path: root/SRC/dtgevc.f

*> \brief \b DTGEVC
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at 
*            http://www.netlib.org/lapack/explore-html/ 
*
*> Download DTGEVC + dependencies 
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtgevc.f"> 
*> [TXT]</a> 
*
*  Definition
*  ==========
*
*       SUBROUTINE DTGEVC( SIDE, HOWMNY, SELECT, N, S, LDS, P, LDP, VL,
*                          LDVL, VR, LDVR, MM, M, WORK, INFO )
* 
*       .. Scalar Arguments ..
*       CHARACTER          HOWMNY, SIDE
*       INTEGER            INFO, LDP, LDS, LDVL, LDVR, M, MM, N
*       ..
*       .. Array Arguments ..
*       LOGICAL            SELECT( * )
*       DOUBLE PRECISION   P( LDP, * ), S( LDS, * ), VL( LDVL, * ),
*      $                   VR( LDVR, * ), WORK( * )
*       ..
*  
*  
*  Purpose
*  =======
*
*>\details \b Purpose:
*>\verbatim
*>
*> DTGEVC computes some or all of the right and/or left eigenvectors of
*> a pair of real matrices (S,P), where S is a quasi-triangular matrix
*> and P is upper triangular.  Matrix pairs of this type are produced by
*> the generalized Schur factorization of a matrix pair (A,B):
*>
*>    A = Q*S*Z**T,  B = Q*P*Z**T
*>
*> as computed by DGGHRD + DHGEQZ.
*>
*> The right eigenvector x and the left eigenvector y of (S,P)
*> corresponding to an eigenvalue w are defined by:
*> 
*>    S*x = w*P*x,  (y**H)*S = w*(y**H)*P,
*> 
*> where y**H denotes the conjugate tranpose of y.
*> The eigenvalues are not input to this routine, but are computed
*> directly from the diagonal blocks of S and P.
*> 
*> This routine returns the matrices X and/or Y of right and left
*> eigenvectors of (S,P), or the products Z*X and/or Q*Y,
*> where Z and Q are input matrices.
*> If Q and Z are the orthogonal factors from the generalized Schur
*> factorization of a matrix pair (A,B), then Z*X and Q*Y
*> are the matrices of right and left eigenvectors of (A,B).
*> 
*>\endverbatim
*
*  Arguments
*  =========
*
*> \param[in] SIDE
*> \verbatim
*>          SIDE is CHARACTER*1
*>          = 'R': compute right eigenvectors only;
*>          = 'L': compute left eigenvectors only;
*>          = 'B': compute both right and left eigenvectors.
*> \endverbatim
*>
*> \param[in] HOWMNY
*> \verbatim
*>          HOWMNY is CHARACTER*1
*>          = 'A': compute all right and/or left eigenvectors;
*>          = 'B': compute all right and/or left eigenvectors,
*>                 backtransformed by the matrices in VR and/or VL;
*>          = 'S': compute selected right and/or left eigenvectors,
*>                 specified by the logical array SELECT.
*> \endverbatim
*>
*> \param[in] SELECT
*> \verbatim
*>          SELECT is LOGICAL array, dimension (N)
*>          If HOWMNY='S', SELECT specifies the eigenvectors to be
*>          computed.  If w(j) is a real eigenvalue, the corresponding
*>          real eigenvector is computed if SELECT(j) is .TRUE..
*>          If w(j) and w(j+1) are the real and imaginary parts of a
*>          complex eigenvalue, the corresponding complex eigenvector
*>          is computed if either SELECT(j) or SELECT(j+1) is .TRUE.,
*>          and on exit SELECT(j) is set to .TRUE. and SELECT(j+1) is
*>          set to .FALSE..
*>          Not referenced if HOWMNY = 'A' or 'B'.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrices S and P.  N >= 0.
*> \endverbatim
*>
*> \param[in] S
*> \verbatim
*>          S is DOUBLE PRECISION array, dimension (LDS,N)
*>          The upper quasi-triangular matrix S from a generalized Schur
*>          factorization, as computed by DHGEQZ.
*> \endverbatim
*>
*> \param[in] LDS
*> \verbatim
*>          LDS is INTEGER
*>          The leading dimension of array S.  LDS >= max(1,N).
*> \endverbatim
*>
*> \param[in] P
*> \verbatim
*>          P is DOUBLE PRECISION array, dimension (LDP,N)
*>          The upper triangular matrix P from a generalized Schur
*>          factorization, as computed by DHGEQZ.
*>          2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks
*>          of S must be in positive diagonal form.
*> \endverbatim
*>
*> \param[in] LDP
*> \verbatim
*>          LDP is INTEGER
*>          The leading dimension of array P.  LDP >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] VL
*> \verbatim
*>          VL is DOUBLE PRECISION array, dimension (LDVL,MM)
*>          On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
*>          contain an N-by-N matrix Q (usually the orthogonal matrix Q
*>          of left Schur vectors returned by DHGEQZ).
*>          On exit, if SIDE = 'L' or 'B', VL contains:
*>          if HOWMNY = 'A', the matrix Y of left eigenvectors of (S,P);
*>          if HOWMNY = 'B', the matrix Q*Y;
*>          if HOWMNY = 'S', the left eigenvectors of (S,P) specified by
*>                      SELECT, stored consecutively in the columns of
*>                      VL, in the same order as their eigenvalues.
*> \endverbatim
*> \verbatim
*>          A complex eigenvector corresponding to a complex eigenvalue
*>          is stored in two consecutive columns, the first holding the
*>          real part, and the second the imaginary part.
*> \endverbatim
*> \verbatim
*>          Not referenced if SIDE = 'R'.
*> \endverbatim
*>
*> \param[in] LDVL
*> \verbatim
*>          LDVL is INTEGER
*>          The leading dimension of array VL.  LDVL >= 1, and if
*>          SIDE = 'L' or 'B', LDVL >= N.
