zhsein.f

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*> \brief \b ZHSEIN
*
*  =========== DOCUMENTATION ===========
*
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*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE ZHSEIN( SIDE, EIGSRC, INITV, SELECT, N, H, LDH, W, VL,
*                          LDVL, VR, LDVR, MM, M, WORK, RWORK, IFAILL,
*                          IFAILR, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          EIGSRC, INITV, SIDE
*       INTEGER            INFO, LDH, LDVL, LDVR, M, MM, N
*       ..
*       .. Array Arguments ..
*       LOGICAL            SELECT( * )
*       INTEGER            IFAILL( * ), IFAILR( * )
*       DOUBLE PRECISION   RWORK( * )
*       COMPLEX*16         H( LDH, * ), VL( LDVL, * ), VR( LDVR, * ),
*      $                   W( * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> ZHSEIN uses inverse iteration to find specified right and/or left
*> eigenvectors of a complex upper Hessenberg matrix H.
*>
*> The right eigenvector x and the left eigenvector y of the matrix H
*> corresponding to an eigenvalue w are defined by:
*>
*>              H * x = w * x,     y**h * H = w * y**h
*>
*> where y**h denotes the conjugate transpose of the vector y.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] SIDE
*> \verbatim
*>          SIDE is CHARACTER*1
*>          = 'R': compute right eigenvectors only;
*>          = 'L': compute left eigenvectors only;
*>          = 'B': compute both right and left eigenvectors.
*> \endverbatim
*>
*> \param[in] EIGSRC
*> \verbatim
*>          EIGSRC is CHARACTER*1
*>          Specifies the source of eigenvalues supplied in W:
*>          = 'Q': the eigenvalues were found using ZHSEQR; thus, if
*>                 H has zero subdiagonal elements, and so is
*>                 block-triangular, then the j-th eigenvalue can be
*>                 assumed to be an eigenvalue of the block containing
*>                 the j-th row/column.  This property allows ZHSEIN to
*>                 perform inverse iteration on just one diagonal block.
*>          = 'N': no assumptions are made on the correspondence
*>                 between eigenvalues and diagonal blocks.  In this
*>                 case, ZHSEIN must always perform inverse iteration
*>                 using the whole matrix H.
*> \endverbatim
*>
*> \param[in] INITV
*> \verbatim
*>          INITV is CHARACTER*1
*>          = 'N': no initial vectors are supplied;
*>          = 'U': user-supplied initial vectors are stored in the arrays
*>                 VL and/or VR.
*> \endverbatim
*>
*> \param[in] SELECT
*> \verbatim
*>          SELECT is LOGICAL array, dimension (N)
*>          Specifies the eigenvectors to be computed. To select the
*>          eigenvector corresponding to the eigenvalue W(j),
*>          SELECT(j) must be set to .TRUE..
*> \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.
*>          If a NaN is detected in H, the routine will return with INFO=-6.
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*>          LDH is INTEGER
*>          The leading dimension of the array H.  LDH >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] W
*> \verbatim
*>          W is COMPLEX*16 array, dimension (N)
*>          On entry, the eigenvalues of H.
*>          On exit, the real parts of W may have been altered since
*>          close eigenvalues are perturbed slightly in searching for
*>          independent eigenvectors.
*> \endverbatim
*>
*> \param[in,out] VL
*> \verbatim
*>          VL is COMPLEX*16 array, dimension (LDVL,MM)
*>          On entry, if INITV = 'U' and SIDE = 'L' or 'B', VL must
*>          contain starting vectors for the inverse iteration for the
*>          left eigenvectors; the starting vector for each eigenvector
*>          must be in the same column in which the eigenvector will be
*>          stored.
*>          On exit, if SIDE = 'L' or 'B', the left eigenvectors
*>          specified by SELECT will be stored consecutively in the
*>          columns of VL, in the same order as their eigenvalues.
*>          If SIDE = 'R', VL is not referenced.
*> \endverbatim
*>
*> \param[in] LDVL
*> \verbatim
*>          LDVL is INTEGER
*>          The leading dimension of the array VL.
*>          LDVL >= max(1,N) if SIDE = 'L' or 'B'; LDVL >= 1 otherwise.
*> \endverbatim
*>
*> \param[in,out] VR
*> \verbatim
*>          VR is COMPLEX*16 array, dimension (LDVR,MM)
*>          On entry, if INITV = 'U' and SIDE = 'R' or 'B', VR must
*>          contain starting vectors for the inverse iteration for the
*>          right eigenvectors; the starting vector for each eigenvector
*>          must be in the same column in which the eigenvector will be
*>          stored.
