cgeesx.f

          SUBROUTINE CGEESX( JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM, W,
         $                   VS, LDVS, RCONDE, RCONDV, WORK, LWORK, RWORK,
         $                   BWORK, INFO )
    *
    *  -- LAPACK driver routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          CHARACTER          JOBVS, SENSE, SORT
          INTEGER            INFO, LDA, LDVS, LWORK, N, SDIM
          REAL               RCONDE, RCONDV
    *     ..
    *     .. Array Arguments ..
          LOGICAL            BWORK( * )
          REAL               RWORK( * )
          COMPLEX            A( LDA, * ), VS( LDVS, * ), W( * ), WORK( * )
    *     ..
    *     .. Function Arguments ..
          LOGICAL            SELECT
          EXTERNAL           SELECT
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  CGEESX computes for an N-by-N complex nonsymmetric matrix A, the
    *  eigenvalues, the Schur form T, and, optionally, the matrix of Schur
    *  vectors Z.  This gives the Schur factorization A = Z*T*(Z**H).
    *
    *  Optionally, it also orders the eigenvalues on the diagonal of the
    *  Schur form so that selected eigenvalues are at the top left;
    *  computes a reciprocal condition number for the average of the
    *  selected eigenvalues (RCONDE); and computes a reciprocal condition
    *  number for the right invariant subspace corresponding to the
    *  selected eigenvalues (RCONDV).  The leading columns of Z form an
    *  orthonormal basis for this invariant subspace.
    *
    *  For further explanation of the reciprocal condition numbers RCONDE
    *  and RCONDV, see Section 4.10 of the LAPACK Users' Guide (where
    *  these quantities are called s and sep respectively).
    *
    *  A complex matrix is in Schur form if it is upper triangular.
    *
    *  Arguments
    *  =========
    *
    *  JOBVS   (input) CHARACTER*1
    *          = 'N': Schur vectors are not computed;
    *          = 'V': Schur vectors are computed.
    *
    *  SORT    (input) CHARACTER*1
    *          Specifies whether or not to order the eigenvalues on the
    *          diagonal of the Schur form.
    *          = 'N': Eigenvalues are not ordered;
    *          = 'S': Eigenvalues are ordered (see SELECT).
    *
    *  SELECT  (external procedure) LOGICAL FUNCTION of one COMPLEX argument
    *          SELECT must be declared EXTERNAL in the calling subroutine.
    *          If SORT = 'S', SELECT is used to select eigenvalues to order
    *          to the top left of the Schur form.
    *          If SORT = 'N', SELECT is not referenced.
    *          An eigenvalue W(j) is selected if SELECT(W(j)) is true.
    *
    *  SENSE   (input) CHARACTER*1
    *          Determines which reciprocal condition numbers are computed.
    *          = 'N': None are computed;
    *          = 'E': Computed for average of selected eigenvalues only;
    *          = 'V': Computed for selected right invariant subspace only;
    *          = 'B': Computed for both.
    *          If SENSE = 'E', 'V' or 'B', SORT must equal 'S'.
    *
    *  N       (input) INTEGER
    *          The order of the matrix A. N >= 0.
    *
    *  A       (input/output) COMPLEX array, dimension (LDA, N)
    *          On entry, the N-by-N matrix A.
    *          On exit, A is overwritten by its Schur form T.
    *
    *  LDA     (input) INTEGER
    *          The leading dimension of the array A.  LDA >= max(1,N).
    *
    *  SDIM    (output) INTEGER
    *          If SORT = 'N', SDIM = 0.
    *          If SORT = 'S', SDIM = number of eigenvalues for which
    *                         SELECT is true.
    *
    *  W       (output) COMPLEX array, dimension (N)
    *          W contains the computed eigenvalues, in the same order
    *          that they appear on the diagonal of the output Schur form T.
    *
    *  VS      (output) COMPLEX array, dimension (LDVS,N)
    *          If JOBVS = 'V', VS contains the unitary matrix Z of Schur
    *          vectors.
    *          If JOBVS = 'N', VS is not referenced.
