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version 1.9, 2011/07/22 07:38:07 version 1.10, 2011/11/21 20:42:58
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   *> \brief \b DLASD4
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download DLASD4 + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasd4.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasd4.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasd4.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DLASD4( N, I, D, Z, DELTA, RHO, SIGMA, WORK, INFO )
   * 
   *       .. Scalar Arguments ..
   *       INTEGER            I, INFO, N
   *       DOUBLE PRECISION   RHO, SIGMA
   *       ..
   *       .. Array Arguments ..
   *       DOUBLE PRECISION   D( * ), DELTA( * ), WORK( * ), Z( * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> This subroutine computes the square root of the I-th updated
   *> eigenvalue of a positive symmetric rank-one modification to
   *> a positive diagonal matrix whose entries are given as the squares
   *> of the corresponding entries in the array d, and that
   *>
   *>        0 <= D(i) < D(j)  for  i < j
   *>
   *> and that RHO > 0. This is arranged by the calling routine, and is
   *> no loss in generality.  The rank-one modified system is thus
   *>
   *>        diag( D ) * diag( D ) +  RHO * Z * Z_transpose.
   *>
   *> where we assume the Euclidean norm of Z is 1.
   *>
   *> The method consists of approximating the rational functions in the
   *> secular equation by simpler interpolating rational functions.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] N
   *> \verbatim
   *>          N is INTEGER
   *>         The length of all arrays.
   *> \endverbatim
   *>
   *> \param[in] I
   *> \verbatim
   *>          I is INTEGER
   *>         The index of the eigenvalue to be computed.  1 <= I <= N.
   *> \endverbatim
   *>
   *> \param[in] D
   *> \verbatim
   *>          D is DOUBLE PRECISION array, dimension ( N )
   *>         The original eigenvalues.  It is assumed that they are in
   *>         order, 0 <= D(I) < D(J)  for I < J.
   *> \endverbatim
   *>
   *> \param[in] Z
   *> \verbatim
   *>          Z is DOUBLE PRECISION array, dimension ( N )
   *>         The components of the updating vector.
   *> \endverbatim
   *>
   *> \param[out] DELTA
   *> \verbatim
   *>          DELTA is DOUBLE PRECISION array, dimension ( N )
   *>         If N .ne. 1, DELTA contains (D(j) - sigma_I) in its  j-th
   *>         component.  If N = 1, then DELTA(1) = 1.  The vector DELTA
   *>         contains the information necessary to construct the
   *>         (singular) eigenvectors.
   *> \endverbatim
   *>
   *> \param[in] RHO
   *> \verbatim
   *>          RHO is DOUBLE PRECISION
   *>         The scalar in the symmetric updating formula.
   *> \endverbatim
   *>
   *> \param[out] SIGMA
   *> \verbatim
   *>          SIGMA is DOUBLE PRECISION
   *>         The computed sigma_I, the I-th updated eigenvalue.
   *> \endverbatim
   *>
   *> \param[out] WORK
   *> \verbatim
   *>          WORK is DOUBLE PRECISION array, dimension ( N )
   *>         If N .ne. 1, WORK contains (D(j) + sigma_I) in its  j-th
   *>         component.  If N = 1, then WORK( 1 ) = 1.
   *> \endverbatim
   *>
   *> \param[out] INFO
   *> \verbatim
   *>          INFO is INTEGER
   *>         = 0:  successful exit
   *>         > 0:  if INFO = 1, the updating process failed.
   *> \endverbatim
   *
   *> \par Internal Parameters:
   *  =========================
   *>
   *> \verbatim
   *>  Logical variable ORGATI (origin-at-i?) is used for distinguishing
   *>  whether D(i) or D(i+1) is treated as the origin.
   *>
   *>            ORGATI = .true.    origin at i
   *>            ORGATI = .false.   origin at i+1
   *>
   *>  Logical variable SWTCH3 (switch-for-3-poles?) is for noting
   *>  if we are working with THREE poles!
   *>
   *>  MAXIT is the maximum number of iterations allowed for each
   *>  eigenvalue.
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee 
   *> \author Univ. of California Berkeley 
   *> \author Univ. of Colorado Denver 
   *> \author NAG Ltd. 