*> \endverbatim
*>
*> \param[in,out] VR
*> \verbatim
*>          VR is DOUBLE PRECISION array, dimension (LDVR,MM)
*>          On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
*>          contain an N-by-N matrix Z (usually the orthogonal matrix Z
*>          of right Schur vectors returned by DHGEQZ).
*> \endverbatim
*> \verbatim
*>          On exit, if SIDE = 'R' or 'B', VR contains:
*>          if HOWMNY = 'A', the matrix X of right eigenvectors of (S,P);
*>          if HOWMNY = 'B' or 'b', the matrix Z*X;
*>          if HOWMNY = 'S' or 's', the right eigenvectors of (S,P)
*>                      specified by SELECT, stored consecutively in the
*>                      columns of VR, in the same order as their
*>                      eigenvalues.
*> \endverbatim
*> \verbatim
*>          A complex eigenvector corresponding to a complex eigenvalue
*>          is stored in two consecutive columns, the first holding the
*>          real part and the second the imaginary part.
*>          
*>          Not referenced if SIDE = 'L'.
*> \endverbatim
*>
*> \param[in] LDVR
*> \verbatim
*>          LDVR is INTEGER
*>          The leading dimension of the array VR.  LDVR >= 1, and if
*>          SIDE = 'R' or 'B', LDVR >= N.
*> \endverbatim
*>
*> \param[in] MM
*> \verbatim
*>          MM is INTEGER
*>          The number of columns in the arrays VL and/or VR. MM >= M.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*>          M is INTEGER
*>          The number of columns in the arrays VL and/or VR actually
*>          used to store the eigenvectors.  If HOWMNY = 'A' or 'B', M
*>          is set to N.  Each selected real eigenvector occupies one
*>          column and each selected complex eigenvector occupies two
*>          columns.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (6*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit.
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*>          > 0:  the 2-by-2 block (INFO:INFO+1) does not have a complex
*>                eigenvalue.
*> \endverbatim
*>
*
*  Authors
*  =======
*
*> \author Univ. of Tennessee 
*> \author Univ. of California Berkeley 
*> \author Univ. of Colorado Denver 
*> \author NAG Ltd. 
*
*> \date November 2011
*
*> \ingroup doubleGEcomputational
*
*
*  Further Details
*  ===============
*>\details \b Further \b Details
*> \verbatim
*>
*>  Allocation of workspace:
*>  ---------- -- ---------
*>
*>     WORK( j ) = 1-norm of j-th column of A, above the diagonal
*>     WORK( N+j ) = 1-norm of j-th column of B, above the diagonal
*>     WORK( 2*N+1:3*N ) = real part of eigenvector
*>     WORK( 3*N+1:4*N ) = imaginary part of eigenvector
*>     WORK( 4*N+1:5*N ) = real part of back-transformed eigenvector
*>     WORK( 5*N+1:6*N ) = imaginary part of back-transformed eigenvector
*>
*>  Rowwise vs. columnwise solution methods:
*>  ------- --  ---------- -------- -------
*>
*>  Finding a generalized eigenvector consists basically of solving the
*>  singular triangular system
*>
*>   (A - w B) x = 0     (for right) or:   (A - w B)**H y = 0  (for left)
*>
*>  Consider finding the i-th right eigenvector (assume all eigenvalues
*>  are real). The equation to be solved is:
*>       n                   i
*>  0 = sum  C(j,k) v(k)  = sum  C(j,k) v(k)     for j = i,. . .,1
*>      k=j                 k=j
*>
*>  where  C = (A - w B)  (The components v(i+1:n) are 0.)
*>
*>  The "rowwise" method is:
*>
*>  (1)  v(i) := 1
*>  for j = i-1,. . .,1:
*>                          i
*>      (2) compute  s = - sum C(j,k) v(k)   and
*>                        k=j+1
*>
*>      (3) v(j) := s / C(j,j)
*>
*>  Step 2 is sometimes called the "dot product" step, since it is an
*>  inner product between the j-th row and the portion of the eigenvector
*>  that has been computed so far.
*>
*>  The "columnwise" method consists basically in doing the sums
*>  for all the rows in parallel.  As each v(j) is computed, the
*>  contribution of v(j) times the j-th column of C is added to the
*>  partial sums.  Since FORTRAN arrays are stored columnwise, this has
*>  the advantage that at each step, the elements of C that are accessed
*>  are adjacent to one another, whereas with the rowwise method, the
*>  elements accessed at a step are spaced LDS (and LDP) words apart.
*>
*>  When finding left eigenvectors, the matrix in question is the
*>  transpose of the one in storage, so the rowwise method then
*>  actually accesses columns of A and B at each step, and so is the
*>  preferred method.
*>
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE DTGEVC( SIDE, HOWMNY, SELECT, N, S, LDS, P, LDP, VL,
     $                   LDVL, VR, LDVR, MM, M, WORK, INFO )
*
*  -- LAPACK computational routine (version 3.3.1) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     November 2011
*
*     .. Scalar Arguments ..
      CHARACTER          HOWMNY, SIDE
      INTEGER            INFO, LDP, LDS, LDVL, LDVR, M, MM, N
*     ..
*     .. Array Arguments ..
      LOGICAL            SELECT( * )
      DOUBLE PRECISION   P( LDP, * ), S( LDS, * ), VL( LDVL, * ),
     $                   VR( LDVR, * ), WORK( * )
*     ..
*
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE, SAFETY
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0,
     $                   SAFETY = 1.0D+2 )
*     ..
*     .. Local Scalars ..
      LOGICAL            COMPL, COMPR, IL2BY2, ILABAD, ILALL, ILBACK,
     $                   ILBBAD, ILCOMP, ILCPLX, LSA, LSB
      INTEGER            I, IBEG, IEIG, IEND, IHWMNY, IINFO, IM, ISIDE,
     $                   J, JA, JC, JE, JR, JW, NA, NW
      DOUBLE PRECISION   ACOEF, ACOEFA, ANORM, ASCALE, BCOEFA, BCOEFI,
     $                   BCOEFR, BIG, BIGNUM, BNORM, BSCALE, CIM2A,
     $                   CIM2B, CIMAGA, CIMAGB, CRE2A, CRE2B, CREALA,
     $                   CREALB, DMIN, SAFMIN, SALFAR, SBETA, SCALE,
     $                   SMALL, TEMP, TEMP2, TEMP2I, TEMP2R, ULP, XMAX,
     $                   XSCALE
*     ..
