*> \brief <b> SGEESX 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
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*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sgeesx.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE SGEESX( JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM,
* WR, WI, VS, LDVS, RCONDE, RCONDV, WORK, LWORK,
* IWORK, LIWORK, BWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBVS, SENSE, SORT
* INTEGER INFO, LDA, LDVS, LIWORK, LWORK, N, SDIM
* REAL RCONDE, RCONDV
* ..
* .. Array Arguments ..
* LOGICAL BWORK( * )
* INTEGER IWORK( * )
* REAL A( LDA, * ), VS( LDVS, * ), WI( * ), WORK( * ),
* $ WR( * )
* ..
* .. Function Arguments ..
* LOGICAL SELECT
* EXTERNAL SELECT
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> SGEESX computes for an N-by-N real nonsymmetric matrix A, the
*> eigenvalues, the real Schur form T, and, optionally, the matrix of
*> Schur vectors Z. This gives the Schur factorization A = Z*T*(Z**T).
*>
*> Optionally, it also orders the eigenvalues on the diagonal of the
*> real 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 real matrix is in real Schur form if it is upper quasi-triangular
*> with 1-by-1 and 2-by-2 blocks. 2-by-2 blocks will be standardized in
*> the form
*> [ a b ]
*> [ c a ]
*>
*> where b*c < 0. The eigenvalues of such a block are a +- sqrt(bc).
*> \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 two REAL arguments
*> SELECT must be declared EXTERNAL in the calling subroutine.
*> If SORT = 'S', SELECT is used to select eigenvalues to sort
*> to the top left of the Schur form.
*> If SORT = 'N', SELECT is not referenced.
*> An eigenvalue WR(j)+sqrt(-1)*WI(j) is selected if
*> SELECT(WR(j),WI(j)) is true; i.e., if either one of a
*> complex conjugate pair of eigenvalues is selected, then both
*> are. Note that a selected complex eigenvalue may no longer
*> satisfy SELECT(WR(j),WI(j)) = .TRUE. after ordering, since
*> ordering may change the value of complex eigenvalues
*> (especially if the eigenvalue is ill-conditioned); in this
*> case INFO may be set to N+3 (see INFO below).
*> \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 REAL array, dimension (LDA, N)
*> On entry, the N-by-N matrix A.
*> On exit, A is overwritten by its real 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 (after sorting)
*> for which SELECT is true. (Complex conjugate
*> pairs for which SELECT is true for either
*> eigenvalue count as 2.)
*> \endverbatim
*>
*> \param[out] WR
*> \verbatim
*> WR is REAL array, dimension (N)
*> \endverbatim
*>
*> \param[out] WI
*> \verbatim
*> WI is REAL array, dimension (N)
*> WR and WI contain the real and imaginary parts, respectively,
*> of the computed eigenvalues, in the same order that they
*> appear on the diagonal of the output Schur form T. Complex
*> conjugate pairs of eigenvalues appear consecutively with the
*> eigenvalue having the positive imaginary part first.
*> \endverbatim
*>
*> \param[out] VS
*> \verbatim
*> VS is REAL array, dimension (LDVS,N)
*> If JOBVS = 'V', VS contains the orthogonal 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 REAL 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(1,3*N).
*> Also, if SENSE = 'E' or 'V' or 'B',
*> LWORK >= N+2*SDIM*(N-SDIM), where SDIM is the number of
*> selected eigenvalues computed by this routine. Note that
*> N+2*SDIM*(N-SDIM) <= N+N*N/2. Note also that an error is only
*> returned if LWORK < max(1,3*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 bounds on the optimal sizes of the
*> arrays WORK and IWORK, returns these values as the first
*> entries of the WORK and IWORK arrays, and no error messages
*> related to LWORK or LIWORK are 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.
*> LIWORK >= 1; if SENSE = 'V' or 'B', LIWORK >= SDIM*(N-SDIM).
*> Note that SDIM*(N-SDIM) <= N*N/4. Note also that an error is
*> only returned if LIWORK < 1, but if SENSE = 'V' or 'B' this
*> may not be large enough.
*>
*> If LIWORK = -1, then a workspace query is assumed; the
*> routine only calculates upper bounds on the optimal sizes of
*> the arrays WORK and IWORK, returns these values as the first
*> entries of the WORK and IWORK arrays, and no error messages
*> related to LWORK or LIWORK are issued by XERBLA.
*> \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 WR and WI
*> 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 SGEESX( JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM,
$ WR, WI, VS, LDVS, RCONDE, RCONDV, WORK, LWORK,
$ IWORK, LIWORK, 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, LIWORK, LWORK, N, SDIM
REAL RCONDE, RCONDV
* ..
* .. Array Arguments ..