*>          On exit, if SIDE = 'R' or 'B', the right eigenvectors
*>          specified by SELECT will be stored consecutively in the
*>          columns of VR, in the same order as their eigenvalues.
*>          If SIDE = 'L', VR is not referenced.
*> \endverbatim
*>
*> \param[in] LDVR
*> \verbatim
*>          LDVR is INTEGER
*>          The leading dimension of the array VR.
*>          LDVR >= max(1,N) if SIDE = 'R' or 'B'; LDVR >= 1 otherwise.
*> \endverbatim
*>
*> \param[in] MM
*> \verbatim
*>          MM is INTEGER
*>          The number of columns in the arrays VL and/or VR. MM >= M.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*>          M is INTEGER
*>          The number of columns in the arrays VL and/or VR required to
*>          store the eigenvectors (= the number of .TRUE. elements in
*>          SELECT).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is COMPLEX*16 array, dimension (N*N)
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*>          RWORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] IFAILL
*> \verbatim
*>          IFAILL is INTEGER array, dimension (MM)
*>          If SIDE = 'L' or 'B', IFAILL(i) = j > 0 if the left
*>          eigenvector in the i-th column of VL (corresponding to the
*>          eigenvalue w(j)) failed to converge; IFAILL(i) = 0 if the
*>          eigenvector converged satisfactorily.
*>          If SIDE = 'R', IFAILL is not referenced.
*> \endverbatim
*>
*> \param[out] IFAILR
*> \verbatim
*>          IFAILR is INTEGER array, dimension (MM)
*>          If SIDE = 'R' or 'B', IFAILR(i) = j > 0 if the right
*>          eigenvector in the i-th column of VR (corresponding to the
*>          eigenvalue w(j)) failed to converge; IFAILR(i) = 0 if the
*>          eigenvector converged satisfactorily.
*>          If SIDE = 'L', IFAILR is not referenced.
*> \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 = i, i is the number of eigenvectors which
*>                failed to converge; see IFAILL and IFAILR for further
*>                details.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hsein
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  Each eigenvector is normalized so that the element of largest
*>  magnitude has magnitude 1; here the magnitude of a complex number
*>  (x,y) is taken to be |x|+|y|.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE zhsein( SIDE, EIGSRC, INITV, SELECT, N, H, LDH, W, VL,
     $                   LDVL, VR, LDVR, MM, M, WORK, RWORK, IFAILL,
     $                   IFAILR, 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          EIGSRC, INITV, SIDE
      INTEGER            INFO, LDH, LDVL, LDVR, M, MM, N
*     ..
*     .. Array Arguments ..
      LOGICAL            SELECT( * )
      INTEGER            IFAILL( * ), IFAILR( * )
      DOUBLE PRECISION   RWORK( * )
      COMPLEX*16         H( LDH, * ), VL( LDVL, * ), VR( LDVR, * ),
     $                   w( * ), work( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      COMPLEX*16         ZERO
      PARAMETER          ( ZERO = ( 0.0d+0, 0.0d+0 ) )
      DOUBLE PRECISION   RZERO
      parameter( rzero = 0.0d+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            BOTHV, FROMQR, LEFTV, NOINIT, RIGHTV
      INTEGER            I, IINFO, K, KL, KLN, KR, KS, LDWORK
      DOUBLE PRECISION   EPS3, HNORM, SMLNUM, ULP, UNFL
      COMPLEX*16         CDUM, WK
*     ..
*     .. External Functions ..
      LOGICAL            LSAME, DISNAN
      DOUBLE PRECISION   DLAMCH, ZLANHS
      EXTERNAL           lsame, dlamch, zlanhs, disnan
*     ..
*     .. External Subroutines ..
      EXTERNAL           xerbla, zlaein
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, dble, dimag, max
*     ..
*     .. Statement Functions ..
      DOUBLE PRECISION   CABS1
*     ..
*     .. Statement Function definitions ..
      cabs1( cdum ) = abs( dble( cdum ) ) + abs( dimag( cdum ) )
*     ..
*     .. Executable Statements ..
*
*     Decode and test the input parameters.
*
      bothv = lsame( side, 'B' )
      rightv = lsame( side, 'R' ) .OR. bothv
      leftv = lsame( side, 'L' ) .OR. bothv
*
      fromqr = lsame( eigsrc, 'Q' )
*
      noinit = lsame( initv, 'N' )
*
*     Set M to the number of columns required to store the selected
*     eigenvectors.
