```      SUBROUTINE SGEEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, WR, WI,
\$                   VL, LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM,
\$                   RCONDE, RCONDV, WORK, LWORK, IWORK, INFO )
*
*  -- LAPACK driver routine (version 3.1) --
*     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
*     November 2006
*
*     .. Scalar Arguments ..
CHARACTER          BALANC, JOBVL, JOBVR, SENSE
INTEGER            IHI, ILO, INFO, LDA, LDVL, LDVR, LWORK, N
REAL               ABNRM
*     ..
*     .. Array Arguments ..
INTEGER            IWORK( * )
REAL               A( LDA, * ), RCONDE( * ), RCONDV( * ),
\$                   SCALE( * ), VL( LDVL, * ), VR( LDVR, * ),
\$                   WI( * ), WORK( * ), WR( * )
*     ..
*
*  Purpose
*  =======
*
*  SGEEVX computes for an N-by-N real nonsymmetric matrix A, the
*  eigenvalues and, optionally, the left and/or right eigenvectors.
*
*  Optionally also, it computes a balancing transformation to improve
*  the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
*  SCALE, and ABNRM), reciprocal condition numbers for the eigenvalues
*  (RCONDE), and reciprocal condition numbers for the right
*  eigenvectors (RCONDV).
*
*  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.
*
*  Balancing a matrix means permuting the rows and columns to make it
*  more nearly upper triangular, and applying a diagonal similarity
*  transformation D * A * D**(-1), where D is a diagonal matrix, to
*  make its rows and columns closer in norm and the condition numbers
*  of its eigenvalues and eigenvectors smaller.  The computed
*  reciprocal condition numbers correspond to the balanced matrix.
*  Permuting rows and columns will not change the condition numbers
*  (in exact arithmetic) but diagonal scaling will.  For further
*  explanation of balancing, see section 4.10.2 of the LAPACK
*  Users' Guide.
*
*  Arguments
*  =========
*
*  BALANC  (input) CHARACTER*1
*          Indicates how the input matrix should be diagonally scaled
*          and/or permuted to improve the conditioning of its
*          eigenvalues.
*          = 'N': Do not diagonally scale or permute;
*          = 'P': Perform permutations to make the matrix more nearly
*                 upper triangular. Do not diagonally scale;
*          = 'S': Diagonally scale the matrix, i.e. replace A by
*                 D*A*D**(-1), where D is a diagonal matrix chosen
*                 to make the rows and columns of A more equal in
*                 norm. Do not permute;
*          = 'B': Both diagonally scale and permute A.
*
*          Computed reciprocal condition numbers will be for the matrix
*          after balancing and/or permuting. Permuting does not change
*          condition numbers (in exact arithmetic), but balancing does.
*
*  JOBVL   (input) CHARACTER*1
*          = 'N': left eigenvectors of A are not computed;
*          = 'V': left eigenvectors of A are computed.
*          If SENSE = 'E' or 'B', JOBVL must = 'V'.
*
*  JOBVR   (input) CHARACTER*1
*          = 'N': right eigenvectors of A are not computed;
*          = 'V': right eigenvectors of A are computed.
*          If SENSE = 'E' or 'B', JOBVR must = 'V'.
*
*  SENSE   (input) CHARACTER*1
*          Determines which reciprocal condition numbers are computed.
*          = 'N': None are computed;
*          = 'E': Computed for eigenvalues only;
*          = 'V': Computed for right eigenvectors only;
*          = 'B': Computed for eigenvalues and right eigenvectors.
*
*          If SENSE = 'E' or 'B', both left and right eigenvectors
*          must also be computed (JOBVL = 'V' and JOBVR = 'V').
*
*  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 has been overwritten.  If JOBVL = 'V' or
*          JOBVR = 'V', A contains the real Schur form of the balanced
*          version of the input matrix A.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  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.  Complex
*          conjugate pairs of eigenvalues will appear consecutively
*          with the eigenvalue having the positive imaginary part
*          first.
*
*  VL      (output) REAL 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) REAL 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, and if
*          JOBVR = 'V', LDVR >= N.
*
*  ILO     (output) INTEGER
*  IHI     (output) INTEGER
*          ILO and IHI are integer values determined when A was
*          balanced.  The balanced A(i,j) = 0 if I > J and
*          J = 1,...,ILO-1 or I = IHI+1,...,N.