   *
   *> \date November 2011
   *
   *> \ingroup auxOTHERauxiliary
   *
   *> \par Contributors:
   *  ==================
   *>
   *>     Ren-Cang Li, Computer Science Division, University of California
   *>     at Berkeley, USA
   *>
   *  =====================================================================
       SUBROUTINE DLASD4( N, I, D, Z, DELTA, RHO, SIGMA, WORK, INFO )        SUBROUTINE DLASD4( N, I, D, Z, DELTA, RHO, SIGMA, WORK, INFO )
 *  *
 *  -- LAPACK auxiliary routine (version 3.3.1) --  *  -- LAPACK auxiliary routine (version 3.4.0) --
 *  -- 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..--
 *  -- April 2011                                                      --  *     November 2011
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       INTEGER            I, INFO, N        INTEGER            I, INFO, N
Line 13 Line 166
       DOUBLE PRECISION   D( * ), DELTA( * ), WORK( * ), Z( * )        DOUBLE PRECISION   D( * ), DELTA( * ), WORK( * ), Z( * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  This subroutine computes the square root of the I-th updated  
 *  eigenvalue of a positive symmetric rank-one modification to  
 *  a positive diagonal matrix whose entries are given as the squares  
 *  of the corresponding entries in the array d, and that  
 *  
 *         0 <= D(i) < D(j)  for  i < j  
 *  
 *  and that RHO > 0. This is arranged by the calling routine, and is  
 *  no loss in generality.  The rank-one modified system is thus  
 *  
 *         diag( D ) * diag( D ) +  RHO *  Z * Z_transpose.  
 *  
 *  where we assume the Euclidean norm of Z is 1.  
 *  
 *  The method consists of approximating the rational functions in the  
 *  secular equation by simpler interpolating rational functions.  
 *  
 *  Arguments  
 *  =========  
 *  
 *  N      (input) INTEGER  
 *         The length of all arrays.  
 *  
 *  I      (input) INTEGER  
 *         The index of the eigenvalue to be computed.  1 <= I <= N.  
 *  
 *  D      (input) DOUBLE PRECISION array, dimension ( N )  
 *         The original eigenvalues.  It is assumed that they are in  
 *         order, 0 <= D(I) < D(J)  for I < J.  
 *  
 *  Z      (input) DOUBLE PRECISION array, dimension ( N )  
 *         The components of the updating vector.  
 *  
 *  DELTA  (output) DOUBLE PRECISION array, dimension ( N )  
 *         If N .ne. 1, DELTA contains (D(j) - sigma_I) in its  j-th  
 *         component.  If N = 1, then DELTA(1) = 1.  The vector DELTA  
 *         contains the information necessary to construct the  
 *         (singular) eigenvectors.  
 *  
 *  RHO    (input) DOUBLE PRECISION  
 *         The scalar in the symmetric updating formula.  
 *  
 *  SIGMA  (output) DOUBLE PRECISION  
 *         The computed sigma_I, the I-th updated eigenvalue.  
 *  
 *  WORK   (workspace) DOUBLE PRECISION array, dimension ( N )  
 *         If N .ne. 1, WORK contains (D(j) + sigma_I) in its  j-th  
 *         component.  If N = 1, then WORK( 1 ) = 1.  
 *  
 *  INFO   (output) INTEGER  
 *         = 0:  successful exit  
 *         > 0:  if INFO = 1, the updating process failed.  
 *  
 *  Internal Parameters  
 *  ===================  
 *  
 *  Logical variable ORGATI (origin-at-i?) is used for distinguishing  
 *  whether D(i) or D(i+1) is treated as the origin.  
 *  
 *            ORGATI = .true.    origin at i  
 *            ORGATI = .false.   origin at i+1  
 *  
 *  Logical variable SWTCH3 (switch-for-3-poles?) is for noting  
 *  if we are working with THREE poles!  
 *  
 *  MAXIT is the maximum number of iterations allowed for each  
 *  eigenvalue.  
 *  
 *  Further Details  
 *  ===============  
 *  
 *  Based on contributions by  
 *     Ren-Cang Li, Computer Science Division, University of California  
 *     at Berkeley, USA  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
Removed from v.1.9  
changed lines
  Added in v.1.10

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