    *
    *  LDVS    (input) INTEGER
    *          The leading dimension of the array VS.  LDVS >= 1, and if
    *          JOBVS = 'V', LDVS >= N.
    *
    *  RCONDE  (output) REAL
    *          If SENSE = 'E' or 'B', RCONDE contains the reciprocal
    *          condition number for the average of the selected eigenvalues.
    *          Not referenced if SENSE = 'N' or 'V'.
    *
    *  RCONDV  (output) REAL
    *          If SENSE = 'V' or 'B', RCONDV contains the reciprocal
    *          condition number for the selected right invariant subspace.
    *          Not referenced if SENSE = 'N' or 'E'.
    *
    *  WORK    (workspace/output) COMPLEX array, dimension (MAX(1,LWORK))
    *          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
    *
    *  LWORK   (input) INTEGER
    *          The dimension of the array WORK.  LWORK >= max(1,2*N).
    *          Also, if SENSE = 'E' or 'V' or 'B', LWORK >= 2*SDIM*(N-SDIM),
    *          where SDIM is the number of selected eigenvalues computed by
    *          this routine.  Note that 2*SDIM*(N-SDIM) <= N*N/2. Note also
    *          that an error is only returned if LWORK < max(1,2*N), but if
    *          SENSE = 'E' or 'V' or 'B' this may not be large enough.
    *          For good performance, LWORK must generally be larger.
    *
    *          If LWORK = -1, then a workspace query is assumed; the routine
    *          only calculates upper bound on the optimal size of the
    *          array WORK, returns this value as the first entry of the WORK
    *          array, and no error message related to LWORK is issued by
    *          XERBLA.
    *
    *  RWORK   (workspace) REAL array, dimension (N)
    *
    *  BWORK   (workspace) LOGICAL array, dimension (N)
    *          Not referenced if SORT = 'N'.
    *
    *  INFO    (output) INTEGER
    *          = 0: successful exit
    *          < 0: if INFO = -i, the i-th argument had an illegal value.
    *          > 0: if INFO = i, and i is
    *             <= N: the QR algorithm failed to compute all the
    *                   eigenvalues; elements 1:ILO-1 and i+1:N of W
    *                   contain those eigenvalues which have converged; if
    *                   JOBVS = 'V', VS contains the transformation which
    *                   reduces A to its partially converged Schur form.
    *             = N+1: the eigenvalues could not be reordered because some
    *                   eigenvalues were too close to separate (the problem
    *                   is very ill-conditioned);
    *             = N+2: after reordering, roundoff changed values of some
    *                   complex eigenvalues so that leading eigenvalues in
    *                   the Schur form no longer satisfy SELECT=.TRUE.  This
    *                   could also be caused by underflow due to scaling.
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          REAL               ZERO, ONE
          PARAMETER          ( ZERO = 0.0E0, ONE = 1.0E0 )
    *     ..
    *     .. Local Scalars ..
          LOGICAL            SCALEA, WANTSB, WANTSE, WANTSN, WANTST,
         $                   WANTSV, WANTVS
          INTEGER            HSWORK, I, IBAL, ICOND, IERR, IEVAL, IHI, ILO,
         $                   ITAU, IWRK, LWRK, MAXWRK, MINWRK
          REAL               ANRM, BIGNUM, CSCALE, EPS, SMLNUM
    *     ..
    *     .. Local Arrays ..
          REAL               DUM( 1 )
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           CCOPY, CGEBAK, CGEBAL, CGEHRD, CHSEQR, CLACPY,
         $                   CLASCL, CTRSEN, CUNGHR, SLABAD, SLASCL, XERBLA
    *     ..
    *     .. External Functions ..
          LOGICAL            LSAME
          INTEGER            ILAENV
          REAL               CLANGE, SLAMCH
          EXTERNAL           LSAME, ILAENV, CLANGE, SLAMCH
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          MAX, SQRT
    *     ..
    *     .. Executable Statements ..
