ztgsen.f.html

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      SUBROUTINE <a name="ZTGSEN.1"></a><a href="ztgsen.f.html#ZTGSEN.1">ZTGSEN</a>( IJOB, WANTQ, WANTZ, SELECT, N, A, LDA, B, LDB,
     $                   ALPHA, 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="ZLACN2.9"></a><a href="zlacn2.f.html#ZLACN2.1">ZLACN2</a> in place of <a name="ZLACON.9"></a><a href="zlacon.f.html#ZLACON.1">ZLACON</a>, 10 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   DIF( * )
      COMPLEX*16         A( LDA, * ), ALPHA( * ), B( LDB, * ),
     $                   BETA( * ), 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="ZTGSEN.28"></a><a href="ztgsen.f.html#ZTGSEN.1">ZTGSEN</a> reorders the generalized Schur decomposition of a complex
</span><span class="comment">*</span><span class="comment">  matrix pair (A, B) (in terms of an unitary 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 pair (A,B). The leading
</span><span class="comment">*</span><span class="comment">  columns of Q and Z form unitary bases of the corresponding left and
</span><span class="comment">*</span><span class="comment">  right eigenspaces (deflating subspaces). (A, B) must be in
</span><span class="comment">*</span><span class="comment">  generalized Schur canonical form, that is, A and B are both 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="ZTGSEN.37"></a><a href="ztgsen.f.html#ZTGSEN.1">ZTGSEN</a> also computes the generalized eigenvalues
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">           w(j)= ALPHA(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, the routine computes 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">
</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. To
</span><span class="comment">*</span><span class="comment">          select an eigenvalue w(j), SELECT(j) must be set to
</span><span class="comment">*</span><span class="comment">          .TRUE..
</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) COMPLEX*16 array, dimension(LDA,N)
</span><span class="comment">*</span><span class="comment">          On entry, the upper triangular matrix A, in generalized
</span><span class="comment">*</span><span class="comment">          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) COMPLEX*16 array, dimension(LDB,N)
</span><span class="comment">*</span><span class="comment">          On entry, the upper triangular matrix B, in generalized
</span><span class="comment">*</span><span class="comment">          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">  ALPHA   (output) COMPLEX*16 array, dimension (N)
</span><span class="comment">*</span><span class="comment">  BETA    (output) COMPLEX*16 array, dimension (N)
</span><span class="comment">*</span><span class="comment">          The diagonal elements of A and B, respectively,
</span><span class="comment">*</span><span class="comment">          when the pair (A,B) has been reduced to generalized Schur
</span><span class="comment">*</span><span class="comment">          form.  ALPHA(i)/BETA(i) i=1,...,N are the generalized
</span><span class="comment">*</span><span class="comment">          eigenvalues.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Q       (input/output) COMPLEX*16 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 unitary
</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">          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) COMPLEX*16 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 unitary
</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
</span><span class="comment">*</span><span class="comment">          eigenspaces, (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">          eigenspace 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, 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, computed using reversed
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