zstedc.f

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*> \brief \b ZSTEDC
*
*  =========== DOCUMENTATION ===========
*
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*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE ZSTEDC( COMPZ, N, D, E, Z, LDZ, WORK, LWORK, RWORK,
*                          LRWORK, IWORK, LIWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          COMPZ
*       INTEGER            INFO, LDZ, LIWORK, LRWORK, LWORK, N
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   D( * ), E( * ), RWORK( * )
*       COMPLEX*16         WORK( * ), Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> ZSTEDC computes all eigenvalues and, optionally, eigenvectors of a
*> symmetric tridiagonal matrix using the divide and conquer method.
*> The eigenvectors of a full or band complex Hermitian matrix can also
*> be found if ZHETRD or ZHPTRD or ZHBTRD has been used to reduce this
*> matrix to tridiagonal form.
*>
*> This code makes very mild assumptions about floating point
*> arithmetic. It will work on machines with a guard digit in
*> add/subtract, or on those binary machines without guard digits
*> which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2.
*> It could conceivably fail on hexadecimal or decimal machines
*> without guard digits, but we know of none.  See DLAED3 for details.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] COMPZ
*> \verbatim
*>          COMPZ is CHARACTER*1
*>          = 'N':  Compute eigenvalues only.
*>          = 'I':  Compute eigenvectors of tridiagonal matrix also.
*>          = 'V':  Compute eigenvectors of original Hermitian matrix
*>                  also.  On entry, Z contains the unitary matrix used
*>                  to reduce the original matrix to tridiagonal form.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The dimension of the symmetric tridiagonal matrix.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          D is DOUBLE PRECISION array, dimension (N)
*>          On entry, the diagonal elements of the tridiagonal matrix.
*>          On exit, if INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*>          E is DOUBLE PRECISION array, dimension (N-1)
*>          On entry, the subdiagonal elements of the tridiagonal matrix.
*>          On exit, E has been destroyed.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*>          Z is COMPLEX*16 array, dimension (LDZ,N)
*>          On entry, if COMPZ = 'V', then Z contains the unitary
*>          matrix used in the reduction to tridiagonal form.
*>          On exit, if INFO = 0, then if COMPZ = 'V', Z contains the
*>          orthonormal eigenvectors of the original Hermitian matrix,
*>          and if COMPZ = 'I', Z contains the orthonormal eigenvectors
*>          of the symmetric tridiagonal matrix.
*>          If  COMPZ = 'N', then Z is not referenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>          The leading dimension of the array Z.  LDZ >= 1.
*>          If eigenvectors are desired, then LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.
*>          If COMPZ = 'N' or 'I', or N <= 1, LWORK must be at least 1.
*>          If COMPZ = 'V' and N > 1, LWORK must be at least N*N.
*>          Note that for COMPZ = 'V', then if N is less than or
*>          equal to the minimum divide size, usually 25, then LWORK need
*>          only be 1.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the optimal sizes of the WORK, RWORK and
*>          IWORK arrays, returns these values as the first entries of
*>          the WORK, RWORK and IWORK arrays, and no error message
*>          related to LWORK or LRWORK or LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*>          RWORK is DOUBLE PRECISION array, dimension (MAX(1,LRWORK))
*>          On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK.
*> \endverbatim
*>
*> \param[in] LRWORK
*> \verbatim
*>          LRWORK is INTEGER
*>          The dimension of the array RWORK.
*>          If COMPZ = 'N' or N <= 1, LRWORK must be at least 1.
*>          If COMPZ = 'V' and N > 1, LRWORK must be at least
*>                         1 + 3*N + 2*N*lg N + 4*N**2 ,
*>                         where lg( N ) = smallest integer k such
*>                         that 2**k >= N.
*>          If COMPZ = 'I' and N > 1, LRWORK must be at least
*>                         1 + 4*N + 2*N**2 .
*>          Note that for COMPZ = 'I' or 'V', then if N is less than or
*>          equal to the minimum divide size, usually 25, then LRWORK
*>          need only be max(1,2*(N-1)).
*>
*>          If LRWORK = -1, then a workspace query is assumed; the
*>          routine only calculates the optimal sizes of the WORK, RWORK
*>          and IWORK arrays, returns these values as the first entries
*>          of the WORK, RWORK and IWORK arrays, and no error message
*>          related to LWORK or LRWORK or LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
*>          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
*> \endverbatim
*>
*> \param[in] LIWORK
*> \verbatim
*>          LIWORK is INTEGER
*>          The dimension of the array IWORK.
*>          If COMPZ = 'N' or N <= 1, LIWORK must be at least 1.
*>          If COMPZ = 'V' or N > 1,  LIWORK must be at least
*>                                    6 + 6*N + 5*N*lg N.
*>          If COMPZ = 'I' or N > 1,  LIWORK must be at least
*>                                    3 + 5*N .
