cggesx.f.html

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      SUBROUTINE <a name="CGGESX.1"></a><a href="cggesx.f.html#CGGESX.1">CGGESX</a>( 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 )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  -- LAPACK driver routine (version 3.1) --
</span><span class="comment">*</span><span class="comment">     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
</span><span class="comment">*</span><span class="comment">     November 2006
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     .. Scalar Arguments ..
</span>      CHARACTER          JOBVSL, JOBVSR, SENSE, SORT
      INTEGER            INFO, LDA, LDB, LDVSL, LDVSR, LIWORK, LWORK, N,
     $                   SDIM
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Array Arguments ..
</span>      LOGICAL            BWORK( * )
      INTEGER            IWORK( * )
      REAL               RCONDE( 2 ), RCONDV( 2 ), RWORK( * )
      COMPLEX            A( LDA, * ), ALPHA( * ), B( LDB, * ),
     $                   BETA( * ), VSL( LDVSL, * ), VSR( LDVSR, * ),
     $                   WORK( * )
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Function Arguments ..
</span>      LOGICAL            SELCTG
      EXTERNAL           SELCTG
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Purpose
</span><span class="comment">*</span><span class="comment">  =======
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  <a name="CGGESX.31"></a><a href="cggesx.f.html#CGGESX.1">CGGESX</a> computes for a pair of N-by-N complex nonsymmetric matrices
</span><span class="comment">*</span><span class="comment">  (A,B), the generalized eigenvalues, the complex Schur form (S,T),
</span><span class="comment">*</span><span class="comment">  and, optionally, the left and/or right matrices of Schur vectors (VSL
</span><span class="comment">*</span><span class="comment">  and VSR).  This gives the generalized Schur factorization
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">       (A,B) = ( (VSL) S (VSR)**H, (VSL) T (VSR)**H )
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  where (VSR)**H is the conjugate-transpose of VSR.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Optionally, it also orders the eigenvalues so that a selected cluster
</span><span class="comment">*</span><span class="comment">  of eigenvalues appears in the leading diagonal blocks of the upper
</span><span class="comment">*</span><span class="comment">  triangular matrix S and the upper triangular matrix T; computes
</span><span class="comment">*</span><span class="comment">  a reciprocal condition number for the average of the selected
</span><span class="comment">*</span><span class="comment">  eigenvalues (RCONDE); and computes a reciprocal condition number for
</span><span class="comment">*</span><span class="comment">  the right and left deflating subspaces corresponding to the selected
</span><span class="comment">*</span><span class="comment">  eigenvalues (RCONDV). The leading columns of VSL and VSR then form
</span><span class="comment">*</span><span class="comment">  an orthonormal basis for the corresponding left and right eigenspaces
</span><span class="comment">*</span><span class="comment">  (deflating subspaces).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  A generalized eigenvalue for a pair of matrices (A,B) is a scalar w
</span><span class="comment">*</span><span class="comment">  or a ratio alpha/beta = w, such that  A - w*B is singular.  It is
</span><span class="comment">*</span><span class="comment">  usually represented as the pair (alpha,beta), as there is a
</span><span class="comment">*</span><span class="comment">  reasonable interpretation for beta=0 or for both being zero.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  A pair of matrices (S,T) is in generalized complex Schur form if T is
</span><span class="comment">*</span><span class="comment">  upper triangular with non-negative diagonal and S is upper
</span><span class="comment">*</span><span class="comment">  triangular.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Arguments
</span><span class="comment">*</span><span class="comment">  =========
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  JOBVSL  (input) CHARACTER*1
</span><span class="comment">*</span><span class="comment">          = 'N':  do not compute the left Schur vectors;
</span><span class="comment">*</span><span class="comment">          = 'V':  compute the left Schur vectors.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  JOBVSR  (input) CHARACTER*1
</span><span class="comment">*</span><span class="comment">          = 'N':  do not compute the right Schur vectors;
</span><span class="comment">*</span><span class="comment">          = 'V':  compute the right Schur vectors.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  SORT    (input) CHARACTER*1
</span><span class="comment">*</span><span class="comment">          Specifies whether or not to order the eigenvalues on the
</span><span class="comment">*</span><span class="comment">          diagonal of the generalized Schur form.
