LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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◆ zggesx()

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

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

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
[in]JOBVSL
!>          JOBVSL is CHARACTER*1
!>          = 'N':  do not compute the left Schur vectors;
!>          = 'V':  compute the left Schur vectors.
!>
[in]JOBVSR
!>          JOBVSR is CHARACTER*1
!>          = 'N':  do not compute the right Schur vectors;
!>          = 'V':  compute the right Schur vectors.
!>
[in]SORT
!>          SORT is CHARACTER*1
!>          Specifies whether or not to order the eigenvalues on the
!>          diagonal of the generalized Schur form.
!>          = 'N':  Eigenvalues are not ordered;
!>          = 'S':  Eigenvalues are ordered (see SELCTG).
!>
[in]SELCTG
!>          SELCTG is a LOGICAL FUNCTION of 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).
!>
[in]SENSE
!>          SENSE is CHARACTER*1
!>          Determines which reciprocal condition numbers are computed.
!>          = 'N': None are computed;
!>          = 'E': Computed for average of selected eigenvalues only;
!>          = 'V': Computed for selected deflating subspaces only;
!>          = 'B': Computed for both.
!>          If SENSE = 'E', 'V', or 'B', SORT must equal 'S'.
!>
[in]N
!>          N is INTEGER
!>          The order of the matrices A, B, VSL, and VSR.  N >= 0.
!>
[in,out]A
!>          A is 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.
!>
[in]LDA
!>          LDA is INTEGER
!>          The leading dimension of A.  LDA >= max(1,N).
!>
[in,out]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.
!>
[in]LDB
!>          LDB is INTEGER
!>          The leading dimension of B.  LDB >= max(1,N).
!>
[out]SDIM
!>          SDIM is INTEGER
!>          If SORT = 'N', SDIM = 0.
!>          If SORT = 'S', SDIM = number of eigenvalues (after sorting)
!>          for which SELCTG is true.
!>
[out]ALPHA
!>          ALPHA is COMPLEX*16 array, dimension (N)
!>
[out]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).
!>
[out]VSL
!>          VSL is COMPLEX*16 array, dimension (LDVSL,N)
!>          If JOBVSL = 'V', VSL will contain the left Schur vectors.
!>          Not referenced if JOBVSL = 'N'.
!>
[in]LDVSL
!>          LDVSL is INTEGER
!>          The leading dimension of the matrix VSL. LDVSL >=1, and
!>          if JOBVSL = 'V', LDVSL >= N.
!>
[out]VSR
!>          VSR is COMPLEX*16 array, dimension (LDVSR,N)
!>          If JOBVSR = 'V', VSR will contain the right Schur vectors.
!>          Not referenced if JOBVSR = 'N'.
!>
[in]LDVSR
!>          LDVSR is INTEGER
!>          The leading dimension of the matrix VSR. LDVSR >= 1, and
!>          if JOBVSR = 'V', LDVSR >= N.
!>
[out]RCONDE
!>          RCONDE is DOUBLE PRECISION array, dimension ( 2 )
!>          If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2) contain the
!>          reciprocal condition numbers for the average of the selected
!>          eigenvalues.
!>          Not referenced if SENSE = 'N' or 'V'.
!>
[out]RCONDV
!>          RCONDV is DOUBLE PRECISION array, dimension ( 2 )
!>          If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2) contain the
!>          reciprocal condition number for the selected deflating
!>          subspaces.
!>          Not referenced if SENSE = 'N' or 'E'.
!>
[out]WORK
!>          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
!>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
!>
[in]LWORK
!>          LWORK is INTEGER
!>          The dimension of the array WORK.
!>          If N = 0, LWORK >= 1, else if SENSE = 'E', 'V', or 'B',
!>          LWORK >= MAX(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.
!>
[out]RWORK
!>          RWORK is DOUBLE PRECISION array, dimension ( 8*N )
!>          Real workspace.
!>
[out]IWORK
!>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
!>          On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.
!>
[in]LIWORK
!>          LIWORK is INTEGER
!>          The dimension of the array IWORK.
!>          If SENSE = 'N' or N = 0, LIWORK >= 1, otherwise
!>          LIWORK >= N+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.
!>
[out]BWORK
!>          BWORK is LOGICAL array, dimension (N)
!>          Not referenced if SORT = 'N'.
!>
[out]INFO
!>          INFO is INTEGER
!>          = 0:  successful exit
!>          < 0:  if INFO = -i, the i-th argument had an illegal value.
!>          = 1,...,N:
!>                The QZ iteration failed.  (A,B) are not in Schur
!>                form, but 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.