OpenBLAS/lapack-netlib/SRC/cggev.f

*> \brief <b> CGGEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
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*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE CGGEV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA,
*                         VL, LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO )
*
*       .. 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( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> 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).
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBVL
*> \verbatim
*>          JOBVL is CHARACTER*1
*>          = 'N':  do not compute the left generalized eigenvectors;
*>          = 'V':  compute the left generalized eigenvectors.
*> \endverbatim
*>
*> \param[in] JOBVR
*> \verbatim
*>          JOBVR is CHARACTER*1
*>          = 'N':  do not compute the right generalized eigenvectors;
*>          = 'V':  compute the right generalized eigenvectors.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrices A, B, VL, and VR.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is COMPLEX array, dimension (LDA, N)
*>          On entry, the matrix A in the pair (A,B).
*>          On exit, A has been overwritten.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is COMPLEX array, dimension (LDB, N)
*>          On entry, the matrix B in the pair (A,B).
*>          On exit, B has been overwritten.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] ALPHA
*> \verbatim
*>          ALPHA is COMPLEX array, dimension (N)
*> \endverbatim
*>
*> \param[out] BETA
*> \verbatim
*>          BETA is 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).
*> \endverbatim
*>
*> \param[out] VL
*> \verbatim
*>          VL is 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'.
*> \endverbatim
*>
*> \param[in] LDVL
*> \verbatim
*>          LDVL is INTEGER
*>          The leading dimension of the matrix VL. LDVL >= 1, and
*>          if JOBVL = 'V', LDVL >= N.
*> \endverbatim
*>
*> \param[out] VR
*> \verbatim
*>          VR is 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'.
*> \endverbatim
*>
*> \param[in] LDVR
*> \verbatim
*>          LDVR is INTEGER
*>          The leading dimension of the matrix VR. LDVR >= 1, and
*>          if JOBVR = 'V', LDVR >= N.
*> \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).
*>          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.
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*>          RWORK is REAL array, dimension (8*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is 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 CHGEQZ,
*>                =N+2: error return from CTGEVC.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ggev
*
*  =====================================================================
      SUBROUTINE CGGEV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA,
     $                  VL, LDVL, VR, LDVR, WORK, LWORK, RWORK, 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          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( * )
*     ..
*
*  =====================================================================
*
*     .. 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, XERBLA
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      REAL               CLANGE, SLAMCH, SROUNDUP_LWORK
      EXTERNAL           LSAME, ILAENV, CLANGE, SLAMCH, SROUNDUP_LWORK
*     ..
*     .. 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 ) = SROUNDUP_LWORK(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
      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
*
   70 CONTINUE
*
      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 )
*
      WORK( 1 ) = SROUNDUP_LWORK(LWKOPT)
      RETURN
*
*     End of CGGEV
*
      END