*
*  SCALE   (output) REAL array, dimension (N)
*          Details of the permutations and scaling factors applied
*          when balancing A.  If P(j) is the index of the row and column
*          interchanged with row and column j, and D(j) is the scaling
*          factor applied to row and column j, then
*          SCALE(J) = P(J),    for J = 1,...,ILO-1
*                   = D(J),    for J = ILO,...,IHI
*                   = P(J)     for J = IHI+1,...,N.
*          The order in which the interchanges are made is N to IHI+1,
*          then 1 to ILO-1.
*
*  ABNRM   (output) REAL
*          The one-norm of the balanced matrix (the maximum
*          of the sum of absolute values of elements of any column).
*
*  RCONDE  (output) REAL array, dimension (N)
*          RCONDE(j) is the reciprocal condition number of the j-th
*          eigenvalue.
*
*  RCONDV  (output) REAL array, dimension (N)
*          RCONDV(j) is the reciprocal condition number of the j-th
*          right eigenvector.
*
*  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.   If SENSE = 'N' or 'E',
*          LWORK >= max(1,2*N), and if JOBVL = 'V' or JOBVR = 'V',
*          LWORK >= 3*N.  If SENSE = 'V' or 'B', LWORK >= N*(N+6).
*          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.
*
*  IWORK   (workspace) INTEGER array, dimension (2*N-2)
*          If SENSE = 'N' or 'E', not referenced.
*
*  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 or condition numbers
*                have been computed; elements 1:ILO-1 and i+1:N of WR
*                and WI contain eigenvalues which have converged.
*
*  =====================================================================
*
*     .. Parameters ..
REAL               ZERO, ONE
PARAMETER          ( ZERO = 0.0E0, ONE = 1.0E0 )
*     ..
*     .. Local Scalars ..
LOGICAL            LQUERY, SCALEA, WANTVL, WANTVR, WNTSNB, WNTSNE,
\$                   WNTSNN, WNTSNV
CHARACTER          JOB, SIDE
INTEGER            HSWORK, I, ICOND, IERR, ITAU, IWRK, K, MAXWRK,
\$                   MINWRK, NOUT
REAL               ANRM, BIGNUM, CS, CSCALE, EPS, R, SCL, SMLNUM,
\$                   SN
*     ..
*     .. Local Arrays ..
LOGICAL            SELECT( 1 )
REAL               DUM( 1 )
*     ..
*     .. External Subroutines ..
EXTERNAL           SGEBAK, SGEBAL, SGEHRD, SHSEQR, SLABAD, SLACPY,
\$                   SLARTG, SLASCL, SORGHR, SROT, SSCAL, STREVC,
\$                   STRSNA, XERBLA
*     ..
*     .. External Functions ..
LOGICAL            LSAME
INTEGER            ILAENV, ISAMAX
REAL               SLAMCH, SLANGE, SLAPY2, SNRM2
EXTERNAL           LSAME, ILAENV, ISAMAX, SLAMCH, SLANGE, SLAPY2,
\$                   SNRM2
*     ..
*     .. 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' )
WNTSNN = LSAME( SENSE, 'N' )
WNTSNE = LSAME( SENSE, 'E' )
WNTSNV = LSAME( SENSE, 'V' )
WNTSNB = LSAME( SENSE, 'B' )
IF( .NOT.( LSAME( BALANC, 'N' ) .OR. LSAME( BALANC, 'S' ) .OR.
\$    LSAME( BALANC, 'P' ) .OR. LSAME( BALANC, 'B' ) ) ) THEN
INFO = -1
ELSE IF( ( .NOT.WANTVL ) .AND. ( .NOT.LSAME( JOBVL, 'N' ) ) ) THEN
INFO = -2
ELSE IF( ( .NOT.WANTVR ) .AND. ( .NOT.LSAME( JOBVR, 'N' ) ) ) THEN
INFO = -3
ELSE IF( .NOT.( WNTSNN .OR. WNTSNE .OR. WNTSNB .OR. WNTSNV ) .OR.
\$         ( ( WNTSNE .OR. WNTSNB ) .AND. .NOT.( WANTVL .AND.
\$         WANTVR ) ) ) THEN
INFO = -4
ELSE IF( N.LT.0 ) THEN
INFO = -5
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDVL.LT.1 .OR. ( WANTVL .AND. LDVL.LT.N ) ) THEN
INFO = -11
ELSE IF( LDVR.LT.1 .OR. ( WANTVR .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.