    *
    *     Test the input arguments
    *
          INFO = 0
          WANTVS = LSAME( JOBVS, 'V' )
          WANTST = LSAME( SORT, 'S' )
          WANTSN = LSAME( SENSE, 'N' )
          WANTSE = LSAME( SENSE, 'E' )
          WANTSV = LSAME( SENSE, 'V' )
          WANTSB = LSAME( SENSE, 'B' )
          IF( ( .NOT.WANTVS ) .AND. ( .NOT.LSAME( JOBVS, 'N' ) ) ) THEN
             INFO = -1
          ELSE IF( ( .NOT.WANTST ) .AND. ( .NOT.LSAME( SORT, 'N' ) ) ) THEN
             INFO = -2
          ELSE IF( .NOT.( WANTSN .OR. WANTSE .OR. WANTSV .OR. WANTSB ) .OR.
         $         ( .NOT.WANTST .AND. .NOT.WANTSN ) ) THEN
             INFO = -4
          ELSE IF( N.LT.0 ) THEN
             INFO = -5
          ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
             INFO = -7
          ELSE IF( LDVS.LT.1 .OR. ( WANTVS .AND. LDVS.LT.N ) ) THEN
             INFO = -11
          END IF
    *
    *     Compute workspace
    *      (Note: Comments in the code beginning "Workspace:" describe the
    *       minimal amount of real workspace needed at that point in the
    *       code, as well as the preferred amount for good performance.
    *       CWorkspace refers to complex workspace, and RWorkspace to real
    *       workspace. NB refers to the optimal block size for the
    *       immediately following subroutine, as returned by ILAENV.
    *       HSWORK refers to the workspace preferred by CHSEQR, as
    *       calculated below. HSWORK is computed assuming ILO=1 and IHI=N,
    *       the worst case.
    *       If SENSE = 'E', 'V' or 'B', then the amount of workspace needed
    *       depends on SDIM, which is computed by the routine CTRSEN later
    *       in the code.)
    *
          IF( INFO.EQ.0 ) THEN
             IF( N.EQ.0 ) THEN
                MINWRK = 1
                LWRK = 1
             ELSE
                MAXWRK = N + N*ILAENV( 1, 'CGEHRD', ' ', N, 1, N, 0 )
                MINWRK = 2*N
    *
                CALL CHSEQR( 'S', JOBVS, N, 1, N, A, LDA, W, VS, LDVS,
         $             WORK, -1, IEVAL )
                HSWORK = WORK( 1 )
    *
                IF( .NOT.WANTVS ) THEN
                   MAXWRK = MAX( MAXWRK, HSWORK )
                ELSE
                   MAXWRK = MAX( MAXWRK, N + ( N - 1 )*ILAENV( 1, 'CUNGHR',
         $                       ' ', N, 1, N, -1 ) )
                   MAXWRK = MAX( MAXWRK, HSWORK )
                END IF
                LWRK = MAXWRK
                IF( .NOT.WANTSN )
         $         LWRK = MAX( LWRK, ( N*N )/2 )
             END IF
             WORK( 1 ) = LWRK
    *
             IF( LWORK.LT.MINWRK ) THEN
                INFO = -15
             END IF
          END IF
    *
          IF( INFO.NE.0 ) THEN
             CALL XERBLA( 'CGEESX', -INFO )
             RETURN
          END IF
    *
    *     Quick return if possible
    *
          IF( N.EQ.0 ) THEN
             SDIM = 0
             RETURN
          END IF
    *
    *     Get machine constants
    *
          EPS = SLAMCH( 'P' )
          SMLNUM = SLAMCH( 'S' )
          BIGNUM = ONE / SMLNUM
          CALL SLABAD( SMLNUM, BIGNUM )
          SMLNUM = SQRT( SMLNUM ) / EPS
          BIGNUM = ONE / SMLNUM
    *
    *     Scale A if max element outside range [SMLNUM,BIGNUM]
    *
          ANRM = CLANGE( 'M', N, N, A, LDA, DUM )
          SCALEA = .FALSE.
          IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
             SCALEA = .TRUE.
             CSCALE = SMLNUM
          ELSE IF( ANRM.GT.BIGNUM ) THEN
             SCALEA = .TRUE.