LOGICAL BWORK( * )
INTEGER IWORK( * )
REAL A( LDA, * ), VS( LDVS, * ), WI( * ), WORK( * ),
$ WR( * )
* ..
* .. Function Arguments ..
LOGICAL SELECT
EXTERNAL SELECT
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 )
* ..
* .. Local Scalars ..
LOGICAL CURSL, LASTSL, LQUERY, LST2SL, SCALEA, WANTSB,
$ WANTSE, WANTSN, WANTST, WANTSV, WANTVS
INTEGER HSWORK, I, I1, I2, IBAL, ICOND, IERR, IEVAL,
$ IHI, ILO, INXT, IP, ITAU, IWRK, LWRK, LIWRK,
$ MAXWRK, MINWRK
REAL ANRM, BIGNUM, CSCALE, EPS, SMLNUM
* ..
* .. Local Arrays ..
REAL DUM( 1 )
* ..
* .. External Subroutines ..
EXTERNAL SCOPY, SGEBAK, SGEBAL, SGEHRD,
$ SHSEQR,
$ SLACPY, SLASCL, SORGHR, SSWAP, STRSEN, XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
REAL SLAMCH, SLANGE, SROUNDUP_LWORK
EXTERNAL LSAME, ILAENV, SLAMCH,
$ SLANGE, 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 .OR. LIWORK.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 = -12
END IF
*
* Compute workspace
* (Note: Comments in the code beginning "RWorkspace:" describe the
* minimal amount of real workspace needed at that point in the
* code, as well as the preferred amount for good performance.
* IWorkspace refers to integer workspace.
* NB refers to the optimal block size for the immediately
* following subroutine, as returned by ILAENV.
* HSWORK refers to the workspace preferred by SHSEQR, 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 STRSEN later
* in the code.)
*
IF( INFO.EQ.0 ) THEN
LIWRK = 1
IF( N.EQ.0 ) THEN
MINWRK = 1
LWRK = 1
ELSE
MAXWRK = 2*N + N*ILAENV( 1, 'SGEHRD', ' ', N, 1, N, 0 )
MINWRK = 3*N
*
CALL SHSEQR( 'S', JOBVS, N, 1, N, A, LDA, WR, WI, VS,
$ LDVS,
$ WORK, -1, IEVAL )
HSWORK = INT( WORK( 1 ) )
*
IF( .NOT.WANTVS ) THEN
MAXWRK = MAX( MAXWRK, N + HSWORK )
ELSE
MAXWRK = MAX( MAXWRK, 2*N + ( N - 1 )*ILAENV( 1,
$ 'SORGHR', ' ', N, 1, N, -1 ) )
MAXWRK = MAX( MAXWRK, N + HSWORK )
END IF
LWRK = MAXWRK
IF( .NOT.WANTSN )
$ LWRK = MAX( LWRK, N + ( N*N )/2 )
IF( WANTSV .OR. WANTSB )
$ LIWRK = ( N*N )/4
END IF
IWORK( 1 ) = LIWRK
WORK( 1 ) = SROUNDUP_LWORK(LWRK)
*
IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
INFO = -16
ELSE IF( LIWORK.LT.1 .AND. .NOT.LQUERY ) THEN
INFO = -18
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SGEESX', -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 = SLANGE( '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 SLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR )
*
* Permute the matrix to make it more nearly triangular
* (RWorkspace: need N)
*
IBAL = 1
CALL SGEBAL( 'P', N, A, LDA, ILO, IHI, WORK( IBAL ), IERR )
*
* Reduce to upper Hessenberg form
* (RWorkspace: need 3*N, prefer 2*N+N*NB)
*
ITAU = N + IBAL
IWRK = N + ITAU
CALL SGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ),
$ LWORK-IWRK+1, IERR )
*
IF( WANTVS ) THEN
*
* Copy Householder vectors to VS
*
CALL SLACPY( 'L', N, N, A, LDA, VS, LDVS )
*
* Generate orthogonal matrix in VS
* (RWorkspace: need 3*N-1, prefer 2*N+(N-1)*NB)
*
CALL SORGHR( 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
* (RWorkspace: need N+1, prefer N+HSWORK (see comments) )
*
IWRK = ITAU
