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## dggesx()

subroutine dggesx | ( | character | jobvsl,
|  | character | jobvsr,
|  | character | sort,
|  | external | selctg,
|  | character | sense,
|  | integer | n,
|  | double precision, dimension( lda, * ) | a,
|  | integer | lda,
|  | double precision, dimension( ldb, * ) | b,
|  | integer | ldb,
|  | integer | sdim,
|  | double precision, dimension( * ) | alphar,
|  | double precision, dimension( * ) | alphai,
|  | double precision, dimension( * ) | beta,
|  | double precision, dimension( ldvsl, * ) | vsl,
|  | integer | ldvsl,
|  | double precision, dimension( ldvsr, * ) | vsr,
|  | integer | ldvsr,
|  | double precision, dimension( 2 ) | rconde,
|  | double precision, dimension( 2 ) | rcondv,
|  | double precision, dimension( * ) | work,
|  | integer | lwork,
|  | integer, dimension( * ) | iwork,
|  | integer | liwork,
|  | logical, dimension( * ) | bwork,
|  | integer | info )

DGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices

Download DGGESX + dependencies [TGZ] [ZIP] [TXT]

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
      !>  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

  [in]

  JOBVSL

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

  [in]

  JOBVSR

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

  [in]

  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).
      !>

  [in]

  SELCTG

      !>          SELCTG is a 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.
      !>

  [in]

  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'.
      !>

  [in]

  N

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

  [in,out]

  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.
      !>

  [in]

  LDA

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

  [in,out]

  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.
      !>

  [in]

  LDB

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

  [out]

  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.)
      !>

  [out]

  ALPHAR

      !>          ALPHAR is DOUBLE PRECISION array, dimension (N)
      !>

  [out]

  ALPHAI

      !>          ALPHAI is DOUBLE PRECISION array, dimension (N)
      !>

  [out]

  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).
      !>

  [out]

  VSL

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

  [in]

  LDVSL

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

  [out]

  VSR

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

  [in]

  LDVSR

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

  [out]

  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'.
      !>

  [out]

  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'.
      !>

  [out]

  WORK

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

  [in]

  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.
      !>

  [out]

  IWORK

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

  [in]

  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.
      !>

  [out]

  BWORK

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

  [out]

  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.

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.
      !>