             CSCALE = BIGNUM
          END IF
          IF( SCALEA )
         $   CALL CLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR )
    *
    *
    *     Permute the matrix to make it more nearly triangular
    *     (CWorkspace: none)
    *     (RWorkspace: need N)
    *
          IBAL = 1
          CALL CGEBAL( 'P', N, A, LDA, ILO, IHI, RWORK( IBAL ), IERR )
    *
    *     Reduce to upper Hessenberg form
    *     (CWorkspace: need 2*N, prefer N+N*NB)
    *     (RWorkspace: none)
    *
          ITAU = 1
          IWRK = N + ITAU
          CALL CGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ),
         $             LWORK-IWRK+1, IERR )
    *
          IF( WANTVS ) THEN
    *
    *        Copy Householder vectors to VS
    *
             CALL CLACPY( 'L', N, N, A, LDA, VS, LDVS )
    *
    *        Generate unitary matrix in VS
    *        (CWorkspace: need 2*N-1, prefer N+(N-1)*NB)
    *        (RWorkspace: none)
    *
             CALL CUNGHR( N, ILO, IHI, VS, LDVS, WORK( ITAU ), WORK( IWRK ),
         $                LWORK-IWRK+1, IERR )
          END IF
    *
          SDIM = 0
    *
    *     Perform QR iteration, accumulating Schur vectors in VS if desired
    *     (CWorkspace: need 1, prefer HSWORK (see comments) )
    *     (RWorkspace: none)
    *
          IWRK = ITAU
          CALL CHSEQR( 'S', JOBVS, N, ILO, IHI, A, LDA, W, VS, LDVS,
         $             WORK( IWRK ), LWORK-IWRK+1, IEVAL )
          IF( IEVAL.GT.0 )
         $   INFO = IEVAL
    *
    *     Sort eigenvalues if desired
    *
          IF( WANTST .AND. INFO.EQ.0 ) THEN
             IF( SCALEA )
         $      CALL CLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, W, N, IERR )
             DO 10 I = 1, N
                BWORK( I ) = SELECT( W( I ) )
       10    CONTINUE
    *
    *        Reorder eigenvalues, transform Schur vectors, and compute
    *        reciprocal condition numbers
    *        (CWorkspace: if SENSE is not 'N', need 2*SDIM*(N-SDIM)
    *                     otherwise, need none )
    *        (RWorkspace: none)
    *
             CALL CTRSEN( SENSE, JOBVS, BWORK, N, A, LDA, VS, LDVS, W, SDIM,
         $                RCONDE, RCONDV, WORK( IWRK ), LWORK-IWRK+1,
         $                ICOND )
             IF( .NOT.WANTSN )
         $      MAXWRK = MAX( MAXWRK, 2*SDIM*( N-SDIM ) )
             IF( ICOND.EQ.-14 ) THEN
    *
    *           Not enough complex workspace
    *
                INFO = -15
             END IF
          END IF
    *
          IF( WANTVS ) THEN
    *
    *        Undo balancing
    *        (CWorkspace: none)
    *        (RWorkspace: need N)
    *
             CALL CGEBAK( 'P', 'R', N, ILO, IHI, RWORK( IBAL ), N, VS, LDVS,
         $                IERR )
          END IF
    *
          IF( SCALEA ) THEN
    *
    *        Undo scaling for the Schur form of A
    *
             CALL CLASCL( 'U', 0, 0, CSCALE, ANRM, N, N, A, LDA, IERR )
             CALL CCOPY( N, A, LDA+1, W, 1 )
             IF( ( WANTSV .OR. WANTSB ) .AND. INFO.EQ.0 ) THEN
                DUM( 1 ) = RCONDV
                CALL SLASCL( 'G', 0, 0, CSCALE, ANRM, 1, 1, DUM, 1, IERR )
                RCONDV = DUM( 1 )
             END IF
          END IF
    *
          WORK( 1 ) = MAXWRK
          RETURN
    *
    *     End of CGEESX
    *
          END