*     .. Local Arrays ..
      DOUBLE PRECISION   BDIAG( 2 ), SUM( 2, 2 ), SUMS( 2, 2 ),
     $                   SUMP( 2, 2 )
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           LSAME, DLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEMV, DLABAD, DLACPY, DLAG2, DLALN2, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Decode and Test the input parameters
*
      IF( LSAME( HOWMNY, 'A' ) ) THEN
         IHWMNY = 1
         ILALL = .TRUE.
         ILBACK = .FALSE.
      ELSE IF( LSAME( HOWMNY, 'S' ) ) THEN
         IHWMNY = 2
         ILALL = .FALSE.
         ILBACK = .FALSE.
      ELSE IF( LSAME( HOWMNY, 'B' ) ) THEN
         IHWMNY = 3
         ILALL = .TRUE.
         ILBACK = .TRUE.
      ELSE
         IHWMNY = -1
         ILALL = .TRUE.
      END IF
*
      IF( LSAME( SIDE, 'R' ) ) THEN
         ISIDE = 1
         COMPL = .FALSE.
         COMPR = .TRUE.
      ELSE IF( LSAME( SIDE, 'L' ) ) THEN
         ISIDE = 2
         COMPL = .TRUE.
         COMPR = .FALSE.
      ELSE IF( LSAME( SIDE, 'B' ) ) THEN
         ISIDE = 3
         COMPL = .TRUE.
         COMPR = .TRUE.
      ELSE
         ISIDE = -1
      END IF
*
      INFO = 0
      IF( ISIDE.LT.0 ) THEN
         INFO = -1
      ELSE IF( IHWMNY.LT.0 ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -4
      ELSE IF( LDS.LT.MAX( 1, N ) ) THEN
         INFO = -6
      ELSE IF( LDP.LT.MAX( 1, N ) ) THEN
         INFO = -8
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DTGEVC', -INFO )
         RETURN
      END IF
*
*     Count the number of eigenvectors to be computed
*
      IF( .NOT.ILALL ) THEN
         IM = 0
         ILCPLX = .FALSE.
         DO 10 J = 1, N
            IF( ILCPLX ) THEN
               ILCPLX = .FALSE.
               GO TO 10
            END IF
            IF( J.LT.N ) THEN
               IF( S( J+1, J ).NE.ZERO )
     $            ILCPLX = .TRUE.
            END IF
            IF( ILCPLX ) THEN
               IF( SELECT( J ) .OR. SELECT( J+1 ) )
     $            IM = IM + 2
            ELSE
               IF( SELECT( J ) )
     $            IM = IM + 1
            END IF
   10    CONTINUE
      ELSE
         IM = N
      END IF
*
*     Check 2-by-2 diagonal blocks of A, B
*
      ILABAD = .FALSE.
      ILBBAD = .FALSE.
      DO 20 J = 1, N - 1
         IF( S( J+1, J ).NE.ZERO ) THEN
            IF( P( J, J ).EQ.ZERO .OR. P( J+1, J+1 ).EQ.ZERO .OR.
     $          P( J, J+1 ).NE.ZERO )ILBBAD = .TRUE.
            IF( J.LT.N-1 ) THEN
               IF( S( J+2, J+1 ).NE.ZERO )
     $            ILABAD = .TRUE.
            END IF
         END IF
   20 CONTINUE
*
      IF( ILABAD ) THEN
         INFO = -5
      ELSE IF( ILBBAD ) THEN
         INFO = -7
      ELSE IF( COMPL .AND. LDVL.LT.N .OR. LDVL.LT.1 ) THEN
         INFO = -10
      ELSE IF( COMPR .AND. LDVR.LT.N .OR. LDVR.LT.1 ) THEN
         INFO = -12
      ELSE IF( MM.LT.IM ) THEN
         INFO = -13
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DTGEVC', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      M = IM
      IF( N.EQ.0 )
     $   RETURN
*
*     Machine Constants
*
      SAFMIN = DLAMCH( 'Safe minimum' )
      BIG = ONE / SAFMIN
      CALL DLABAD( SAFMIN, BIG )
      ULP = DLAMCH( 'Epsilon' )*DLAMCH( 'Base' )
      SMALL = SAFMIN*N / ULP
      BIG = ONE / SMALL
      BIGNUM = ONE / ( SAFMIN*N )
*
*     Compute the 1-norm of each column of the strictly upper triangular
*     part (i.e., excluding all elements belonging to the diagonal
*     blocks) of A and B to check for possible overflow in the
*     triangular solver.
*
      ANORM = ABS( S( 1, 1 ) )
      IF( N.GT.1 )
     $   ANORM = ANORM + ABS( S( 2, 1 ) )
      BNORM = ABS( P( 1, 1 ) )
      WORK( 1 ) = ZERO
      WORK( N+1 ) = ZERO
*
      DO 50 J = 2, N
         TEMP = ZERO
         TEMP2 = ZERO
         IF( S( J, J-1 ).EQ.ZERO ) THEN
            IEND = J - 1
         ELSE
            IEND = J - 2
         END IF
         DO 30 I = 1, IEND
            TEMP = TEMP + ABS( S( I, J ) )
            TEMP2 = TEMP2 + ABS( P( I, J ) )
   30    CONTINUE
         WORK( J ) = TEMP
         WORK( N+J ) = TEMP2
         DO 40 I = IEND + 1, MIN( J+1, N )
            TEMP = TEMP + ABS( S( I, J ) )
            TEMP2 = TEMP2 + ABS( P( I, J ) )
   40    CONTINUE
         ANORM = MAX( ANORM, TEMP )
         BNORM = MAX( BNORM, TEMP2 )
   50 CONTINUE
*
      ASCALE = ONE / MAX( ANORM, SAFMIN )
      BSCALE = ONE / MAX( BNORM, SAFMIN )
*
*     Left eigenvectors
*
      IF( COMPL ) THEN
         IEIG = 0
*
*        Main loop over eigenvalues
*
         ILCPLX = .FALSE.
