```      SUBROUTINE DGEEV( JOBVL, JOBVR, N, A, LDA, WR, WI, VL, LDVL, VR,
\$                  LDVR, WORK, LWORK, 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, LDVL, LDVR, LWORK, N
*     ..
*     .. Array Arguments ..
DOUBLE PRECISION   A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ),
\$                   WI( * ), WORK( * ), WR( * )
*     ..
*
*  Purpose
*  =======
*
*  DGEEV computes for an N-by-N real nonsymmetric matrix A, the
*  eigenvalues and, optionally, the left and/or right eigenvectors.
*
*  The right eigenvector v(j) of A satisfies
*                   A * v(j) = lambda(j) * v(j)
*  where lambda(j) is its eigenvalue.
*  The left eigenvector u(j) of A satisfies
*                u(j)**H * A = lambda(j) * u(j)**H
*  where u(j)**H denotes the conjugate transpose of u(j).
*
*  The computed eigenvectors are normalized to have Euclidean norm
*  equal to 1 and largest component real.
*
*  Arguments
*  =========
*
*  JOBVL   (input) CHARACTER*1
*          = 'N': left eigenvectors of A are not computed;
*          = 'V': left eigenvectors of A are computed.
*
*  JOBVR   (input) CHARACTER*1
*          = 'N': right eigenvectors of A are not computed;
*          = 'V': right eigenvectors of A are computed.
*
*  N       (input) INTEGER
*          The order of the matrix A. N >= 0.
*
*  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
*          On entry, the N-by-N matrix A.
*          On exit, A has been overwritten.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  WR      (output) DOUBLE PRECISION array, dimension (N)
*  WI      (output) DOUBLE PRECISION array, dimension (N)
*          WR and WI contain the real and imaginary parts,
*          respectively, of the computed eigenvalues.  Complex
*          conjugate pairs of eigenvalues appear consecutively
*          with the eigenvalue having the positive imaginary part
*          first.
*
*  VL      (output) DOUBLE PRECISION array, dimension (LDVL,N)
*          If JOBVL = 'V', the left eigenvectors u(j) are stored one
*          after another in the columns of VL, in the same order
*          as their eigenvalues.
*          If JOBVL = 'N', VL is not referenced.
*          If the j-th eigenvalue is real, then u(j) = VL(:,j),
*          the j-th column of VL.
*          If the j-th and (j+1)-st eigenvalues form a complex
*          conjugate pair, then u(j) = VL(:,j) + i*VL(:,j+1) and
*          u(j+1) = VL(:,j) - i*VL(:,j+1).
*
*  LDVL    (input) INTEGER
*          The leading dimension of the array VL.  LDVL >= 1; if
*          JOBVL = 'V', LDVL >= N.
*
*  VR      (output) DOUBLE PRECISION array, dimension (LDVR,N)
*          If JOBVR = 'V', the right eigenvectors v(j) are stored one
*          after another in the columns of VR, in the same order
*          as their eigenvalues.
*          If JOBVR = 'N', VR is not referenced.
*          If the j-th eigenvalue is real, then v(j) = VR(:,j),
*          the j-th column of VR.
*          If the j-th and (j+1)-st eigenvalues form a complex
*          conjugate pair, then v(j) = VR(:,j) + i*VR(:,j+1) and
*          v(j+1) = VR(:,j) - i*VR(:,j+1).
*
*  LDVR    (input) INTEGER
*          The leading dimension of the array VR.  LDVR >= 1; if
*          JOBVR = 'V', LDVR >= N.
*
*  WORK    (workspace/output) DOUBLE PRECISION 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), and
*          if JOBVL = 'V' or JOBVR = 'V', LWORK >= 4*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.
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value.
*          > 0:  if INFO = i, the QR algorithm failed to compute all the
*                eigenvalues, and no eigenvectors have been computed;
*                elements i+1:N of WR and WI contain eigenvalues which
*                have converged.
*
*  =====================================================================
*
*     .. Parameters ..
DOUBLE PRECISION   ZERO, ONE
PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0 )
*     ..
*     .. Local Scalars ..
LOGICAL            LQUERY, SCALEA, WANTVL, WANTVR
CHARACTER          SIDE
INTEGER            HSWORK, I, IBAL, IERR, IHI, ILO, ITAU, IWRK, K,
\$                   MAXWRK, MINWRK, NOUT
DOUBLE PRECISION   ANRM, BIGNUM, CS, CSCALE, EPS, R, SCL, SMLNUM,
\$                   SN
*     ..
*     .. Local Arrays ..
LOGICAL            SELECT( 1 )
DOUBLE PRECISION   DUM( 1 )
*     ..
*     .. External Subroutines ..
EXTERNAL           DGEBAK, DGEBAL, DGEHRD, DHSEQR, DLABAD, DLACPY,
\$                   DLARTG, DLASCL, DORGHR, DROT, DSCAL, DTREVC,
\$                   XERBLA
*     ..
*     .. External Functions ..
LOGICAL            LSAME
INTEGER            IDAMAX, ILAENV
DOUBLE PRECISION   DLAMCH, DLANGE, DLAPY2, DNRM2
EXTERNAL           LSAME, IDAMAX, ILAENV, DLAMCH, DLANGE, DLAPY2,
\$                   DNRM2
*     ..
