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NAME

       dggesx.f -

SYNOPSIS

   Functions/Subroutines
       subroutine dggesx (JOBVSL, JOBVSR, SORT, SELCTG, SENSE, N, A, LDA, B, LDB, SDIM, ALPHAR,
           ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, RCONDE, RCONDV, WORK, LWORK, IWORK, LIWORK,
           BWORK, INFO)
            DGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur
           vectors for GE matrices

Function/Subroutine Documentation

   subroutine dggesx (characterJOBVSL, characterJOBVSR, characterSORT, logical, externalSELCTG,
       characterSENSE, integerN, double precision, dimension( lda, * )A, integerLDA, double
       precision, dimension( ldb, * )B, integerLDB, integerSDIM, double precision, dimension( *
       )ALPHAR, double precision, dimension( * )ALPHAI, double precision, dimension( * )BETA,
       double precision, dimension( ldvsl, * )VSL, integerLDVSL, double precision, dimension(
       ldvsr, * )VSR, integerLDVSR, double precision, dimension( 2 )RCONDE, double precision,
       dimension( 2 )RCONDV, double precision, dimension( * )WORK, integerLWORK, integer,
       dimension( * )IWORK, integerLIWORK, logical, dimension( * )BWORK, integerINFO)
        DGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur
       vectors for GE matrices

       Purpose:

            DGGESX computes for a pair of N-by-N real nonsymmetric matrices
            (A,B), the generalized eigenvalues, the real Schur form (S,T), and,
            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; computes
            a reciprocal condition number for the average of the selected
            eigenvalues (RCONDE); and computes a reciprocal condition number for
            the right and left deflating subspaces corresponding to the selected
            eigenvalues (RCONDV). The leading columns of VSL and VSR then form
            an orthonormal basis for the corresponding left and right eigenspaces
            (deflating subspaces).

            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 interpretation for beta=0 or for 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
            triangular 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
            "standardized" 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
            complex conjugate pair of generalized eigenvalues.

       Parameters:
           JOBVSL

                     JOBVSL is CHARACTER*1
                     = 'N':  do not compute the left Schur vectors;
                     = 'V':  compute the left Schur vectors.

           JOBVSR

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

           SORT

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

           SELCTG

                     SELCTG is procedure) LOGICAL FUNCTION of three DOUBLE PRECISION arguments
                     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 a selected complex eigenvalue may no longer satisfy
                     SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) = .TRUE. after ordering,
                     since ordering may change the value of complex eigenvalues
                     (especially if the eigenvalue is ill-conditioned), in this
                     case INFO is set to N+3.

           SENSE

                     SENSE is CHARACTER*1
                     Determines which reciprocal condition numbers are computed.
                     = 'N' : None are computed;
                     = 'E' : Computed for average of selected eigenvalues only;
                     = 'V' : Computed for selected deflating subspaces only;
                     = 'B' : Computed for both.
                     If SENSE = 'E', 'V', or 'B', SORT must equal 'S'.

           N

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

           A

                     A is 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

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

           B

                     B is 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

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

           SDIM

                     SDIM is 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.)

           ALPHAR

                     ALPHAR is DOUBLE PRECISION array, dimension (N)

           ALPHAI

                     ALPHAI is DOUBLE PRECISION array, dimension (N)

           BETA

                     BETA is DOUBLE PRECISION array, dimension (N)
                     On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
                     be the generalized 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
                     positive, then the j-th and (j+1)-st eigenvalues are a
                     complex conjugate 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 comparable with norm(A) in magnitude, and BETA always
                     less than and usually comparable with norm(B).

           VSL

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

           LDVSL

                     LDVSL is INTEGER
                     The leading dimension of the matrix VSL. LDVSL >=1, and
                     if JOBVSL = 'V', LDVSL >= N.

           VSR

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

           LDVSR

                     LDVSR is INTEGER
                     The leading dimension of the matrix VSR. LDVSR >= 1, and
                     if JOBVSR = 'V', LDVSR >= N.

           RCONDE

                     RCONDE is DOUBLE PRECISION array, dimension ( 2 )
                     If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2) contain the
                     reciprocal condition numbers for the average of the selected
                     eigenvalues.
                     Not referenced if SENSE = 'N' or 'V'.

           RCONDV

                     RCONDV is DOUBLE PRECISION array, dimension ( 2 )
                     If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2) contain the
                     reciprocal condition numbers for the selected deflating
                     subspaces.
                     Not referenced if SENSE = 'N' or 'E'.

           WORK

                     WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
                     On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

           LWORK

                     LWORK is INTEGER
                     The dimension of the array WORK.
                     If N = 0, LWORK >= 1, else if SENSE = 'E', 'V', or 'B',
                     LWORK >= max( 8*N, 6*N+16, 2*SDIM*(N-SDIM) ), else
                     LWORK >= max( 8*N, 6*N+16 ).
                     Note that 2*SDIM*(N-SDIM) <= N*N/2.
                     Note also that an error is only returned if
                     LWORK < max( 8*N, 6*N+16), but if SENSE = 'E' or 'V' or 'B'
                     this may not be large enough.

                     If LWORK = -1, then a workspace query is assumed; the routine
                     only calculates the bound on the optimal size of the WORK
                     array and the minimum size of the IWORK array, returns these
                     values as the first entries of the WORK and IWORK arrays, and
                     no error message related to LWORK or LIWORK is issued by
                     XERBLA.

           IWORK

                     IWORK is INTEGER array, dimension (MAX(1,LIWORK))
                     On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.

           LIWORK

                     LIWORK is INTEGER
                     The dimension of the array IWORK.
                     If SENSE = 'N' or N = 0, LIWORK >= 1, otherwise
                     LIWORK >= N+6.

                     If LIWORK = -1, then a workspace query is assumed; the
                     routine only calculates the bound on the optimal size of the
                     WORK array and the minimum size of the IWORK array, returns
                     these values as the first entries of the WORK and IWORK
                     arrays, and no error message related to LWORK or LIWORK is
                     issued by XERBLA.

           BWORK

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

           INFO

                     INFO is 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.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           November 2011

       Further Details:

             An approximate (asymptotic) bound on the average absolute error of
             the selected eigenvalues is

                  EPS * norm((A, B)) / RCONDE( 1 ).

             An approximate (asymptotic) bound on the maximum angular error in
             the computed deflating subspaces is

                  EPS * norm((A, B)) / RCONDV( 2 ).

             See LAPACK User's Guide, section 4.11 for more information.

       Definition at line 363 of file dggesx.f.

Author

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