cggev.f

          SUBROUTINE CGGEV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA,
         $                  VL, LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO )
    *
    *  -- LAPACK driver routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          CHARACTER          JOBVL, JOBVR
          INTEGER            INFO, LDA, LDB, LDVL, LDVR, LWORK, N
    *     ..
    *     .. Array Arguments ..
          REAL               RWORK( * )
          COMPLEX            A( LDA, * ), ALPHA( * ), B( LDB, * ),
         $                   BETA( * ), VL( LDVL, * ), VR( LDVR, * ),
         $                   WORK( * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  CGGEV computes for a pair of N-by-N complex nonsymmetric matrices
    *  (A,B), the generalized eigenvalues, and optionally, the left and/or
    *  right generalized eigenvectors.
    *
    *  A generalized eigenvalue for a pair of matrices (A,B) is a scalar
    *  lambda or a ratio alpha/beta = lambda, such that A - lambda*B is
    *  singular. It is usually represented as the pair (alpha,beta), as
    *  there is a reasonable interpretation for beta=0, and even for both
    *  being zero.
    *
    *  The right generalized eigenvector v(j) corresponding to the
    *  generalized eigenvalue lambda(j) of (A,B) satisfies
    *
    *               A * v(j) = lambda(j) * B * v(j).
    *
    *  The left generalized eigenvector u(j) corresponding to the
    *  generalized eigenvalues lambda(j) of (A,B) satisfies
    *
    *               u(j)**H * A = lambda(j) * u(j)**H * B
    *
    *  where u(j)**H is the conjugate-transpose of u(j).
    *
    *  Arguments
    *  =========
    *
    *  JOBVL   (input) CHARACTER*1
    *          = 'N':  do not compute the left generalized eigenvectors;
    *          = 'V':  compute the left generalized eigenvectors.
    *
    *  JOBVR   (input) CHARACTER*1
    *          = 'N':  do not compute the right generalized eigenvectors;
    *          = 'V':  compute the right generalized eigenvectors.
    *
    *  N       (input) INTEGER
    *          The order of the matrices A, B, VL, and VR.  N >= 0.
    *
    *  A       (input/output) COMPLEX array, dimension (LDA, N)
    *          On entry, the matrix A in the pair (A,B).
    *          On exit, A has been overwritten.
    *
    *  LDA     (input) INTEGER
    *          The leading dimension of A.  LDA >= max(1,N).
    *
    *  B       (input/output) COMPLEX array, dimension (LDB, N)
    *          On entry, the matrix B in the pair (A,B).
    *          On exit, B has been overwritten.
    *
    *  LDB     (input) INTEGER
    *          The leading dimension of B.  LDB >= max(1,N).
    *
    *  ALPHA   (output) COMPLEX array, dimension (N)
    *  BETA    (output) COMPLEX array, dimension (N)
    *          On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the
    *          generalized eigenvalues.
    *
    *          Note: the quotients ALPHA(j)/BETA(j) may easily over- or
    *          underflow, and BETA(j) may even be zero.  Thus, the user
    *          should avoid naively computing the ratio alpha/beta.
    *          However, ALPHA will be always less than and usually
    *          comparable with norm(A) in magnitude, and BETA always less
    *          than and usually comparable with norm(B).
    *
    *  VL      (output) COMPLEX array, dimension (LDVL,N)
    *          If JOBVL = 'V', the left generalized eigenvectors u(j) are
    *          stored one after another in the columns of VL, in the same
    *          order as their eigenvalues.
    *          Each eigenvector is scaled so the largest component has
    *          abs(real part) + abs(imag. part) = 1.
    *          Not referenced if JOBVL = 'N'.
    *
    *  LDVL    (input) INTEGER
    *          The leading dimension of the matrix VL. LDVL >= 1, and
    *          if JOBVL = 'V', LDVL >= N.
    *
    *  VR      (output) COMPLEX array, dimension (LDVR,N)
    *          If JOBVR = 'V', the right generalized eigenvectors v(j) are
    *          stored one after another in the columns of VR, in the same
    *          order as their eigenvalues.
    *          Each eigenvector is scaled so the largest component has
    *          abs(real part) + abs(imag. part) = 1.
    *          Not referenced if JOBVR = 'N'.