*     .. Intrinsic Functions ..
INTRINSIC          MAX, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments
*
INFO = 0
LQUERY = ( LWORK.EQ.-1 )
WANTVL = LSAME( JOBVL, 'V' )
WANTVR = LSAME( JOBVR, 'V' )
IF( ( .NOT.WANTVL ) .AND. ( .NOT.LSAME( JOBVL, 'N' ) ) ) THEN
INFO = -1
ELSE IF( ( .NOT.WANTVR ) .AND. ( .NOT.LSAME( JOBVR, 'N' ) ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDVL.LT.1 .OR. ( WANTVL .AND. LDVL.LT.N ) ) THEN
INFO = -9
ELSE IF( LDVR.LT.1 .OR. ( WANTVR .AND. LDVR.LT.N ) ) THEN
INFO = -11
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.
*       HSWORK refers to the workspace preferred by DHSEQR, as
*       calculated below. HSWORK is computed assuming ILO=1 and IHI=N,
*       the worst case.)
*
IF( INFO.EQ.0 ) THEN
IF( N.EQ.0 ) THEN
MINWRK = 1
MAXWRK = 1
ELSE
MAXWRK = 2*N + N*ILAENV( 1, 'DGEHRD', ' ', N, 1, N, 0 )
IF( WANTVL ) THEN
MINWRK = 4*N
MAXWRK = MAX( MAXWRK, 2*N + ( N - 1 )*ILAENV( 1,
\$                       'DORGHR', ' ', N, 1, N, -1 ) )
CALL DHSEQR( 'S', 'V', N, 1, N, A, LDA, WR, WI, VL, LDVL,
\$                WORK, -1, INFO )
HSWORK = WORK( 1 )
MAXWRK = MAX( MAXWRK, N + 1, N + HSWORK )
MAXWRK = MAX( MAXWRK, 4*N )
ELSE IF( WANTVR ) THEN
MINWRK = 4*N
MAXWRK = MAX( MAXWRK, 2*N + ( N - 1 )*ILAENV( 1,
\$                       'DORGHR', ' ', N, 1, N, -1 ) )
CALL DHSEQR( 'S', 'V', N, 1, N, A, LDA, WR, WI, VR, LDVR,
\$                WORK, -1, INFO )
HSWORK = WORK( 1 )
MAXWRK = MAX( MAXWRK, N + 1, N + HSWORK )
MAXWRK = MAX( MAXWRK, 4*N )
ELSE
MINWRK = 3*N
CALL DHSEQR( 'E', 'N', N, 1, N, A, LDA, WR, WI, VR, LDVR,
\$                WORK, -1, INFO )
HSWORK = WORK( 1 )
MAXWRK = MAX( MAXWRK, N + 1, N + HSWORK )
END IF
MAXWRK = MAX( MAXWRK, MINWRK )
END IF
WORK( 1 ) = MAXWRK
*
IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
INFO = -13
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEEV ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
*     Quick return if possible
*
IF( N.EQ.0 )
\$   RETURN
*
*     Get machine constants
*
EPS = DLAMCH( 'P' )
SMLNUM = DLAMCH( 'S' )
BIGNUM = ONE / SMLNUM
SMLNUM = SQRT( SMLNUM ) / EPS
BIGNUM = ONE / SMLNUM
*
*     Scale A if max element outside range [SMLNUM,BIGNUM]
*
ANRM = DLANGE( '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 DLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR )
*
*     Balance the matrix
*     (Workspace: need N)
*
IBAL = 1
CALL DGEBAL( 'B', N, A, LDA, ILO, IHI, WORK( IBAL ), IERR )
*
*     Reduce to upper Hessenberg form
*     (Workspace: need 3*N, prefer 2*N+N*NB)
*
ITAU = IBAL + N
IWRK = ITAU + N
CALL DGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ),
\$             LWORK-IWRK+1, IERR )
*
IF( WANTVL ) THEN
*
*        Want left eigenvectors
*        Copy Householder vectors to VL
*
SIDE = 'L'
CALL DLACPY( 'L', N, N, A, LDA, VL, LDVL )
*
*        Generate orthogonal matrix in VL
*        (Workspace: need 3*N-1, prefer 2*N+(N-1)*NB)
*
CALL DORGHR( N, ILO, IHI, VL, LDVL, WORK( ITAU ), WORK( IWRK ),
\$                LWORK-IWRK+1, IERR )
*
*        Perform QR iteration, accumulating Schur vectors in VL
*        (Workspace: need N+1, prefer N+HSWORK (see comments) )
*
IWRK = ITAU
CALL DHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, WR, WI, VL, LDVL,
\$                WORK( IWRK ), LWORK-IWRK+1, INFO )
*
IF( WANTVR ) THEN
*
*           Want left and right eigenvectors
*           Copy Schur vectors to VR
*
SIDE = 'B'
CALL DLACPY( 'F', N, N, VL, LDVL, VR, LDVR )
END IF
*
ELSE IF( WANTVR ) THEN
*
