


DGGES(3f)      LAPACK driver routine (version 3.1)      DGGES(3f)


NNAAMMEE
       DGGES - for a pair of N-by-N real nonsymmetric matrices (A,B),

SSYYNNOOPPSSIISS
       SUBROUTINE DGGES( JOBVSL,  JOBVSR,  SORT,  SELCTG,  N,  A, LDA, B, LDB,
                         SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL,  VSR,  LDVSR,
                         WORK, LWORK, BWORK, INFO )

           CHARACTER     JOBVSL, JOBVSR, SORT

           INTEGER       INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N, SDIM

           LOGICAL       BWORK( * )

           DOUBLE        PRECISION  A(  LDA, * ), ALPHAI( * ), ALPHAR( * ), B(
                         LDB, * ), BETA( * ), VSL( LDVSL, * ), VSR(  LDVSR,  *
                         ), WORK( * )

           LOGICAL       SELCTG

           EXTERNAL      SELCTG

PPUURRPPOOSSEE
       DGGES  computes  for a pair of N-by-N real nonsymmetric matrices (A,B),
       the generalized eigenvalues, the generalized  real  Schur  form  (S,T),
       optionally,  the  left  and/or right matrices of Schur vectors (VSL and
       VSR). This gives the generalized Schur factorization

                (A,B) = ( (VSL)*S*(VSR)**T, (VSL)*T*(VSR)**T )

       Optionally, it also orders the eigenvalues so that a  selected  cluster
       of  eigenvalues  appears  in  the  leading diagonal blocks of the upper
       quasi-triangular matrix S and the upper triangular matrix T.The leading
       columns  of  VSL  and VSR then form an orthonormal basis for the corre-
       sponding left and right eigenspaces (deflating subspaces).

       (If only the generalized eigenvalues are needed, use the  driver  DGGEV
       instead, which is faster.)

       A  generalized eigenvalue for a pair of matrices (A,B) is a scalar w or
       a ratio alpha/beta = w, such that  A - w*B is singular.  It is  usually
       represented  as  the pair (alpha,beta), as there is a reasonable inter-
       pretation for beta=0 or both being zero.

       A pair of matrices (S,T) is in generalized real  Schur  form  if  T  is
       upper triangular with non-negative diagonal and S is block upper trian-
       gular with 1-by-1 and 2-by-2 blocks.  1-by-1 blocks correspond to  real
       generalized  eigenvalues,  while  2-by-2 blocks of S will be "standard-
       ized" by making the corresponding elements of T have the form:
               [  a  0  ]
               [  0  b  ]

       and the pair of corresponding 2-by-2 blocks in S and T will have a com-
       plex conjugate pair of generalized eigenvalues.



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DGGES(3f)      LAPACK driver routine (version 3.1)      DGGES(3f)


AARRGGUUMMEENNTTSS
       JOBVSL  (input) CHARACTER*1
               = 'N':  do not compute the left Schur vectors;
               = 'V':  compute the left Schur vectors.

       JOBVSR  (input) CHARACTER*1
               = 'N':  do not compute the right Schur vectors;
               = 'V':  compute the right Schur vectors.

       SORT    (input) CHARACTER*1
               Specifies whether or not to order the eigenvalues on the diago-
               nal of the generalized Schur form.  = 'N':  Eigenvalues are not
               ordered;
               = 'S':  Eigenvalues are ordered (see SELCTG);

       SELCTG   (external procedure) LOGICAL FUNCTION of three DOUBLE
       PRECISION argu- ments
               SELCTG must be declared EXTERNAL in the calling subroutine.  If
               SORT = 'N', SELCTG is not referenced.  If SORT = 'S', SELCTG is
               used to select eigenvalues to sort to the top left of the Schur
               form.  An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is  selected
               if  SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either
               one of a complex conjugate pair  of  eigenvalues  is  selected,
               then both complex eigenvalues are selected.

               Note  that  in  the  ill-conditioned  case,  a selected complex
               eigenvalue may no  longer  satisfy  SELCTG(ALPHAR(j),ALPHAI(j),
               BETA(j))  =  .TRUE. after ordering. INFO is to be set to N+2 in
               this case.

