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.

  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
    !>  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.
    !>

  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
    !>  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.
    !>

  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.

Author

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