SRC/sggesx.f(3)

NAME
SRC/sggesx.f

SYNOPSIS

  Functions/Subroutines
subroutine sggesx (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)
SGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices

Function/Subroutine Documentation

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

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

Purpose:

     SGGESX 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 a LOGICAL FUNCTION of three REAL 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 REAL 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 REAL 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 REAL array, dimension (N)

    ALPHAI

              ALPHAI is REAL array, dimension (N)

    BETA

              BETA is REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 SHGEQZ
                    =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 STGSEN.

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.

Definition at line 361 of file sggesx.f.

Author

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