    *
    *  LDVR    (input) INTEGER
    *          The leading dimension of the matrix VR. LDVR >= 1, and
    *          if JOBVR = 'V', LDVR >= N.
    *
    *  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).
    *          For good performance, LWORK must generally be larger.
    *
    *          If LWORK = -1, then a workspace query is assumed; the routine
    *          only calculates the optimal size of the WORK array, returns
    *          this value as the first entry of the WORK array, and no error
    *          message related to LWORK is issued by XERBLA.
    *
    *  RWORK   (workspace/output) REAL array, dimension (8*N)
    *
    *  INFO    (output) INTEGER
    *          = 0:  successful exit
    *          < 0:  if INFO = -i, the i-th argument had an illegal value.
    *          =1,...,N:
    *                The QZ iteration failed.  No eigenvectors have been
    *                calculated, but ALPHA(j) and BETA(j) should be
    *                correct for j=INFO+1,...,N.
    *          > N:  =N+1: other then QZ iteration failed in SHGEQZ,
    *                =N+2: error return from STGEVC.
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          REAL               ZERO, ONE
          PARAMETER          ( ZERO = 0.0E0, ONE = 1.0E0 )
          COMPLEX            CZERO, CONE
          PARAMETER          ( CZERO = ( 0.0E0, 0.0E0 ),
         $                   CONE = ( 1.0E0, 0.0E0 ) )
    *     ..
    *     .. Local Scalars ..
          LOGICAL            ILASCL, ILBSCL, ILV, ILVL, ILVR, LQUERY
          CHARACTER          CHTEMP
          INTEGER            ICOLS, IERR, IHI, IJOBVL, IJOBVR, ILEFT, ILO,
         $                   IN, IRIGHT, IROWS, IRWRK, ITAU, IWRK, JC, JR,
         $                   LWKMIN, LWKOPT
          REAL               ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS,
         $                   SMLNUM, TEMP
          COMPLEX            X
    *     ..
    *     .. Local Arrays ..
          LOGICAL            LDUMMA( 1 )
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           CGEQRF, CGGBAK, CGGBAL, CGGHRD, CHGEQZ, CLACPY,
         $                   CLASCL, CLASET, CTGEVC, CUNGQR, CUNMQR, SLABAD,
         $                   XERBLA
    *     ..
    *     .. External Functions ..
          LOGICAL            LSAME
          INTEGER            ILAENV
          REAL               CLANGE, SLAMCH
          EXTERNAL           LSAME, ILAENV, CLANGE, SLAMCH
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          ABS, AIMAG, MAX, REAL, SQRT
    *     ..
    *     .. Statement Functions ..
          REAL               ABS1
    *     ..
    *     .. Statement Function definitions ..
          ABS1( X ) = ABS( REAL( X ) ) + ABS( AIMAG( X ) )
    *     ..
    *     .. Executable Statements ..
    *
    *     Decode the input arguments
    *
          IF( LSAME( JOBVL, 'N' ) ) THEN
             IJOBVL = 1
             ILVL = .FALSE.
          ELSE IF( LSAME( JOBVL, 'V' ) ) THEN
             IJOBVL = 2
             ILVL = .TRUE.
          ELSE
             IJOBVL = -1
             ILVL = .FALSE.
          END IF
    *
          IF( LSAME( JOBVR, 'N' ) ) THEN
             IJOBVR = 1
             ILVR = .FALSE.
          ELSE IF( LSAME( JOBVR, 'V' ) ) THEN
             IJOBVR = 2
             ILVR = .TRUE.
          ELSE
             IJOBVR = -1
             ILVR = .FALSE.
          END IF
          ILV = ILVL .OR. ILVR
    *
    *     Test the input arguments
    *
          INFO = 0
          LQUERY = ( LWORK.EQ.-1 )
          IF( IJOBVL.LE.0 ) THEN
             INFO = -1
          ELSE IF( IJOBVR.LE.0 ) THEN
             INFO = -2
          ELSE IF( N.LT.0 ) THEN
             INFO = -3
          ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
             INFO = -5
          ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
             INFO = -7
          ELSE IF( LDVL.LT.1 .OR. ( ILVL .AND. LDVL.LT.N ) ) THEN
             INFO = -11
          ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN
             INFO = -13
          END IF
    *
    *     Compute workspace
    *      (Note: Comments in the code beginning "Workspace:" describe the
    *       minimal amount of workspace needed at that point in the code,
    *       as well as the preferred amount for good performance.