         DO 220 JE = 1, N
*
*           Skip this iteration if (a) HOWMNY='S' and SELECT=.FALSE., or
*           (b) this would be the second of a complex pair.
*           Check for complex eigenvalue, so as to be sure of which
*           entry(-ies) of SELECT to look at.
*
            IF( ILCPLX ) THEN
               ILCPLX = .FALSE.
               GO TO 220
            END IF
            NW = 1
            IF( JE.LT.N ) THEN
               IF( S( JE+1, JE ).NE.ZERO ) THEN
                  ILCPLX = .TRUE.
                  NW = 2
               END IF
            END IF
            IF( ILALL ) THEN
               ILCOMP = .TRUE.
            ELSE IF( ILCPLX ) THEN
               ILCOMP = SELECT( JE ) .OR. SELECT( JE+1 )
            ELSE
               ILCOMP = SELECT( JE )
            END IF
            IF( .NOT.ILCOMP )
     $         GO TO 220
*
*           Decide if (a) singular pencil, (b) real eigenvalue, or
*           (c) complex eigenvalue.
*
            IF( .NOT.ILCPLX ) THEN
               IF( ABS( S( JE, JE ) ).LE.SAFMIN .AND.
     $             ABS( P( JE, JE ) ).LE.SAFMIN ) THEN
*
*                 Singular matrix pencil -- return unit eigenvector
*
                  IEIG = IEIG + 1
                  DO 60 JR = 1, N
                     VL( JR, IEIG ) = ZERO
   60             CONTINUE
                  VL( IEIG, IEIG ) = ONE
                  GO TO 220
               END IF
            END IF
*
*           Clear vector
*
            DO 70 JR = 1, NW*N
               WORK( 2*N+JR ) = ZERO
   70       CONTINUE
*                                                 T
*           Compute coefficients in  ( a A - b B )  y = 0
*              a  is  ACOEF
*              b  is  BCOEFR + i*BCOEFI
*
            IF( .NOT.ILCPLX ) THEN
*
*              Real eigenvalue
*
               TEMP = ONE / MAX( ABS( S( JE, JE ) )*ASCALE,
     $                ABS( P( JE, JE ) )*BSCALE, SAFMIN )
               SALFAR = ( TEMP*S( JE, JE ) )*ASCALE
               SBETA = ( TEMP*P( JE, JE ) )*BSCALE
               ACOEF = SBETA*ASCALE
               BCOEFR = SALFAR*BSCALE
               BCOEFI = ZERO
*
*              Scale to avoid underflow
*
               SCALE = ONE
               LSA = ABS( SBETA ).GE.SAFMIN .AND. ABS( ACOEF ).LT.SMALL
               LSB = ABS( SALFAR ).GE.SAFMIN .AND. ABS( BCOEFR ).LT.
     $               SMALL
               IF( LSA )
     $            SCALE = ( SMALL / ABS( SBETA ) )*MIN( ANORM, BIG )
               IF( LSB )
     $            SCALE = MAX( SCALE, ( SMALL / ABS( SALFAR ) )*
     $                    MIN( BNORM, BIG ) )
               IF( LSA .OR. LSB ) THEN
                  SCALE = MIN( SCALE, ONE /
     $                    ( SAFMIN*MAX( ONE, ABS( ACOEF ),
     $                    ABS( BCOEFR ) ) ) )
                  IF( LSA ) THEN
                     ACOEF = ASCALE*( SCALE*SBETA )
                  ELSE
                     ACOEF = SCALE*ACOEF
                  END IF
                  IF( LSB ) THEN
                     BCOEFR = BSCALE*( SCALE*SALFAR )
                  ELSE
                     BCOEFR = SCALE*BCOEFR
                  END IF
               END IF
               ACOEFA = ABS( ACOEF )
               BCOEFA = ABS( BCOEFR )
*
*              First component is 1
*
               WORK( 2*N+JE ) = ONE
               XMAX = ONE
            ELSE
*
*              Complex eigenvalue
*
               CALL DLAG2( S( JE, JE ), LDS, P( JE, JE ), LDP,
     $                     SAFMIN*SAFETY, ACOEF, TEMP, BCOEFR, TEMP2,
     $                     BCOEFI )
               BCOEFI = -BCOEFI
               IF( BCOEFI.EQ.ZERO ) THEN
                  INFO = JE
                  RETURN
               END IF
*
*              Scale to avoid over/underflow
*
               ACOEFA = ABS( ACOEF )
               BCOEFA = ABS( BCOEFR ) + ABS( BCOEFI )
               SCALE = ONE
               IF( ACOEFA*ULP.LT.SAFMIN .AND. ACOEFA.GE.SAFMIN )
     $            SCALE = ( SAFMIN / ULP ) / ACOEFA
               IF( BCOEFA*ULP.LT.SAFMIN .AND. BCOEFA.GE.SAFMIN )
     $            SCALE = MAX( SCALE, ( SAFMIN / ULP ) / BCOEFA )
               IF( SAFMIN*ACOEFA.GT.ASCALE )
     $            SCALE = ASCALE / ( SAFMIN*ACOEFA )
               IF( SAFMIN*BCOEFA.GT.BSCALE )
     $            SCALE = MIN( SCALE, BSCALE / ( SAFMIN*BCOEFA ) )
               IF( SCALE.NE.ONE ) THEN
                  ACOEF = SCALE*ACOEF
                  ACOEFA = ABS( ACOEF )
                  BCOEFR = SCALE*BCOEFR
                  BCOEFI = SCALE*BCOEFI
                  BCOEFA = ABS( BCOEFR ) + ABS( BCOEFI )
               END IF
*
*              Compute first two components of eigenvector
*
               TEMP = ACOEF*S( JE+1, JE )
               TEMP2R = ACOEF*S( JE, JE ) - BCOEFR*P( JE, JE )
               TEMP2I = -BCOEFI*P( JE, JE )
               IF( ABS( TEMP ).GT.ABS( TEMP2R )+ABS( TEMP2I ) ) THEN
                  WORK( 2*N+JE ) = ONE
                  WORK( 3*N+JE ) = ZERO
                  WORK( 2*N+JE+1 ) = -TEMP2R / TEMP
                  WORK( 3*N+JE+1 ) = -TEMP2I / TEMP
               ELSE
                  WORK( 2*N+JE+1 ) = ONE
                  WORK( 3*N+JE+1 ) = ZERO
                  TEMP = ACOEF*S( JE, JE+1 )
                  WORK( 2*N+JE ) = ( BCOEFR*P( JE+1, JE+1 )-ACOEF*
     $                             S( JE+1, JE+1 ) ) / TEMP
                  WORK( 3*N+JE ) = BCOEFI*P( JE+1, JE+1 ) / TEMP
               END IF
               XMAX = MAX( ABS( WORK( 2*N+JE ) )+ABS( WORK( 3*N+JE ) ),
     $                ABS( WORK( 2*N+JE+1 ) )+ABS( WORK( 3*N+JE+1 ) ) )
            END IF
*
            DMIN = MAX( ULP*ACOEFA*ANORM, ULP*BCOEFA*BNORM, SAFMIN )
*
*                                           T
*           Triangular solve of  (a A - b B)  y = 0
*
*                                   T
*           (rowwise in  (a A - b B) , or columnwise in (a A - b B) )
*
            IL2BY2 = .FALSE.
