*> \brief <b> CGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download CGEESX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cgeesx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cgeesx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cgeesx.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE CGEESX( JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM, W,
* VS, LDVS, RCONDE, RCONDV, WORK, LWORK, RWORK,
* BWORK, INFO )
*
* .. 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
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> 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.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBVS
*> \verbatim
*> JOBVS is CHARACTER*1
*> = 'N': Schur vectors are not computed;
*> = 'V': Schur vectors are computed.
*> \endverbatim
*>
*> \param[in] SORT
*> \verbatim
*> SORT is 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).
*> \endverbatim
*>
*> \param[in] SELECT
*> \verbatim
*> SELECT is a 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.
*> \endverbatim
*>
*> \param[in] SENSE
*> \verbatim
*> SENSE is 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'.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is COMPLEX array, dimension (LDA, N)
*> On entry, the N-by-N matrix A.
*> On exit, A is overwritten by its Schur form T.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] SDIM
*> \verbatim
*> SDIM is INTEGER
*> If SORT = 'N', SDIM = 0.
*> If SORT = 'S', SDIM = number of eigenvalues for which
*> SELECT is true.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is 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.
*> \endverbatim
*>
*> \param[out] VS
*> \verbatim
*> VS is COMPLEX array, dimension (LDVS,N)
*> If JOBVS = 'V', VS contains the unitary matrix Z of Schur
*> vectors.
*> If JOBVS = 'N', VS is not referenced.
*> \endverbatim
*>
*> \param[in] LDVS
*> \verbatim
*> LDVS is INTEGER
*> The leading dimension of the array VS. LDVS >= 1, and if
*> JOBVS = 'V', LDVS >= N.
*> \endverbatim
*>
*> \param[out] RCONDE
*> \verbatim
*> RCONDE is 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'.
*> \endverbatim
*>
*> \param[out] RCONDV
*> \verbatim
*> RCONDV is 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'.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is COMPLEX 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. 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.
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*> RWORK is REAL array, dimension (N)
*> \endverbatim
*>
*> \param[out] BWORK
*> \verbatim
*> BWORK is LOGICAL array, dimension (N)
*> Not referenced if SORT = '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 = 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.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup geesx
*
* =====================================================================
SUBROUTINE CGEESX( JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM,
$ W,
$ VS, LDVS, RCONDE, RCONDV, WORK, LWORK, RWORK,
$ BWORK, INFO )
*
* -- LAPACK driver 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 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
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY, 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, SLASCL, XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
REAL CLANGE, SLAMCH, SROUNDUP_LWORK
EXTERNAL LSAME, ILAENV, CLANGE, SLAMCH,
$ SROUNDUP_LWORK
* ..
* .. 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' )
LQUERY = ( LWORK.EQ.-1 )
*
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 = INT( 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 ) = SROUNDUP_LWORK(LWRK)
*
IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
INFO = -15
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'CGEESX', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
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
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 ) = SROUNDUP_LWORK(MAXWRK)
RETURN
*
* End of CGEESX
*
END