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      SUBROUTINE <a name="DTGSEN.1"></a><a href="dtgsen.f.html#DTGSEN.1">DTGSEN</a>( IJOB, WANTQ, WANTZ, SELECT, N, A, LDA, B, LDB,
     $                   ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, M, PL,
     $                   PR, DIF, WORK, LWORK, IWORK, LIWORK, INFO )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  -- LAPACK routine (version 3.1.1) --
</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">     January 2007
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     Modified to call <a name="DLACN2.9"></a><a href="dlacn2.f.html#DLACN2.1">DLACN2</a> in place of <a name="DLACON.9"></a><a href="dlacon.f.html#DLACON.1">DLACON</a>, 5 Feb 03, SJH.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     .. Scalar Arguments ..
</span>      LOGICAL            WANTQ, WANTZ
      INTEGER            IJOB, INFO, LDA, LDB, LDQ, LDZ, LIWORK, LWORK,
     $                   M, N
      DOUBLE PRECISION   PL, PR
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Array Arguments ..
</span>      LOGICAL            SELECT( * )
      INTEGER            IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
     $                   B( LDB, * ), BETA( * ), DIF( * ), Q( LDQ, * ),
     $                   WORK( * ), Z( LDZ, * )
<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="DTGSEN.28"></a><a href="dtgsen.f.html#DTGSEN.1">DTGSEN</a> reorders the generalized real Schur decomposition of a real
</span><span class="comment">*</span><span class="comment">  matrix pair (A, B) (in terms of an orthonormal equivalence trans-
</span><span class="comment">*</span><span class="comment">  formation Q' * (A, B) * Z), so that a selected cluster of eigenvalues
</span><span class="comment">*</span><span class="comment">  appears in the leading diagonal blocks of the upper quasi-triangular
</span><span class="comment">*</span><span class="comment">  matrix A and the upper triangular B. The leading columns of Q and
</span><span class="comment">*</span><span class="comment">  Z form orthonormal bases of the corresponding left and right eigen-
</span><span class="comment">*</span><span class="comment">  spaces (deflating subspaces). (A, B) must be in generalized real
</span><span class="comment">*</span><span class="comment">  Schur canonical form (as returned by <a name="DGGES.35"></a><a href="dgges.f.html#DGGES.1">DGGES</a>), i.e. A is block upper
</span><span class="comment">*</span><span class="comment">  triangular with 1-by-1 and 2-by-2 diagonal blocks. B 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">  <a name="DTGSEN.39"></a><a href="dtgsen.f.html#DTGSEN.1">DTGSEN</a> also computes the generalized eigenvalues
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">              w(j) = (ALPHAR(j) + i*ALPHAI(j))/BETA(j)
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  of the reordered matrix pair (A, B).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Optionally, <a name="DTGSEN.45"></a><a href="dtgsen.f.html#DTGSEN.1">DTGSEN</a> computes the estimates of reciprocal condition
</span><span class="comment">*</span><span class="comment">  numbers for eigenvalues and eigenspaces. These are Difu[(A11,B11),
</span><span class="comment">*</span><span class="comment">  (A22,B22)] and Difl[(A11,B11), (A22,B22)], i.e. the separation(s)
</span><span class="comment">*</span><span class="comment">  between the matrix pairs (A11, B11) and (A22,B22) that correspond to
</span><span class="comment">*</span><span class="comment">  the selected cluster and the eigenvalues outside the cluster, resp.,
</span><span class="comment">*</span><span class="comment">  and norms of &quot;projections&quot; onto left and right eigenspaces w.r.t.
</span><span class="comment">*</span><span class="comment">  the selected cluster in the (1,1)-block.
</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">  IJOB    (input) INTEGER
</span><span class="comment">*</span><span class="comment">          Specifies whether condition numbers are required for the
</span><span class="comment">*</span><span class="comment">          cluster of eigenvalues (PL and PR) or the deflating subspaces
</span><span class="comment">*</span><span class="comment">          (Difu and Difl):
</span><span class="comment">*</span><span class="comment">           =0: Only reorder w.r.t. SELECT. No extras.
</span><span class="comment">*</span><span class="comment">           =1: Reciprocal of norms of &quot;projections&quot; onto left and right
</span><span class="comment">*</span><span class="comment">               eigenspaces w.r.t. the selected cluster (PL and PR).
</span><span class="comment">*</span><span class="comment">           =2: Upper bounds on Difu and Difl. F-norm-based estimate
</span><span class="comment">*</span><span class="comment">               (DIF(1:2)).
</span><span class="comment">*</span><span class="comment">           =3: Estimate of Difu and Difl. 1-norm-based estimate
</span><span class="comment">*</span><span class="comment">               (DIF(1:2)).
</span><span class="comment">*</span><span class="comment">               About 5 times as expensive as IJOB = 2.
</span><span class="comment">*</span><span class="comment">           =4: Compute PL, PR and DIF (i.e. 0, 1 and 2 above): Economic
</span><span class="comment">*</span><span class="comment">               version to get it all.
</span><span class="comment">*</span><span class="comment">           =5: Compute PL, PR and DIF (i.e. 0, 1 and 3 above)
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  WANTQ   (input) LOGICAL
</span><span class="comment">*</span><span class="comment">          .TRUE. : update the left transformation matrix Q;
</span><span class="comment">*</span><span class="comment">          .FALSE.: do not update Q.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  WANTZ   (input) LOGICAL
</span><span class="comment">*</span><span class="comment">          .TRUE. : update the right transformation matrix Z;
</span><span class="comment">*</span><span class="comment">          .FALSE.: do not update Z.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  SELECT  (input) LOGICAL array, dimension (N)
</span><span class="comment">*</span><span class="comment">          SELECT specifies the eigenvalues in the selected cluster.