       N       (input) INTEGER
               The order of the matrices A, B, VSL, and VSR.  N >= 0.

       A       (input/output) DOUBLE PRECISION array, dimension (LDA, N)
               On entry, the first of the pair of matrices.  On  exit,  A  has
               been overwritten by its generalized Schur form S.

       LDA     (input) INTEGER
               The leading dimension of A.  LDA >= max(1,N).

       B       (input/output) DOUBLE PRECISION array, dimension (LDB, N)
               On  entry,  the second of the pair of matrices.  On exit, B has
               been overwritten by its generalized Schur form T.

       LDB     (input) INTEGER
               The leading dimension of B.  LDB >= max(1,N).

       SDIM    (output) INTEGER
               If SORT = 'N', SDIM = 0.  If SORT  =  'S',  SDIM  =  number  of
               eigenvalues (after sorting) for which SELCTG is true.  (Complex
               conjugate pairs for which SELCTG is true for either  eigenvalue
               count as 2.)





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DGGES(3f)      LAPACK driver routine (version 3.1)      DGGES(3f)


       ALPHAR  (output) DOUBLE PRECISION array, dimension (N)
               ALPHAI   (output)  DOUBLE  PRECISION  array, dimension (N) BETA
               (output)  DOUBLE  PRECISION  array,  dimension  (N)  On   exit,
               (ALPHAR(j)  + ALPHAI(j)*i)/BETA(j), j=1,...,N, will be the gen-
               eralized   eigenvalues.    ALPHAR(j)   +    ALPHAI(j)*i,    and
               BETA(j),j=1,...,N  are  the diagonals of the complex Schur form
               (S,T) that would result if the 2-by-2 diagonal  blocks  of  the
               real  Schur  form  of  (A,B) were further reduced to triangular
               form  using  2-by-2  complex   unitary   transformations.    If
               ALPHAI(j)  is  zero, then the j-th eigenvalue is real; if posi-
               tive, then the j-th and (j+1)-st eigenvalues are a complex con-
               jugate pair, with ALPHAI(j+1) negative.

               Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j) may
               easily over- or underflow, and BETA(j) may even be zero.  Thus,
               the  user  should  avoid naively computing the ratio.  However,
               ALPHAR and ALPHAI will be always less than and usually compara-
               ble  with  norm(A)  in magnitude, and BETA always less than and
               usually comparable with norm(B).

       VSL     (output) DOUBLE PRECISION array, dimension (LDVSL,N)
               If JOBVSL = 'V', VSL will contain the left Schur vectors.   Not
               referenced if JOBVSL = 'N'.

       LDVSL   (input) INTEGER
               The leading dimension of the matrix VSL. LDVSL >=1, and if JOB-
               VSL = 'V', LDVSL >= N.

       VSR     (output) DOUBLE PRECISION array, dimension (LDVSR,N)
               If JOBVSR = 'V', VSR will contain the right Schur vectors.  Not
               referenced if JOBVSR = 'N'.

       LDVSR   (input) INTEGER
               The  leading  dimension  of  the matrix VSR. LDVSR >= 1, and if
               JOBVSR = 'V', LDVSR >= 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.  If N = 0, LWORK >= 1, else
               LWORK >= 8*N+16.  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.

       BWORK   (workspace) LOGICAL array, dimension (N)
               Not referenced if SORT = 'N'.





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DGGES(3f)      LAPACK driver routine (version 3.1)      DGGES(3f)


       INFO    (output) INTEGER
               = 0:  successful exit
               < 0:  if INFO = -i, the i-th argument had an illegal value.
               = 1,...,N: The QZ iteration failed.  (A,B)  are  not  in  Schur
               form,  but  ALPHAR(j), ALPHAI(j), and BETA(j) should be correct
               for j=INFO+1,...,N.  > N:  =N+1: other than QZ iteration failed
               in DHGEQZ.
               =N+2: after reordering, roundoff changed values of some complex
               eigenvalues so that  leading  eigenvalues  in  the  Generalized
               Schur  form no longer satisfy SELCTG=.TRUE.  This could also be
               caused due to scaling.  =N+3: reordering failed in DTGSEN.














































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