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## sgges3()

subroutine sgges3 | ( | character | jobvsl,
|  | character | jobvsr,
|  | character | sort,
|  | external | selctg,
|  | 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( * ) | work,
|  | integer | lwork,
|  | logical, dimension( * ) | bwork,
|  | integer | info )

SGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices (blocked algorithm)

Download SGGES3 + dependencies [TGZ] [ZIP] [TXT]

Purpose:

      !>
      !> SGGES3 computes for a pair of N-by-N real nonsymmetric matrices (A,B),
      !> the generalized eigenvalues, the generalized real Schur form (S,T),
      !> 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.The
      !> leading columns of VSL and VSR then form an orthonormal basis for the
      !> corresponding left and right eigenspaces (deflating subspaces).
      !>
      !> (If only the generalized eigenvalues are needed, use the driver
      !> SGGEV 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 or 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

  [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 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 in the ill-conditioned case, a selected complex
      !>          eigenvalue may no longer satisfy SELCTG(ALPHAR(j),ALPHAI(j),
      !>          BETA(j)) = .TRUE. after ordering. INFO is to be set to N+2
      !>          in this case.
      !>

  [in]

  N

      !>          N is INTEGER
      !>          The order of the matrices A, B, VSL, and VSR.  N >= 0.
      !>

  [in,out]

  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.
      !>

  [in]

  LDA

      !>          LDA is INTEGER
      !>          The leading dimension of A.  LDA >= max(1,N).
      !>

  [in,out]

  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.
      !>

  [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.  (Complex conjugate pairs for which
      !>          SELCTG is true for either eigenvalue count as 2.)
      !>

  [out]

  ALPHAR

      !>          ALPHAR is REAL array, dimension (N)
      !>

  [out]

  ALPHAI

      !>          ALPHAI is REAL array, dimension (N)
      !>

  [out]

  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).
      !>

  [out]

  VSL

      !>          VSL is REAL 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 REAL 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 REAL 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 LWORK >= 6*N+16.
      !>          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]

  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 ALPHAR(j), ALPHAI(j), and BETA(j) should
      !>                be correct for j=INFO+1,...,N.
      !>          > N:  =N+1: other than QZ iteration failed in SLAQZ0.
      !>                =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.