</span><span class="comment">*</span><span class="comment">          = 'N':  Eigenvalues are not ordered;
</span><span class="comment">*</span><span class="comment">          = 'S':  Eigenvalues are ordered (see SELCTG).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  SELCTG  (external procedure) LOGICAL FUNCTION of two COMPLEX arguments
</span><span class="comment">*</span><span class="comment">          SELCTG must be declared EXTERNAL in the calling subroutine.
</span><span class="comment">*</span><span class="comment">          If SORT = 'N', SELCTG is not referenced.
</span><span class="comment">*</span><span class="comment">          If SORT = 'S', SELCTG is used to select eigenvalues to sort
</span><span class="comment">*</span><span class="comment">          to the top left of the Schur form.
</span><span class="comment">*</span><span class="comment">          Note that a selected complex eigenvalue may no longer satisfy
</span><span class="comment">*</span><span class="comment">          SELCTG(ALPHA(j),BETA(j)) = .TRUE. after ordering, since
</span><span class="comment">*</span><span class="comment">          ordering may change the value of complex eigenvalues
</span><span class="comment">*</span><span class="comment">          (especially if the eigenvalue is ill-conditioned), in this
</span><span class="comment">*</span><span class="comment">          case INFO is set to N+3 see INFO below).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  SENSE   (input) CHARACTER*1
</span><span class="comment">*</span><span class="comment">          Determines which reciprocal condition numbers are computed.
</span><span class="comment">*</span><span class="comment">          = 'N' : None are computed;
</span><span class="comment">*</span><span class="comment">          = 'E' : Computed for average of selected eigenvalues only;
</span><span class="comment">*</span><span class="comment">          = 'V' : Computed for selected deflating subspaces only;
</span><span class="comment">*</span><span class="comment">          = 'B' : Computed for both.
</span><span class="comment">*</span><span class="comment">          If SENSE = 'E', 'V', or 'B', SORT must equal 'S'.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  N       (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The order of the matrices A, B, VSL, and VSR.  N &gt;= 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  A       (input/output) COMPLEX array, dimension (LDA, N)
</span><span class="comment">*</span><span class="comment">          On entry, the first of the pair of matrices.
</span><span class="comment">*</span><span class="comment">          On exit, A has been overwritten by its generalized Schur
</span><span class="comment">*</span><span class="comment">          form S.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  LDA     (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The leading dimension of A.  LDA &gt;= max(1,N).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  B       (input/output) COMPLEX array, dimension (LDB, N)
</span><span class="comment">*</span><span class="comment">          On entry, the second of the pair of matrices.
</span><span class="comment">*</span><span class="comment">          On exit, B has been overwritten by its generalized Schur
</span><span class="comment">*</span><span class="comment">          form T.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  LDB     (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The leading dimension of B.  LDB &gt;= max(1,N).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  SDIM    (output) INTEGER
</span><span class="comment">*</span><span class="comment">          If SORT = 'N', SDIM = 0.
</span><span class="comment">*</span><span class="comment">          If SORT = 'S', SDIM = number of eigenvalues (after sorting)
</span><span class="comment">*</span><span class="comment">          for which SELCTG is true.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  ALPHA   (output) COMPLEX array, dimension (N)
</span><span class="comment">*</span><span class="comment">  BETA    (output) COMPLEX array, dimension (N)
</span><span class="comment">*</span><span class="comment">          On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the
</span><span class="comment">*</span><span class="comment">          generalized eigenvalues.  ALPHA(j) and BETA(j),j=1,...,N  are
</span><span class="comment">*</span><span class="comment">          the diagonals of the complex Schur form (S,T).  BETA(j) will
</span><span class="comment">*</span><span class="comment">          be non-negative real.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">          Note: the quotients ALPHA(j)/BETA(j) may easily over- or
</span><span class="comment">*</span><span class="comment">          underflow, and BETA(j) may even be zero.  Thus, the user
</span><span class="comment">*</span><span class="comment">          should avoid naively computing the ratio alpha/beta.
</span><span class="comment">*</span><span class="comment">          However, ALPHA will be always less than and usually
</span><span class="comment">*</span><span class="comment">          comparable with norm(A) in magnitude, and BETA always less
</span><span class="comment">*</span><span class="comment">          than and usually comparable with norm(B).
</span><span class="comment">*</span><span class="comment">
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