sgeesx.f

          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 (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, 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
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  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).
    *
    *  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 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).
    *
    *  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) REAL array, dimension (LDA, N)
    *          On entry, the N-by-N matrix A.
    *          On exit, A is overwritten by its real 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 (after sorting)
    *                         for which SELECT is true. (Complex conjugate
    *                         pairs for which SELECT is true for either
    *                         eigenvalue count as 2.)
    *
    *  WR      (output) REAL array, dimension (N)
    *  WI      (output) 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.
    *
    *  VS      (output) REAL array, dimension (LDVS,N)
    *          If JOBVS = 'V', VS contains the orthogonal 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) REAL 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(1,3*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.
    *
    *  IWORK   (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
    *          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
    *
    *  LIWORK  (input) 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.
    *
    *  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 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.
    *
    *  =====================================================================
    *
    *     .. 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, SLABAD,
         $                   SLACPY, SLASCL, SORGHR, SSWAP, STRSEN, XERBLA
    *     ..
    *     .. External Functions ..
          LOGICAL            LSAME
          INTEGER            ILAENV
          REAL               SLAMCH, SLANGE
          EXTERNAL           LSAME, ILAENV, SLAMCH, SLANGE
    *     ..
    *     .. 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 = 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 ) = 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
          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 = 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 )
                         CALL SSWAP( N, VS( 1, I ), 1, VS( 1, I+1 ), 1 )
                         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 ) = MAXWRK
          IF( WANTSV .OR. WANTSB ) THEN
             IWORK( 1 ) = SDIM*(N-SDIM)
          ELSE
             IWORK( 1 ) = 1
          END IF
    *
          RETURN
    *
    *     End of SGEESX
    *
          END