*
            DO 160 J = JE + NW, N
               IF( IL2BY2 ) THEN
                  IL2BY2 = .FALSE.
                  GO TO 160
               END IF
*
               NA = 1
               BDIAG( 1 ) = P( J, J )
               IF( J.LT.N ) THEN
                  IF( S( J+1, J ).NE.ZERO ) THEN
                     IL2BY2 = .TRUE.
                     BDIAG( 2 ) = P( J+1, J+1 )
                     NA = 2
                  END IF
               END IF
*
*              Check whether scaling is necessary for dot products
*
               XSCALE = ONE / MAX( ONE, XMAX )
               TEMP = MAX( WORK( J ), WORK( N+J ),
     $                ACOEFA*WORK( J )+BCOEFA*WORK( N+J ) )
               IF( IL2BY2 )
     $            TEMP = MAX( TEMP, WORK( J+1 ), WORK( N+J+1 ),
     $                   ACOEFA*WORK( J+1 )+BCOEFA*WORK( N+J+1 ) )
               IF( TEMP.GT.BIGNUM*XSCALE ) THEN
                  DO 90 JW = 0, NW - 1
                     DO 80 JR = JE, J - 1
                        WORK( ( JW+2 )*N+JR ) = XSCALE*
     $                     WORK( ( JW+2 )*N+JR )
   80                CONTINUE
   90             CONTINUE
                  XMAX = XMAX*XSCALE
               END IF
*
*              Compute dot products
*
*                    j-1
*              SUM = sum  conjg( a*S(k,j) - b*P(k,j) )*x(k)
*                    k=je
*
*              To reduce the op count, this is done as
*
*              _        j-1                  _        j-1
*              a*conjg( sum  S(k,j)*x(k) ) - b*conjg( sum  P(k,j)*x(k) )
*                       k=je                          k=je
*
*              which may cause underflow problems if A or B are close
*              to underflow.  (E.g., less than SMALL.)
*
*
               DO 120 JW = 1, NW
                  DO 110 JA = 1, NA
                     SUMS( JA, JW ) = ZERO
                     SUMP( JA, JW ) = ZERO
*
                     DO 100 JR = JE, J - 1
                        SUMS( JA, JW ) = SUMS( JA, JW ) +
     $                                   S( JR, J+JA-1 )*
     $                                   WORK( ( JW+1 )*N+JR )
                        SUMP( JA, JW ) = SUMP( JA, JW ) +
     $                                   P( JR, J+JA-1 )*
     $                                   WORK( ( JW+1 )*N+JR )
  100                CONTINUE
  110             CONTINUE
  120          CONTINUE
*
               DO 130 JA = 1, NA
                  IF( ILCPLX ) THEN
                     SUM( JA, 1 ) = -ACOEF*SUMS( JA, 1 ) +
     $                              BCOEFR*SUMP( JA, 1 ) -
     $                              BCOEFI*SUMP( JA, 2 )
                     SUM( JA, 2 ) = -ACOEF*SUMS( JA, 2 ) +
     $                              BCOEFR*SUMP( JA, 2 ) +
     $                              BCOEFI*SUMP( JA, 1 )
                  ELSE
                     SUM( JA, 1 ) = -ACOEF*SUMS( JA, 1 ) +
     $                              BCOEFR*SUMP( JA, 1 )
                  END IF
  130          CONTINUE
*
*                                  T
*              Solve  ( a A - b B )  y = SUM(,)
*              with scaling and perturbation of the denominator
*
               CALL DLALN2( .TRUE., NA, NW, DMIN, ACOEF, S( J, J ), LDS,
     $                      BDIAG( 1 ), BDIAG( 2 ), SUM, 2, BCOEFR,
     $                      BCOEFI, WORK( 2*N+J ), N, SCALE, TEMP,
     $                      IINFO )
               IF( SCALE.LT.ONE ) THEN
                  DO 150 JW = 0, NW - 1
                     DO 140 JR = JE, J - 1
                        WORK( ( JW+2 )*N+JR ) = SCALE*
     $                     WORK( ( JW+2 )*N+JR )
  140                CONTINUE
  150             CONTINUE
                  XMAX = SCALE*XMAX
               END IF
               XMAX = MAX( XMAX, TEMP )
  160       CONTINUE
*
*           Copy eigenvector to VL, back transforming if
*           HOWMNY='B'.