CALL SHSEQR( 'S', JOBVS, N, ILO, IHI, A, LDA, WR, WI, 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 ) THEN
CALL SLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, WR, N, IERR )
CALL SLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, WI, N, IERR )
END IF
DO 10 I = 1, N
BWORK( I ) = SELECT( WR( I ), WI( I ) )
10 CONTINUE
*
* Reorder eigenvalues, transform Schur vectors, and compute
* reciprocal condition numbers
* (RWorkspace: if SENSE is not 'N', need N+2*SDIM*(N-SDIM)
* otherwise, need N )
* (IWorkspace: if SENSE is 'V' or 'B', need SDIM*(N-SDIM)
* otherwise, need 0 )
*
CALL STRSEN( SENSE, JOBVS, BWORK, N, A, LDA, VS, LDVS, WR,
$ WI,
$ SDIM, RCONDE, RCONDV, WORK( IWRK ), LWORK-IWRK+1,
$ IWORK, LIWORK, ICOND )
IF( .NOT.WANTSN )
$ MAXWRK = MAX( MAXWRK, N+2*SDIM*( N-SDIM ) )
IF( ICOND.EQ.-15 ) THEN
*
* Not enough real workspace
*
INFO = -16
ELSE IF( ICOND.EQ.-17 ) THEN
*
* Not enough integer workspace
*
INFO = -18
ELSE IF( ICOND.GT.0 ) THEN
*
* STRSEN failed to reorder or to restore standard Schur form
*
INFO = ICOND + N
END IF
END IF
*
IF( WANTVS ) THEN
*
* Undo balancing
* (RWorkspace: need N)
*
CALL SGEBAK( 'P', 'R', N, ILO, IHI, WORK( IBAL ), N, VS,
$ LDVS,
$ IERR )
END IF
*
IF( SCALEA ) THEN
*
* Undo scaling for the Schur form of A
*
CALL SLASCL( 'H', 0, 0, CSCALE, ANRM, N, N, A, LDA, IERR )
CALL SCOPY( N, A, LDA+1, WR, 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
IF( CSCALE.EQ.SMLNUM ) THEN
*
* If scaling back towards underflow, adjust WI if an
* offdiagonal element of a 2-by-2 block in the Schur form
* underflows.
*
IF( IEVAL.GT.0 ) THEN
I1 = IEVAL + 1
I2 = IHI - 1
CALL SLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WI, N,
$ IERR )
ELSE IF( WANTST ) THEN
I1 = 1
I2 = N - 1
ELSE
I1 = ILO
I2 = IHI - 1
END IF
INXT = I1 - 1
DO 20 I = I1, I2
IF( I.LT.INXT )
$ GO TO 20
IF( WI( I ).EQ.ZERO ) THEN
INXT = I + 1
ELSE
IF( A( I+1, I ).EQ.ZERO ) THEN
WI( I ) = ZERO
WI( I+1 ) = ZERO
ELSE IF( A( I+1, I ).NE.ZERO .AND. A( I, I+1 ).EQ.
$ ZERO ) THEN
WI( I ) = ZERO
WI( I+1 ) = ZERO
IF( I.GT.1 )
$ CALL SSWAP( I-1, A( 1, I ), 1, A( 1, I+1 ),
$ 1 )
IF( N.GT.I+1 )
$ CALL SSWAP( N-I-1, A( I, I+2 ), LDA,
$ A( I+1, I+2 ), LDA )
IF( WANTVS ) THEN
CALL SSWAP( N, VS( 1, I ), 1, VS( 1, I+1 ),
$ 1 )
END IF
A( I, I+1 ) = A( I+1, I )
A( I+1, I ) = ZERO
END IF
INXT = I + 2
END IF
20 CONTINUE
END IF
CALL SLASCL( 'G', 0, 0, CSCALE, ANRM, N-IEVAL, 1,
$ WI( IEVAL+1 ), MAX( N-IEVAL, 1 ), IERR )
END IF
*
IF( WANTST .AND. INFO.EQ.0 ) THEN
*
* Check if reordering successful
*
LASTSL = .TRUE.
LST2SL = .TRUE.
SDIM = 0
IP = 0
DO 30 I = 1, N
CURSL = SELECT( WR( I ), WI( I ) )
IF( WI( I ).EQ.ZERO ) THEN
IF( CURSL )
$ SDIM = SDIM + 1
IP = 0
IF( CURSL .AND. .NOT.LASTSL )
$ INFO = N + 2
ELSE
IF( IP.EQ.1 ) THEN
*
* Last eigenvalue of conjugate pair
*
CURSL = CURSL .OR. LASTSL
LASTSL = CURSL
IF( CURSL )
$ SDIM = SDIM + 2
IP = -1
IF( CURSL .AND. .NOT.LST2SL )
$ INFO = N + 2
ELSE
*
* First eigenvalue of conjugate pair
*
IP = 1
END IF
END IF
LST2SL = LASTSL
LASTSL = CURSL
30 CONTINUE
END IF
*
WORK( 1 ) = SROUNDUP_LWORK(MAXWRK)
IF( WANTSV .OR. WANTSB ) THEN
IWORK( 1 ) = SDIM*(N-SDIM)
ELSE
IWORK( 1 ) = 1
END IF
*
RETURN
*
* End of SGEESX
*
END