*>          Note that for COMPZ = 'I' or 'V', then if N is less than or
*>          equal to the minimum divide size, usually 25, then LIWORK
*>          need only be 1.
*>
*>          If LIWORK = -1, then a workspace query is assumed; the
*>          routine only calculates the optimal sizes of the WORK, RWORK
*>          and IWORK arrays, returns these values as the first entries
*>          of the WORK, RWORK and IWORK arrays, and no error message
*>          related to LWORK or LRWORK or LIWORK is issued by XERBLA.
*> \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 to compute an eigenvalue while
*>                working on the submatrix lying in rows and columns
*>                INFO/(N+1) through mod(INFO,N+1).
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup complex16OTHERcomputational
*
*> \par Contributors:
*  ==================
*>
*> Jeff Rutter, Computer Science Division, University of California
*> at Berkeley, USA
*
*  =====================================================================
      SUBROUTINE zstedc( COMPZ, N, D, E, Z, LDZ, WORK, LWORK, RWORK,
     $                   LRWORK, IWORK, LIWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          COMPZ
      INTEGER            INFO, LDZ, LIWORK, LRWORK, LWORK, N
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   D( * ), E( * ), RWORK( * )
      COMPLEX*16         WORK( * ), Z( LDZ, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE, TWO
      parameter( zero = 0.0d0, one = 1.0d0, two = 2.0d0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY
      INTEGER            FINISH, I, ICOMPZ, II, J, K, LGN, LIWMIN, LL,
     $                   lrwmin, lwmin, m, smlsiz, start
      DOUBLE PRECISION   EPS, ORGNRM, P, TINY
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      DOUBLE PRECISION   DLAMCH, DLANST
      EXTERNAL           lsame, ilaenv, dlamch, dlanst
*     ..
*     .. External Subroutines ..
      EXTERNAL           dlascl, dlaset, dstedc, dsteqr, dsterf, xerbla,
     $                   zlacpy, zlacrm, zlaed0, zsteqr, zswap
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, dble, int, log, max, mod, sqrt
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      info = 0
      lquery = ( lwork.EQ.-1 .OR. lrwork.EQ.-1 .OR. liwork.EQ.-1 )
*
      IF( lsame( compz, 'N' ) ) THEN
         icompz = 0
      ELSE IF( lsame( compz, 'V' ) ) THEN
         icompz = 1
      ELSE IF( lsame( compz, 'I' ) ) THEN
         icompz = 2
      ELSE
         icompz = -1
      END IF
      IF( icompz.LT.0 ) THEN
         info = -1
      ELSE IF( n.LT.0 ) THEN
         info = -2
      ELSE IF( ( ldz.LT.1 ) .OR.
     $         ( icompz.GT.0 .AND. ldz.LT.max( 1, n ) ) ) THEN
         info = -6
      END IF
*
      IF( info.EQ.0 ) THEN
*
*        Compute the workspace requirements
*
         smlsiz = ilaenv( 9, 'ZSTEDC', ' ', 0, 0, 0, 0 )
         IF( n.LE.1 .OR. icompz.EQ.0 ) THEN
            lwmin = 1
            liwmin = 1
            lrwmin = 1
         ELSE IF( n.LE.smlsiz ) THEN
            lwmin = 1
            liwmin = 1
            lrwmin = 2*( n - 1 )
         ELSE IF( icompz.EQ.1 ) THEN
            lgn = int( log( dble( n ) ) / log( two ) )
            IF( 2**lgn.LT.n )
     $         lgn = lgn + 1
            IF( 2**lgn.LT.n )
     $         lgn = lgn + 1
            lwmin = n*n
            lrwmin = 1 + 3*n + 2*n*lgn + 4*n**2
            liwmin = 6 + 6*n + 5*n*lgn
         ELSE IF( icompz.EQ.2 ) THEN
            lwmin = 1
            lrwmin = 1 + 4*n + 2*n**2
            liwmin = 3 + 5*n
         END IF
         work( 1 ) = lwmin
         rwork( 1 ) = lrwmin
         iwork( 1 ) = liwmin
*
         IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
            info = -8
         ELSE IF( lrwork.LT.lrwmin .AND. .NOT.lquery ) THEN
            info = -10
         ELSE IF( liwork.LT.liwmin .AND. .NOT.lquery ) THEN
            info = -12
         END IF
      END IF
*
      IF( info.NE.0 ) THEN
         CALL xerbla( 'ZSTEDC', -info )
         RETURN
      ELSE IF( lquery ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( n.EQ.0 )
     $   RETURN
      IF( n.EQ.1 ) THEN
         IF( icompz.NE.0 )
     $      z( 1, 1 ) = one
         RETURN
      END IF
*
*     If the following conditional clause is removed, then the routine
*     will use the Divide and Conquer routine to compute only the
*     eigenvalues, which requires (3N + 3N**2) real workspace and
*     (2 + 5N + 2N lg(N)) integer workspace.
