sgges.f.html

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      SUBROUTINE <a name="SGGES.1"></a><a href="sgges.f.html#SGGES.1">SGGES</a>( JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB,
     $                  SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR,
     $                  LDVSR, WORK, LWORK, 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, SORT
      INTEGER            INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N, SDIM
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Array Arguments ..
</span>      LOGICAL            BWORK( * )
      REAL               A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
     $                   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="SGGES.27"></a><a href="sgges.f.html#SGGES.1">SGGES</a> computes for a pair of N-by-N real nonsymmetric matrices (A,B),
</span><span class="comment">*</span><span class="comment">  the generalized eigenvalues, the generalized real Schur form (S,T),
</span><span class="comment">*</span><span class="comment">  optionally, the left and/or right matrices of Schur vectors (VSL and
</span><span class="comment">*</span><span class="comment">  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)**T, (VSL)*T*(VSR)**T )
</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">  quasi-triangular matrix S and the upper triangular matrix T.The
</span><span class="comment">*</span><span class="comment">  leading columns of VSL and VSR then form an orthonormal basis for the
</span><span class="comment">*</span><span class="comment">  corresponding left and right eigenspaces (deflating subspaces).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  (If only the generalized eigenvalues are needed, use the driver
</span><span class="comment">*</span><span class="comment">  <a name="SGGEV.41"></a><a href="sggev.f.html#SGGEV.1">SGGEV</a> instead, which is faster.)
</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 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 real Schur form if T is
</span><span class="comment">*</span><span class="comment">  upper triangular with non-negative diagonal and S is block upper
</span><span class="comment">*</span><span class="comment">  triangular with 1-by-1 and 2-by-2 blocks.  1-by-1 blocks correspond
</span><span class="comment">*</span><span class="comment">  to real generalized eigenvalues, while 2-by-2 blocks of S will be
</span><span class="comment">*</span><span class="comment">  &quot;standardized&quot; by making the corresponding elements of T have the
</span><span class="comment">*</span><span class="comment">  form:
</span><span class="comment">*</span><span class="comment">          [  a  0  ]
</span><span class="comment">*</span><span class="comment">          [  0  b  ]
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  and the pair of corresponding 2-by-2 blocks in S and T will have a
</span><span class="comment">*</span><span class="comment">  complex conjugate pair of generalized eigenvalues.
</span><span class="comment">*</span><span class="comment">
</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 three REAL 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">          An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if
</span><span class="comment">*</span><span class="comment">          SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either
</span><span class="comment">*</span><span class="comment">          one of a complex conjugate pair of eigenvalues is selected,
</span><span class="comment">*</span><span class="comment">          then both complex eigenvalues are selected.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">          Note that in the ill-conditioned case, a selected complex
</span><span class="comment">*</span><span class="comment">          eigenvalue may no longer satisfy SELCTG(ALPHAR(j),ALPHAI(j),
</span><span class="comment">*</span><span class="comment">          BETA(j)) = .TRUE. after ordering. INFO is to be set to N+2
</span><span class="comment">*</span><span class="comment">          in this case.
</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) REAL 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) REAL 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.  (Complex conjugate pairs for which
</span><span class="comment">*</span><span class="comment">          SELCTG is true for either eigenvalue count as 2.)
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  ALPHAR  (output) REAL array, dimension (N)
</span><span class="comment">*</span><span class="comment">  ALPHAI  (output) REAL array, dimension (N)
</span><span class="comment">*</span><span class="comment">  BETA    (output) REAL array, dimension (N)
</span><span class="comment">*</span><span class="comment">          On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
</span><span class="comment">*</span><span class="comment">          be the generalized eigenvalues.  ALPHAR(j) + ALPHAI(j)*i,
</span><span class="comment">*</span><span class="comment">          and  BETA(j),j=1,...,N are the diagonals of the complex Schur
</span><span class="comment">*</span><span class="comment">          form (S,T) that would result if the 2-by-2 diagonal blocks of
</span><span class="comment">*</span><span class="comment">          the real Schur form of (A,B) were further reduced to
</span><span class="comment">*</span><span class="comment">          triangular form using 2-by-2 complex unitary transformations.
</span><span class="comment">*</span><span class="comment">          If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
</span><span class="comment">*</span><span class="comment">          positive, then the j-th and (j+1)-st eigenvalues are a
</span><span class="comment">*</span><span class="comment">          complex conjugate pair, with ALPHAI(j+1) negative.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">          Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
</span><span class="comment">*</span><span class="comment">          may easily over- or underflow, and BETA(j) may even be zero.
</span><span class="comment">*</span><span class="comment">          Thus, the user should avoid naively computing the ratio.
</span><span class="comment">*</span><span class="comment">          However, ALPHAR and ALPHAI will be always less than and
</span><span class="comment">*</span><span class="comment">          usually comparable with norm(A) in magnitude, and BETA always
</span><span class="comment">*</span><span class="comment">          less than and usually comparable with norm(B).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  VSL     (output) REAL array, dimension (LDVSL,N)
</span><span class="comment">*</span><span class="comment">          If JOBVSL = 'V', VSL will contain the left Schur vectors.
</span><span class="comment">*</span><span class="comment">          Not referenced if JOBVSL = 'N'.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  LDVSL   (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The leading dimension of the matrix VSL. LDVSL &gt;=1, and
</span><span class="comment">*</span><span class="comment">          if JOBVSL = 'V', LDVSL &gt;= N.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  VSR     (output) REAL array, dimension (LDVSR,N)
</span><span class="comment">*</span><span class="comment">          If JOBVSR = 'V', VSR will contain the right Schur vectors.
</span><span class="comment">*</span><span class="comment">          Not referenced if JOBVSR = 'N'.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  LDVSR   (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The leading dimension of the matrix VSR. LDVSR &gt;= 1, and
</span><span class="comment">*</span><span class="comment">          if JOBVSR = 'V', LDVSR &gt;= N.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
</span><span class="comment">*</span><span class="comment">          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  LWORK   (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The dimension of the array WORK.
</span><span class="comment">*</span><span class="comment">          If N = 0, LWORK &gt;= 1, else LWORK &gt;= max(8*N,6*N+16).
</span><span class="comment">*</span><span class="comment">          For good performance , LWORK must generally be larger.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">          If LWORK = -1, then a workspace query is assumed; the routine
</span><span class="comment">*</span><span class="comment">          only calculates the optimal size of the WORK array, returns
</span><span class="comment">*</span><span class="comment">          this value as the first entry of the WORK array, and no error
</span><span class="comment">*</span><span class="comment">          message related to LWORK is issued by <a name="XERBLA.165"></a><a href="xerbla.f.html#XERBLA.1">XERBLA</a>.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  BWORK   (workspace) LOGICAL array, dimension (N)
</span><span class="comment">*</span><span class="comment">          Not referenced if SORT = 'N'.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  INFO    (output) INTEGER
</span><span class="comment">*</span><span class="comment">          = 0:  successful exit
</span><span class="comment">*</span><span class="comment">          &lt; 0:  if INFO = -i, the i-th argument had an illegal value.
</span><span class="comment">*</span><span class="comment">          = 1,...,N:
</span><span class="comment">*</span><span class="comment">                The QZ iteration failed.  (A,B) are not in Schur
</span><span class="comment">*</span><span class="comment">                form, but ALPHAR(j), ALPHAI(j), and BETA(j) should
</span><span class="comment">*</span><span class="comment">                be correct for j=INFO+1,...,N.
123

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