</span><span class="comment">*</span><span class="comment">          To select a real eigenvalue w(j), SELECT(j) must be set to
</span><span class="comment">*</span><span class="comment">          .TRUE.. To select a complex conjugate pair of eigenvalues
</span><span class="comment">*</span><span class="comment">          w(j) and w(j+1), corresponding to a 2-by-2 diagonal block,
</span><span class="comment">*</span><span class="comment">          either SELECT(j) or SELECT(j+1) or both must be set to
</span><span class="comment">*</span><span class="comment">          .TRUE.; a complex conjugate pair of eigenvalues must be
</span><span class="comment">*</span><span class="comment">          either both included in the cluster or both excluded.
</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 and B. N &gt;= 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  A       (input/output) DOUBLE PRECISION array, dimension(LDA,N)
</span><span class="comment">*</span><span class="comment">          On entry, the upper quasi-triangular matrix A, with (A, B) in
</span><span class="comment">*</span><span class="comment">          generalized real Schur canonical form.
</span><span class="comment">*</span><span class="comment">          On exit, A is overwritten by the reordered matrix A.
</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 the array 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) DOUBLE PRECISION array, dimension(LDB,N)
</span><span class="comment">*</span><span class="comment">          On entry, the upper triangular matrix B, with (A, B) in
</span><span class="comment">*</span><span class="comment">          generalized real Schur canonical form.
</span><span class="comment">*</span><span class="comment">          On exit, B is overwritten by the reordered matrix B.
</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 the array B. LDB &gt;= max(1,N).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  ALPHAR  (output) DOUBLE PRECISION array, dimension (N)
</span><span class="comment">*</span><span class="comment">  ALPHAI  (output) DOUBLE PRECISION array, dimension (N)
</span><span class="comment">*</span><span class="comment">  BETA    (output) DOUBLE PRECISION 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 generalized Schur form of (A,B) were further reduced
</span><span class="comment">*</span><span class="comment">          to triangular form using 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">  Q       (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
</span><span class="comment">*</span><span class="comment">          On entry, if WANTQ = .TRUE., Q is an N-by-N matrix.
</span><span class="comment">*</span><span class="comment">          On exit, Q has been postmultiplied by the left orthogonal
</span><span class="comment">*</span><span class="comment">          transformation matrix which reorder (A, B); The leading M
</span><span class="comment">*</span><span class="comment">          columns of Q form orthonormal bases for the specified pair of
</span><span class="comment">*</span><span class="comment">          left eigenspaces (deflating subspaces).
</span><span class="comment">*</span><span class="comment">          If WANTQ = .FALSE., Q is not referenced.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  LDQ     (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The leading dimension of the array Q.  LDQ &gt;= 1;
</span><span class="comment">*</span><span class="comment">          and if WANTQ = .TRUE., LDQ &gt;= N.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Z       (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
</span><span class="comment">*</span><span class="comment">          On entry, if WANTZ = .TRUE., Z is an N-by-N matrix.
</span><span class="comment">*</span><span class="comment">          On exit, Z has been postmultiplied by the left orthogonal
</span><span class="comment">*</span><span class="comment">          transformation matrix which reorder (A, B); The leading M
</span><span class="comment">*</span><span class="comment">          columns of Z form orthonormal bases for the specified pair of
</span><span class="comment">*</span><span class="comment">          left eigenspaces (deflating subspaces).
</span><span class="comment">*</span><span class="comment">          If WANTZ = .FALSE., Z is not referenced.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  LDZ     (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The leading dimension of the array Z. LDZ &gt;= 1;
</span><span class="comment">*</span><span class="comment">          If WANTZ = .TRUE., LDZ &gt;= N.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  M       (output) INTEGER
</span><span class="comment">*</span><span class="comment">          The dimension of the specified pair of left and right eigen-
</span><span class="comment">*</span><span class="comment">          spaces (deflating subspaces). 0 &lt;= M &lt;= N.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  PL      (output) DOUBLE PRECISION
</span><span class="comment">*</span><span class="comment">  PR      (output) DOUBLE PRECISION
</span><span class="comment">*</span><span class="comment">          If IJOB = 1, 4 or 5, PL, PR are lower bounds on the
</span><span class="comment">*</span><span class="comment">          reciprocal of the norm of &quot;projections&quot; onto left and right
</span><span class="comment">*</span><span class="comment">          eigenspaces with respect to the selected cluster.
</span><span class="comment">*</span><span class="comment">          0 &lt; PL, PR &lt;= 1.
</span><span class="comment">*</span><span class="comment">          If M = 0 or M = N, PL = PR  = 1.
</span><span class="comment">*</span><span class="comment">          If IJOB = 0, 2 or 3, PL and PR are not referenced.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  DIF     (output) DOUBLE PRECISION array, dimension (2).
</span><span class="comment">*</span><span class="comment">          If IJOB &gt;= 2, DIF(1:2) store the estimates of Difu and Difl.
</span><span class="comment">*</span><span class="comment">          If IJOB = 2 or 4, DIF(1:2) are F-norm-based upper bounds on
</span><span class="comment">*</span><span class="comment">          Difu and Difl. If IJOB = 3 or 5, DIF(1:2) are 1-norm-based
</span><span class="comment">*</span><span class="comment">          estimates of Difu and Difl.
</span><span class="comment">*</span><span class="comment">          If M = 0 or N, DIF(1:2) = F-norm([A, B]).
</span><span class="comment">*</span><span class="comment">          If IJOB = 0 or 1, DIF is not referenced.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  WORK    (workspace/output) DOUBLE PRECISION array,
</span><span class="comment">*</span><span class="comment">          dimension (MAX(1,LWORK)) 
</span><span class="comment">*</span><span class="comment">          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
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