*
            IEIG = IEIG + 1
            IF( ILBACK ) THEN
               DO 170 JW = 0, NW - 1
                  CALL DGEMV( 'N', N, N+1-JE, ONE, VL( 1, JE ), LDVL,
     $                        WORK( ( JW+2 )*N+JE ), 1, ZERO,
     $                        WORK( ( JW+4 )*N+1 ), 1 )
  170          CONTINUE
               CALL DLACPY( ' ', N, NW, WORK( 4*N+1 ), N, VL( 1, JE ),
     $                      LDVL )
               IBEG = 1
            ELSE
               CALL DLACPY( ' ', N, NW, WORK( 2*N+1 ), N, VL( 1, IEIG ),
     $                      LDVL )
               IBEG = JE
            END IF
*
*           Scale eigenvector
*
            XMAX = ZERO
            IF( ILCPLX ) THEN
               DO 180 J = IBEG, N
                  XMAX = MAX( XMAX, ABS( VL( J, IEIG ) )+
     $                   ABS( VL( J, IEIG+1 ) ) )
  180          CONTINUE
            ELSE
               DO 190 J = IBEG, N
                  XMAX = MAX( XMAX, ABS( VL( J, IEIG ) ) )
  190          CONTINUE
            END IF
*
            IF( XMAX.GT.SAFMIN ) THEN
               XSCALE = ONE / XMAX
*
               DO 210 JW = 0, NW - 1
                  DO 200 JR = IBEG, N
                     VL( JR, IEIG+JW ) = XSCALE*VL( JR, IEIG+JW )
  200             CONTINUE
  210          CONTINUE
            END IF
            IEIG = IEIG + NW - 1
*
  220    CONTINUE
      END IF
*
*     Right eigenvectors
*
      IF( COMPR ) THEN
         IEIG = IM + 1
*
*        Main loop over eigenvalues
*
         ILCPLX = .FALSE.
         DO 500 JE = N, 1, -1
*
*           Skip this iteration if (a) HOWMNY='S' and SELECT=.FALSE., or
*           (b) this would be the second of a complex pair.
*           Check for complex eigenvalue, so as to be sure of which
*           entry(-ies) of SELECT to look at -- if complex, SELECT(JE)
*           or SELECT(JE-1).
*           If this is a complex pair, the 2-by-2 diagonal block
*           corresponding to the eigenvalue is in rows/columns JE-1:JE
*
            IF( ILCPLX ) THEN
               ILCPLX = .FALSE.
               GO TO 500
            END IF
            NW = 1
            IF( JE.GT.1 ) THEN
               IF( S( JE, JE-1 ).NE.ZERO ) THEN
                  ILCPLX = .TRUE.
                  NW = 2
               END IF
            END IF
            IF( ILALL ) THEN
               ILCOMP = .TRUE.
            ELSE IF( ILCPLX ) THEN
               ILCOMP = SELECT( JE ) .OR. SELECT( JE-1 )
            ELSE
               ILCOMP = SELECT( JE )
            END IF
            IF( .NOT.ILCOMP )
     $         GO TO 500
*
*           Decide if (a) singular pencil, (b) real eigenvalue, or
*           (c) complex eigenvalue.
*
            IF( .NOT.ILCPLX ) THEN
               IF( ABS( S( JE, JE ) ).LE.SAFMIN .AND.
     $             ABS( P( JE, JE ) ).LE.SAFMIN ) THEN
*
*                 Singular matrix pencil -- unit eigenvector
*
                  IEIG = IEIG - 1
                  DO 230 JR = 1, N
                     VR( JR, IEIG ) = ZERO
  230             CONTINUE
                  VR( IEIG, IEIG ) = ONE
                  GO TO 500
               END IF
            END IF
*
*           Clear vector
*
            DO 250 JW = 0, NW - 1
               DO 240 JR = 1, N
                  WORK( ( JW+2 )*N+JR ) = ZERO
  240          CONTINUE
  250       CONTINUE
*
*           Compute coefficients in  ( a A - b B ) x = 0
*              a  is  ACOEF
*              b  is  BCOEFR + i*BCOEFI
*
            IF( .NOT.ILCPLX ) THEN
*
*              Real eigenvalue
*
               TEMP = ONE / MAX( ABS( S( JE, JE ) )*ASCALE,
     $                ABS( P( JE, JE ) )*BSCALE, SAFMIN )
               SALFAR = ( TEMP*S( JE, JE ) )*ASCALE
               SBETA = ( TEMP*P( JE, JE ) )*BSCALE
               ACOEF = SBETA*ASCALE
               BCOEFR = SALFAR*BSCALE
               BCOEFI = ZERO
*
*              Scale to avoid underflow
*
               SCALE = ONE
               LSA = ABS( SBETA ).GE.SAFMIN .AND. ABS( ACOEF ).LT.SMALL
               LSB = ABS( SALFAR ).GE.SAFMIN .AND. ABS( BCOEFR ).LT.
     $               SMALL
               IF( LSA )
     $            SCALE = ( SMALL / ABS( SBETA ) )*MIN( ANORM, BIG )
               IF( LSB )
     $            SCALE = MAX( SCALE, ( SMALL / ABS( SALFAR ) )*
     $                    MIN( BNORM, BIG ) )
               IF( LSA .OR. LSB ) THEN
                  SCALE = MIN( SCALE, ONE /
     $                    ( SAFMIN*MAX( ONE, ABS( ACOEF ),
     $                    ABS( BCOEFR ) ) ) )
                  IF( LSA ) THEN
                     ACOEF = ASCALE*( SCALE*SBETA )
                  ELSE
                     ACOEF = SCALE*ACOEF
                  END IF
                  IF( LSB ) THEN
                     BCOEFR = BSCALE*( SCALE*SALFAR )
                  ELSE
                     BCOEFR = SCALE*BCOEFR
                  END IF
               END IF
               ACOEFA = ABS( ACOEF )
               BCOEFA = ABS( BCOEFR )
*
*              First component is 1
*
               WORK( 2*N+JE ) = ONE
               XMAX = ONE
*
*              Compute contribution from column JE of A and B to sum
*              (See "Further Details", above.)