*     Since on many architectures DSTERF is much faster than any other
*     algorithm for finding eigenvalues only, it is used here
*     as the default. If the conditional clause is removed, then
*     information on the size of workspace needs to be changed.
*
*     If COMPZ = 'N', use DSTERF to compute the eigenvalues.
*
      IF( icompz.EQ.0 ) THEN
         CALL dsterf( n, d, e, info )
         GO TO 70
      END IF
*
*     If N is smaller than the minimum divide size (SMLSIZ+1), then
*     solve the problem with another solver.
*
      IF( n.LE.smlsiz ) THEN
*
         CALL zsteqr( compz, n, d, e, z, ldz, rwork, info )
*
      ELSE
*
*        If COMPZ = 'I', we simply call DSTEDC instead.
*
         IF( icompz.EQ.2 ) THEN
            CALL dlaset( 'Full', n, n, zero, one, rwork, n )
            ll = n*n + 1
            CALL dstedc( 'I', n, d, e, rwork, n,
     $                   rwork( ll ), lrwork-ll+1, iwork, liwork, info )
            DO 20 j = 1, n
               DO 10 i = 1, n
                  z( i, j ) = rwork( ( j-1 )*n+i )
   10          CONTINUE
   20       CONTINUE
            GO TO 70
         END IF
*
*        From now on, only option left to be handled is COMPZ = 'V',
*        i.e. ICOMPZ = 1.
*
*        Scale.
*
         orgnrm = dlanst( 'M', n, d, e )
         IF( orgnrm.EQ.zero )
     $      GO TO 70
*
         eps = dlamch( 'Epsilon' )
*
         start = 1
*
*        while ( START <= N )
*
   30    CONTINUE
         IF( start.LE.n ) THEN
*
*           Let FINISH be the position of the next subdiagonal entry
*           such that E( FINISH ) <= TINY or FINISH = N if no such
*           subdiagonal exists.  The matrix identified by the elements
*           between START and FINISH constitutes an independent
*           sub-problem.
*
            finish = start
   40       CONTINUE
            IF( finish.LT.n ) THEN
               tiny = eps*sqrt( abs( d( finish ) ) )*
     $                    sqrt( abs( d( finish+1 ) ) )
               IF( abs( e( finish ) ).GT.tiny ) THEN
                  finish = finish + 1
                  GO TO 40
               END IF
            END IF
*
*           (Sub) Problem determined.  Compute its size and solve it.
*
            m = finish - start + 1
            IF( m.GT.smlsiz ) THEN
*
*              Scale.
*
               orgnrm = dlanst( 'M', m, d( start ), e( start ) )
               CALL dlascl( 'G', 0, 0, orgnrm, one, m, 1, d( start ), m,
     $                      info )
               CALL dlascl( 'G', 0, 0, orgnrm, one, m-1, 1, e( start ),
     $                      m-1, info )
*
               CALL zlaed0( n, m, d( start ), e( start ), z( 1, start ),
     $                      ldz, work, n, rwork, iwork, info )
               IF( info.GT.0 ) THEN
                  info = ( info / ( m+1 )+start-1 )*( n+1 ) +
     $                   mod( info, ( m+1 ) ) + start - 1
                  GO TO 70
               END IF
*
*              Scale back.
*
               CALL dlascl( 'G', 0, 0, one, orgnrm, m, 1, d( start ), m,
     $                      info )
*
            ELSE
               CALL dsteqr( 'I', m, d( start ), e( start ), rwork, m,
     $                      rwork( m*m+1 ), info )
               CALL zlacrm( n, m, z( 1, start ), ldz, rwork, m, work, n,
     $                      rwork( m*m+1 ) )
               CALL zlacpy( 'A', n, m, work, n, z( 1, start ), ldz )
               IF( info.GT.0 ) THEN
                  info = start*( n+1 ) + finish
                  GO TO 70
               END IF
            END IF
*
            start = finish + 1
            GO TO 30
         END IF
*
*        endwhile
*
*
*        Use Selection Sort to minimize swaps of eigenvectors
*
         DO 60 ii = 2, n
           i = ii - 1
           k = i
           p = d( i )
           DO 50 j = ii, n
              IF( d( j ).LT.p ) THEN
                 k = j
                 p = d( j )
              END IF
   50      CONTINUE
           IF( k.NE.i ) THEN
              d( k ) = d( i )
              d( i ) = p
              CALL zswap( n, z( 1, i ), 1, z( 1, k ), 1 )
           END IF
   60    CONTINUE
      END IF
*
   70 CONTINUE
      work( 1 ) = lwmin
      rwork( 1 ) = lrwmin
      iwork( 1 ) = liwmin
*
      RETURN
*
*     End of ZSTEDC
*
      END