*       HSWORK refers to the workspace preferred by SHSEQR, 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 = N + N*ILAENV( 1, 'SGEHRD', ' ', N, 1, N, 0 )
*
IF( WANTVL ) THEN
CALL SHSEQR( 'S', 'V', N, 1, N, A, LDA, WR, WI, VL, LDVL,
\$                WORK, -1, INFO )
ELSE IF( WANTVR ) THEN
CALL SHSEQR( 'S', 'V', N, 1, N, A, LDA, WR, WI, VR, LDVR,
\$                WORK, -1, INFO )
ELSE
IF( WNTSNN ) THEN
CALL SHSEQR( 'E', 'N', N, 1, N, A, LDA, WR, WI, VR,
\$                LDVR, WORK, -1, INFO )
ELSE
CALL SHSEQR( 'S', 'N', N, 1, N, A, LDA, WR, WI, VR,
\$                LDVR, WORK, -1, INFO )
END IF
END IF
HSWORK = WORK( 1 )
*
IF( ( .NOT.WANTVL ) .AND. ( .NOT.WANTVR ) ) THEN
MINWRK = 2*N
IF( .NOT.WNTSNN )
\$            MINWRK = MAX( MINWRK, N*N+6*N )
MAXWRK = MAX( MAXWRK, HSWORK )
IF( .NOT.WNTSNN )
\$            MAXWRK = MAX( MAXWRK, N*N + 6*N )
ELSE
MINWRK = 3*N
IF( ( .NOT.WNTSNN ) .AND. ( .NOT.WNTSNE ) )
\$            MINWRK = MAX( MINWRK, N*N + 6*N )
MAXWRK = MAX( MAXWRK, HSWORK )
MAXWRK = MAX( MAXWRK, N + ( N - 1 )*ILAENV( 1, 'SORGHR',
\$                       ' ', N, 1, N, -1 ) )
IF( ( .NOT.WNTSNN ) .AND. ( .NOT.WNTSNE ) )
\$            MAXWRK = MAX( MAXWRK, N*N + 6*N )
MAXWRK = MAX( MAXWRK, 3*N )
END IF
MAXWRK = MAX( MAXWRK, MINWRK )
END IF
WORK( 1 ) = MAXWRK
*
IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
INFO = -21
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SGEEVX', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
*     Quick return if possible
*
IF( N.EQ.0 )
\$   RETURN
*
*     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]
*
ICOND = 0
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 )
*
*     Balance the matrix and compute ABNRM
*
CALL SGEBAL( BALANC, N, A, LDA, ILO, IHI, SCALE, IERR )
ABNRM = SLANGE( '1', N, N, A, LDA, DUM )
IF( SCALEA ) THEN
DUM( 1 ) = ABNRM
CALL SLASCL( 'G', 0, 0, CSCALE, ANRM, 1, 1, DUM, 1, IERR )
ABNRM = DUM( 1 )
END IF
*
*     Reduce to upper Hessenberg form
*     (Workspace: need 2*N, prefer N+N*NB)
*
ITAU = 1
IWRK = ITAU + N
CALL SGEHRD( 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 SLACPY( 'L', N, N, A, LDA, VL, LDVL )
*
*        Generate orthogonal matrix in VL
*        (Workspace: need 2*N-1, prefer N+(N-1)*NB)
*
CALL SORGHR( N, ILO, IHI, VL, LDVL, WORK( ITAU ), WORK( IWRK ),
\$                LWORK-IWRK+1, IERR )
*
*        Perform QR iteration, accumulating Schur vectors in VL
*        (Workspace: need 1, prefer HSWORK (see comments) )
*
IWRK = ITAU
CALL SHSEQR( '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 SLACPY( 'F', N, N, VL, LDVL, VR, LDVR )
END IF
*
ELSE IF( WANTVR ) THEN
*
*        Want right eigenvectors
*        Copy Householder vectors to VR
*
SIDE = 'R'
CALL SLACPY( 'L', N, N, A, LDA, VR, LDVR )
*
*        Generate orthogonal matrix in VR
*        (Workspace: need 2*N-1, prefer N+(N-1)*NB)
*
CALL SORGHR( N, ILO, IHI, VR, LDVR, WORK( ITAU ), WORK( IWRK ),
\$                LWORK-IWRK+1, IERR )
*
*        Perform QR iteration, accumulating Schur vectors in VR
*        (Workspace: need 1, prefer HSWORK (see comments) )