*
               DO 260 JR = 1, JE - 1
                  WORK( 2*N+JR ) = BCOEFR*P( JR, JE ) -
     $                             ACOEF*S( JR, JE )
  260          CONTINUE
            ELSE
*
*              Complex eigenvalue
*
               CALL DLAG2( S( JE-1, JE-1 ), LDS, P( JE-1, JE-1 ), LDP,
     $                     SAFMIN*SAFETY, ACOEF, TEMP, BCOEFR, TEMP2,
     $                     BCOEFI )
               IF( BCOEFI.EQ.ZERO ) THEN
                  INFO = JE - 1
                  RETURN
               END IF
*
*              Scale to avoid over/underflow
*
               ACOEFA = ABS( ACOEF )
               BCOEFA = ABS( BCOEFR ) + ABS( BCOEFI )
               SCALE = ONE
               IF( ACOEFA*ULP.LT.SAFMIN .AND. ACOEFA.GE.SAFMIN )
     $            SCALE = ( SAFMIN / ULP ) / ACOEFA
               IF( BCOEFA*ULP.LT.SAFMIN .AND. BCOEFA.GE.SAFMIN )
     $            SCALE = MAX( SCALE, ( SAFMIN / ULP ) / BCOEFA )
               IF( SAFMIN*ACOEFA.GT.ASCALE )
     $            SCALE = ASCALE / ( SAFMIN*ACOEFA )
               IF( SAFMIN*BCOEFA.GT.BSCALE )
     $            SCALE = MIN( SCALE, BSCALE / ( SAFMIN*BCOEFA ) )
               IF( SCALE.NE.ONE ) THEN
                  ACOEF = SCALE*ACOEF
                  ACOEFA = ABS( ACOEF )
                  BCOEFR = SCALE*BCOEFR
                  BCOEFI = SCALE*BCOEFI
                  BCOEFA = ABS( BCOEFR ) + ABS( BCOEFI )
               END IF
*
*              Compute first two components of eigenvector
*              and contribution to sums
*
               TEMP = ACOEF*S( JE, JE-1 )
               TEMP2R = ACOEF*S( JE, JE ) - BCOEFR*P( JE, JE )
               TEMP2I = -BCOEFI*P( JE, JE )
               IF( ABS( TEMP ).GE.ABS( TEMP2R )+ABS( TEMP2I ) ) THEN
                  WORK( 2*N+JE ) = ONE
                  WORK( 3*N+JE ) = ZERO
                  WORK( 2*N+JE-1 ) = -TEMP2R / TEMP
                  WORK( 3*N+JE-1 ) = -TEMP2I / TEMP
               ELSE
                  WORK( 2*N+JE-1 ) = ONE
                  WORK( 3*N+JE-1 ) = ZERO
                  TEMP = ACOEF*S( JE-1, JE )
                  WORK( 2*N+JE ) = ( BCOEFR*P( JE-1, JE-1 )-ACOEF*
     $                             S( JE-1, JE-1 ) ) / TEMP
                  WORK( 3*N+JE ) = BCOEFI*P( JE-1, JE-1 ) / TEMP
               END IF
*
               XMAX = MAX( ABS( WORK( 2*N+JE ) )+ABS( WORK( 3*N+JE ) ),
     $                ABS( WORK( 2*N+JE-1 ) )+ABS( WORK( 3*N+JE-1 ) ) )
*
*              Compute contribution from columns JE and JE-1
*              of A and B to the sums.
*
               CREALA = ACOEF*WORK( 2*N+JE-1 )
               CIMAGA = ACOEF*WORK( 3*N+JE-1 )
               CREALB = BCOEFR*WORK( 2*N+JE-1 ) -
     $                  BCOEFI*WORK( 3*N+JE-1 )
               CIMAGB = BCOEFI*WORK( 2*N+JE-1 ) +
     $                  BCOEFR*WORK( 3*N+JE-1 )
               CRE2A = ACOEF*WORK( 2*N+JE )
               CIM2A = ACOEF*WORK( 3*N+JE )
               CRE2B = BCOEFR*WORK( 2*N+JE ) - BCOEFI*WORK( 3*N+JE )
               CIM2B = BCOEFI*WORK( 2*N+JE ) + BCOEFR*WORK( 3*N+JE )
               DO 270 JR = 1, JE - 2
                  WORK( 2*N+JR ) = -CREALA*S( JR, JE-1 ) +
     $                             CREALB*P( JR, JE-1 ) -
     $                             CRE2A*S( JR, JE ) + CRE2B*P( JR, JE )
                  WORK( 3*N+JR ) = -CIMAGA*S( JR, JE-1 ) +
     $                             CIMAGB*P( JR, JE-1 ) -
     $                             CIM2A*S( JR, JE ) + CIM2B*P( JR, JE )
  270          CONTINUE
            END IF
*
            DMIN = MAX( ULP*ACOEFA*ANORM, ULP*BCOEFA*BNORM, SAFMIN )
*
*           Columnwise triangular solve of  (a A - b B)  x = 0
*
            IL2BY2 = .FALSE.
            DO 370 J = JE - NW, 1, -1
*
*              If a 2-by-2 block, is in position j-1:j, wait until
*              next iteration to process it (when it will be j:j+1)
*
               IF( .NOT.IL2BY2 .AND. J.GT.1 ) THEN
                  IF( S( J, J-1 ).NE.ZERO ) THEN
                     IL2BY2 = .TRUE.