*
      m = 0
      DO 10 k = 1, n
         IF( SELECT( k ) )
     $      m = m + 1
   10 CONTINUE
*
      info = 0
      IF( .NOT.rightv .AND. .NOT.leftv ) THEN
         info = -1
      ELSE IF( .NOT.fromqr .AND. .NOT.lsame( eigsrc, 'N' ) ) THEN
         info = -2
      ELSE IF( .NOT.noinit .AND. .NOT.lsame( initv, 'U' ) ) THEN
         info = -3
      ELSE IF( n.LT.0 ) THEN
         info = -5
      ELSE IF( ldh.LT.max( 1, n ) ) THEN
         info = -7
      ELSE IF( ldvl.LT.1 .OR. ( leftv .AND. ldvl.LT.n ) ) THEN
         info = -10
      ELSE IF( ldvr.LT.1 .OR. ( rightv .AND. ldvr.LT.n ) ) THEN
         info = -12
      ELSE IF( mm.LT.m ) THEN
         info = -13
      END IF
      IF( info.NE.0 ) THEN
         CALL xerbla( 'ZHSEIN', -info )
         RETURN
      END IF
*
*     Quick return if possible.
*
      IF( n.EQ.0 )
     $   RETURN
*
*     Set machine-dependent constants.
*
      unfl = dlamch( 'Safe minimum' )
      ulp = dlamch( 'Precision' )
      smlnum = unfl*( n / ulp )
*
      ldwork = n
*
      kl = 1
      kln = 0
      IF( fromqr ) THEN
         kr = 0
      ELSE
         kr = n
      END IF
      ks = 1
*
      DO 100 k = 1, n
         IF( SELECT( k ) ) THEN
*
*           Compute eigenvector(s) corresponding to W(K).
*
            IF( fromqr ) THEN
*
*              If affiliation of eigenvalues is known, check whether
*              the matrix splits.
*
*              Determine KL and KR such that 1 <= KL <= K <= KR <= N
*              and H(KL,KL-1) and H(KR+1,KR) are zero (or KL = 1 or
*              KR = N).
*
*              Then inverse iteration can be performed with the
*              submatrix H(KL:N,KL:N) for a left eigenvector, and with
*              the submatrix H(1:KR,1:KR) for a right eigenvector.
*
               DO 20 i = k, kl + 1, -1
                  IF( h( i, i-1 ).EQ.zero )
     $               GO TO 30
   20          CONTINUE
   30          CONTINUE
               kl = i
               IF( k.GT.kr ) THEN
                  DO 40 i = k, n - 1
                     IF( h( i+1, i ).EQ.zero )
     $                  GO TO 50
   40             CONTINUE
   50             CONTINUE
                  kr = i
               END IF
            END IF
*
            IF( kl.NE.kln ) THEN
               kln = kl
*
*              Compute infinity-norm of submatrix H(KL:KR,KL:KR) if it
*              has not ben computed before.
*
               hnorm = zlanhs( 'I', kr-kl+1, h( kl, kl ), ldh, rwork )
               IF( disnan( hnorm ) ) THEN
                  info = -6
                  RETURN
               ELSE IF( hnorm.GT.rzero ) THEN
                  eps3 = hnorm*ulp
               ELSE
                  eps3 = smlnum
               END IF
            END IF
*
*           Perturb eigenvalue if it is close to any previous
*           selected eigenvalues affiliated to the submatrix
*           H(KL:KR,KL:KR). Close roots are modified by EPS3.
*
            wk = w( k )
   60       CONTINUE
            DO 70 i = k - 1, kl, -1
               IF( SELECT( i ) .AND. cabs1( w( i )-wk ).LT.eps3 ) THEN
                  wk = wk + eps3
                  GO TO 60
               END IF
   70       CONTINUE
            w( k ) = wk
*
            IF( leftv ) THEN
*
*              Compute left eigenvector.
*
               CALL zlaein( .false., noinit, n-kl+1, h( kl, kl ), ldh,
     $                      wk, vl( kl, ks ), work, ldwork, rwork, eps3,
     $                      smlnum, iinfo )
               IF( iinfo.GT.0 ) THEN
                  info = info + 1
                  ifaill( ks ) = k
               ELSE
                  ifaill( ks ) = 0
               END IF
               DO 80 i = 1, kl - 1
                  vl( i, ks ) = zero
   80          CONTINUE
            END IF
            IF( rightv ) THEN
*
*              Compute right eigenvector.
*
               CALL zlaein( .true., noinit, kr, h, ldh, wk, vr( 1, ks ),
     $                      work, ldwork, rwork, eps3, smlnum, iinfo )
               IF( iinfo.GT.0 ) THEN
                  info = info + 1
                  ifailr( ks ) = k
               ELSE
                  ifailr( ks ) = 0
               END IF
               DO 90 i = kr + 1, n
                  vr( i, ks ) = zero
   90          CONTINUE
            END IF
            ks = ks + 1
         END IF
  100 CONTINUE
*
      RETURN
*
*     End of ZHSEIN
*
      END