1) contain the off-diagonal elements  
 *        of the bidiagonal matrix whose SVD is desired.  
 *        On exit, E is overwritten.  
 *  
 *  WORK  (workspace) DOUBLE PRECISION array, dimension (4*N)  
 *  
 *  INFO  (output) INTEGER  
 *        = 0: successful exit  
 *        < 0: if INFO = -i, the i-th argument had an illegal value  
 *        > 0: the algorithm failed  
 *             = 1, a split was marked by a positive value in E  
 *             = 2, current block of Z not diagonalized after 30*N  
 *                  iterations (in inner while loop)  
 *             = 3, termination criterion of outer while loop not met   
 *                  (program created more than N unreduced blocks)  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
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 *  *
       INFO = 0        INFO = 0
       IF( N.LT.0 ) THEN        IF( N.LT.0 ) THEN
          INFO = -2           INFO = -1
          CALL XERBLA( 'DLASQ1', -INFO )           CALL XERBLA( 'DLASQ1', -INFO )
          RETURN           RETURN
       ELSE IF( N.EQ.0 ) THEN        ELSE IF( N.EQ.0 ) THEN
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       CALL DCOPY( N-1, E, 1, WORK( 2 ), 2 )        CALL DCOPY( N-1, E, 1, WORK( 2 ), 2 )
       CALL DLASCL( 'G', 0, 0, SIGMX, SCALE, 2*N-1, 1, WORK, 2*N-1,        CALL DLASCL( 'G', 0, 0, SIGMX, SCALE, 2*N-1, 1, WORK, 2*N-1,
      $             IINFO )       $             IINFO )
 *           *
 *     Compute the q's and e's.  *     Compute the q's and e's.
 *  *
       DO 30 I = 1, 2*N - 1        DO 30 I = 1, 2*N - 1
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             D( I ) = SQRT( WORK( I ) )              D( I ) = SQRT( WORK( I ) )
    40    CONTINUE     40    CONTINUE
          CALL DLASCL( 'G', 0, 0, SCALE, SIGMX, N, 1, D, N, IINFO )           CALL DLASCL( 'G', 0, 0, SCALE, SIGMX, N, 1, D, N, IINFO )
         ELSE IF( INFO.EQ.2 ) THEN
   *
   *     Maximum number of iterations exceeded.  Move data from WORK
   *     into D and E so the calling subroutine can try to finish
   *
            DO I = 1, N
               D( I ) = SQRT( WORK( 2*I-1 ) )
               E( I ) = SQRT( WORK( 2*I ) )
            END DO
            CALL DLASCL( 'G', 0, 0, SCALE, SIGMX, N, 1, D, N, IINFO )
            CALL DLASCL( 'G', 0, 0, SCALE, SIGMX, N, 1, E, N, IINFO )
       END IF        END IF
 *  *
       RETURN        RETURN
Removed from v.1.4  
changed lines
  Added in v.1.19

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