                     GO TO 370
                  END IF
               END IF
               BDIAG( 1 ) = P( J, J )
               IF( IL2BY2 ) THEN
                  NA = 2
                  BDIAG( 2 ) = P( J+1, J+1 )
               ELSE
                  NA = 1
               END IF
*
*              Compute x(j) (and x(j+1), if 2-by-2 block)
*
               CALL DLALN2( .FALSE., NA, NW, DMIN, ACOEF, S( J, J ),
     $                      LDS, BDIAG( 1 ), BDIAG( 2 ), WORK( 2*N+J ),
     $                      N, BCOEFR, BCOEFI, SUM, 2, SCALE, TEMP,
     $                      IINFO )
               IF( SCALE.LT.ONE ) THEN
*
                  DO 290 JW = 0, NW - 1
                     DO 280 JR = 1, JE
                        WORK( ( JW+2 )*N+JR ) = SCALE*
     $                     WORK( ( JW+2 )*N+JR )
  280                CONTINUE
  290             CONTINUE
               END IF
               XMAX = MAX( SCALE*XMAX, TEMP )
*
               DO 310 JW = 1, NW
                  DO 300 JA = 1, NA
                     WORK( ( JW+1 )*N+J+JA-1 ) = SUM( JA, JW )
  300             CONTINUE
  310          CONTINUE
*
*              w = w + x(j)*(a S(*,j) - b P(*,j) ) with scaling
*
               IF( J.GT.1 ) THEN
*
*                 Check whether scaling is necessary for sum.
*
                  XSCALE = ONE / MAX( ONE, XMAX )
                  TEMP = ACOEFA*WORK( J ) + BCOEFA*WORK( N+J )
                  IF( IL2BY2 )
     $               TEMP = MAX( TEMP, ACOEFA*WORK( J+1 )+BCOEFA*
     $                      WORK( N+J+1 ) )
                  TEMP = MAX( TEMP, ACOEFA, BCOEFA )
                  IF( TEMP.GT.BIGNUM*XSCALE ) THEN
*
                     DO 330 JW = 0, NW - 1
                        DO 320 JR = 1, JE
                           WORK( ( JW+2 )*N+JR ) = XSCALE*
     $                        WORK( ( JW+2 )*N+JR )
  320                   CONTINUE
  330                CONTINUE
                     XMAX = XMAX*XSCALE
                  END IF
*
*                 Compute the contributions of the off-diagonals of
*                 column j (and j+1, if 2-by-2 block) of A and B to the
*                 sums.
*
*
                  DO 360 JA = 1, NA
                     IF( ILCPLX ) THEN
                        CREALA = ACOEF*WORK( 2*N+J+JA-1 )
                        CIMAGA = ACOEF*WORK( 3*N+J+JA-1 )
                        CREALB = BCOEFR*WORK( 2*N+J+JA-1 ) -
     $                           BCOEFI*WORK( 3*N+J+JA-1 )
                        CIMAGB = BCOEFI*WORK( 2*N+J+JA-1 ) +
     $                           BCOEFR*WORK( 3*N+J+JA-1 )
                        DO 340 JR = 1, J - 1
                           WORK( 2*N+JR ) = WORK( 2*N+JR ) -
     $                                      CREALA*S( JR, J+JA-1 ) +
     $                                      CREALB*P( JR, J+JA-1 )
                           WORK( 3*N+JR ) = WORK( 3*N+JR ) -
     $                                      CIMAGA*S( JR, J+JA-1 ) +
     $                                      CIMAGB*P( JR, J+JA-1 )
  340                   CONTINUE
                     ELSE
                        CREALA = ACOEF*WORK( 2*N+J+JA-1 )
                        CREALB = BCOEFR*WORK( 2*N+J+JA-1 )
                        DO 350 JR = 1, J - 1
                           WORK( 2*N+JR ) = WORK( 2*N+JR ) -
     $                                      CREALA*S( JR, J+JA-1 ) +
     $                                      CREALB*P( JR, J+JA-1 )
  350                   CONTINUE
                     END IF
  360             CONTINUE
               END IF
*
               IL2BY2 = .FALSE.
  370       CONTINUE
*
*           Copy eigenvector to VR, back transforming if
*           HOWMNY='B'.
*
            IEIG = IEIG - NW
            IF( ILBACK ) THEN
*
               DO 410 JW = 0, NW - 1
                  DO 380 JR = 1, N
                     WORK( ( JW+4 )*N+JR ) = WORK( ( JW+2 )*N+1 )*
     $                                       VR( JR, 1 )
  380             CONTINUE
*
*                 A series of compiler directives to defeat
*                 vectorization for the next loop
*
*
                  DO 400 JC = 2, JE
                     DO 390 JR = 1, N
                        WORK( ( JW+4 )*N+JR ) = WORK( ( JW+4 )*N+JR ) +
     $                     WORK( ( JW+2 )*N+JC )*VR( JR, JC )
  390                CONTINUE
  400             CONTINUE
  410          CONTINUE
*
               DO 430 JW = 0, NW - 1
                  DO 420 JR = 1, N
                     VR( JR, IEIG+JW ) = WORK( ( JW+4 )*N+JR )
  420             CONTINUE
  430          CONTINUE
*
               IEND = N
            ELSE
               DO 450 JW = 0, NW - 1
                  DO 440 JR = 1, N
                     VR( JR, IEIG+JW ) = WORK( ( JW+2 )*N+JR )
  440             CONTINUE
  450          CONTINUE
*
               IEND = JE
            END IF
*
*           Scale eigenvector
*
            XMAX = ZERO
            IF( ILCPLX ) THEN
               DO 460 J = 1, IEND
                  XMAX = MAX( XMAX, ABS( VR( J, IEIG ) )+
     $                   ABS( VR( J, IEIG+1 ) ) )
  460          CONTINUE
            ELSE
               DO 470 J = 1, IEND
                  XMAX = MAX( XMAX, ABS( VR( J, IEIG ) ) )
  470          CONTINUE
            END IF
*
            IF( XMAX.GT.SAFMIN ) THEN
               XSCALE = ONE / XMAX
               DO 490 JW = 0, NW - 1
                  DO 480 JR = 1, IEND
                     VR( JR, IEIG+JW ) = XSCALE*VR( JR, IEIG+JW )
  480             CONTINUE
  490          CONTINUE
            END IF
  500    CONTINUE
      END IF
*
      RETURN
*
*     End of DTGEVC
*
      END