*        Want right eigenvectors
*        Copy Householder vectors to VR
*
SIDE = 'R'
CALL DLACPY( 'L', N, N, A, LDA, VR, LDVR )
*
*        Generate orthogonal matrix in VR
*        (Workspace: need 3*N-1, prefer 2*N+(N-1)*NB)
*
CALL DORGHR( N, ILO, IHI, VR, LDVR, WORK( ITAU ), WORK( IWRK ),
\$                LWORK-IWRK+1, IERR )
*
*        Perform QR iteration, accumulating Schur vectors in VR
*        (Workspace: need N+1, prefer N+HSWORK (see comments) )
*
IWRK = ITAU
CALL DHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, WR, WI, VR, LDVR,
\$                WORK( IWRK ), LWORK-IWRK+1, INFO )
*
ELSE
*
*        Compute eigenvalues only
*        (Workspace: need N+1, prefer N+HSWORK (see comments) )
*
IWRK = ITAU
CALL DHSEQR( 'E', 'N', N, ILO, IHI, A, LDA, WR, WI, VR, LDVR,
\$                WORK( IWRK ), LWORK-IWRK+1, INFO )
END IF
*
*     If INFO > 0 from DHSEQR, then quit
*
IF( INFO.GT.0 )
\$   GO TO 50
*
IF( WANTVL .OR. WANTVR ) THEN
*
*        Compute left and/or right eigenvectors
*        (Workspace: need 4*N)
*
CALL DTREVC( SIDE, 'B', SELECT, N, A, LDA, VL, LDVL, VR, LDVR,
\$                N, NOUT, WORK( IWRK ), IERR )
END IF
*
IF( WANTVL ) THEN
*
*        Undo balancing of left eigenvectors
*        (Workspace: need N)
*
CALL DGEBAK( 'B', 'L', N, ILO, IHI, WORK( IBAL ), N, VL, LDVL,
\$                IERR )
*
*        Normalize left eigenvectors and make largest component real
*
DO 20 I = 1, N
IF( WI( I ).EQ.ZERO ) THEN
SCL = ONE / DNRM2( N, VL( 1, I ), 1 )
CALL DSCAL( N, SCL, VL( 1, I ), 1 )
ELSE IF( WI( I ).GT.ZERO ) THEN
SCL = ONE / DLAPY2( DNRM2( N, VL( 1, I ), 1 ),
\$               DNRM2( N, VL( 1, I+1 ), 1 ) )
CALL DSCAL( N, SCL, VL( 1, I ), 1 )
CALL DSCAL( N, SCL, VL( 1, I+1 ), 1 )
DO 10 K = 1, N
WORK( IWRK+K-1 ) = VL( K, I )**2 + VL( K, I+1 )**2
10          CONTINUE
K = IDAMAX( N, WORK( IWRK ), 1 )
CALL DLARTG( VL( K, I ), VL( K, I+1 ), CS, SN, R )
CALL DROT( N, VL( 1, I ), 1, VL( 1, I+1 ), 1, CS, SN )
VL( K, I+1 ) = ZERO
END IF
20    CONTINUE
END IF
*
IF( WANTVR ) THEN
*
*        Undo balancing of right eigenvectors
*        (Workspace: need N)
*
CALL DGEBAK( 'B', 'R', N, ILO, IHI, WORK( IBAL ), N, VR, LDVR,
\$                IERR )
*
*        Normalize right eigenvectors and make largest component real
*
DO 40 I = 1, N
IF( WI( I ).EQ.ZERO ) THEN
SCL = ONE / DNRM2( N, VR( 1, I ), 1 )
CALL DSCAL( N, SCL, VR( 1, I ), 1 )
ELSE IF( WI( I ).GT.ZERO ) THEN
SCL = ONE / DLAPY2( DNRM2( N, VR( 1, I ), 1 ),
\$               DNRM2( N, VR( 1, I+1 ), 1 ) )
CALL DSCAL( N, SCL, VR( 1, I ), 1 )
CALL DSCAL( N, SCL, VR( 1, I+1 ), 1 )
DO 30 K = 1, N
WORK( IWRK+K-1 ) = VR( K, I )**2 + VR( K, I+1 )**2
30          CONTINUE
K = IDAMAX( N, WORK( IWRK ), 1 )
CALL DLARTG( VR( K, I ), VR( K, I+1 ), CS, SN, R )
CALL DROT( N, VR( 1, I ), 1, VR( 1, I+1 ), 1, CS, SN )
VR( K, I+1 ) = ZERO
END IF
40    CONTINUE
END IF
*
*     Undo scaling if necessary
*
50 CONTINUE
IF( SCALEA ) THEN
CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1, WR( INFO+1 ),
\$                MAX( N-INFO, 1 ), IERR )
CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1, WI( INFO+1 ),
\$                MAX( N-INFO, 1 ), IERR )
IF( INFO.GT.0 ) THEN
CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WR, N,
\$                   IERR )
CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WI, N,
\$                   IERR )
END IF
END IF
*
WORK( 1 ) = MAXWRK
RETURN
*
*     End of DGEEV
*
END

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