    *       NB refers to the optimal block size for the immediately
    *       following subroutine, as returned by ILAENV. The workspace is
    *       computed assuming ILO = 1 and IHI = N, the worst case.)
    *
          IF( INFO.EQ.0 ) THEN
             LWKMIN = MAX( 1, 2*N )
             LWKOPT = MAX( 1, N + N*ILAENV( 1, 'CGEQRF', ' ', N, 1, N, 0 ) )
             LWKOPT = MAX( LWKOPT, N +
         $                 N*ILAENV( 1, 'CUNMQR', ' ', N, 1, N, 0 ) )
             IF( ILVL ) THEN
                LWKOPT = MAX( LWKOPT, N +
         $                 N*ILAENV( 1, 'CUNGQR', ' ', N, 1, N, -1 ) )
             END IF
             WORK( 1 ) = LWKOPT
    *
             IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY )
         $      INFO = -15
          END IF
    *
          IF( INFO.NE.0 ) THEN
             CALL XERBLA( 'CGGEV ', -INFO )
             RETURN
          ELSE IF( LQUERY ) THEN
             RETURN
          END IF
    *
    *     Quick return if possible
    *
          IF( N.EQ.0 )
         $   RETURN
    *
    *     Get machine constants
    *
          EPS = SLAMCH( 'E' )*SLAMCH( 'B' )
          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, RWORK )
          ILASCL = .FALSE.
          IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
             ANRMTO = SMLNUM
             ILASCL = .TRUE.
          ELSE IF( ANRM.GT.BIGNUM ) THEN
             ANRMTO = BIGNUM
             ILASCL = .TRUE.
          END IF
          IF( ILASCL )
         $   CALL CLASCL( 'G', 0, 0, ANRM, ANRMTO, N, N, A, LDA, IERR )
    *
    *     Scale B if max element outside range [SMLNUM,BIGNUM]
    *
          BNRM = CLANGE( 'M', N, N, B, LDB, RWORK )
          ILBSCL = .FALSE.
          IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
             BNRMTO = SMLNUM
             ILBSCL = .TRUE.
          ELSE IF( BNRM.GT.BIGNUM ) THEN
             BNRMTO = BIGNUM
             ILBSCL = .TRUE.
          END IF
          IF( ILBSCL )
         $   CALL CLASCL( 'G', 0, 0, BNRM, BNRMTO, N, N, B, LDB, IERR )
    *
    *     Permute the matrices A, B to isolate eigenvalues if possible
    *     (Real Workspace: need 6*N)
    *
          ILEFT = 1
          IRIGHT = N + 1
          IRWRK = IRIGHT + N
          CALL CGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, RWORK( ILEFT ),
         $             RWORK( IRIGHT ), RWORK( IRWRK ), IERR )
    *
    *     Reduce B to triangular form (QR decomposition of B)
    *     (Complex Workspace: need N, prefer N*NB)
    *
          IROWS = IHI + 1 - ILO
          IF( ILV ) THEN
             ICOLS = N + 1 - ILO
          ELSE
             ICOLS = IROWS
          END IF
          ITAU = 1
          IWRK = ITAU + IROWS
          CALL CGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
         $             WORK( IWRK ), LWORK+1-IWRK, IERR )
    *
    *     Apply the orthogonal transformation to matrix A
    *     (Complex Workspace: need N, prefer N*NB)
    *
          CALL CUNMQR( 'L', 'C', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
         $             WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWRK ),
         $             LWORK+1-IWRK, IERR )
    *
    *     Initialize VL
    *     (Complex Workspace: need N, prefer N*NB)
    *
          IF( ILVL ) THEN
             CALL CLASET( 'Full', N, N, CZERO, CONE, VL, LDVL )
             IF( IROWS.GT.1 ) THEN
                CALL CLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
         $                   VL( ILO+1, ILO ), LDVL )
             END IF
             CALL CUNGQR( IROWS, IROWS, IROWS, VL( ILO, ILO ), LDVL,
         $                WORK( ITAU ), WORK( IWRK ), LWORK+1-IWRK, IERR )
          END IF
    *
    *     Initialize VR
    *
          IF( ILVR )
         $   CALL CLASET( 'Full', N, N, CZERO, CONE, VR, LDVR )
    *
    *     Reduce to generalized Hessenberg form
    *
          IF( ILV ) THEN
    *
    *        Eigenvectors requested -- work on whole matrix.
