ggesx(3)

NAME
ggesx - ggesx: Schur form, expert

SYNOPSIS

  Functions
subroutine cggesx (jobvsl, jobvsr, sort, selctg, sense, n, a, lda, b, ldb, sdim, alpha, beta, vsl, ldvsl, vsr, ldvsr, rconde, rcondv, work, lwork, rwork, iwork, liwork, bwork, info)
CGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices
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
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
subroutine zggesx (jobvsl, jobvsr, sort, selctg, sense, n, a, lda, b, ldb, sdim, alpha, beta, vsl, ldvsl, vsr, ldvsr, rconde, rcondv, work, lwork, rwork, iwork, liwork, bwork, info)
ZGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices

Detailed Description

Function Documentation

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

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

Purpose:

     CGGESX computes for a pair of N-by-N complex nonsymmetric matrices
     (A,B), the generalized eigenvalues, the complex 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)**H, (VSL) T (VSR)**H )
     where (VSR)**H is the conjugate-transpose of VSR.
     Optionally, it also orders the eigenvalues so that a selected cluster
     of eigenvalues appears in the leading diagonal blocks of the upper
     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 complex Schur form if T is
     upper triangular with non-negative diagonal and S is upper
     triangular.

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 two COMPLEX 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.
              Note that a selected complex eigenvalue may no longer satisfy
              SELCTG(ALPHA(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 see INFO below).

    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 COMPLEX 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 COMPLEX 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.

    ALPHA

              ALPHA is COMPLEX array, dimension (N)

    BETA

              BETA is COMPLEX array, dimension (N)
              On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the
              generalized eigenvalues.  ALPHA(j) and BETA(j),j=1,...,N  are
              the diagonals of the complex Schur form (S,T).  BETA(j) will
              be non-negative real.
              Note: the quotients ALPHA(j)/BETA(j) may easily over- or
              underflow, and BETA(j) may even be zero.  Thus, the user
              should avoid naively computing the ratio alpha/beta.
              However, ALPHA 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 COMPLEX 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 COMPLEX 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 number for the selected deflating
              subspaces.
              Not referenced if SENSE = 'N' or 'E'.

    WORK

              WORK is COMPLEX 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(1,2*N,2*SDIM*(N-SDIM)), else
              LWORK >= MAX(1,2*N).  Note that 2*SDIM*(N-SDIM) <= N*N/2.
              Note also that an error is only returned if
              LWORK < MAX(1,2*N), 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.

    RWORK

              RWORK is REAL array, dimension ( 8*N )
              Real workspace.

    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 WORK.
              If SENSE = 'N' or N = 0, LIWORK >= 1, otherwise
              LIWORK >= N+2.
              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 ALPHA(j) and BETA(j) should be correct for
                    j=INFO+1,...,N.
              > N:  =N+1: other than QZ iteration failed in CHGEQZ
                    =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 CTGSEN.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 326 of file cggesx.f.

  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

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 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.

    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.

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 dggesx.f.

  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.

  subroutine zggesx (character jobvsl, character jobvsr, character sort, external selctg, character sense, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, integer sdim, complex*16, dimension( * ) alpha, complex*16, dimension( * ) beta, complex*16, dimension( ldvsl, * ) vsl, integer ldvsl, complex*16, dimension( ldvsr, * ) vsr, integer ldvsr, double precision, dimension( 2 ) rconde, double precision, dimension( 2 ) rcondv, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, integer, dimension( * ) iwork, integer liwork, logical, dimension( * ) bwork, integer info)

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

Purpose:

     ZGGESX computes for a pair of N-by-N complex nonsymmetric matrices
     (A,B), the generalized eigenvalues, the complex 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)**H, (VSL) T (VSR)**H )
     where (VSR)**H is the conjugate-transpose of VSR.
     Optionally, it also orders the eigenvalues so that a selected cluster
     of eigenvalues appears in the leading diagonal blocks of the upper
     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 complex Schur form if T is
     upper triangular with non-negative diagonal and S is upper
     triangular.

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 two COMPLEX*16 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.
              Note that a selected complex eigenvalue may no longer satisfy
              SELCTG(ALPHA(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 see INFO below).

    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 COMPLEX*16 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 COMPLEX*16 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.

    ALPHA

              ALPHA is COMPLEX*16 array, dimension (N)

    BETA

              BETA is COMPLEX*16 array, dimension (N)
              On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the
              generalized eigenvalues.  ALPHA(j) and BETA(j),j=1,...,N  are
              the diagonals of the complex Schur form (S,T).  BETA(j) will
              be non-negative real.
              Note: the quotients ALPHA(j)/BETA(j) may easily over- or
              underflow, and BETA(j) may even be zero.  Thus, the user
              should avoid naively computing the ratio alpha/beta.
              However, ALPHA 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 COMPLEX*16 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 COMPLEX*16 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 number for the selected deflating
              subspaces.
              Not referenced if SENSE = 'N' or 'E'.

    WORK

              WORK is COMPLEX*16 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(1,2*N,2*SDIM*(N-SDIM)), else
              LWORK >= MAX(1,2*N).  Note that 2*SDIM*(N-SDIM) <= N*N/2.
              Note also that an error is only returned if
              LWORK < MAX(1,2*N), 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.

    RWORK

              RWORK is DOUBLE PRECISION array, dimension ( 8*N )
              Real workspace.

    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+2.
              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 ALPHA(j) and BETA(j) should be correct for
                    j=INFO+1,...,N.
              > N:  =N+1: other than QZ iteration failed in ZHGEQZ
                    =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 ZTGSEN.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 326 of file zggesx.f.

Author

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