*
IWRK = ITAU
CALL SHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, WR, WI, VR, LDVR,
\$                WORK( IWRK ), LWORK-IWRK+1, INFO )
*
ELSE
*
*        Compute eigenvalues only
*        If condition numbers desired, compute Schur form
*
IF( WNTSNN ) THEN
JOB = 'E'
ELSE
JOB = 'S'
END IF
*
*        (Workspace: need 1, prefer HSWORK (see comments) )
*
IWRK = ITAU
CALL SHSEQR( JOB, 'N', N, ILO, IHI, A, LDA, WR, WI, VR, LDVR,
\$                WORK( IWRK ), LWORK-IWRK+1, INFO )
END IF
*
*     If INFO > 0 from SHSEQR, then quit
*
IF( INFO.GT.0 )
\$   GO TO 50
*
IF( WANTVL .OR. WANTVR ) THEN
*
*        Compute left and/or right eigenvectors
*        (Workspace: need 3*N)
*
CALL STREVC( SIDE, 'B', SELECT, N, A, LDA, VL, LDVL, VR, LDVR,
\$                N, NOUT, WORK( IWRK ), IERR )
END IF
*
*     Compute condition numbers if desired
*     (Workspace: need N*N+6*N unless SENSE = 'E')
*
IF( .NOT.WNTSNN ) THEN
CALL STRSNA( SENSE, 'A', SELECT, N, A, LDA, VL, LDVL, VR, LDVR,
\$                RCONDE, RCONDV, N, NOUT, WORK( IWRK ), N, IWORK,
\$                ICOND )
END IF
*
IF( WANTVL ) THEN
*
*        Undo balancing of left eigenvectors
*
CALL SGEBAK( BALANC, 'L', N, ILO, IHI, SCALE, 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 / SNRM2( N, VL( 1, I ), 1 )
CALL SSCAL( N, SCL, VL( 1, I ), 1 )
ELSE IF( WI( I ).GT.ZERO ) THEN
SCL = ONE / SLAPY2( SNRM2( N, VL( 1, I ), 1 ),
\$               SNRM2( N, VL( 1, I+1 ), 1 ) )
CALL SSCAL( N, SCL, VL( 1, I ), 1 )
CALL SSCAL( N, SCL, VL( 1, I+1 ), 1 )
DO 10 K = 1, N
WORK( K ) = VL( K, I )**2 + VL( K, I+1 )**2
10          CONTINUE
K = ISAMAX( N, WORK, 1 )
CALL SLARTG( VL( K, I ), VL( K, I+1 ), CS, SN, R )
CALL SROT( 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
*
CALL SGEBAK( BALANC, 'R', N, ILO, IHI, SCALE, 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 / SNRM2( N, VR( 1, I ), 1 )
CALL SSCAL( N, SCL, VR( 1, I ), 1 )
ELSE IF( WI( I ).GT.ZERO ) THEN
SCL = ONE / SLAPY2( SNRM2( N, VR( 1, I ), 1 ),
\$               SNRM2( N, VR( 1, I+1 ), 1 ) )
CALL SSCAL( N, SCL, VR( 1, I ), 1 )
CALL SSCAL( N, SCL, VR( 1, I+1 ), 1 )
DO 30 K = 1, N
WORK( K ) = VR( K, I )**2 + VR( K, I+1 )**2
30          CONTINUE
K = ISAMAX( N, WORK, 1 )
CALL SLARTG( VR( K, I ), VR( K, I+1 ), CS, SN, R )
CALL SROT( 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 SLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1, WR( INFO+1 ),
\$                MAX( N-INFO, 1 ), IERR )
CALL SLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1, WI( INFO+1 ),
\$                MAX( N-INFO, 1 ), IERR )
IF( INFO.EQ.0 ) THEN
IF( ( WNTSNV .OR. WNTSNB ) .AND. ICOND.EQ.0 )
\$         CALL SLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, RCONDV, N,
\$                      IERR )
ELSE
CALL SLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WR, N,
\$                   IERR )
CALL SLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WI, N,
\$                   IERR )
END IF
END IF
*
WORK( 1 ) = MAXWRK
RETURN
*
*     End of SGEEVX
*
END

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