zlaein.f

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*> \brief \b ZLAEIN computes a specified right or left eigenvector of an upper Hessenberg matrix by inverse iteration.
*
*  =========== DOCUMENTATION ===========
*
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*            http://www.netlib.org/lapack/explore-html/
*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE ZLAEIN( RIGHTV, NOINIT, N, H, LDH, W, V, B, LDB, RWORK,
*                          EPS3, SMLNUM, INFO )
*
*       .. Scalar Arguments ..
*       LOGICAL            NOINIT, RIGHTV
*       INTEGER            INFO, LDB, LDH, N
*       DOUBLE PRECISION   EPS3, SMLNUM
*       COMPLEX*16         W
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   RWORK( * )
*       COMPLEX*16         B( LDB, * ), H( LDH, * ), V( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> ZLAEIN uses inverse iteration to find a right or left eigenvector
*> corresponding to the eigenvalue W of a complex upper Hessenberg
*> matrix H.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] RIGHTV
*> \verbatim
*>          RIGHTV is LOGICAL
*>          = .TRUE. : compute right eigenvector;
*>          = .FALSE.: compute left eigenvector.
*> \endverbatim
*>
*> \param[in] NOINIT
*> \verbatim
*>          NOINIT is LOGICAL
*>          = .TRUE. : no initial vector supplied in V
*>          = .FALSE.: initial vector supplied in V.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix H.  N >= 0.
*> \endverbatim
*>
*> \param[in] H
*> \verbatim
*>          H is COMPLEX*16 array, dimension (LDH,N)
*>          The upper Hessenberg matrix H.
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*>          LDH is INTEGER
*>          The leading dimension of the array H.  LDH >= max(1,N).
*> \endverbatim
*>
*> \param[in] W
*> \verbatim
*>          W is COMPLEX*16
*>          The eigenvalue of H whose corresponding right or left
*>          eigenvector is to be computed.
*> \endverbatim
*>
*> \param[in,out] V
*> \verbatim
*>          V is COMPLEX*16 array, dimension (N)
*>          On entry, if NOINIT = .FALSE., V must contain a starting
*>          vector for inverse iteration; otherwise V need not be set.
*>          On exit, V contains the computed eigenvector, normalized so
*>          that the component of largest magnitude has magnitude 1; here
*>          the magnitude of a complex number (x,y) is taken to be
*>          |x| + |y|.
*> \endverbatim
*>
*> \param[out] B
*> \verbatim
*>          B is COMPLEX*16 array, dimension (LDB,N)
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*>          RWORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[in] EPS3
*> \verbatim
*>          EPS3 is DOUBLE PRECISION
*>          A small machine-dependent value which is used to perturb
*>          close eigenvalues, and to replace zero pivots.
*> \endverbatim
*>
*> \param[in] SMLNUM
*> \verbatim
*>          SMLNUM is DOUBLE PRECISION
*>          A machine-dependent value close to the underflow threshold.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          = 1:  inverse iteration did not converge; V is set to the
*>                last iterate.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup laein
*
*  =====================================================================
      SUBROUTINE zlaein( RIGHTV, NOINIT, N, H, LDH, W, V, B, LDB, RWORK,
     $                   EPS3, SMLNUM, INFO )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      LOGICAL            NOINIT, RIGHTV
      INTEGER            INFO, LDB, LDH, N
      DOUBLE PRECISION   EPS3, SMLNUM
      COMPLEX*16         W
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   RWORK( * )
      COMPLEX*16         B( LDB, * ), H( LDH, * ), V( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, TENTH
      parameter( one = 1.0d+0, tenth = 1.0d-1 )
      COMPLEX*16         ZERO
      parameter( zero = ( 0.0d+0, 0.0d+0 ) )
*     ..
*     .. Local Scalars ..
      CHARACTER          NORMIN, TRANS
      INTEGER            I, IERR, ITS, J
      DOUBLE PRECISION   GROWTO, NRMSML, ROOTN, RTEMP, SCALE, VNORM
      COMPLEX*16         CDUM, EI, EJ, TEMP, X
*     ..
*     .. External Functions ..
      INTEGER            IZAMAX
      DOUBLE PRECISION   DZASUM, DZNRM2
      COMPLEX*16         ZLADIV
      EXTERNAL           izamax, dzasum, dznrm2, zladiv
*     ..
*     .. External Subroutines ..
      EXTERNAL           zdscal, zlatrs
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, dble, dimag, max, sqrt
*     ..
*     .. Statement Functions ..
      DOUBLE PRECISION   CABS1
*     ..
*     .. Statement Function definitions ..
      cabs1( cdum ) = abs( dble( cdum ) ) + abs( dimag( cdum ) )
*     ..
