slaed1.f

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*> \brief \b SLAED1 used by sstedc. Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used when the original matrix is tridiagonal.
*
*  =========== DOCUMENTATION ===========
*
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*            http://www.netlib.org/lapack/explore-html/ 
*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE SLAED1( N, D, Q, LDQ, INDXQ, RHO, CUTPNT, WORK, IWORK,
*                          INFO )
* 
*       .. Scalar Arguments ..
*       INTEGER            CUTPNT, INFO, LDQ, N
*       REAL               RHO
*       ..
*       .. Array Arguments ..
*       INTEGER            INDXQ( * ), IWORK( * )
*       REAL               D( * ), Q( LDQ, * ), WORK( * )
*       ..
*  
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> SLAED1 computes the updated eigensystem of a diagonal
*> matrix after modification by a rank-one symmetric matrix.  This
*> routine is used only for the eigenproblem which requires all
*> eigenvalues and eigenvectors of a tridiagonal matrix.  SLAED7 handles
*> the case in which eigenvalues only or eigenvalues and eigenvectors
*> of a full symmetric matrix (which was reduced to tridiagonal form)
*> are desired.
*>
*>   T = Q(in) ( D(in) + RHO * Z*Z**T ) Q**T(in) = Q(out) * D(out) * Q**T(out)
*>
*>    where Z = Q**T*u, u is a vector of length N with ones in the
*>    CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.
*>
*>    The eigenvectors of the original matrix are stored in Q, and the
*>    eigenvalues are in D.  The algorithm consists of three stages:
*>
*>       The first stage consists of deflating the size of the problem
*>       when there are multiple eigenvalues or if there is a zero in
*>       the Z vector.  For each such occurence the dimension of the
*>       secular equation problem is reduced by one.  This stage is
*>       performed by the routine SLAED2.
*>
*>       The second stage consists of calculating the updated
*>       eigenvalues. This is done by finding the roots of the secular
*>       equation via the routine SLAED4 (as called by SLAED3).
*>       This routine also calculates the eigenvectors of the current
*>       problem.
*>
*>       The final stage consists of computing the updated eigenvectors
*>       directly using the updated eigenvalues.  The eigenvectors for
*>       the current problem are multiplied with the eigenvectors from
*>       the overall problem.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>         The dimension of the symmetric tridiagonal matrix.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          D is REAL array, dimension (N)
*>         On entry, the eigenvalues of the rank-1-perturbed matrix.
*>         On exit, the eigenvalues of the repaired matrix.
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*>          Q is REAL array, dimension (LDQ,N)
*>         On entry, the eigenvectors of the rank-1-perturbed matrix.
*>         On exit, the eigenvectors of the repaired tridiagonal matrix.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*>          LDQ is INTEGER
*>         The leading dimension of the array Q.  LDQ >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] INDXQ
*> \verbatim
*>          INDXQ is INTEGER array, dimension (N)
*>         On entry, the permutation which separately sorts the two
*>         subproblems in D into ascending order.
*>         On exit, the permutation which will reintegrate the
*>         subproblems back into sorted order,
*>         i.e. D( INDXQ( I = 1, N ) ) will be in ascending order.
*> \endverbatim
*>
*> \param[in] RHO
*> \verbatim
*>          RHO is REAL
*>         The subdiagonal entry used to create the rank-1 modification.
*> \endverbatim
*>
*> \param[in] CUTPNT
*> \verbatim
*>          CUTPNT is INTEGER
*>         The location of the last eigenvalue in the leading sub-matrix.
*>         min(1,N) <= CUTPNT <= N/2.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is REAL array, dimension (4*N + N**2)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER 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:  if INFO = 1, an eigenvalue did not converge
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee 
*> \author Univ. of California Berkeley 
*> \author Univ. of Colorado Denver 
*> \author NAG Ltd. 
*
*> \date September 2012
*
*> \ingroup auxOTHERcomputational
*
*> \par Contributors:
*  ==================
*>
*> Jeff Rutter, Computer Science Division, University of California
*> at Berkeley, USA \n
*>  Modified by Francoise Tisseur, University of Tennessee
*>
*  =====================================================================
      SUBROUTINE slaed1( N, D, Q, LDQ, INDXQ, RHO, CUTPNT, WORK, IWORK,
     $                   info )
*
*  -- LAPACK computational routine (version 3.4.2) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     September 2012
*
*     .. Scalar Arguments ..
      INTEGER            cutpnt, info, ldq, n
      REAL               rho
*     ..
*     .. Array Arguments ..
      INTEGER            indxq( * ), iwork( * )
      REAL               d( * ), q( ldq, * ), work( * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      INTEGER            coltyp, cpp1, i, idlmda, indx, indxc, indxp,
     $                   iq2, is, iw, iz, k, n1, n2
*     ..
*     .. External Subroutines ..
      EXTERNAL           scopy, slaed2, slaed3, slamrg, xerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          max, min
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      info = 0
*
      IF( n.LT.0 ) THEN
         info = -1
      ELSE IF( ldq.LT.max( 1, n ) ) THEN
         info = -4
      ELSE IF( min( 1, n / 2 ).GT.cutpnt .OR. ( n / 2 ).LT.cutpnt ) THEN
         info = -7
      END IF
      IF( info.NE.0 ) THEN
         CALL xerbla( 'SLAED1', -info )
         return
      END IF
*
*     Quick return if possible
*
      IF( n.EQ.0 )
     $   return
*
*     The following values are integer pointers which indicate
*     the portion of the workspace
*     used by a particular array in SLAED2 and SLAED3.
*
      iz = 1
      idlmda = iz + n
      iw = idlmda + n
      iq2 = iw + n
*
      indx = 1
      indxc = indx + n
      coltyp = indxc + n
      indxp = coltyp + n
*
*
*     Form the z-vector which consists of the last row of Q_1 and the
*     first row of Q_2.
*
      CALL scopy( cutpnt, q( cutpnt, 1 ), ldq, work( iz ), 1 )
      cpp1 = cutpnt + 1
      CALL scopy( n-cutpnt, q( cpp1, cpp1 ), ldq, work( iz+cutpnt ), 1 )
*
*     Deflate eigenvalues.
*
      CALL slaed2( k, n, cutpnt, d, q, ldq, indxq, rho, work( iz ),
     $             work( idlmda ), work( iw ), work( iq2 ),
     $             iwork( indx ), iwork( indxc ), iwork( indxp ),
     $             iwork( coltyp ), info )
*
      IF( info.NE.0 )
     $   go to 20
*
*     Solve Secular Equation.
*
      IF( k.NE.0 ) THEN
         is = ( iwork( coltyp )+iwork( coltyp+1 ) )*cutpnt +
     $        ( iwork( coltyp+1 )+iwork( coltyp+2 ) )*( n-cutpnt ) + iq2
         CALL slaed3( k, n, cutpnt, d, q, ldq, rho, work( idlmda ),
     $                work( iq2 ), iwork( indxc ), iwork( coltyp ),
     $                work( iw ), work( is ), info )
         IF( info.NE.0 )
     $      go to 20
*
*     Prepare the INDXQ sorting permutation.
*
         n1 = k
         n2 = n - k
         CALL slamrg( n1, n2, d, 1, -1, indxq )
      ELSE
         DO 10 i = 1, n
            indxq( i ) = i
   10    continue
      END IF
*
   20 continue
      return
*
*     End of SLAED1
*
      END