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## zgges3()

subroutine zgges3 | ( | character | jobvsl,
|  | character | jobvsr,
|  | character | sort,
|  | external | selctg,
|  | 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,
|  | complex*16, dimension( * ) | work,
|  | integer | lwork,
|  | double precision, dimension( * ) | rwork,
|  | logical, dimension( * ) | bwork,
|  | integer | info )

ZGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices (blocked algorithm)

Download ZGGES3 + dependencies [TGZ] [ZIP] [TXT]

Purpose:

      !>
      !> ZGGES3 computes for a pair of N-by-N complex nonsymmetric matrices
      !> (A,B), the generalized eigenvalues, the generalized complex Schur
      !> form (S, T), and optionally left and/or right 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. The leading
      !> columns of VSL and VSR then form an unitary basis for the
      !> corresponding left and right eigenspaces (deflating subspaces).
      !>
      !> (If only the generalized eigenvalues are needed, use the driver
      !> ZGGEV instead, which is faster.)
      !>
      !> 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, and even for both being zero.
      !>
      !> A pair of matrices (S,T) is in generalized complex Schur form if S
      !> and T are upper triangular and, in addition, the diagonal elements
      !> of T are non-negative real numbers.
      !>

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.
      !>          An eigenvalue ALPHA(j)/BETA(j) is selected if
      !>          SELCTG(ALPHA(j),BETA(j)) is true.
      !>
      !>          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+2 (See INFO below).
      !>

  [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), j=1,...,N  and  BETA(j),
      !>          j=1,...,N  are the diagonals of the complex Schur form (A,B)
      !>          output by ZGGES3. The  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]

  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. LWORK >= MAX(1,2*N)
      !>          For good performance, LWORK must generally be larger.
      !>
      !>          If LWORK = -1, then a workspace query is assumed; the routine
      !>          only calculates the optimal size of the WORK array, returns
      !>          this value as the first entry of the WORK array, and no error
      !>          message related to LWORK is issued by XERBLA.
      !>

  [out]

  RWORK

      !>          RWORK is DOUBLE PRECISION array, dimension (8*N)
      !>

  [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 ZLAQZ0
      !>                =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.