*     .. Executable Statements ..
*
      info = 0
*
*     GROWTO is the threshold used in the acceptance test for an
*     eigenvector.
*
      rootn = sqrt( dble( n ) )
      growto = tenth / rootn
      nrmsml = max( one, eps3*rootn )*smlnum
*
*     Form B = H - W*I (except that the subdiagonal elements are not
*     stored).
*
      DO 20 j = 1, n
         DO 10 i = 1, j - 1
            b( i, j ) = h( i, j )
   10    CONTINUE
         b( j, j ) = h( j, j ) - w
   20 CONTINUE
*
      IF( noinit ) THEN
*
*        Initialize V.
*
         DO 30 i = 1, n
            v( i ) = eps3
   30    CONTINUE
      ELSE
*
*        Scale supplied initial vector.
*
         vnorm = dznrm2( n, v, 1 )
         CALL zdscal( n, ( eps3*rootn ) / max( vnorm, nrmsml ), v, 1 )
      END IF
*
      IF( rightv ) THEN
*
*        LU decomposition with partial pivoting of B, replacing zero
*        pivots by EPS3.
*
         DO 60 i = 1, n - 1
            ei = h( i+1, i )
            IF( cabs1( b( i, i ) ).LT.cabs1( ei ) ) THEN
*
*              Interchange rows and eliminate.
*
               x = zladiv( b( i, i ), ei )
               b( i, i ) = ei
               DO 40 j = i + 1, n
                  temp = b( i+1, j )
                  b( i+1, j ) = b( i, j ) - x*temp
                  b( i, j ) = temp
   40          CONTINUE
            ELSE
*
*              Eliminate without interchange.
*
               IF( b( i, i ).EQ.zero )
     $            b( i, i ) = eps3
               x = zladiv( ei, b( i, i ) )
               IF( x.NE.zero ) THEN
                  DO 50 j = i + 1, n
                     b( i+1, j ) = b( i+1, j ) - x*b( i, j )
   50             CONTINUE
               END IF
            END IF
   60    CONTINUE
         IF( b( n, n ).EQ.zero )
     $      b( n, n ) = eps3
*
         trans = 'N'
*
      ELSE
*
*        UL decomposition with partial pivoting of B, replacing zero
*        pivots by EPS3.
*
         DO 90 j = n, 2, -1
            ej = h( j, j-1 )
            IF( cabs1( b( j, j ) ).LT.cabs1( ej ) ) THEN
*
*              Interchange columns and eliminate.
*
               x = zladiv( b( j, j ), ej )
               b( j, j ) = ej
               DO 70 i = 1, j - 1
                  temp = b( i, j-1 )
                  b( i, j-1 ) = b( i, j ) - x*temp
                  b( i, j ) = temp
   70          CONTINUE
            ELSE
*
*              Eliminate without interchange.
*
               IF( b( j, j ).EQ.zero )
     $            b( j, j ) = eps3
               x = zladiv( ej, b( j, j ) )
               IF( x.NE.zero ) THEN
                  DO 80 i = 1, j - 1
                     b( i, j-1 ) = b( i, j-1 ) - x*b( i, j )
   80             CONTINUE
               END IF
            END IF
   90    CONTINUE
         IF( b( 1, 1 ).EQ.zero )
     $      b( 1, 1 ) = eps3
*
         trans = 'C'
*
      END IF
*
      normin = 'N'
      DO 110 its = 1, n
*
*        Solve U*x = scale*v for a right eigenvector
*          or U**H *x = scale*v for a left eigenvector,
*        overwriting x on v.
*
         CALL zlatrs( 'Upper', trans, 'Nonunit', normin, n, b, ldb, v,
     $                scale, rwork, ierr )
         normin = 'Y'
*
*        Test for sufficient growth in the norm of v.
*
         vnorm = dzasum( n, v, 1 )
         IF( vnorm.GE.growto*scale )
     $      GO TO 120
*
*        Choose new orthogonal starting vector and try again.
*
         rtemp = eps3 / ( rootn+one )
         v( 1 ) = eps3
         DO 100 i = 2, n
            v( i ) = rtemp
  100    CONTINUE
         v( n-its+1 ) = v( n-its+1 ) - eps3*rootn
  110 CONTINUE
*
*     Failure to find eigenvector in N iterations.
*
      info = 1
*
  120 CONTINUE
*
*     Normalize eigenvector.
*
      i = izamax( n, v, 1 )
      CALL zdscal( n, one / cabs1( v( i ) ), v, 1 )
*
      RETURN
*
*     End of ZLAEIN
*
      END