    *
             CALL CGGHRD( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL,
         $                LDVL, VR, LDVR, IERR )
          ELSE
             CALL CGGHRD( 'N', 'N', IROWS, 1, IROWS, A( ILO, ILO ), LDA,
         $                B( ILO, ILO ), LDB, VL, LDVL, VR, LDVR, IERR )
          END IF
    *
    *     Perform QZ algorithm (Compute eigenvalues, and optionally, the
    *     Schur form and Schur vectors)
    *     (Complex Workspace: need N)
    *     (Real Workspace: need N)
    *
          IWRK = ITAU
          IF( ILV ) THEN
             CHTEMP = 'S'
          ELSE
             CHTEMP = 'E'
          END IF
          CALL CHGEQZ( CHTEMP, JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB,
         $             ALPHA, BETA, VL, LDVL, VR, LDVR, WORK( IWRK ),
         $             LWORK+1-IWRK, RWORK( IRWRK ), IERR )
          IF( IERR.NE.0 ) THEN
             IF( IERR.GT.0 .AND. IERR.LE.N ) THEN
                INFO = IERR
             ELSE IF( IERR.GT.N .AND. IERR.LE.2*N ) THEN
                INFO = IERR - N
             ELSE
                INFO = N + 1
             END IF
             GO TO 70
          END IF
    *
    *     Compute Eigenvectors
    *     (Real Workspace: need 2*N)
    *     (Complex Workspace: need 2*N)
    *
          IF( ILV ) THEN
             IF( ILVL ) THEN
                IF( ILVR ) THEN
                   CHTEMP = 'B'
                ELSE
                   CHTEMP = 'L'
                END IF
             ELSE
                CHTEMP = 'R'
             END IF
    *
             CALL CTGEVC( CHTEMP, 'B', LDUMMA, N, A, LDA, B, LDB, VL, LDVL,
         $                VR, LDVR, N, IN, WORK( IWRK ), RWORK( IRWRK ),
         $                IERR )
             IF( IERR.NE.0 ) THEN
                INFO = N + 2
                GO TO 70
             END IF
    *
    *        Undo balancing on VL and VR and normalization
    *        (Workspace: none needed)
    *
             IF( ILVL ) THEN
                CALL CGGBAK( 'P', 'L', N, ILO, IHI, RWORK( ILEFT ),
         $                   RWORK( IRIGHT ), N, VL, LDVL, IERR )
                DO 30 JC = 1, N
                   TEMP = ZERO
                   DO 10 JR = 1, N
                      TEMP = MAX( TEMP, ABS1( VL( JR, JC ) ) )
       10          CONTINUE
                   IF( TEMP.LT.SMLNUM )
         $            GO TO 30
                   TEMP = ONE / TEMP
                   DO 20 JR = 1, N
                      VL( JR, JC ) = VL( JR, JC )*TEMP
       20          CONTINUE
       30       CONTINUE
             END IF
             IF( ILVR ) THEN
                CALL CGGBAK( 'P', 'R', N, ILO, IHI, RWORK( ILEFT ),
         $                   RWORK( IRIGHT ), N, VR, LDVR, IERR )
                DO 60 JC = 1, N
                   TEMP = ZERO
                   DO 40 JR = 1, N
                      TEMP = MAX( TEMP, ABS1( VR( JR, JC ) ) )
       40          CONTINUE
                   IF( TEMP.LT.SMLNUM )
         $            GO TO 60
                   TEMP = ONE / TEMP
                   DO 50 JR = 1, N
                      VR( JR, JC ) = VR( JR, JC )*TEMP
       50          CONTINUE
       60       CONTINUE
             END IF
          END IF
    *
    *     Undo scaling if necessary
    *
          IF( ILASCL )
         $   CALL CLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHA, N, IERR )
    *
          IF( ILBSCL )
         $   CALL CLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR )
    *
       70 CONTINUE
          WORK( 1 ) = LWKOPT
    *
          RETURN
    *
    *     End of CGGEV
    *
          END