zggev.f

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*> \brief <b> ZGGEV 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/
*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE ZGGEV( 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 ..
*       DOUBLE PRECISION   RWORK( * )
*       COMPLEX*16         A( LDA, * ), ALPHA( * ), B( LDB, * ),
*      $                   BETA( * ), VL( LDVL, * ), VR( LDVR, * ),
*      $                   WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> ZGGEV 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*16 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*16 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*16 array, dimension (N)
*> \endverbatim
*>
*> \param[out] BETA
*> \verbatim
*>          BETA is COMPLEX*16 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*16 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*16 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*16 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 DOUBLE PRECISION 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 ZHGEQZ,
*>                =N+2: error return from ZTGEVC.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ggev
*
*  =====================================================================
      SUBROUTINE zggev( 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 ..
      DOUBLE PRECISION   RWORK( * )
      COMPLEX*16         A( LDA, * ), ALPHA( * ), B( LDB, * ),
     $                   beta( * ), vl( ldvl, * ), vr( ldvr, * ),
     $                   work( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      parameter( zero = 0.0d0, one = 1.0d0 )
      COMPLEX*16         CZERO, CONE
      parameter( czero = ( 0.0d0, 0.0d0 ),
     $                   cone = ( 1.0d0, 0.0d0 ) )
*     ..
*     .. 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
      DOUBLE PRECISION   ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS,
     $                   smlnum, temp
      COMPLEX*16         X
*     ..
*     .. Local Arrays ..
      LOGICAL            LDUMMA( 1 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           xerbla, zgeqrf, zggbak, zggbal, zgghrd,
     $                   zhgeqz,
     $                   zlacpy, zlascl, zlaset, ztgevc, zungqr, zunmqr
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      DOUBLE PRECISION   DLAMCH, ZLANGE
      EXTERNAL           lsame, ilaenv, dlamch, zlange
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, dble, dimag, max, sqrt
*     ..
*     .. Statement Functions ..
      DOUBLE PRECISION   ABS1
*     ..
*     .. Statement Function definitions ..
      abs1( x ) = abs( dble( x ) ) + abs( dimag( 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, 'ZGEQRF', ' ', n, 1, n,
     $                 0 ) )
         lwkopt = max( lwkopt, n +
     $                 n*ilaenv( 1, 'ZUNMQR', ' ', n, 1, n, 0 ) )
         IF( ilvl ) THEN
            lwkopt = max( lwkopt, n +
     $                    n*ilaenv( 1, 'ZUNGQR', ' ', 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( 'ZGGEV ', -info )
         RETURN
      ELSE IF( lquery ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( n.EQ.0 )
     $   RETURN
*
*     Get machine constants
*
      eps = dlamch( 'E' )*dlamch( 'B' )
      smlnum = dlamch( 'S' )
      bignum = one / smlnum
      smlnum = sqrt( smlnum ) / eps
      bignum = one / smlnum
*
*     Scale A if max element outside range [SMLNUM,BIGNUM]
*
      anrm = zlange( '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 zlascl( 'G', 0, 0, anrm, anrmto, n, n, a, lda, ierr )
*
*     Scale B if max element outside range [SMLNUM,BIGNUM]
*
      bnrm = zlange( '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 zlascl( '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 zggbal( '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 zgeqrf( 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 zunmqr( '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 zlaset( 'Full', n, n, czero, cone, vl, ldvl )
         IF( irows.GT.1 ) THEN
            CALL zlacpy( 'L', irows-1, irows-1, b( ilo+1, ilo ), ldb,
     $                   vl( ilo+1, ilo ), ldvl )
         END IF
         CALL zungqr( irows, irows, irows, vl( ilo, ilo ), ldvl,
     $                work( itau ), work( iwrk ), lwork+1-iwrk, ierr )
      END IF
*
*     Initialize VR
*
      IF( ilvr )
     $   CALL zlaset( 'Full', n, n, czero, cone, vr, ldvr )
*
*     Reduce to generalized Hessenberg form
*
      IF( ilv ) THEN
*
*        Eigenvectors requested -- work on whole matrix.
*
         CALL zgghrd( jobvl, jobvr, n, ilo, ihi, a, lda, b, ldb, vl,
     $                ldvl, vr, ldvr, ierr )
      ELSE
         CALL zgghrd( '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 zhgeqz( 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 ztgevc( 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 zggbak( '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 zggbak( '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 zlascl( 'G', 0, 0, anrmto, anrm, n, 1, alpha, n, ierr )
*
      IF( ilbscl )
     $   CALL zlascl( 'G', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
*
      work( 1 ) = lwkopt
      RETURN
*
*     End of ZGGEV
*
      SUBROUTINE zggev( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA, …
      END